Superanalysis
Mathematics and Its Applications
Managing Editor.
M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 470
Superanalysis by
Andrei Khrennikov Department of Mathematics, Statistics and Computer Sciences, University of Vdxjo, Vdxjo, Sweden
and Department of Mathematics, Moscow State University of Electronic Engineering, Zelenograd, Moscow, Russia
KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON /LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 0-7923-5607-1
Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid free paper
This is a completely updated and revised translation of the original Russian work of the same title. Nauka, Moscow 01997
All Rights Reserved 01999 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner
Printed in the Netherlands.
This book is dedicated to Professor Vasilii Vladimirov.
Table of Contents
Introduction I
1
Analysis on a Superspace over Banach Superalgebras 1.
Differential Calculus
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2. Cauchy-Riemann Conditions and the Condition of A. . Linearity of Derivatives . . . . 3. Integral Calculus 4. Integration of Differential Forms of Commuting Variables . . . 5. Review of the Development of Superanalysis 6. Unsolved Problems and Possible Generalizations . .
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II
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Generalized Functions on a Superspace Locally Convex Superalgebras and Supermodules 2. Analytic Generalized Functions on the Vladimirov-Vo. . . . lovich Superspace . 3. Fourier Transformation of Superanalytic Generalized . . . Functions . 4. Superanalog of the Theory of Schwartz Distributions 5. Theorem of Existence of a Fundamental Solution . . 6. Unsolved Problems and Possible Generalizations . . .
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51
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60 63 74 92 100 106
Table of Contents
viii
III Distribution Theory on an Infinite-Dimensional Superspace 1.
2. 3. 4. 5.
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Polylinear Algebra over Commutative Supermodules Banach Supermodules . . Hilbert Supermodules . Duality of Topological Supermodules . Differential Calculus on a Superspace over Topological Supermodules . Analytic Distributions on a Superspace over Topological Supermodules Gaussian and Feynman Distributions . Unsolved Problems and Possible Generalizations .
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IV
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Pseudo differential Operators in Superanalysis
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158 166 180
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183 197
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Fundamentals of the Probability Theory on a Superspace 227 . Limit Theorems on a Superspace . 2. Random Processes on a Superspace 3. Axiomatics of the Probability Theory over Superalgebras . . . 4. Unsolved Problems and Possible Generalizations .
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VI
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1. Pseudo differential Operators Calculus . 2. The Correspondence Principle . 3. The Feynman-Kac Formula for the Symbol of the Evo. . . lution Operator . 4. Unsolved Problems and Possible Generalizations . .
110 116 130 141
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Non-Archimedean Superanalysis .
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258 264 267 269
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270
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244 254
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Differentiable and Analytic Functions . 2. Generalized Functions . 3. Laplace Transformation . 4. Gaussian Distributions . . 5. Duhamel non-Archimedean Integral. Chronological Exponent . . . . . . 1.
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227 240
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ix
Table of Contents
Cauchy Problem for Partial Differential Equations with . . 273 . . . Variable Coefficients 7. Non-Archimedean Supersymmetrical Quantum Mechan6.
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ics ............................ Trotter Formula for non-Archimedean Banach Alge-
. . . . bras 9. Volkenborn Distribution on a non-Archimedean Super. . space . 10. Infinite-Dimensional non-Archimedean Superanalysis 11. Unsolved Problems and Possible Generalizations . .
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VII Noncommutative Analysis 1.
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VIII Applications in Physics 1.
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278 279 283 289
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Differential Calculus on a Superspace over a Noncom. . mutative Banach Algebra Differential Calculus on Noncommutative Banach Al. gebras and Modules Generalized Functions of Noncommuting Variables .
276
294 298 309
313
. . . Quantization in Hilbert Supermodules Transition Amplitudes and Distributions on the Space
314
of Schwinger Sources
315
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References
329
Index
345
Introduction
The foundations of the theory of functions of commuting and anticommuting variables were laid in the well-known work A Note to the Quantum Dynamical Principle by J. Schwinger [172] published in 1953.
Schwinger presented the analysis for commuting and anticommuting variables on the physical level of strictness. He assumed that there existed a set of points (which was later called a superspace) on which commuting and anticommuting coordinates were given and a differential calculus was constructed. This set was similar in many respects to Newton's differential calculus. However, the superspace was not
defined on the mathematical level of strictness (although the work [172] contained a remark concerning the construction of a superspace,
namely, it was proposed to define a superspace as a subset of the algebra of quantum field operators). A problem arose of constructing a mathematical formalism adequate to Schwinger's theory. Investigations in this direction were stimulated by applications in physics in which functions dependent on commuting and anticommuting variables would play an increasingly important part. The first mathematical formalism that made it possible to operate with commuting and anticommuting coordinates was Martin's algebraic formalism proposed in 1959 [114, 115]. Martin did not follow the way paved by Schwinger, neither did he try to give a mathematical definition of superanalysis - a set of superpoints with commuting and anticommuting coordinates. Instead, he developed a purely algebraic theory in which the "functions" of anticommuting variables were
2
Introduction
defined as elements of Grassmann algebra (an algebra with anticommuting generators). The derivatives of these elements with respect to anticommuting generators were defined according to algebraic laws, and nothing like Newton's analysis arose when Martin's approach was used.
Later, during the next twenty years, the algebraic apparatus developed by Martin was used in all mathematical works. We must point out here the considerable contribution made by F. A. Berezin, G I. Kac, D. A. Leites, B. Kostant. In their works, they constructed a new division of mathematics which can naturally be called an algebraic superanalysis. Following the example of physicists, researchers called the investigations carried out with the use of commuting and anticommuting coordinates supermathematics; all mathematical objects that appeared in supermathematics were called superobjects, although, of course, there is nothing "super" in supermathematics. However, despite the great achievements in algebraic superanalysis, this formalism could not be regarded as a generalization to the case of commuting and anticommuting variables from the ordinary Newton analysis. What is more, Schwinger's formalism was still used in practically all physical works, on an intuitive level, and physicists regarded functions of anticommuting variables as "real functions" maps of sets and not as elements of Grassmann algebras. In 1974,
Salam and Strathdee proposed a very apt name for a set of superpoints. They called this set a superspace. Psychologically, physicists associated the introduction of the term "superspace" with the fact that this set was defined on the mathematical level of strictness, and, after the works by Salam and Strathdee [123] and by Wess and Zumino [184] were published, the superspace became a foundation for the most important physical theories. A paradoxical situation took shape by the end of the 1970s, namely, mathematicians continued an active development of an algebraic superanalysis whereas physicists used a different formalism which was considerably simpler and visual (there was no need to use here, as it was done by mathematicians, the language of algebraic geometry and the theory of bundles). It was clear that the use of such words as a
Introduction
3
ringed space and a structural bundle could not elucidate anything in physical theory and only made a very simple intuitive formalism more complicated. A serious problem arose in algebraic superanalysis in connection
with the construction of supersymmetric theories (D. Yu. Gel'fand, E. S. Likhtman, 1971; V. P. Akulov, D.V. Volkov, 1974; J. Wess, B. Zumino, 1974). The transformation of supersymmetry (SUSY) includes a SUSY parameter e which is a constant for a fixed transformation of SUSY, and not an ordinary constant but a constant anticommuting with the other anticommuting coordinates. However, there are no anticommuting constants in algebraic superanalysis. Here the concept of a constant is mixed up with that of a function since a "function" is a constant, an element of Grassmann algebra. However, from physical considerations, it was necessary to distinguish in SUSY between the constant e and anticommuting variables. For the first time,
this problem was subjected to a detailed discussion by J. Dell and 1. Smolin in 1979 [28]. This was, apparently, the first work in which the authors pointed out the difficulties that arose in superanalysis in connection with the attempts to use it for the description of SUSY. Moreover, a purely mathematical problem arose in algebraic superanalysis which was very disturbing. It was a problem of a change of variables in the Berezin integral. The simplest changes of variables (such, for instance, as those encountered in Rudakov's example, see [42]) led to senseless answers. Because of all these problems (the nonagreement of formalisms used by mathematicians and physicists, anticommuting constants in SUSY, a change of variables in Berezin's integral), some mathemati-
cians and physicists tried to realize, on the mathematical level of strictness, Schwinger's idea concerning a set of superpoints. Several mathematical models of superspace were proposed. The first model was constructed by Batchelor in 1979 [82]. However, this model, constructed as a point realization corresponding to the graded Kostant manifold, did not answer the idea that physicists had of a superspace, and, despite the beautiful mathematical theory, was discarded.
Introduction
4
Practically at the same time as Batchelor published his article (and, perhaps, a little earlier), De Witt wrote his book Supermanifolds. Although this book was published only in 1984 [27], many mathematicians and physicists got acquainted with it in 1979. References to this book can be found, for instance, in [119]. De Witt proposed his model of a set of superpoints based on Grassmann's infinite-dimensional algebra. De Witt constructed a well developed theory (differential and integral calculus, differential geometry, generalized functions). However, De Witt's model of a superspace had one drawback, namely, the topology that he proposed was not a segregated topology.
Models of superspaces endowed with an ordinary topology of a Banach space were proposed by Rogers [119, 120] and Vladimirov and Volovich [19, 20]. Roger's models of a superspace were based on Grassmann algebras endowed with normed topology. Vladimirov and Volovich constructed a superanalysis over an arbitrary (supercommutative) Banach superalgebra. The Vladimirov-Volovich superanalysis is invariant with respect to the choice of Banach superalgebra. Moreover, they not only achieved the greatest generality of mathematical constructions but also realized the following principle (the VladimirovVolovich principle of superinvariance of physical theories): all physical formalisms must be invariant with respect to the choice of a supercommutative Banach algebra that serves as the basis for the superspace.
I have analyzed practically all applications of superanalysis in Any formalism can be realized over an arbitrary Banach
physics.
superalgebra. It is natural to call the analysis developed in the works of De Witt, Rogers, Vladimirov, and Volovich a functional superanalysis [65]. It
is an analysis of "real functions" of commuting and anticommuting variables - maps of a set of superpoints called a superspace. The functional superanalysis is a mathematical realization of Schwinger's formalism of 1953 whereas the models of a superspace of De Witt, Rogers, Vladimirov, and Volovich are different mathematical models of what Salam and Strathdee called a superspace. The first chapter of this book is devoted to the Vladimirov-Volovich superanalysis. In this chapter we consider differential and integral cal-
Introduction
5
culus on a superspace over a Banach superalgebra. The other chapters constitute the exposition of my D. Sc. dissertation, 1990 [68]. My contribution to superanalysis consists of (1) the theory of generalized functions, (2) the theory of pseudo differential operators, (3) an infinite-dimensional superanalysis, (4) the theory of generalized functions on infinite-dimensional superspaces and its applications to functional integration, (5) probability theory on a superspace. The book also includes a number of applications of functional superanalysis to the quantum theory of a field and a string. These models are considered only schematically. However, I hope that hav-
ing read this book, any specialist in the quantum theory of a field and a string and of gravitation will be able to use easily functional superanalysis in his research. It should be emphasized once again that one of the main advantages of functional superanalysis is its simplicity and visuality. As for mathematicians, functional analysis constitutes for them a whole field of new unsolved problems. Although the fundamentals of superanalysis are similar to those of ordinary mathematical analysis, new nontrivial mathematical constructions arise in its further development. We can formulate here a number of general problems whose solution would lead to the creation of new mathematical theories such as, for instance, the construction of a spectral theory of self-adjoint operators in Hilbert supermodules. Nothing has been done yet in this direction. A large number of mathematicians and physicists took part in the discussion of the results exposed in this book. I want to use the opportunity to express my deepest gratitude to all of them. I feel myself especially indebted to V. S. Vladimirov, I. V. Volovich, B. S. De Witt, C. De Witt-Morette, O. G. Smolyanov, A. A. Slavnov, Yu. V. Egorov, Yu. A. Dubinskii, V. I. Ogievetskii, R. Cianci, T. Hida. I am also very grateful to my wife, Olga Shustova, for her constant support.
Chapter I
Analysis on a Superspace over Banach Superalgebras
Here we follow the works by Vladimirov and Volovich [19, 20].
1.
Differential Calculus 1.1. Superspace over a commutative Banach superalgebra.
Recall that a linear space L is Z2-graded if it is represented as a direct
sum of two subspaces L = L° ® L1. The elements of the spaces L° and L1 are homogeneous. The parity is defined in the graded space L = L° ® L1 if it is said, in addition, that the elements of one of these spaces are even and those of the other space are odd. We assume, in what follows, that L° is a subspace consisting of even elements and L1 is a subspace consisting of odd elements. For the element f E L = L° ®L1 we denote by f ° and f 1 its even and odd components. The symbols 7r° and ir1 denote the projectors onto L° and L1. A parity function is introduced on the Z2-graded space, namely,
Jal =0ifaEL°and Jal =1 ifaEL1. In the Z2-graded space L = L° ® L1 we introduce a parity automorphism a: L -+ L by setting a (f) = (-1) I f I f for homogeneous elements. Note that a2 = 1 and a(f) = f if and only if f E L°. A superalgebra is a Z2-graded space A = A° ® Al on which a structure is introduced of an associative algebra with a unit e and
Chapter I. Analysis on a Superspace
8
even multiplication operation (i.e., the product of two even and two odd elements is an even element and the product of an even element by
an odd one is an odd element: jabl = lal + JbI (mod 2). In particular, the subspace Ao is a subalgebra of the algebra A. Everywhere in this book, linear spaces are considered over a field
K = R or C. The non-Archimedean superanalysis is presented in Chap. VI. The supercommutator [a, b} of the homogeneous elements a and b from the superalgebra A is defined by the relation (1.1)
[a, b} = ab - (-1)161 Iblba
The supercommutator is extended to nonhomogeneous elements by linearity.
The superalgebra A = AO ® Al is said to be (super) commutative if, for the arbitrary homogeneous elements a, b E A, we have [a, b} = 0.
(1.2)
We introduce an annihilator of the set of odd elements (A1-annihila-
tor) by setting 'A1 = {\ E A: \A1 = 0}. In the sequel, the concept of an A1-annihilator will play an important role when we construct the theory of generalized functions and harmonic analysis on a superspace. In this book, we use the abbreviation CSA for (super) commutative superalgebra.
Example 1.1. A finite-dimensional Grassmann algebra (an exterior algebra) is CSA Gn = Gn(g1, ..., qn) whose elements have the form [[nom
f = fo +
E fi1...ikQi1 ...qik L k=1 il<...
r
(1.3)
where e = 1 is a unit, q1, ..., qn are anticommuting generators qiqj _ -q,qi (and, in particular, q; = 0), and the coefficients fil...ik E K. The monomials {eo = 1, ei = qil ...qik }, i = (i1 < ... < ik) form a basis in the algebra Gn, dim Gn = 2n. The basis of the even subspace Gn,o consists of even monomials and the basis of the odd subspace Gn,1 consists of odd monomials (the parity function is defined by the
1. Differential Calculus
9
relation Iqi, ...qik I = k mod 2); G = G,,,o ® G,,,1. It follows from the anticommutativity of the generators qi that the even monomials and, consequently, the elements of the subalgebra G,,,o commute with all elements of the algebra G,,; the odd monomials and, consequently, the elements of G,,,1 anticommute with one another. The annihilator of the odd subspace 1Gn,1 = KQ1...q,,. The Grassmann algebra G can be realized as the interior algebra A 'K by a correspondence qj H dxj (qjqi H dxi A dxi). Note that in the sequel the part of the coordinates on a superspace will be played not by the numerical coefficients fo,..., fl...n in decom-
position (1.3) with respect to the basis, but by an even and an odd component of the variable f = f ° ® f 1, f ° E G,,,o, f 1 E G,,,1.
This example can be generalized in two directions. First, we can take an arbitrary associative commutative algebra B with unit instead of the field K, and then G,, (B) = A B = {f = (1.3): fi,...ik E B}. Second, we can consider an infinite number of anticommuting generators {qj}-1: 00
(I/
t
Goo(B) = A°Ob =
I 111
f=
1l
E fi1...ikgil...gik fil...ik c B1.
k=0il<...
The algebra GA(B) = A°°B is an infinite-dimensional linear space. Different subalgebras of the algebra Go,,(B) are the main model examples of CSA used in the theory of generalized functions and harmonic analysis on a superspace. The CSA A is Banach if A is a Banach algebra (I I f 9I I< 11f 11 I g Hell = 1), the direct sum A = Ao®A1 is topological (i.e., the projectors rr,, are continuous). On the Banach CSA A we can always introduce an equivalent norm for which JJx0 ®xl11 = Ilx0JJ + JJx'M, 117roll = MiriM = 11Q11 = 1. In what
follows, we consider norms satisfying this condition. Everywhere in the sequel the symbol A = A0 ® Al is used only to denote Banach CSA.
Definition 1.1. The superspace over CSA A = A0 ® Al is a K-
Chapter I. Analysis on a Superspace
10
linear space
KK'm=Ao
xA'=Aox...xAoxA1x...xA1. n
m
The superspace KA'm is a Banach space with the norm IlxIJ = max IIxj ll. The superspace KAn'm is a A0-module, but (note!) not a i A-module.
The structure of the Banach algebra in the Grassmann algebra Gn n If,, is introduced with the aid of the norm I fil...ik I for f
=k=1Eii<...
having form (1.3). If B is a Banach algebra with a norm II JIB, then n Gn (B) is a Banach algebra with a norm f I I= I fit ...ik I I B
i
I
k=0 it < ...
We denote by G100 (B) a subalgebra of the algebra G. (B), namely, C'00
= l f E Gm(B)
EE 00
IIfil...ikJIB < 00
k=0 it <...
G' (B) is a Banach CSA.
We call the CSA G1 00 = G' (K) a Rogers superalgebra (it was widely used by A. Rogers in her works).
1.2. Superdifferentiability. Recall that the mapping F from a Banach space El into a Banach space E2 is Frechet differentiable if
F(x + h) = F(x) + F'(x)(h) + o(h),
(1.4)
where F'(x): El -> E2 is a linear continuous operator and l o(IIo(h)II/IJhIJ) = 0.
h
The superspace is a Banach space, and therefore we can consider Frechet differentiability for the mapping F: Kn'm -+ KK". The class of Frechet differentiable mappings of Banach superspaces is not adequate to physical applications. Here the situation is similar
to the transition from a real analysis on R2 to an analysis on C,
1. Differential Calculus
11
namely, not every differentiable mapping f : R2 -+ R2 is "recognized" as complex differentiable (C-differentiable). In the class C1(R2) we distinguish a subclass A(C) consisting of holomorphic functions. In
order to distinguish the class A(C) in C'(R 2), we can use either the Cauchy-Riemann condition or the C-linearity of the derivative of the function f E C1(R2). The superanalogs of the Cauchy-Riemann condition and C-linearity are nontrivial, and therefore we shall first give a simple definition of superdifferentiability and then consider the analogy with the complex analysis.
Definition 1.2. The mapping F: U -> A, where U is an open subset of the superspace KA'm, is said to be superdifferentiable (Sdifferentiable) at the point x E U if n+m aF
F(x + h) = F(x) + E ax, (x)h; + o(h),
(1.5)
where az (x), j = 1, ..., n + m, are elements of A. Thus the mapping F is S-differentiable if it is F rechet differentiable and the derivative is defined by the operators of multiplication by the elements of CSA: n+m
F'(x)(h) = j=1
aF
ax 7 (x)hi,
h E KK'm.
Let us discuss the question concerning the uniqueness of derivatives. Supose that in addition to representation (1.5) we have a representation
nE F(x + h) = F(x) + Then
aF
(x)
ax, (x) h; + o(h).
- aF (x) I h; = 0,
j = 1, ..., n + m.
Chapter I. Analysis on a Superspace
12
Setting h; = e, j = 1, ..., n, we obtain az (x) =
2 8xj
The derivatives with respect to even variables are uniquely defined. For F odd variables we have (ay (x) axj (x)) = 0 for any element E A1.
--
(x).
Consequently, the derivatives with respect to odd variables are not uniquely defined, but with an accuracy to within an additive constant from -A1. For instance, for a superspace over the finite-dimensional Grasmann algebra Gn the derivatives with respect to odd variables are defined with an accuracy to within an additive constant of the form cQ1...gn, c E K.
Therefore we interpret the derivative with respect to an odd variable as an equivalence class mod ('A1).
If the CSA A is such that the A1-annihilator is trivial, 'A1 = 0, then the derivatives with respect to odd variables are uniquely defined.
In practically all applications use is made of CSA with a trivial Alannihilator. It will be shown in Chap. II that all CSA G100 (B) (and, in particular, Rogers CSA) have trivial A1-annihilator. Reasoning by analogy, we infer that the differentiation operation is linear, i.e., linear with respect to even variables and linear modulo -LAI with respect to odd variables:
19x j
(a1F1 + a2F2) = a119x1 + a2 19x2 ,
a1, a2 E A.
The S-differentiability of a function with values in the superspace Kip (or Kip, where A' is a Banach CSA, A' j A) is defined by analogy, componentwise. The derivatives defined by (1.5) are known as right-hand derivatives. By analogy, we can introduce left-hand derivatives, but then we must replace representation (1.5) by n+m
F(x + h) = F(x) + E h;
aF
(x) + o(h).
(1.5a)
j=1
The right-hand and left-hand derivatives with respect to the variables x1, .., xn coincide, but those with respect to the variables xn+1, ..., xn+m are, in general, different. When necessary, we shall use the
1. Differential Calculus
13
symbol aR to denote the right-hand derivative and the symbol aL to denote the left-hand derivative. In the sequel, we repeatedly use the following formula that relates the right-hand and left-hand derivatives with respect to odd variables: aLF axi
-
(1)1+IFI
aRF axe ,
(1.6)
j = n + 1, ..., n + m. We shall prove this formula (illustrating the rules of operating with a parity function). Using representation (1.5a), we have
JFJ _
h; aLF aL F = I hj l + ax; ax;
=1+
aL F
axi
All operations involving a parity function are considered mod 2, h, aLF = (_1)Ih,IIOLF/axjIOLFh.i =
(-1)1+IFIaLFhj
ax;
ax;
ax;
Relation (1.6) holds for functions F that assume homogeneous values. It can be applied to functions with values in A with the aid of the decomposition F = F' (D F1, where F°: KK,m -> Ao, F': KK,m -+ Al.
Example 1.2. Let F: K11'2 -+
aRF
F(x, 01i 02) = x30102. Then
aLF
ax - ax = 3x201, 02,
aRF
aLF
a0, =-a01
3
=-x
02i
aRF a02
aLF
=-ae2 =-x
3
01.
Example 1.3. Suppose that x belongs to the group of invertible elements Inv(A°) in the comutative Banach algebra A° and F(x) _ X_'. Then aaxl = x-2 (the set Inv(A°) is open).
Example 1.4. Let 0 = (f;) be canonical numerical coordinates in a finite-dimensional linear space Gp,1, i = (i1 < ... < i2k+1), see F;(0) = fi. Then (1.3). We shall consider functions F;: K j -+ these functions are differentiable being mappings of finite-dimensional (K-)linear spaces, but not S-differentiable.
Chapter I. Analysis on a Superspace
14
Note that for finite-dimensional (K-)linear spaces, Frechet differentiability coincides with the ordinary differentiability. Since the S-differentiability implies Frechet differentiability in Banach spaces, it follows that all theorems of differential calculus in Banach spaces [37] hold with a replacement of linear continuous operators (Frechet derivatives) by operators of multiplication by the elements of the algebra A. In the theorems that follow, we denote by U, V, W the neighborRA2'm2, zo E hoods of the points xo E RAl'ml , yo E RA3'm3
Theorem 1.1 (the chain rule). If the functions f : U -+ RA 2'm2 and g: V -+ RA3'm3 are S-differentiable at the points xo and yo = f (xo), then the composite function cp = g o f : U -+ RA3im3 is Sdiferentiable at the point xo and cp'(xo) = 9'(yo)f'(xo) The only distinguishing feature of Frechet differentiability is that the derivatives cp'(xo), g'(xo), and f'(xo) are matrices composed by elements of the algebra A.
Theorem 1.2 (on the differentiability of an implicit function). Let the function F: U x V -+ RA3,m3 (n2+m2 = n3+m3) be continuous at the point (xo, yo) and F(xo, yo) = 0. If there exist partial S-derivatives aF/ay and OF/ax which are continuous at the point (xo, yo) and the matrix aF/ay(xo, yo) is invertible, then there exists an implicit function y = f (x) which is S-differentiable at the point x0 and f'(xo)
(OF
ay (xo, yo))
-1
ax
(xo,
yo)
The existence of an implicit function follows from the classical theorem for Banach spaces. The only distinguishing feature is that the derivative f'(x) is a matrix composed by elements from A obtained by the multiplication of the matrices -(ay (xo, yo))-1 and a (xo, yo). By analogy, we can formulate a theorem of an inverse function.
Theorem 1.3 (Leibniz formula). Suppose that the functions f, 9: U -+ RA2im2 are S-differentiable at the point x0, and g(xo) is a
1. Differential Calculus
15
homogeneous element in A (i.e., g(xo) E Ao or g(xo) E A1). Then the formula axi (fg)(xo) = aR
f(xo)-Rg(x0)
axi
+
(-1)Ixill9(xo)IaRf
axi (xo)g(xo)
holds true.
Proof. Using (1.5), we obtain f(xo + h)g(xo + h) = f(xo)g(xo)
n+m
+
[f(xo)(xO)hi + axf (xo)hig(xo) + o(h).
i=1
It remains to point out that hig(xo) = (-1)Ih:ll9(xo)Ig(xo)hi. It should be emphasized that all relations containing derivatives with respect to odd variables are relations mod 1A1.
Example 1.5 (counterexample to Leibniz rule). Let A = G, g1...gn_19, 0 E Gn,1, g(O) = qn, and then W(O) = f (9)g(O) - 0. f (9) = Leibniz formula for co gives P" (9) a e (9)qn = -gl...gn. It remains to note that g1...gn E1 G., 1.
Example 1.6 (counterexample to a chain rule). Let A = Gn, f (9) = gl...Qn_19, g(O) = q,,9, and then cp(9) = f (g(9)) = 0. The chain rule for cp gives 19
ae (B) = ae
(e) ae (9) = Q1...gn.
If the A1-annihilator is trivial, then all these subtleties are absent. We can define by induction higher-order S-derivatives. We denote by Cn(U) (- Cn(U, R1"P)) the space of functions f : U -+ R" which
are n times continuously differentiable; by Sn(U) (= Sn(U, R',P)) we denote the subspace Cn (U) consisting of n times continuously S-differentiable functions; and by SP,9(U) we denote the space of functions which have continuous S-derivatives of order p with respect to even variables and of order q with respect to odd variables.
Chapter I. Analysis on a Superspace
16
By analogy, we introduce spaces C°°, S°°, S°°,9, Sp"°°. We use the symbol P(KA'm, A') to denote the space of polynomial functions p: n
KAn,m -+ A', where A' is a Banach CSA, A C A': p(x) _ E p.x" 101=0
where pa E A', x° = xi 1...xn+m ; aj = 0, 1 for j = n + 1, ..., n + m. Throughout this book we interpret the integral of a function with values in a Banach space as a Bochner integral (see, e.g., [33]). For continuous functions of a real argument (which are mainly the functions that we are considering), the Bochner integral coincides with a Riemannian integral for vector-valued functions (the convergence of Riemannian sums is considered in the norm of a Banach space). In that follows, it is more convenient to denote even and odd variables by different symbols, namely, x = (y, 9), where y E Ao, OE Am. We use the symbol T (a, , r, p), a E An , e E Am, r E R+, P C R+ to denote a polydisk: Ilyj - ajlj < rj, MMB3 - jll < Pi.
Theorem 1.4 (Taylor's formula). Let f E S"(T(a, , r, p)), and then, for all x = (y, 0) from T (a, l;, r, p), Taylor's formula
f (y, 9) = E ICI+IR1=0
a y°`
I f (a, t)(x - a)'(9 - e)A + rp(y, 0), 00
0 as y --- a, 0 holds true. - a, 9 Now if f E SP+1(T (a, , r, p)), then the remainder has the form
where 11 rp(y, 9) 11 /11 (y
rp(y, 9)
_
1 (1
f 1.1+IQ1=P+1
t)p acf+a
p
LJo
1
f (a + ty, l; + t9) dtJ
x (y - a)"(9 - l;)'. Here, as usual, a = (a1, ..., an), Q =
n
(01, ..., Qm), lal = E aj, j=1
l al = E Qj, a! = a1!...On!, i=1
a°`
01
aya = ayi 1 ...
°^
ap
apl
aAm a9fi1...
ayn"
a9A
a9mm
1. Differential Calculus
17
andf=0,1. In order to prove Taylor's formula, it suffices to employ Taylor's formula in Banach spaces. Taylor's formula implies
Proposition 1.1. If, for every y E T(a, r), the function f (y, ) belongs to the class Sp+1(T(e, p)), then the following Taylor formula is valid for the variables 9: aep f
f (y, B) _
(y, 0 (0 - W + rp(y, 0)
101=o
rp(y, 0)
E
[
t)p 90
I (1 pi
of f (y, e + t9) dt] (9 -1;)p.
IAI=p+1
In particular, if p > m, then rp(y, 0) - 0 and the function f (y, 9) is a polynomial with respect to 9 of degree m:
f (y, 0) _ E 5-9Q f (y, 0(0 - )Q. 1101=0
Thus S7) (U) = S""°° (U) for q > m + 1; Sm+1(RO,m) - P(Am, A),
i.e., every m + 1 times differentiable function of m anticommuting variables has the form aef (0 -
f (0) _ 101=0
The question arises whether there can exist S-differentiable functions which nonpolynomially depend on 0. As we can see from the following example, functions of this kind do exist.
Example 1.7. Suppose that A = G2, f : R°'1 -+ A, and f (0) _ W(r1, r2)g1g2, 9 = rlgl+r2Q2 E A1i (,(r1, r2) is an arbitrary real function
of the class C'. The function f (0) belongs to the class S1. Thus, for finite-dimensional Grassmann algebras, an m times Sdifferentiable function of m anticommuting variables may not be a polynomial. As to other classes of CSA, the situation is not clear.
Chapter I. Analysis on a Superspace
18
In Secs. 1.3, 1.4, 1.5, and 1.8 we assume that the A1-annihilator is trivial.
1.3. Superanalyticity. The function f : U -> A, where U is an open set in the superspace KA'm, is S-analytic at the point xo E U if, in a certain neighborhood of the point xo, the function f can be expanded in a power series f (x) =
ff(x - xo)Q,
fQ E A,
IIf,IIRQ < 00
Ilf IIR =
a
Q
for a sufficiently small radius R = (R1, ..., Rn+,n) E R+m
The S-analytic function belongs to the class S°° and the (left) coefficients fQ =
,
g (xo).
In the variables x = (y, 0) E KA'm the function f can be represented as
m
f (y, B) = E fQ (y) (0 - Bo)Q, IIQ11=0
where fQ(y) are S-analytic functions of commuting variables. Note that every analytic function f (t), t E Kn, can be canonically extended to a S-analytic function f (y), y c A. The definition of S-analyticity on AO is an ordinary definition of analyticity on a commutative Banach algebra. 1.4. Group of translations of a superspace. Canonical commutation relations on a superspace. Let us consider the representation of a group of translations of the superspace RA" in the functional space S°O; p(h): S°° -+ S°°, p(h) (f) (x) = f (x + h), h E RA'm. The (left) infinitesimal operators of this representation are
S-derivatives a , j = 1, ..., n + m. We introduce operators of left momenta pjL(f) (x) = 4 az, (x) and left coordinates gjL(f) (x) = xj f (x). The operators of pulses and coordinates on the superspace RA'm satisfy the canonical commutation relations (CCR) [PjL, PkL} = 0,
[ jL, QkL} = 0,
l
{PjL, 4kL} = -iht5kj.
In particular, the square of an odd derivative is zero.
(1.7)
1. Differential Calculus
19
By analogy, we introduce the right operators of the coordinate qjR and momentum J5jR.
1.5. Transformations of supersymmetry (SUSY). The SUSY transformations mix up commuting and anticommuting coordinates. The simplest SUSY transformation is an affine transformation of the superspace TE: R n -+ x'µ = xµ + Eµ9µ,
9µ = 9µ + Eµ,
(1.8)
where xµ E Ao, 9µ E A1, p = 1, ..., n, and Eµ E Al are anticommuting constants, namely, the parameters of SUSY. Let us consider the representation of SUSY in the functional space S°°; p(TE): So0 -4 S00, p(TE)(f)(x, 0) = f oTE(x, 9). We shall calculate infinitesimal generators of SUSY: P(TE) (f) (x, 0)
Eµ9µ µ=1
af (7xµ
- f (x, 9) 'ad f
61
(x, 9) + E µ=1
(x, 9) + O(E)
C79µ
n
_
EµDµ(f)(x,9)+O(E), µ=1
a where D. = 9µ s a + ae Note that Dµ =a2µ , i.e., the generators of SUSY D,L, p = 1, ..., n, are square roots of translation generators with respect to commuting Y
coordinates.
1.6. Supermanifolds. A manifold M with a model Banach space KA'm and S-smooth (or S-analytic) atlas is known as a supermanifold. The supermanifold M is locally homeomorphic to an open set in the superspace KK'm
The dimension (n, m) of the model superspace KA'm is the dimension of the supermanifold M. Note that a supermainfold M of dimension (n, m) can have an infinite dimension over a field K. For
Chapter I. Analysis on a Superspace
20
instance, for a supermanifold M with a model space K"' (Rogers supermanifold), every one of the local coordinates xj E A0, j = 1, ..., n, Oj E Al, J= 1, ..., m, has the form 00
xj = E E
xja1...Cf2kQckl*_q(k2k,
k=0 Ol <... «2k 03 0 1 k=0 al <...
{xj,,, Bj,,} are local coordinates of an infinite-dimensional (Banach) manifold over a field K. 1.7. Lie supergroups. A Lie group which is a supermanifold with a S-smooth group operation is called a Lie supergroup.
1.8. Superconformality. Riemannian supersurfaces. Let us consider a complex S-analytic supermanifold M of dimension (1, 1); z is a commuting local coordinate and 0 is an anticommuting local coordinate. It follows from the S-analyticity of M that the transition functions have the form z = f (z) + Bcp(z),
B = i(z) + Bg(z),
(1.9)
where f, o, cp, g are S-analytic functions of the commuting coordinate z. In every map we can consider a SUSY transformation: z' = z + EB,
0' = 9 + E. Upon the transition (1.9) from one map to another, the SUSY generator D = as + U , D2 = aZ is transformed according to the law
D = (D9)D + (Di - OD9)D2.
(1.10)
The transition functions (1.9) are said to be superconformal if D =
(DO)D, i.e., the second term in (1.10) is zero. Consequently, Di =
9D9 or cp = go, g2 = d + d Superconformal transition functions .
have the form
f +9o V '
'+9 dz+ z.
(1.11)
The supermanifold M with the superconformal transition functions (1.11) is called a Riemannian supersurface.
2. Cauchy-Riemann Conditions
2.
21
Cauchy-Riemann Conditions and the Condition of A-Linearity of Derivatives 2.1. Examples of Cauchy-Riemann conditions for com-
mutative algebras. It should be pointed out, first, that if A° = R (A1 = 0), then condition (1.5) of S-differentiability of the function f defined on R"0 = R" coincides with the definition of ordinary differentiability.
Example 2.1 (complex analysis). Suppose that A° has two generators e° = 1 and el = i, i2 = -1, Al = 0, A = A° (R"° = A° = C) and let f : A° -+ A° so that f (z) = f (x + iy) = u(x, y) + iv (x, y), where u, v, x, y are real. Then condition (1.5) of S-differentiability of f (z) with respect to the variable z = x + iy is equivalent to the = a au = - av or = 0. Cauchy-Riemann conditions au ay' ay - ax a2 ax Example 2.2 (hyperbolic analysis). Under the conditions of Example 2.1, we assume that e° = 1, el = j, j2 = 1. In this case, conditon (1.5) of S-differentiability is equivalent to the Cauchy-Riemann
= a au = a hyperbolic conditions: au ax - ay' ay - ax Note the following essential difference: in Example 2.1, the CauchyRiemann conditions lead to a Laplace equation whereas in Example 2.2 they lead to a wave equation. Example 2.3. Under the conditions of Example 2.1, we assume
that e° = 1, el = j, j2 = 0. In this case, the Cauchy-Riemann conditions assume the form ay = 0, ay = a Hence, we find that u(x, y) = u(x), v(x, y) = yu'(x) + v(x), where u and v are arbitrary .
real functions, u E C2, V E C1.
In Examples 2.2 and 2.3, the Cauchy-Riemann conditions, together with the Cl-smoothness, are equivalent to S-differentiability. However, in contrast to complex analysis, here conditions of the Cauchy-Riemann type do not imply analyticity.
2.2. The Cauchy-Riemann conditions in superanalysis. Theorem 2.1. For the function f to be S-differentiable in U C RA'm, it is necessary that it be Frechet differentiable in U and satisfy
Chapter I. Analysis on a Superspace
22
in U the following conditions of the Cauchy-Riemann type:
f'(x)(u)v
- (-1)lull"I ff(x)(v)u = 0,
f'(x)(uv) = f'(x)(u)v,
u E RAm,
u,v E R;;m;
(2.1)
v E Ao+m;
(2.2)
or, in components,
of
(-1)'UiIIVila (x)(vi)ui = 0, axi (x)(ui)vi -
(2.3)
i = 1, ..., n + m;
luil = Ivil = Ixil,
a ()(uivi) = a (x)(ui)vi,
(2.4)
jail _ Ixil, i = 1, ..., n + m. For m = 0, condition (2.1) is sufficient. Moreover, it is equivalent to condition (2.2) which, in turn, is equivalent to the conditions Ivil = 0,
axi (x)
Of
(ui) = a
(x) (e) ui,
i = 1, ..., n;
(2.5)
the relations axi
(x) =
(x) (e),
i = 1, ..., n;
(2.6)
hold true.
Proof. Necessity. Using the fact that the Frechet derivative is a multiplication operator for S-differentiable mapping, we have a (x)(ui)
=
(2.7) af(x)ui,
Of
Of (x) (vi) =
(x)vi
(2.8)
for any vectors u = (ul) ..., un+m), v = (vl) ..., Vn+m) E RA ,m. Hence, postmultiplying relation (2.7) by vi and (2.8) by ui, subtracting one from the other, and using the law of (super)commutativity in CSA A, we get condition (2.1). By analogy, we get condition (2.2).
2. Cauchy-Riemann Conditions
23
Sufficiency. Let n = 0. Setting ui = e in (2.4), we get (2.5). Hence follows the S-differentiability of f. Conditions (2.4) follow from conditions (2.5)
a
(x)(Uivi) =
a
(x)(e)uivi =
8x (x)(ui)vi.
Corollary 2.1. If the function f (x) is S-diferentiable in U C AO and the S-derivative - (x) is Frechet differentiable in U for a certain k, then the function a (x) is S-diferentiable in U. 2.3. Cauchy-Riemann conditions for finite-dimensional CSA. Let A = A0 ® Al be a finite-dimensional CSA with basis eo = e, ..., eN and structural components Cep: N
eaep =
(2.9)
C.pe.r. 'Y=O
The homogeneous element x E A has the form x = > taea, Iea a
(x)(ea) = a (x) and setting ui = >uaiea, uai E R, we obtain a (x)(ui) _ E - (x)uai. The Cauchy-Riemann type IxI.
Denoting
a
conditions (2.3), (2.4) assume, respectively, the form
I
(f-_cxec
f (x)el uaiv(i = 0, pi
- (-1)leaI keI
i
E ( at ai (x)(eaep) ap
8t ai (x)ep)
0,
jepl = 0,
whence, because of arbitrariness of the numbers uai, vpi, we take into account (2.9) and infer that
ati
(x)ep - (-1)Ieol leal
(x)ea = 0,
(2.10)
At
a E C-10 ry
(x) = atyi
(x)ep, ai
jepj = 0.
(2.11)
Chapter I. Analysis on a Superspace
24
Relation (2.8) assumes the form af
at.i (x)
af(x)ea,
1xil = le.1.
(2.12)
=
2.4. Cauchy-Riemann conditions for finite-dimensional Grassmann algebras. In accordance with Theorem 2.1, the CauchyRiemann conditions (2.10), (2.11) are necessary and sufficient for Sdifferentiability in an even case and necessary in an odd case. Let us prove that for finite-dimensional Grassmann algebras Gn conditions (2.10), (2.11) are also sufficient. We shall need the following algebraic lemma.
Lemma 2.1. ([19, p.
Let q1, ..., qn be generators of the
17]).
Grassmann algebra Gn, n > 2, and let the elements A1i ..., An from Gn satisfy the relations
Aiq,+A;q;=0, i,j=1,...,n. Then there exists an element F E Gn such that
Ai = Fei,
i = 1, ..., n.
Theorem 2.2. For the function f = f (x, B): U -4 A, where A = Gn, to be S-differentiable in U C A, it is necessary and sufficient that the following Cauchy-Riemann type conditions be fulfilled in U:
of
of
k is even,
axQl...Qk = ax0 af af q+ q; = 0, aei ae;
af
af
ao"i...Ckk
ae.l
q02
i, j =1, ..., n, k is odd,
here
x = E xal...Qkq l...gQk E Gn,O; k is even
(2.13) (2.14)
(2.15)
2. Cauchy-Riemann Conditions
x=
25
OQ1...akgal...gak E G,,,°.
k is odd
Proof. The necessity of conditions (2.13), (2.14), and (2.15) and the sufficiency of conditions (2.13) (a purely even case) follow from Theorem 2.1 in accordance with relations (2.10), (2.11) with the use of the canonical basis via the generators of the algebra Gn. Let us prove the sufficiency of conditions (2.14), (2.15) (a purely odd case).
By virtue of Lemma 2.1, it follows from (2.14) that there exists a function F = F(x, 0) E A such that a = Fe,, i = 1, ..., n. Hence, it follows from (2.15) that of
= 1" ga1...Qak.
aeal...ak
Corollary 2.2. If A is the Grassmann algebra Gn, the function A is S-diferentiable in U, and its derivative a (x) f : U C RA'm (x) is S-diferentiable in U. is Frechet differentiable in U, then
a
Conditions (2.13), (2.14), (2.15) of the Cauchy-Riemann type can be represented in the following symmetric equivalent form: 19f
axal...ak
qpl ... gQ9 -
aaaf
ak
of
a1131...0, gA1...ggp +
of aeal...ak
nn p f
qal ...qak = 0,
p, k are even,
Qal ...qak = 0,
p, k are odd,
Op
of
qa,+l ...qak
7
k is odd.
aeal...a,
2.5. The condition of A-linearity of an S-derivative on the module An+'. It follows from the definition of an S-derivative (1.5) that the mapping f : R''0 = Ao A is S-differentiable if and only if f is Frechet differentiable and the R-linear operator f'(xo): Ao -+ A is Ao-linear. However, for the mappings f : R m -+ A, the A°-linearity of the Frechet derivative is not sufficient for S-differentiability (see the Cauchy-Riemann conditions).
Chapter I. Analysis on a Superspace
26
Example 2.4. Let A = G1, and then AO = R, Al = Rq, q2 = 0. The mapping f : Al -+ R, f (xq) = x, x c R is Ao-linear but not S-differentiable. By analogy with complex analysis, it is natural to impose the condition of A-linearity on the Frechet derivative. However, as has been
pointed out, a superspace is not a A-module. There appears an analysis on the pair (Rn'm, An+m) = (superspace, A-module).
Theorem 2.3. The mapping f : R m -+ A is S-differentiable at the point xo if and only if f is Frechet differentiable and the Fr6chet derivative f'(xo) can be extended to the A-linear operator from An+m into A.
Proof. 1. The operator of multiplication by an element of the algebra A is a A-linear operator from A into A.
2. Let A: A -+ A be a A-linear operator. Then A(x) = A(e)x, where A(e) E A.
3.
Integral Calculus 3.1. Integration with respect to anticommuting variables.
The integral over the superspace An is defined as a A-linear functional (a generalized function), which is invariant with respect to shifts, on the space of test functions P(An, A) (on the space of polynomials = the space of analytic functions = the space of infinitely diferentiable functions). The following axiomatic definition of an integral is proposed [20]. An integral is a mapping I: P(Ai , A) -4 A satisfying the following conditions. 1. A-linearity (on the left): I (p f + \g) = µI (f) + \I(g), µ, A E A,
f, g E P. 2. Translational invariance: I (fe) = I(f), where fe(B) = f (0 + t;) for all
e A, f E P.
Let us assume that an integral I (f) of this kind exists. We denote I(0E) = I, where E belongs to the set of multiindices Nn = Nn(A) {E _ (E1, ..., En); Ej = 0, 1, 9E = 0"...0'- jt 0}. We set L1E1 _ {S;
3. Integral Calculus
27
= 0E, 0 E All. The symbols L1 and 1L denote the right and left annihilators of the subset L of the CSA A; L1 = {A E A; LA = 0}, 1L = {A E A; AL = 0}. Let f c P(Ai , A), f (0) _ pE0E, where EENN
PC = as (0) E A/1L1El. By virtue of condition 1, we have
fE
I (f) = E AE, IE,
(3.1)
P.
EENn
For this integral to be a single-valued functional (the lack of unique-
ness appears because the coefficients pE are defined with an accuracy to within an additive constant from the annihilator 'L1E1), it is necessary and sufficient that the elements IE satisfy the relations I(pEBE) = pEIE = 0 if pE E 1LIEI. Thus, we have conditions IE
E (1LIE1) -,
(3.2)
E E Nn.
Note that, obviously, L1E1 C ('LlE1)1. Furthermore, by virtue of condition 2, we obtain CEIE = E PEI((0 + )E) EENn
EENN
E PETE + E EENn
EENn
E
CE
±,9`0'
,
IE-OrI>1,o<E
whence we derive
E
pE
EENn
floE
= 0.
IE-oI>1,o<E
From this relation, using the arbitrariness of pE E A and E A1, we get relations p,I C' = 0, JE - al > 1, o < E; o, e E Nn, which are equivalent to the relations (since A contains the unit e) IE = 0, E A1i E E N, JEJ < n - 1. In other words, IEE1A1i We have thus proved
EENn,
Je
(3.3)
Chapter I. Analysis on a Superspace
28
Theorem 3.1. For an integral I (f) satisfying conditions 1 and 2 to exist, it is necessary and sufficient that a set of elements IE, E E N, be defined satisfying conditions (3.2), (3.3). This integral is defined by relation (3.1).
If the multiindex (1, ..., 1) E Nn (i.e., the CSA A is such that 0, say A = Gn), then the element I(1, ..., 1) = I(01 ... On) = 81...On K. play a special part, namely, icn must satisfy only condition (3.2), n E ('La)', and, hence, possess a larger arbitrariness than the other IE .
The case
IE=O,
EENn,
IEI
(3.4)
is an important special case which we shall bear in mind in what follows. In this case, integral (3.1) has the form an
I (.f) = 891...'
(O),n,
fE
P.
(3.5)
Remark 3.1. In the theory of differential equations on a superspace use is made of CSA in which the A1-annihilator is trivial, i.e., 'A1 = Al = 0. For these CSA, all the statements presented above become trivial: since 'L1E1 = 0, it follows that condition (3.2) imposed on the coefficients IE disappears (IE E A); it follows from condition (3.3) that the only integral over the superspace An (in the case 'A1 = Ai = 0) is integral (3.5). Note that in the CSA A with a trivial A1-annihilator we have 01...On 0 0 for any n. Integral (3.5) can be represented in the form I (f) = J(f )icn, where an f
J(f) =
(3.6)
891...89n (0) .
Since the derivative is defined with an accuracy to within an additive
constant from the annihilator 1'Ln, it follows that J: P -+ A/1Ln is a single-valued mapping. This mapping also satisfies conditions 1 and 2, and therefore we shall also call it an integral and denote J(f) = f .f (Oi, ..., On) d91...do,, = An 1
f
.f (9) d9.
3. Integral Calculus
29
Properties of integral I(f): 1. f 01...0,d91...dOn=1. 2. The integral of the derivative a
J a9
j = 1, ..., n.
(9) d91...dOn = 0,
3. Integration by parts: if g(O) is a homogeneous function, then f f (0) a8 (9) d9 = (-1)191+1 f
ofj (0)g(0) d9.
4. A linear change of variables: if 0 =
A-1 exists, then
f f (0) dO = (det A)-1 f f (AC) dC.
Proof.
ff
(3.7)
an
d = ael ..
aen
(0) det A. 1
5. f f (9) d9 = f [.f f (01, ..., Ok, Ok+l, ..., 9n) d91...dOk] d9k+1...dOn.
It follows from the property 5 that the calculation of an n-fold integral reduces to a calculation of simple integrals according to the rules
f d9 = 0,
fOdO=1.
6. The 6-function of anticommuting variables f f (0) (01
- e1)...(on - n) d91...dOn = f (0
This relation allows us to introduce a 6-function
b(9-0 =: (81 -0...(On -W
(3.8)
Chapter I. Analysis on a Superspace
30
7. An odd change of variables. Let 0 = O(ff) be an odd change of variables of the class Sn+1(An) such that 0(0) = 0 and the element det a{ If-o is invertible in A. Then the formula f f (0) dO =
ff(0(e))det'()de,
f c P,
(3.9)
for a change of variables is valid.
Remark 3.2. It should be pointed out, first of all, that all eleof the matrix ao are even and therefore commute. Next, ments it is veried, as usual, that the square matrix with elements from A0, A = 1Ja1 -1, is invertible if and only if det A is invertible. Therefore the matrix a (0) is invertible. However, the matrix 11
ao
W=
and the matrix quently, there exists
(0) +
W-
'90(o)
aO- (0)) consists of nilpotent elements. Conse-
)(o)
-1
(a (e))
ao
(go)
=I-
-1
-1
(go)
-1
)
where I is an identity n x n matrix. Moreover, expanding it in a geometric series, we find that the elements of the matrix belong to P(Ai, A). The formula for a change of variables can be proved by induction (see Fikhtengolts [60]). For n = 1, the change of variables has the form 9 = <, where a c AO and there exists a-1. In this case, (3.8) = (3.7).
Remark 3.3. Integral (3.5) can be extended as a A-linear continuous operator to the whole space C(A?) endowed with the topology of uniform convergence on compact sets. This can be done with the aid of the relation 1
1
I (f) = f...ff(e1t1...,etn)t1...tdt1...dt, -1
-1
3. Integral Calculus
for instance, with nn
31 s
"
3.2. Integration with respect to commuting variables. Let us consider the field C as an algebra AO = C over R. Then the integral I = fy f (z) dz along the curve 'y, z = z(t) E C is defined as a t2
line integral of the second kind; I = f f (z(t))z'(t) dt. It is natural to ti
generalize this definition to the integral along the curve 'y, z = z(t) E AO in the arbitrary commutative algebra A0. Let G be a domain in R" and the mapping cp: G -+ Ao be of class C'. The surface a = W(G) is called an n-dimensional singular surface in Ao (we use the term "singular" since W is defined on a domain in Rn and not in Ao). The integral of the function f (x1...xn) over the n-dimensional singular surface a is defined by the (ordinary) relation
1 f (x) dx - f f o `p(y) det aV (y) dy The integral possesses "good" properties if f E S1.
3.3. Integration with respect to commuting and anticommuting variables. Upon a change of variables on the superspace Rn'' block matrices of the form
M= CA DB
(3.10)
appear, where the matrices A and D have even matrix elements and B and C have odd matrix elements. We shall first find out conditions for invertibility of matrices of the form (3.10). Using the well-known formulas for block matrices,
01 M= A CI
1
M-1 =
I
A-1B
0 D-CA-1B
I B 0 D
I -A-'B(D 0
(D - CA-1B)-1
CA-1B)-1
A - BD-1C 0 D-1C A-1
0
CA-1 I
I
Chapter I. Analysis on a Superspace
32
(A - BD-1C)-1
0
-D-1C(A - BD-1C)-1 I
I -BD-1 0
D-1
we infer that the matrix M is invertible if and only if the matrices A and D - CA-1B or D and A - BD-1C are invertible. The latter matrices, because of the evenness of their elements, are invertible if and only if their detrminants are invertible. For what follows, we shall need a quantity which is called a superdeterminant of the matrix M and denoted by sdet M:
sdet M = det A det-1(D - CA-1B) = det (A - BD-1C) det-1D. For the sake of comparison with the formula for the superdeterminant of the matrix M, we give a formula for the ordinary determinant of the block matrix M where the matrices A, B, C, and D have commuting elements:
det M = det A det (D - CA-1B) = det (A
- BD-1C) det D.
Note that the superdeterminant of a matrix is multiplicative, i.e., sdet (M1M2) = sdet M1 sdet M2. The proof of this fact for Grassmann algebras can be found, for instance, in [4]. Since this proof is of a purely algebraic character, it can be immediately generalized to the derivatives of CSA. We shall also need the obvious generalization of the S-differential calculus to the mappings f : 0 -+ where 0 is a domain not in RA'"`, but in ro x rm, where ro and r1 are R-linear subspaces of AO and A1. The case I'o = R, r1 = Al is of a special interest to us. Let X = X (Y) be S-smooth mapping of the domain 0 C ro x rm in
In the coordinates X = (x, 0), Y = (y, ) this mapping can be written as x = x(y, ), 0 = O(y, ). The Jacobi matrix of this mapping has the block structure (3.10), where A = ay, B =
a
.
Using the conditions of invertibility of block matrices
and the arguments similar to Remark 3.2, we find that the matrix J(y, 0 is invertible for all Y = (y, ) E 0 if and only if det Aj£-o and det (D - CA-1B&-o or det and det (A - BD-1C)lc=o are invertible.
33
3. Integral Calculus
Let Y = Y(Z) be S-smooth mapping of the domain 0' c ro x rm into the domain O. In the coordinates Y = (y, ) and Z = (z, 77) we have y = y(z, 77), = e(z, 77). Then, by the chain rule we have ax _ ax". We assume that the Jacobi matrix ax is invertible, and then the matrices ax and az are also invertible. Furthermore, sdet az = sdet aY sdet a
If X = X (Y) = X (y, e) belongs to the class
Sl,m+l (O, RA''"),
r0 = Ao, r1 = A1, then m
m
x(y,e) _ E XE(y)(E, I£I=0
0(y,0) _ i 0E(yW,
(3.11)
IEI=0
where BE, xE E S1. Suppose that mapping (3.11) is one-to-one and the
Jacobi matrix is invertible. Then this mapping is called a change of variables on a superspace. Let G be a bounded oriented domain in R" and the S-smooth mapping cp: G x Am -4 Rn'm is defined (i.e., ro = R, r1 = A1),
cp(GxAm)=M. The collection (cp, M, G), or simply M = M(cp, G), is called an (n, m)-dimensional singular supersurface. In the coordinates cp(t, 77) = (x(t, 77), B(t, 77)), a singular superX = cp(t, 77), where surface can be written as M = [X E (t, 77) E G x Am].
We assume that the Jacobi matrix J = J(c2) of the mapping cp is invertible. Let 0 be a neighborhood of M in Rn'm and let a function f : 0 -4 A
be given. The integral of the function f over a singular supersurface is the expression fMf(X)dX = 1M f (x, B) dx dB
=
fAr
[f f ((t, 77)) sdet J() (t, 7) dt] d77.
The integral exists and the formula for a change of integration order fG [fnm f (cp(t,
77)) sdet J(cp) (t, r7)
d77]
dt
Chapter I. Analysis on a Superspace
34
am a77i...877m
=
f
fG f (W (t, 77)) sdet J(W) (t, 77) dt
am
G al71...a77m
[f (W (t,
n)) sdet J(cp) (t, 77) ] dt
is valid if, for every t E G, the function f (cp(t, 77)) belongs to the class Sm+1 with respect to 77 and there exists an integral JG f (cp(t, r7))
sdet J(cp) (t, r7) dt.
This definition does not depend on the reparametrization of a singular supersurface in the following sense. Let t = t(t') be a diffeomorphism of the domain G' onto G with a
Jacobi matrix at/at' which is continuous in G' and det (-,) > 0 and 77 = 77(t', 77'), and let G' x Am -+ Am be a S-differentiable function being invertible from the class S"+1 with respect to if, with det and the quantity M = [X E RA'm; X = cp o Q(t', 77'), (t', r7') E G' x Ai ], where the mapping Q(t', 77') = (t(t'), 77(t', 77')); G' x Am --> G x Am.
Performing the indicated change of variables with respect to 77 for every fixed t E G on the right-hand side of (3.12) and using relation (3.8) for a change of variables in the integral with respect to odd variables and then performing a change of variables t = t(t') (this is an ordinary change of variables in R"), we obtain fn
'1"
f [IGI
r7')))
sdet J(cp)(o,(t', rj )) det-1
877
det
a17
at dt', drj at
1
fAm
[f f(
po o,(t', rl )) sdet J(cp o o) (t', T7') dtl drj J
where we used the relations
sdet J(u) = det-1 sdet
J(W) (o (t', 77'))
det &,
sdet J(a) (t', 77') = sdet
J(cp o o) (t', 77") .
Suppose that a change of variables X = F(Y), Y = (y, ) is defined and let M = M((p, G) be a singular supersurface; M1 = M1(cp1, G) is
3. Integral Calculus
35
a singular supersurface induced by the mapping F from M: M, = F-1(M), cp1 = F-1 o cp. Then the formula fM f (X) dX = fMi f (F(Y)) sdet aY dY
(3.13)
for a change of variables is valid.
The sufficient condition for the existence of integrals and their equality is that m
f (x, 0) = E ff
(x)0',
ICI-0
where fE E Sm+1-IEI.
In order to prove relation (3.13), it suffices to use expression (3.12)
for writing out the integrals over the singular supersurfaces M and M, and employ the theorems of the derivative of a composite function and an inverse function.
Example 3.1. Let G = (a, b) C R, cp(t) = te, M = M(cp, G) (a, b)e C A0; F: AO -+ A0, F(y) = y - a, a E A0, and then M, = F-1(M) = (ae + a, be + a) _ {y E Ao: y = a + se, s E (a, b) }. Relation (3.13) yields be+a
be
fea
x dx =
fae+a
(y - a) dy.
Example 3.2. Let G = (a, b) C R, cp(t, 01, 02) _ (te, 01 02), M = (ae, be) x A2 c RA 2; F: RA2 -4 RA2, F(y,.1, e2) _ (y + U2, e1, e2), and then sdet ar = 1 and relation (3.13) yields i
be
fA2
x dx dO, d02 = j f i
befae
e-(if2
(y +
dy d 1 dC2 = 0.
Chapter I. Analysis on a Superspace
36
Example 3.3. Let G = (a, b) C R, co(t, 0) = (te, 0), M = F(y, (y - i ), 77 = const E Al, and (ae, be) x Al C ,
then sdet ay = 1 and relation (3.13) yields fAl
f
6e
ae
fl
x dx dO =
be+i
J ae+nf
(y -
dy d = 0.
Note that when an algebraic approach to superanalysis is used (Martin [114], Berezin [4]), these changes of variables serve as counterexamples (see Sec. 5.10). The integral calculus on singular supersurfaces can be directly generalized to singular supermanifolds: a singular supermanifold is locally homeomorphic to a domain in the space RI x All.
3.4. Newton-Leibniz formula. The Newton-Leibniz formula LB f (x) dx = F(B) - F(A),
(3.14)
where F(x) is an antiderivative of the function f (x), holds for the continuous function f (x) of a real variable. Similarly, let f (z) be an analytic function of a complex variable and let ryAB be a contour connecting the points A and B in C. Then the integral fYAB f (z) dz fA f (z) dz does not depend on the choice of a contour and formula (3.14) holds true. The Newton-Leibniz formula can be generalized to the contour integrals in A0. Let the algebra AO be representable in the form AO = Re ® CO, where Co is a subalgebra (say, A = G'). We use the symbol 1(C0) to denote nil-radical (a set of nilpotent elements) of the algebra Co. We can extend every function f E C°° (R, A) to the algebra UO = Re (D N(Co) with the use of Taylor's formula
AX) = E f (n!
t) a",
(3.15)
n=0
where x = to + a, t E R, a E N(CO). If we provide the algebra UO with a suitable pseudotopology (see Chap. II), then the function f (x) will be infinitely S-differentiable.
3. Integral Calculus
37
Let F(t) be an antiderivative of the function f (t), t E R. Then
f (n- I) (t)an
F(x) = F(t) +
(3.16)
'
n=1
In what follows, we shall only consider contours which lie entirely in the algebra UO.
Theorem 3.2 (De Witt). Let f E C°° (R, A). Then the integral fA f (x) dx does not depend on the contour that connects the points A, B E U0, and the Newton-Leibniz formula (3.14) holds true.
Proof. Let A = a + a, B= b + /3, a, b c R, a,# E N(CO); 'AB (t) = ((p (t), V) (t)) E uO = Re ® N(CO). Then B
fA f (x) dx = f of f (w (t) +V) (t)){'(t) + '(t)} dt,
dt =
J ' f ((p(t) +
E
(n-1) (b)on]
f (n-1) ((p(t)) d n(t)
n=1
00
f
00 n=1
f' 1
- Li _ f (n-1) (a)an] n=1
00
-
n!
dt
jol
(we have used formula (3.15)). Next we have
f =
f
tj
f (cp(t) + 0(t))0(t) dt
tj
00
f(i(t)) dp(t) +
,
dt.
f of
n=1
The final result is LB
+
00
f (x)
nl f(n-1)
dx = f O1 f (V (t)) dcp(t) 00
(b)on]
-I
1
f (n-1) (a)an]
Chapter I. Analysis on a Superspace
38
= F(B) - F(A) (we have used relation (3.16)).
Example 3.4. Let us calculate the Gauss integral (a is a nilpotent +00
element): I = f exp{-2 (1 + a)zxz} dx. In this integral, we make a
change of variables y = (1 + a)x; I = (1 + a)-' fy exp{-y2/2} dy, where the contour -y: v = te, u = at, t E R, v E Re, u E N(C0). Let us consider the integral along the closed contour rN = rN U rN U FN;
rN: v = te, u = at; rN: v = Ne, u = at; r
:
v = te, u = 0;
0 < t < N (clockwise). Then we have frN exp{-yz/2} dy = 0, and, consequently, z jexp{-
2
} dy=f
z
z
+00
eXp{-2
} dy-2NmofNexp
2
dy.
However,
f
z
exp -
N
Nz
21 J
dy = exp
-2 f
N
PN (t) dt
where pN(t) is a polynomial of degree k = k(a), i.e.,
limo fNexp and I =
2 dy = 0
27x/(1 + a).
3.5. Calculating Gauss integrals. Let the matrix M have a block structure: A C M= -C* B
where the matrices A = (n x n) and B = (2k x 2k) consist of even elements, the matrix C = (n x 2k) consists of odd elements, the matrix A is symmetric, and the matrix B is skew-symmetric: A = A',
B = -B*. The choice of parities of the elements of the matrix M ensures the even-valuedness of the bilinear form corresponding to the
3. Integral Calculus
39
matrix M; this is also equivalent to the fact that the linear operator corresponding to the matrix M maps the superspace R',2k into itself. Let the elements of the matrices A and B belong to the algebra UO, i.e., A = a + a, B = b + /3, where a and b are numerical matrices and a and Q are matrices with nilpotent elements. We shall assume, in addition, that the matrices A and B are invertible (since a and 0 are nilpotent, this is equivalent to the invertibility of the matrices a and b) and to the fact that the matrix a is positive definite, a > 0. As the first step in the calculation of the Gauss integral
I=
f
exp { - 2 (X, MX) } dX J
R^exAik
we make a change of variables x = x+A-'CO, 0 = 0; the Jacobi matrix II
of this change has the form J l =
0
A
IC
,
and, consequently,
sdet Jl = 1. In the new coordinates, the Gauss integral has the form
I = f exp {_(x,MX)} dX, sl
where S1 is a singular surface in U x Aik obtained from R"e x Aik as a result of the change and M is a diagonal block matrix
M=
A
0
0 B + C"A-1C
Since the matrix a is real, symmetric, and positive, there exists an orthogonal (numerical) matrix 01, det 01 = 1, such that O1aOi = diag()1i..., an), )j > 0. Similarly, for the matrix b there exists an orthogonal matrix 02, det 02 = 1, such that O2bO2 = diag I
0
_µl
p1 0
0
_µk
Pk 0
We make a new change of variables X = LX, where
L = diag(Ol (I + a-'a)1/2,
O2[I + b-1(Q + C'A-1C)]112
Chapter I. Analysis on a Superspace
40
(the square roots are defined by their binomial expansions). For this change
sdet J2 = det (I + a-1a)1/2det [I + b-1(Q + C'A-1C)]1/2
and the Gauss integral reduces to the form
I=f exp{-2[A12i + ... +.\n2n] [µ1e102 + ... + µke2k-102k]} sdet J2 1 dX.
The integral with respect to anticommuting variables can be immediately calculated. To calculate an integral with respect to commuting variables, we must use the multidimensional analog of the reasoning given in Example 3.4. Thus, we obtain I = (27r)n/2(det a)-1/2(det b)1/2(sdet J2) -1
_
(27r)n/2(det A)-1/2[det (B + C'A-1C)]1/2
= (2ir)/2(sdet M)-1/2 For the sequel, we shall need one more property of a bilinear form corresponding to the matrix M. The bilinear form is symmetric, i.e., (Y, MX) == (X, MY) for X, Y E RA,2k. Indeed, using the symmetries and the parities of the matrices A, B, C, we obtain (Y, MX) =
yiAijxj + E yiCijOj + E ej (-C?j)xi + E (IBijej O Bijej = (X, MY).
E xiAijyj + E Oj (-Cij)yi + >
Let us calculate the Fourier transformation of the Gauss "measure"
f =
exp { l
f exp l-
2 (X, MX) + i(X, Y) }
(X
J
dX (27r)nsdet M-1
- iM-1Y, M(X - iM-1Y)) -
(M-1Y, y) J
2
41
3. Integral Calculus
x
dX
= exp
(27r)nsdet M-1
l
2
-1(M-1Y, Y) } J
.
Here we have used the symmetry of bilinear form
(M-1Y, MX) _ (X, MM-1Y) = (X, Y).
Remark 3.4. The even number of odd variables in the Gauss integral is caused by the requirement of the nondegeneracy of the matrix B. Remark 3.5. With respect to odd variables, the Gauss "measure" is a generalized function (see Chap. II). It stands to reason that it can be extended as a A-linear continuous functional to the space C(Aik) (see Remark 3.3). However, such an extension is not unique since the set of polynomials of anticommuting variables is not dense in the space of continuous functions. It should also be pointed out that all calculations become considerably more difficult if the quadratic form is not even-valued. Let us consider a simple example.
Example 3.5. Let 'y = -yo + -r', 0 = 0° + 01 E A, the elements 'y° and /3° being invertible. We calculate the Gauss integral with respect to anticommuting variables
f exp{-91ry92 - 83/394 + i 1: 4
I=
d91...d94.
j-1
It is more convenient to calculate the Gauss integral
I = f exp{-0102'y-93940+i
}d91...d94. vi i-1
(which easily reduces to the preceding one). J = f (1 + 2191)...(1 + 2494) 1
X
(1-o1o2-o3o4/3+o1o2o3o4P)
d91...d94,
Chapter I. Analysis on a Superspace
42
p = -y/3 + /37. Furthermore, we have
J=
2
+ f(1O12O2e3O3e4O4 + 60160203040 +e303e40401027) d91 d02 d03 d04
_ [1 + e1C2e3e42P-1 7P-1/3
Lemma 3.1. /3p-17 +
Proof
e1e22/3P-1
-
e3e427P-1]2
= 1.
p = (2/3° y° + /31.y° + 71/30);
p-1
= 2-1(,0yo)-1(1 + (/jo7o)-1(170 + 71/30))-1
= 2-1(0o7o)-1(1 - (/3170 + y10o)(0o yo)-1) Furthermore, /31)(/3°7°)-1(1
/32P-17 = (/30 +
- (/317° + 71/3°)(/3°7°) 1)(7° + 7')
+/31(/3°)-1(7°)-1)(1- /3'(/3°)-'
=1_
- 71(7°)-')(7° +7')
/31(0)-171(70)-1
Similarly, 72p-1/3 = 1 - 71(70)-1/31(/30)-1 Thus we have /32p-17 + 72p-1/3
= 2 + 71/31(/3°7°)
1
-
71/31(/3°7°)-1 = 2.
Using this lemma, we obtain 3S427P-'} = 1 - e1e22/3P
+7P-'/3P-1] =
1
-
U427P-1
2,Jp-1
Setting /3 = a(/3), 7 = a(7), p = a(p), where or is an automorphism of parity, we find that
I = ----
2u(7P-')e354}.
4. Integration of Differential forms
43
If the elements y and Q are even, then p = 20-y and
I = ayexp {-
2
_
exp
M=diag1( 0
{(M-'e
0)'(
11
1
)}
det M,
Q 00)).
0
The question concerning the calculation of the Fourier transform of the quadratic exponent in the case where the quadratic form assumes noncommuting values remains open.
It should be pointed out that when the number of odd variables is odd, the Fourier transform of the quadratic exponent may be not a quadratic exponent.
Example 3.6. fexp{_[0102 + 0283 + 0103] + i[Bi6 + 02e2 + 03e3]} d91d02d03 = i(6 C2 3 - e1 + e2 - 6) -
4.
Integration of Differential Forms of Commuting Variables 4.1. Definition of S-forms. Let A be a commutative Banach
algebra with identity e. The theory of A-linear differential superforms (S-forms) in algebra A exposed below is constructed by analogy with the theory of ordinary differential forms in a Banach space exposed, for instance, by Cartan in his book [35]. The main difference is that instead of the R-linearity for ordinary differential forms we require the fulfilment of the condition of A-linearity for S-forms. Recall that ordinary differential forms of degree p introduced in the domain 0 of the Banach space E with values in the Banach space F are introduces as mappings of 0 into the set of R-polylinear skewsymmetric mappings of the space E9 = E x ... x E in F. We shall denote the value of the form w of degree p at the point x E 0 on the vectors y2 E E, j = 1, p, by w(x yl yP).
Chapter I. Analysis on a Superspace
44
For S-forms of in commuting variables the role of E will be played
by the space Am and the role of F by the arbitrary Banach CSA A=AoED A1 with A0 j A. We denote by GP = Gp(Am, A) a set of A-polylinear skew-symmetric
mappings of the space Am x ... x Am = Amp into A, i.e., f c Gp is equivalent to (1) f (yl, ..., ayj + Qz4, ..., yp) = of (yl, ..., y4, ..., yp) + of (yl) ..., Z47 yp), y4, z4 E Am, a, Q E A, y°(P) = E(a) f (yl, ..., yP), where E(or) is the signature (2) f
of the permutation a: (1, ...,p) -+ (a(1), ..., a(p)). The mapping w: 0 -* Gp is an S-form of degree p defined on an open set 0 C Am with values in A. We denote by w(x; y', ..., yP) the
value of the S-form w at the point x E 0 on the vectors yJ E Am, j = 1, .... The S-form w of degree p belongs to the class S'(0) if the function w: 0 -+ Gp is k times continuously S-differentiable. By c(pk) (0; A) we denote the set of all S-forms of degree p in 0 with values in A of the class Sk(O). Example 4.1. An S-form of degree 1 is a mapping 0 -+ L(Am, A), where L(Am, A) is a space of A-linear mappings of Am into A.
It follows from the definition that every S-form is, at the same time, an ordinary differential form. We shall widely use the following criterion which connects ordinary differential forms with S-forms.
Criterion 4.1. For the differential form w to be an S-form, it is necessary and sufficient for it to be A-linear with respect to y', i.e., that it satisfy the condition w(x; hy1, ..., yp) = hw(x; y1, ..., yp),
h E A.
As usual, we introduce differential dxj: dx,(y) = yj, y = (yi, , ym) E Am as well as an outer product of differentials dx1 A dx; (yl, y2) _ dx; (yl) dx, (y2) dxi (y2) dxi (y') = y: y2 - y2 1; similarly,
-
dx,1 A ... A dxip(yl,..., yP) = det IIy pIIQ,A=1.
4. Integration of Differential forms
45
Theorem 4.1. Every S-form w E
q(k) (O; A) can be uniquely
represented in the canonical form
w=
E
wi1...i9(x)dxi1 A ... A dxip,
(4.1)
1
where the functions wi,...jP: 0 -4 A belong to the class Sk(O).
The proof repeats the standard proof given in [35].
Corollary 4.1. 1 ,°) (O; A) = 0, p > m. Corollary 4.2. The condition w E Qpk) (0; A) is equivalent to the conditions wil...i, E Sk(O),
p = 0, 1, ..., M.
Example 4.2. Let A be a finite-dimensional commutative Banach algebra with basis e° = 1, e1, ..., eN so that Am can be identified with Rm(N+1) Therefore the S-form w of degree p can be represented by relation (4.1), where N
N
xj = E eauia, uia E R;
dx3 = E eaduia, i = 1, ..., m,
and the forms duia are canonical differentials on Rm(N+1) . Note that ordinary differential forms in Rm(N+1) of degree p can be written as
w=
wila,...ipap (x)dui,a, A ... A dui9ap.
They vanish for p > m(N + 1). Therefore we can see that not every ordinary differential form is an S-form. For instance, the differentials duia are not S-forms. These differentials are not A-linear. Let, for instance, A be an even part of the Grassmann algebra with two generators Q1i q2: y = to + t1g1g2. Then duo(y) = to and du°(yg1g2) = 0. If A = C, then we get the ordinary theory of C-linear differential forms, in which case the S-smoothness is equivalent to holomorphy.
Chapter I. Analysis on a Superspace
46
The outer product of the S-forms a E p(k) (O; A) and /3 E Q(k) (O; A) is defined as an S-form a A,3 E SZp+q (O; A) according to the formula
aA/3= i ail
... :p
(x)/391 ..(x)dxtl A ... Adxty Adx71 A ... AdXja.
i l <...
11< .<1q
An outer product is associative, and if the coefficients of a and /3 are homogeneous elements from A of parity jal and 1/31, then a A,3 =
(-1)nq+lQl H01/3
A a.
4.2. Exterior S-differentiation. The exterior S-differential of the S-form w E OP (O; A) defined by relation (4.1) is the S-form dw c S2p+q1) (O; A) defined by the relation m
dw = m 3=1i1<...
wil...ip
dxj A dxil A ... A dxip,
C7xj
awax.'p
are S-derivatives. This definition (with the use of standard arguments) implies the following properties of the operation of exterior S-differentiation: where
d2=0, d(aA/3) =daA/3+(-1)paAd/3, a E 1 ,1) (O; A),
/3 E
0(1) (0, A).
If the algebra A is finite-dimensional, then every S-form is an ordinary form (with respect to the differentials dui,,) and we can define the ordinary exterior differential of this form. We can show that this definition is equivalent to the preceding one. The same refers to Cartan's exterior differential [35] in Banach spaces (see [20]).
The S-form w is said to be closed in 0 if dw = 0 and exact if w = da, where a is an S-form. We can point out the following generalization of the Poincare theorem.
Theorem 4.2. Let the S-form w E cPk) (O; A), and let k > 1, p > 1 and dw = 0 in 0, where 0 is a star domain with respect to the
47
4. Integration of Differential forms
origin. Then there exists an S-form a E SZpkll (0; A) such that da = w, i.e., every locally closed S-form is exact. Proof. Let the S-form w be defined by expression (4.1). Then the S-form fi
dt
[(_1)k_1xikdxj, X
n ... A dxipJ
a= P
11
(4.2)
k=1
is a form of the class
A) and da = w.
4.3. The preimage of an S-form. Let A' be a commutative Banach algebra and 0' be a domain in (A')m'. Suppose that we are given a function x = W(y) of the class Sk+1(0') which maps 0' into O C Am. For every S-form w E S2pk)(O; A) defined by (4.1) we shall determine the S-form*w E q(k) (O'; A) from the formula W*w
=
wi, ...ip (W (y)) dWi1(y) A ... A dWip (y) it <...
The S-form cp*w is the preimage of the S-form w under the mapping W. By analogy, we determine the preimage cp*w of the S-form w from SZpkl (0; A) under the mapping W: N -4 0, where N is a q-dimensional (real) manifold of the class Ck and W is a mapping of the class Ck+1 What we have said also concerns finite-dimensional manifolds with boundary and manifolds with pseudoboundary (in particular, manifolds in R" with picewise-smooth boundary). In the case A' = R, (A' )q = Rq, the manifold N is locally homeomorphic relative to the domain 0' in Rq, and, for the commutative Banach algebra R the S-differentiability coincides with ordinary differentiability and the classes Sk coincide with Ck, and the S-form W*w is an ordinary differential form (with values in the Banach space A) on the manifold N. 4.4. Integrals taken over singular manifolds. Somewhat
earlier we considered integrals of an S-form of degree m over mdimensional singular surfaces. This definition can be immediately generalized to S-forms of an arbitrary degree.
Chapter I. Analysis on a Superspace
48
Let N be a p-dimensional (real) oriented manifold belonging to the class C' and the mapping cp: N -f 0 C Am belong to the class C1. The collection (M = cp(N), cp, N) or simply M is a p-dimensional singular manifold. It should be pointed out in passing that every pdimensional manifold M C Am is a p-dimensional singular manifold relative to an identity mapping. Suppose that we are given an S-form w c SZp°)(0; A). The integral over the S-form w with respect to the singular manifold M is fM W =
cp`w.
(4.3)
JN
The integral on the right-hand side of (4.3) of the differential form cp'w
with respect to the manifold N must be understood in the ordinary sense.
Integral (4.3) does not depend on the choice of the pair (cp, N) in the class of equivalent pairs. The pairs (cp, N) and (w,, N,) are said to be equivalent if M = W(N) = cp1(N,) and there exists a diffeo-
morphism f : N -* N, such that cp, = cp o f (see [35]). Therefore, fN cp'w = fN1 cpiw. We can interpret this relation as a formula for a change of variables. Let us consider this in greater detail. Let g be a mapping of the class S'(O1) of the neighborhood 01 C Am of the singular manifold M, = cp, (N,) into the singular manifold M = W(N) in Am and satisfy the relation cp = g o cp, o f. Suppose that we are given an S-form of class SZp' (O; A), where 0 is the neighborhood of M in Am and g'w is the preimage of the S-form w under the mapping g. Then the relation fm w = fMl g`w is valid.
Example 4.3. Let N be a domain in RP, the algebra A be finitedimensional with basis a°, ..., en (see Example 4.2), and let the S-form w be defined by relation (4.1). Let W map N onto M C Am = Rm(n+1) Then
W=> fm
n
il<...
/
f
N
/
A x e°l ...eam dcpil a'1 (t) A ... A d7oj,,QP (t),
,
where dcpi(t) =
at, (t) dt 1
+ ... +
a"'-° (t) atp
dt.p
4. Integration of Differential forms
49
If the algebra A is infinite-dimensional, then the sum over a is infinite.
4.5. Stokes formula. Let N be a compact oriented p-dimensional C'-manifold with pseudoboundary 8N, the orientation of 8N being induced by that of N, and let M = cp(N) be a compact singular manifold in Atm with pseudoboundary 8M = cp(8N). Suppose that we are given an S-form w of the class Q(')1(O; A), where 0 is the neighborhood of M. Then the following generalization of Stokes' formula holds true:
M
dw =
w.
aM
By the definition of an integral with respect to the singular manifold of an S-form, this Stokes' formula immediately reduces to Stokes' formula for ordinary differential forms with values in a Banach space [35]:
fM dw
= fN *(dw) = JN
f8N P*w = IP(M) W = faN W-
Note that an S-form of degree m in local coordinates has the form w = f (x1, ..., xm)dx1 A ... A dxm and, consequently, dw = 0. Using Stokes' formula, we obtain
Generalization of the Cauchy-Poincare theorem from complex analysis to the case of any real commutative Banach algebra A. If an S-form of degree m is S-diferentiable in 0 C Am and M is an m-dimensional singular manifold which is a pseudoboundary of an (m + 1)-dimensional compact singular manifold contained in 0, then
fMf(1,,m)1tm = 0.
(4.4)
As a consequence we find that the integral fm w does not depend on the manifold M for all homological in 0 manifolds with the indicated properties. Note the following valid
50
Chapter I. Analysis on a Superspace
Generalization of Morera theorem. If the function f (x) is Frechet differentiable in 0 and satisfies (4.4) for any manifold M with the indicated properties, then f is S -differentiable in 0.
This statement can be proved by analogy with complex analysis. Indeed, let m = 1 and xo E 0. Then the function F(x) = f,, f (x) dx is defined and S-differentiable in the neighborhood of the point xo, and F'(x) = f (x). It remains to use Corollary 2.1. Remark 4.1. If A is an (n + 1)-dimensional algebra, then formula (4.4) has sense only for (m + 1) < m(n + 1), i.e., for n > 1; thus the generalization of the Cauchy-Poincare theorem holds for any finitedimensional real algebra A of dimension higher than or equal to 2 (i.e., except for A = R).
It is interesting to compare, for m = 1, the results obtained with those from Sec. 3.4. It follows from the Cauchy-Poincare theorem that the Newton-Leibniz formula holds for any function from class
S' and there is no restrictions of the type of nilpotency on the path of integration. However, De Witt theorem does not follow from this. In De Witt theorem we can consider the extension of any function f E C°° to the algebra UO. However, generally speaking, functions of this kind cannot be extended to the whole Banach algebra A. Therefore a necessity arises of constructing an analysis on the Vladimirov-Volovich superspace not only over Banach CSA but also over arbitrary topological CSA (in particular, over CSA in which the even part has the form Uo, see Chap. II). Since all entire analytic functions can be extended to entire Sanalytic functions on Ao, the Cauchy-Poincare theorem on the Vladimirov-Volovich superspace makes it possible to extend essentially the class of Gaussian integrals over a superspace that we are considering (cf. Secs. 3.4, 3.5). In the commutative Banach algebra A, for any a c A there exists a limit lim jja"jjl/n = p(a) called a spectral radius [13]. The element a E A with p(a) = 0 is said to be quasinilpotent [13]. For instance, for the CSA Gam, G0,0 = Re ® Co all elements of the subalgebra Co are quasinilpotent (see [119]).
5. Review of Superanalysis
51
Numerous examples of the CSA A = AO ® Al in which the subalgebra Co consists of quasinilpotent elements were constructed in [99].
Example 4.4. Let us calculate the Gaussian integral from Example 3.4 in the case where a is a quasinilpotent element. We shall show, in the first place, that this integral converges. Note that for the quasinilpotent element Q for any E > 0 we have Ile°x211 < CEef22,
CE > 0,
x E R.
It follows from this estimate that the integral converges and that the integral along the path 1,N in Example 3.4 tends to zero. Consequently, for the quasinilpotent a we get the same answer as in Example 3.4.
Repeating the calculations from Sec. 3.5, we calculate the Gaussian integral for block matrices M in which A = a + a and B = b + Q, and a and ,Q are matrices with quasinilpotent elements.
Remark 4.2. If x is a quasinilpotent element, then the function (1 + x)' is correctly defined by its Taylor series. The same refers to the function ln(1 + x). For elements which are not nilpotent Gaussian integrals may diverge.
5.
Review of the Development of Superanalysis
5.1. Model of De Witt superspace [27]. In this model, the CSA A = G,,(K), n = 1, ..., oo. The superspace Kin is endowed with the (not Hausdorff) topology induced from K' by the canonical projector E: Kin -+ K' (a body projector, see Chap. II).
5.2. Model of Rogers superspace [119]. In this model, the CSA A = Gn or G. The superspace K'" is endowed with a Banach topology (see Sec. 1.1).
5.3. Model of Batchelor supermanifold [83]. The model space for the Batchelor supermanifold is the De Witt superspace. However, the definition of the S-smoothness for transition functions is different from Definition 1.2 used in the works by De Witt, Vladimirov,
Chapter I. Analysis on a Superspace
52
Volovich, Rogers. In the work [83] a mapping is said to be S-smooth if it is approximated by polynomials with coefficients from the field K (and not from the CSA A).
5.4. Model of Jadzyk and Pilch superspace [98]. The works by Jadzyk and Pilch were based on Roger's article [119]. Instead of Grassmann algebras, the authors of [98] considered Banach-Grassmann
algebras. The Banach-Grassmann algebra is the CSA A = Ao ® Al satisfying the following conditions. 1. For any continuous Ao-linear mapping f : A,. -4 A, (r, s = 0, 1) there exists a unique element u E A,+, such that I I u I I= I I f I I and f(A)=u.1,AEA,.
2. Ao = K®A', 1ju+vjj = 1jull + jjvjj, u E K, v E A', where Ao is a Banach subalgebra of the algebra Ao generated by even products of elements from Al. Condition 2 is technical in nature and Condition 1 is the basis of the Jadzyk and Pilch differential calculus. By virtue of this condition, the Ao-linearity implies that the operator f is a multiplication operator by the element A. However, the space KA'm is a Ao-module, and therefore the S-differentiability on the superspace over the Banach-Grassmann algebras can be defined as follows: the mapping F: KAn'm -4 A is Frechet differentiable, F'(x): KAn'm -> A is a Ao-linear operator (cf. Theorem 2.3). Note that GI is a Banach-Grassmann algebra.
5.5. Model of Boyer and Gitler superspace [79]. The article [79] continues the investigations of Rogers. Boyer and Gitler proposed that in the definition of the S-differentiability, not only the Z2-grading
must be taken into account in the Grassmann algebra, but also the grading corresponding to the degree of the monomials e; = g11...gi,, e{ k: Gn = GI + ... + Gn; G° = K (in this case, Gn o = GnP GnP+l).
Gnj =
n
P
P
The ideals IP = GP + ... + Gn and the quotient algebras G$') _ n Gn/IP are introduced. The S-differential calculus is developed for the mappings f : KG ` -> GnP). In the work [79], the authors obtained
5. Review of Superanalysis
53
the Cauchy-Riemann conditions in superanalysis which were similar to those proposed by Vladimirov and Volovich.
5.6. Model of Kobayashi and Nagamashi superspace [101, 102]. The authors of [101] introduced a superspace over a-symmetric
Grassmann algebras, where a: r x r -+ K is a sign function on the finite Abelian group r = ro ® F1, where Fo = {a E F: cr(a, a) = 1},
F1 = {a E F: 01(a, a) = -1}.
5.7. Martin-Berezin algebraic superanalysis [114, 115, 3, 4]. This approach to superanalysis was proposed in 1959 by Martin and, independently, by Berezin early in the 1960s. The Martin-Berezin theory is of a purely algebraic character. The elements of the Grassmann algebra Gn are considered to be functions of anticommuting variables in the algebraic theory of Martin and Berezin. For the element E = (1.3) use is made of the symbolic notation e = E(ql,..., qn). The derivatives of the element e of the Grassmann algebra Gn with respect to the generators of this algebra are defined according to algebraic rules. A very good exposition of algebraic superanalysis can be found in Berezin's monograph [4].
5.8. Algebraic supermanifolds. The theory of algebraic supermanifolds (graded manifolds) was developed by many eminent mathematicians (see, e.g., [114, 4, 40, 42]). In classical geometry, there exist two approaches to the study of the geometry of manifolds. Traditionally, differential geometry regarded
a space as a primary object, but, at the same time, there existed a point of view of algebraic geometry according to which the geometry of space was studied via the algebraic structure of its bundle of functions. The theory of algebraic supermanifolds is based on the formalism of algebraic geometry. For a smooth real manifold M, the bundle of functions consists of algebras C°° (U), where U is a system of open subsets of the manifold
M. In order to obtain an algebraic supermanifold, we must extend the algebras C°°(U) to the algebras A(U) containing anticommuting
Chapter I. Analysis on a Superspace
54
elements. Thus, an algebraic supermanifold (a graded manifold) is an n-dimensional smooth manifold M with a bundle of CSA A(U). It is customary to consider CSA of the form A(U) = C°°(U) ®G(U), where G(U) are Grassmann algebras.
5.9. Local realization of an algebraic supermanifold. "Since a supermanifold does not consist of points" according to Berezin (see [4, p. 20]), the use of algebraic supermanifolds led to a loss of geometrical visuality of physical theories. Batchelor was the first to give a local construction for an algebraic supermanifold (see [82]). She suggested to consider, for a smooth manifold M, the exterior vector
bundle Ext (M), namely, a bundle with fiber AIR. The following statement was proved in [82]. Let I'(Ext (M)) be a pencil of sections of a vector bundle Ext (M).
Then any graded manifold (Kostant [40]) over M is isomorphic for r(Ext (M)) of a certain exterior algebra AIR.
5.10. Nonequivalence of the two approaches to superanalysis. Here are the main differences between the algebraic superanalysis (it is more natural to call it a superalgebra) and the analysis on the superspace K A" presented in this chapter. 1. In algebraic superanalysis all "functions" of anicommuting variables (elements of Grassmann algebra) have a polynomial form (1.3). There are no nonpolynomial continuous or differentiable "functions." 2. In algebraic superanalysis there are no anticommuting constants. Any odd Grassmannian element E under an algebraic approach is a "function" E = E(ql, ..., qn). In superdifferential geometry, this leads to the existence of supermanifolds over KA'm with bundles of functions which are more extensive than the bundle of graded manifold considered in Sec. 5.8. Graded manifolds coincide [83] with supermanifolds over the model of De Witt superspace (5.1) with transition functions from Sec. 5.3. The topology of graded manifolds in the anticommuting sector is trivial. 3. In the commutative sector the Martin-Berezin analysis is an analysis on Rn and not on By virtue of item 2, it is impossible to realize the SUSY transformation [95, 75, 84, 124] in the algebraic approach to superanalysis A0.
5. Review of Superanalysis
55
since the SUSY parameters E;' are the same "variables" as the coordinates P. However, the interpretation of SUSY as the transformation of variables 0'`, E'` is meaningless (a detailed discussion of the reason why SUSY cannot be realized in the algebraic approach can be found in the article [82]).
Item 3 implies that Berezin's integral with respect to commuting variables is an integral over R" and not over Ao A. An integral of this kind possesses a number of "pathological" properties which make its application in physics impossible. For instance, the theorem of a change of variables in Berezin's integral is valid only for functions with compact supports on R". One of the first counterexamples for functions with noncompact supports was constructed by Rudakov (see [42]). b
Counterexample 1. In the integral I1 = f x dx = a (b2 - a2) we a make a change of variables x' = x + a, where a is a fixed nilpotent element. In accordance with the laws of algebraic superanalysis [4], the integration limits do not change because the real part of the variable does not change (since a is nilpotent):
f x dx = f (x - a) dx = b
x: I1 =
b
a
a
1
2
(b2
- a2) - a(b - a). b
Counterexample 2. In the integral 12 = f f f x d91d92dx = 0 we a make a change of variables x' = x + 0102, 0 = 03 Then .
b
12
(a - b). = f af f (x - 0102)d91d92dx = b
Counterexample 3. In the integral 13 = f f x d0dx = 0 we make a a change of variables x' = x + 770, 0' = 0, where 77 is a fixed odd element. Then b
I3 = f f(x_i70)dodx=zi(a_b). a
constructed by Vladimirov, Volovich, De Witt, In the integral and Rogers these changes of variables are correct.
Chapter I. Analysis on a Superspace
56
6.
Unsolved Problems and Possible Generalizations
1. The sufficiency for the Cauchy-Riemann conditions for CSA which are not finite-dimensional Grassmann algebras. The following special cases are of a considerable interest. 1.1. The sufficiency of the Cauchy-Riemann conditions for finitedimensional CSA which are not Grassmannian. 1.2.
The sufficiency of the Cauchy-Riemann conditions for the
infinite-dimensional Grassmann algebra of Rogers.
2. Construction on a supermanifold of an integral calculus similar to the integral calculus with respect to singular supermanifolds. Remarks Any finite-dimensional Grassmann algebra G,ti can be realized as a matrix algebra. Therefore, for finite-dimensional Grassmann algebras superanalysis is included into the general theory of functions of matrices. In this connection, we must point out the works by Lappo-Danilevskii [41] in which the general theory of analytic functions of several matrices was constructed.
Chapter II
Generalized Functions on a Superspace
Generalized functions were introduced by Dirac [29] in his quantomechanical research in which the famous 6-function was systematically
used. The fundamentals of the mathematical theory of generalized functions were laid by Sobolev ([131 (1936)]) and used to solve the Cauchy problem for hyperbolic equations. Schwartz ([69 (1950)] and [51]) gave a systematic exposition of the theory of generalized functions and indicated a number of its important applications. Generalized functions appeared in connection with quantum physics problems. This relationship served and continues to serve as the basis for the further development of the theory of generalized functions. Practically all new divisions of the theory of generalized functions emerged from physical problems. In particular, the works by N. N. Bogolyubov exerted a considerable influence on the development of new divisions of the theory of generalized functions. Bogolyubov was the first to show the fundamental role played by generalized functions in the description of the local interactions of elementary particles (see [10]). Later, generalized functions were widely used for constructing the axiomatic quantized field theory [7, 8].
In connection with the applications to the quantized field theory, we must point out the works by Vladimirov [15, 17], Vladimirov, Drozhzhinov, Zavyalov [18], Streater and Wightman [57], and Jost [34].
Chapter II. Generalized Functions
58
The further development of the quantum field theory (superfield theory, superstring theory, supergravitation [25, 27, 28, 40, 76, 123]) led to the theory of generalized functions on a superspace. Just as in the ordinary theory of generalized functions, physicists were the first here. For the first time generalized functions on a superspace (on the physical level of exposition) were considered in De Witt's monograph [27] (1984).
The first mathematical theory of generalized functions on a superspace was proposed in [141, 144] (1986-87). De Witt's generalized functions are analogs of Schwartz distribu-
tions (although De Witt did not deal with problems of functional analysis on a superspace and did not introduce spaces of test and generalized functions, it is clear conceptually that the mathematical realization [27] leads to the supertheory of L. Schwartz). In my works [65, 68, 146-148] I constructed the theory of analytic generalized functions. Another essential distinction between De Witt's theory of generalized functions and my theory is the class of CSA over which generalized functions are considered. I have introduced generalized functions on the Vladimirov-Volovich superspace over an arbitrary Banach CSA with a trivial annihilator of the odd part. In the monograph [27] generalized functions are introduced over a non-Banach CSA in which all even elements with zero numerical part are nilpotent. Note that one of the fundamental problems of the theory of generalized functions is the problem of the choice of CSA. For certain CSA the theory of generalized functions is essentially simplified and for other CSA practically insurmountable difficulties arise. In particular, restricting our consideration to the CSA A = AO G Al with a trivial A1-annihilator [65, 68, 146-148], we obtain a simple theory of generalized functions on a superspace whose exposition differs but little from the standard theory of generalized functions. The choice of a CSA is closely connected with the class of problems being studied. The type of a CSA is a new parameter of the theory of generalized functions; when solving a specific problem, we choose a CSA which is adequate to the problem at hand.
In the standard theory of generalized functions the part of this
59
parameter is played by the number field over which the theory of generalized functions is constructed. The theory of generalized functions
over the field of real functions is adequate to one type of problems and that over the field of complex numbers is adequate to some other type of problems. Theories of the generalized functions over the field of p-adic numbers arise in applications to mathematical physics (see [21, 72, 66]). It stands to reason that we can consider CSA not only over fields of real and complex numbers. The analysis on a superspace over an arbitrary locally compact number field was developed in [19]. The theory of generalized functions on a superspace over an arbitrary non-Archimedean number field (not necessarily locally compact) was proposed in [163].
Nagamashi and Kobayashi [103] constructed a theory of generalized functions on CSA which is a strict inductive limit of finitedimensional Grassmann algebras (this algebra is non-Banach).
In [152, 153] I proposed a theory of generalized functions on a pseudotopological CSA which is an inductive limit of nilpotent sets. The theory of generalized functions over Banach CSA with a trivial annihilator of the odd part is most adequate to Caushy's problem for linear differential equations on a superspace, in particular, in the framework of this theory the existence of a fundamental solution of Cauchy's problem on a superspace was proved for an arbitrary linear differential equation with constant coefficients [146]. The theory of generalized functions over CSA [152, 153] is more adequate to the problem of the fundamental solution of a linear differential operator on a superspace. In particular, the theorem of existence of fundamental solution of a linear differential operator with constant coefficients on a superspace was proved in the framework of this theory [153]. In contrast to the real case, in a supercase there exist differential operators which do not have a fundamental solution.
60
Chapter II. Generalized Functions
1.
Locally Convex Superalgebras and Supermodules
1.1. Locally convex superalgebras. A topological linear space is a linear space E over a field K (K = R, C) endowed with topology relative to which the algebraic operations ((x, y) H x+y, E x E -+ E; (A, x) H Ax, K x E -+ E) are continuous. The topological linear space E is locally convex if there exists, in this space, a basis of the neighborhoods of zero {U} consisting of convex sets (neighborhoods of the
form {a + U} form the basis of convex neighborhoods of the point a c E). Just as in a normed space, the topology can be defined with the aid of a system of prenorms (a prenorm is a map x H 11x11 satisfying the same axioms as a norm, except for the fact that, generally speaking, IIxII = 0 does not imply x = 0). We shall use the symbol FE to denote the system of prenorms { } which defines the topology in the locally convex space E. A topological algebra is the algebra A endowed with a topology relative to which the algebraic operations are continuous. A topological algebra is said to be locally convex if its topology is locally convex. It follows from the continuity of the multiplication operation that for any prenorm II II E IFA there exists a prenorm ' E rA such that llxyll < CIIxII'llyll' (the constant C depends on
Let the CSA A = AO ® Al be a topological algebra, where the direct sum AO ® Al is topological (i.e., the projectors 7r,,: A -4 A,, are continuous). A = Ao ® Al is called a topological CSA. If the topology in A is locally convex, then A is called a locally convex CSA. A locally convex space whose topology is metrizable is known as
a Frechet space. This is equivalent to the fact that the system I'E is countable. Frechet algebras and Frechet CSA can be defined by analogy.
1.2. Locally convex supermodules. A supermodule is a Z2graded space M = Mo®M1 on which a structure of a two-sided module over the CSA A = Ao ® Al is defined with an even multiplication operation by the elements of the CSA A = Ao®A1 (Iabl = lal Ibl, mod 2, for homogeneous elements from the algebra and the module). The su-
1. Locally Convex Superalgebras
61
percommutator [a, b} of homogeneous elements from the algebra and the module is defined by relation (1.1), Chap. I. The supermodule M = Mo ® M1 is said to be (super) commutative if, for arbitrary homogeneous elements [a, b} = 0 (i.e., the even elements of the algebra
commute with all elements of the module and the odd elements of the algebra anticommute with the odd elements of the module). Instead of the term a (super) commutative supermodule we shall use its abbreviation CSM. A CSM is said to be unital if ex = x for all x E M (e is an identity in CSA). In this book we consider only unital CSM. The concept of a CSM is equivalent to the concept of the representation of a CSA A = A0®A1 in a Z2-graded linear space M = Mo®M1. In this case we consider representations consistent with the structure of parity. Topological CSM over topological CSA are defined in a natural
way, as well as locally convex CSM over locally convex CSA and Frechet CSM over Frechet CSA.
We shall not give here any examples of CSM, they will appear later, in the theory of generalized functions on a superspace.
1.3. Conjugate supermodule. The successive exposition of the duality theory for CSM is given in Chap. III. Here we introduce only the main concepts which are necessary for constructing the theory of generalized functions. Let M = Mo ® Ml be a CSM. The linear functional 1R: M -+ A
is said to be right A-linear if IR(mA) = lR(m)A for any m E M, A E A. Left A-linear functionals 1L can be introduced by analogy. On the space of functionals which are right A-linear we introduce the structure of a module setting (AIR)(m) = AIR(m), (IRA)(m) = lR(Am), A E A, m E M. By analogy we introduce the structure of a module on the space of functionals which are left A-linear. A functional l is even if 1 1 (m) I= 0 for Im l = 0 and 11(m) I= 1 for Im l = 1. A functional l is odd if 1l(m)I = 0 for Iml = 1 and 1l(m)I = 1 for Iml = 0.
A Z2-grading and a parity function are introduced on the space of functionals which are right (or left) A-linear. These spaces become CSM over the CSA A = Ao ® A1.
Chapter II. Generalized Functions
62
A is a right A-linear It can be noted that if 1R = 1R ® AR: M functional, then the functional IL = IL ® 1i defined by 10
= 10
IL' (m') = lR(m°);
1L(m') = -lk(ml).
(1.1)
where Im° _ laI = Ili = a, a = 0, 1, is a left A-linear. Similarly, every functional which is left A-linear defines a right A-linear functional. Identifying the spaces of left and right A-linear functionals, we obtain a CSM M' = MO* ® M1 which is an algebraic conjugate of the CSM M = MO ® M1. Considering continuous A-linear functionals on the topological CSM M = MO ® M1, we get a CSM M' = MO '(D M1' which is a topological conjugate of the CSM M = MO ® M1. 1.4. Differential calculus for mappings from a subspace
into a supermodule. Let KA'm' = Ao x Am be a superspace over a topological CSA A = AO ® Al and let M = MO ® M1 be a CSM over the CSA A = Ao ® A1. The following definition of S-differentiability is a trivial generalization of Definition 1.2 from Chap. I. This generalization proceeds in two directions. First, the functions assume
values not in a CSA but in a CSM, and, consequently, the derivatives are multiplication operators by elements from the CSM. Second, generally speaking, there is no norm on a superspace, and therefore we must replace Frechet differentiability by some differentiability in a topological linear space [54]. By way of example, we shall consider differentiability with respect to a system of bounded subsets [54].
Definition 1.1. The mapping f : U -+ M, where U is an open subset of KA'm, is S-differentiable at a point x E U if representation (1.5), Chap. I, is valid, where the partial derivatives are elements of the CSM M and the remainder o(h) satisfies the condition m o(th)/t = 0 uniformly on any bounded subset B of the superspace Kn'm (t E K). Recall that the set B in the topological vector space E is bounded if, for any neighborhood V if zero in E, there exists a A > 0 such that B C AV. If E is normed, then the bounded subsets coincide with subsets bounded in norm and the differentiability with respect to the system of bounded subsets coincides with the Frechet differentiability.
2. Analytic Generalized Functions
2.
63
Analytic Generalized Functions on the Vladimirov-Volovich Superspace 2.1. Superalgebras with a trivial annihilator of the odd
part. As was pointed out in the introduction to this chapter, CSA with this property serve as the basis for the theory of generalized functions on the Vladimirov-Volovich superspace.
Theorem 1.1. Let A = AO ® Al be a CSA with a trivial A1annihilator, and there exist odd elements whose product is nonzero. Then the subspace Al is infinite-dimensional.
Proof. Assume that Al is finite-dimensional and let a1, ..., am be the basis in the linear space A1. We shall prove by induction that
aiaj =0foralli,j: 1. Since a,...am E' Al always, it follows that a,...am = 0. 2. Assume that for any collection j1 < ... < jk: aj1...aik = 0. 3. Let us show that it follows from item 2 that for any collection 71 < ... < jk-1: a,j,...ajk_, = 0. Assume that there exists a,1...ajk_, 0. Let A = c1a1 +... + cmam m be an arbitrary element of A1. Then a,j,...a7k_1A = > cjaj,...ajk_laj. 3=1
If aj = a 1, then a ... ajk_, = 0 since a = 0.
If a3
ail, then
aj,...ajk_, = 0 by virtue of item 2. Everywhere in what follows in this chapter we assume that in the Banach CSA A = AO ® AO the A1-annihilator is trivial or the CSA A is a commutative Banach algebra (i.e., A = A0). The main model example of a CSA with a trivial A1-annihilator is the Banach exterior algebra Gl(B). Let us prove this simple assertion. Let
=
[
f - k=OJjl<...<7k [ 00
>1
1
1
fj1...7kQJ1...Qjk E G1 00,1
(B)'
Multiplying f by the generator qj, we find that fil...3k = 0 for all 31, .., jk which are not equal to j. It remains to use the fact that the number of anticommuting generators is infinite.
Chapter II. Generalized Functions
64
2.2. Spaces of test S-analytic functions. We use the symbol to denote the space of entire S-analytic functions f : C" -4 A. Every entire S-analytic function can be expanded in a power series XV3fap,
f (x, 9) =
fop E A,
ap
where z = (x, B) E CA "`, x° = x11...xn^ , a; = 0, 1, 2, ..., 00 = BA1...Bmm, Q3 = 0,1, and IIf IIR = E R°II fa,pII < oo for all R E R+. aQ
The topology in the space {II
-
is defined by the system of norms
IIR}, R? > 0; the graduation is introduced in a natural way:
Aa(CA'm) _ If E A(Cn'm): f (Cnm) C A0}, a = 0, 1.
Proposition 2.1. The space A(C m) = A° (C m) ® A, (Cn'm) is a Frechet CSA. Moreover, we have an estimate Ilf9IIR <_ IIf IIRII9IIR
(2.1)
In particular, A(Cn'm) is a Frechet CSM over the Banach CSA A=AO ®A,. It should be emphasized that, as was assumed in Chap. I, the norm of the Banach CSA A = A° ® Al satisfies the condition IIx° ®x1II = 11x0 Ii -f
lxi
Proposition 2.2. The differentiation operators aa, and azs , i = 1, ..., m; j = 1, ..., n and the shift operators f H fh, fh(z) = f (z + h), h E CA 'm are continuous in the space A(Cn'm).
These propositions can be immediately proved with the aid of the estimates of power series. An entire S-analytic function of the commuting variables f: Cn'0 A is called a function of the first order of growth if the following estimate is satisfied for it: If (x)II < Ce'llxll,
C = Cf > 0,
a = of > 0.
(2.2)
The space of entire S-analytic functions of commuting variables which have the first order of growth is denoted by E(Cn'°).
2. Analytic Generalized Functions
65
Proposition 2.3. The space
E(Cnm) _ {f E A(Cnm): f (x,9) _
fp(x)9Q, fo(x) E E(CA°)}. 10 1=0
Using Proposition 2.3, we find that the space E(Cn'm) consists of entire S-analytic functions satisfying the following estimate of the derivatives: as+pf
axaaop (°)
< CRa,
C = C2,
R = R;.
(2.3)
We denote by ER(Cn'm) the subspace of the space consisting of functions satisfying estimate (2.3) for a fixed R = (R1, ...,
R; > 0. This space endowed with the norm as+13f 1
1
1f1
1
1 R = SUP
R-a
axaaea (°)
is a Banach CSM over the Banach CSA A. The space E(CA m) is endowed with a topology of an inductive limit: Jim R-oo
Proposition 2.4. The functional space E(C m) is a complete local convex CSA.
66
Chapter II. Generalized Functions
The proof reduces to an estimation of the series in the norm of the Banach space ER(C'A''").
Remark 2.1. All estimates for S-analytic functions on a superspace which assume values in a CSA reduce to the case of functions dependent only on commuting variables and assuming values in a CSM. Indeed, we can regard every function f (x, 0) which is infinitely S-differentiable with respect to anticommuting variables as a function from KK'° into the commutative Banach module M° = P(Ar, A) -4 P(Am, A); moreover, if f c Sk'°°(KA'm), then (over A°), f : f E Sk(KK'0, P(Am, A)). Thus, the peculiarity of anticommuting variables is practically of no use in estimations. We must only bear in mind that the norm of the continuous operator aB, : P(Am, A) --+ P(Am, A) is equal to unity.
Remark 2.2. However, the situation is so simple only for CSA with a trivial A1-annihilator for which the coefficients of power series are uniquely defined.
Proposition 2.5. The differentiation and shift operators are continuous in the functional space E(C"m).
2.3. Spaces of generalized S-analytic functions. The functional spaces and (locally convex complete CSM over the CSA A) are usually chosen as spaces of test functions on the Vladimirov-Volovich superspace CA'm; the conjugate spaces and (spaces of A-linear continuous functionals) are spaces of generalized functions. By virtue of Propositions 2.2 and 2.5, we can define generalized derivatives on a superspace. We introduce a shift of the argument of a generalized function according to the relation (Uh, f) = (u, f_h), h E CA''", fh(z) = f (z + h). Let us consider the mapping cp from the superspace Cn'm into the CSM A'(Cn'm) endowed with a topology of pointwise convergence (weak topology) W(h) = nh. Direct calculations show that
((o)h'' 80 f ) - (u' Rf )
h3,
2. Analytic Generalized Functions
67
i.e., (3(0), f) = (-1)Ifl+1(u, a). Using relation (1.6) from Chap. ). It is natural to define the generI, we have (a(0), f) _ (u, -L 80, alized derivative a as the value at zero of the derivative of the Sdifferentiable mapping cp: CA 'm -* A'(CAm), i.e., (a , f) = (u, ae ); similarly, (AML, f)
(u, a )
The operators e : A'(Cn'm) -+ A'(Cnm) are left A-linear as well The same arguments hold as the operators e : A(Cn'm) -+
for the left-hand derivatives (f) a (a, u)
.
Nothing changes upon the replacement of the space of generalized functions A'(Cp'm) by E'(Cn'm).
The product of a generalized function u by a test function cp is defined by the relations (ucp, f) = (u, cp f ), (f, cpu) = (f co, u).
Remark 2.3. It is very convenient to use both the right and the left realization of a generalized function.
We introduce the Dirac b-function on the superspace KA'm, assuming, as usual, that (b, f) = f (0); the Dirac 6-function is an even generalized function, I J I = 0, and, consequently, (b, f) = (f , J). Using the definition of generalized derivatives on a superspace, we obtain a
lal
"
a aaOIf(°)
f,a -891b =(-1)
Proposition 2.6. The space of generalized functions coincides with the space of differential operators of infinite order. 00
m
u
a° as
L b(X, 0)u"p,
a1=0 101=0
a aep
for which I
I Jul I IR = sup 11u,,,311 R-'a! < oo a'O
for a certain R = R,,.
Chapter II. Generalized Functions
68
Proposition 2.7. The space of generalized functions E'(Cn'm) coincides with the space of differential operators of infinite order (2.3) for which (2.6) IIUIIR = E IIuaiII R' < oo Or0
for all R. Let us prove, for instance, Proposition 2.7. Suppose that the funcR = (R1, ..., Rn). It follows from the definition tional u E
of inductive topology that the series E > Ra. x°`90 converges in the 1.1=0101=0
to the function fR(x, 9). The continuity of the functional u implies that the series space
00
m
(-1)1 R' u.#
(fR, u) = E
(2.7)
1.1=0101=0
converges. The convergence of series (2.7) for any R yields (2.6). The system of norms {II IIR} defines the Frechet topology in the CSM E'(Cnm). We introduce spaces of generalized functions A' (C m) = {u E A'(Cn'm): I U I IR < oo}; they are Banach CSM. The space of generalized functions is endowed with a topology of inductive limit: A'(Cn'm) = lim ind A' (Cn'm); A'(C m) is a complete locally convex I
CSM.
Propositions 2.6 and 2.7 can be used to prove
Proposition 2.8. The spaces A"(Cpm) and E"(Cn'm) which are conjugates of the spaces of generalized functions A'(Cn'm) and E'(Cn'm) coincide with the spaces of the test functions and E(Cn'm).
Complete proofs of Propositions 2.6-2.8 can be found in [146]. Furthermore, we shall use the symbol of an integral
f w(x, 9) p(dxdO) = (W, p)
(f p(dxdO) cp(x, 0)
cp))
to denote the action of a generalized function on a test function.
2. Analytic Generalized Functions
69
Theorem 2.1. The mapping z H µZ from the superspace CA 'm into the space of generalized functions A'(Cn'm) (or E'(Cp'm)) is Scomplete for any generalized function p E A'(CA'm) (or p E
Proof. Using the fact that for any test function f its Taylor series converges in the space of test functions, we obtain (µ=, f) _ (µ, f-=)
i.. V-
(-1)101+101
a!
00
xaea
as
aL
ax° aemm
ao p1
f
0
µ' axa ao#' ...a9p1 f)xaO'(-1)IkI+If1101
_
3
a.
axea9aR
mm
...
m
n Jal
_ E ail j=1
aOx"90' f ) 1
1131
E oi. i=1
It remains to verify that the power series µy,9) =
1
,a
x° 9o
cfo
converges absolutely in the space of generalized functions (or E'(CAm)). By virtue of Remark 2.1, here again everything reduces to a commutative case.
2.4. A direct product and convolution of generalized functions. The direct product of the generalized functions 1L1 and µ2 from 'm2 is the spaces A i(CAn"mi ) and A '(CAn2 'mz ) or E '(CAn"mi ) and E ' (CAnz) correctly defined by the relation
f µ1 ®p2(dx dy d9 <) ep(x, y, 9, = f µ1(dxd9) f µ2(dyde) cp(x, y, 0, C).
(2.8)
The convolution of the generalized functions µ1, µ2 from the space A'(CAm) (or E'(CA'm)) is correctly defined by the relation
f µl * p2(dxd9) cp(x, 9)
Chapter II. Generalized Functions
70
= f,ii®µ2(dxdydOdt;)cp(x+y,9+e)
(2.9)
Theorem 2.2. The operations of the direct multiplication and convolution of generalized functions are supercommutative:
[µ, v}® = µ ® v -
[µ,v}.
=µ*v-
(-1)11,1 I-I
v 0 µ = 0;
(-1)lµllvly*IL =0.
Proof. Let us verify, for instance, that the supercommutator [µ, v}® is zero. We shall consider the case IpI = I v I = 1. It suffices to prove that Ker [p, v}® contains all functions c,(x, y) 0, 0 = g(x, O) f (y, l;), where g and f are homogeneous elements. For these cp we have (µ(9 v, AP) = (µ,(v,co)) _ (µ,(v,9)f)
(-l)11(µ,f)(v,9) _
(-1)02(v,9)(µ,f)
_ (-1)Q2(v,9(µ,f)) _ (-1)13(v, (µ,f9)) = (-1)14 (v, (µ, 9f )), wherea1 = If I(1-I9I), a2 = al+(1-191)(1-If 1), a3 = a2+191(1-If 1), a4 = a3 + If 11g1 . It remains to note that a4 = 1.
Corollary 2.1. The spaces of generalized functions A'(CAm) and E'(CA'm) are topological convolution CSA.
We shall prove the continuity of the operation of direct multiplication (and that of convolution) with the use of the Fourier transformation operator. Note the formula for differentiation of a convolution: 89.(µ*v) =µ*
By.
(2.10)
With respect to even variables a convolution can be differentiated in an ordinary way. Let us verify relation (2.10):
f
'9R
(µ * v) (dxd0) f (x, 0)
2. Analytic Generalized Functions
=
71
f µ * v(dxd9) "aojL'
= f p(dxd9) f v(dy<)
(X, B)
ad (x + y, B + e)
fp(dxdO)f(dyde)f(x+y,O+e). The direct product and the convolution defined by relations (2.8), (2.9) are the right-sided multiplication 0 = OR and the right-sided convolution * = *R. The left-sided multiplication ®L and the leftsided convolution *L can be introduced by analogy: f co(x, y, B, )µ ®L v(dx dy dO <)
=f
(f (P (x, y, 0, )p(dxd9))v(dyde);
f W(x, 9)µ *L v(dxd0) = f co(x + y, 0 + )µ ®L v(dx dy dB <). The left-sided convolution can be differentiated with the use of the relation aL 09Lµ *L v. (µ *L v) = a0j
ae;
The convolution of the generalized function p and the test function p is correctly defined by the relation µ * co(x, 9) =
f(dyd)co(x_y,0_e),
((P * µ(x, 0) = fco(x
- y, 9 - t)µ(dyde))
Let us verify that the mapping *: E(CA'm) x E'(Cnm) --> E(Cn'm) is continuous. Let cP E E E'(Cp'm). Then, by virtue of Proposition 2.7, we have a
µ= aQ
5(x,0)µ"3, aap x e
Chapter H. Generalized Functions
72
with I I µ I I R = E I I µ.a I I R' < oo for any R. Consequently,
W * µ(x) 0) = E
< R-
note that IIIa ao I
I
a. aR
axa aep O(x, B)A.O; R.
Therefore
I O* µI I IR 511 IWIIIRE II µ.01l Ra =I I I(PI I IR IIµIIR. cp
We shall obtain relations for the differentiation of the convolution of a generalized and a test function. We begin with showing that relation (1.6) from Chap. I that connects the right-hand and left-hand derivatives remains valid for generalized derivatives.
Recall that every functional u E M', where M is a topological CSM, results from the identification of the functionals UR and UL which
are right and left A-linear respectively, and, by virtue of (1.1),
(u, f) = (uR, f) =
(-1)Iu1IfI(f,
uL) = (-1)1uI vi(f, u).
Suppose now that u is a generalized function, µ = ae ; µR and µL are the right-hand and left-hand realizations of M. The definition of the right-hand generalized derivative gives its right-hand realization: (µR, f) = (u, as ). Let us find its left-hand realization:
(f, µL) _ (-1)Ifl IPI(zR, f) _
_
(_)IfI(iul+1)+IuI(1f1+1) 1
(-1)IfI(IuI+1)
l u, ad \
\ ae/
(ad'u a
0.
Thus, the left-hand realization of the generalized derivative according to the formula
f
aRU\ _ (-1)I"I+1 '
090
ao u l \ aRf
ao
acts
2. Analytic Generalized Functions
73
i.e., B = (-1)IuI+1 e . For instance, ae directly verify this relation:
-e
We can also
l
= (-1)IVI a8 , (P) = (-1)Iwl a8 (0);
l (P7
\
/
a8 aB /
= ae (o) = -(-1)Iwl ae (o).
The relation connecting the right-hand and left-hand generalized derivatives can also be obtained in a different way. Let us consider the (0) and shift mapping co(z): z H uZ. Then, by definition, e =
e = a (O). The same computation as were carried out in Chap. I show that relation (1.6) from Chap. I holds for maps with values in CSM as well. Let us now derive a formula for the differentiation of the convolu-
tion of a test and a generalized function: 3R(co * u)
_
(-1)Iwl+lul+1aR(W * u)
80
ae
_
(-1)IWI+lul+1
J
ae
(-1)lul f a (0 - e)u(d ) _
(-1)I"I+1 f a (0 - e)u(de)
_ (-1)Iul+l f 0(0 - ) a (de). We have thus obtained the formula aR `°'u = `P ae
By analogy, we can prove the formula a` ae
'v
definition immediately yields the formulas u) = &W
ae
ae * u'
aR(u * co) 190
=u
* aRu
ae
= e * W. aRW
* ae
The
Chapter II. Generalized Functions
74
3.
Fourier Transformation of Superanalytic Generalized Functions 3.1. Properties of the Fourier transformation. The Fourier
transformation of the generalized function u E A'(CAm) is defined by the relation .F(u)(x, 0) = f u(dydl;) exp{i(y, x) + i(e, 0)}, n
m
E t;,9,
where (x, y) = E y7x.i, 9=1
j=1
Theorem 3.1. The Fourier transformation .F:A'(CA'm) -+E(C m) is the isomorphism of locally convex CSM.
Proof. We shall show that the triviality of the A1-annihilator implies that Ker.F = {0}. We shall restrict the consideration to the case n = 0, m = 1. Here e=ee = 1 + ie9. Assume that u E Ker.F. Then .F(u)(0) _ (u, 1) + i(u, ) 0 = 0. Consequently, (u, 1) = 0 and (u, )O = 0 for all 0 E A1. It remains to use the fact that 'Al = 0. In the general case, the reasoning is the same. In order to complete the proof, we must apply Proposition 2.6. Note that the Fourier transformation operator, which is an even operator, transforms even generalized functions into even test functions and odd generalized functions into odd test functions. Denoting by .F' the conjugate of the operator F, we get a scheme of Fourier transformation on the superspace CA 'm: A'(CA'm)
E(CA'm),
A(CA'm)
E'(CA'm).
Note that .F'(v)(y, ) = f exp{i(y, x) + i0)}v(dxd0). A Fourier transformation on a superspace possesses all properties of an ordinary Fourier transformation (if the right-hand and lefthand derivatives and the postmultiplication and premultiplication by a derivative are correctly taken into account).
3. Fourier Transformation
75
1. The right product and the convolution of generalized functions are transformed into the product of Fourier transforms.
2. a .P(u) = i.P(ue ); .P(a ) = i.F(u)03. .P'(v) = iF(03v); F'(B) = ie,F'(v). 3. The ordinary formulas are valid for even variables. Using the fact that the spaces of test functions A(C"') and are locally convex CSA, we find that the spaces of generalized functions and A'(C"m) are locally convex convolution CSA.
3.2. Fundamental solution of a Cauchy problem on a superspace. We consider a Cauchy problem on C"' with the evolution parameter t from AO
0 < j < p - 1,
at (0, x, 0) =cps (x, B),
(3.2)
where P(at, ax, 00) is a differential operator (with right-hand derivatives with respect to 0) with constant coefficients (the multiplication by which is also a postmultiplication) aj,,,p E A which is of order p with respect to ae ;
P
a
at ax ae
P
u(t, x, 0) = E E
av a a
R
;-o aA ati axa aea
u(t, x, 0)aj,,p.
We shall study the Cauchy problem both in the space of test functions and in the space of generalized functions. The fundamental solution of the Cauchy problem (3.1), (3.2) is the solution of the Cauchy problem in the space of generalized functions: P (c7t' ax' 090
ati
= 0,
0
- j <- P -
w) = 0, 2,-1 (0) P_1
(3.3)
J..
(3.4)
Chapter II. Generalized Functions
76
Theorem 3.2. Let a be a fundamental solution of the Cauchy problem (3.1), (3.2) and let cps (x, 0), 0 < j < p - 1, be test functions. Then the function
ap-;-1
p--1
u(t, x, 0) = u Wj(x' 0) * 0-tp-7-1 (t) i=o
is a solution of the Cauchy problem (3.1), (3.2) in the space of test functions.
Proof. To distinguish the algebraic substance of this theorem, we shall consider the Cauchy problem (3.1), (3.2) in a purely anticommutative case (for p = 1): 09U
M at (t, 0) = X01
aRU aBA
(t, 0) a#,
u(0, 0) = co(0)
Then we have
at
fi(t) =
* as t)
-
m OR(
- (P * E aRU(t)ao
L. aeQ iAi=o
t (t))-a#.
oo 'goo
1Ai
By analogy, with the aid of formula (2.10), we can prove
Theorem 3.3.
be a fundamental solution of the Cauchy problem (3.1), (3.2) and let cps (x, 0), 0 < j < p - 1, be generalized functions. Then function (3.5) is a solution of the Cauchy problem (3.1), (3.2) in the space of generalized functions. Let
Let us now consider the Cauchy problems (3.1), (3.2) and (3.3), (3.4) for differential operators with left-hand derivatives with respect to 0 and the premultiplication by the coefficients:
P a a a
(at' ax' 190
P-1
u(t, X' 8) _ ,_o
a a° aL aBAu(t, x, B).
a'°Qatj
arc,
3. Fourier Transformation
77
Then the solution of the Cauchy problem (3.1), (3.2) in the space of test functions is expressed in terms of the fundamental solution with the aid of the formula P-- ap-j-1G " (t
x 0)
= j=0 atP-j-Sl (t)
and in the space of generalized functions with the aid of the formula P-1-1 aP-j-1t
u(t, x, 0)
= j-O atP-j-S1 (t) *L cpj.
(3.7)
3.3. The classes of well-posedness of the Cauchy problem for A-linear differential equations with constant coefficients. Recall that the space of test or generalized functions is known as the class of well-posedness of the Cauchy problem if there exists in this space a unique solution of the Cauchy problem which continuously depends on the initial conditions. On the superspace Cp'm we consider a Cauchy problem with the evolution parameter t c Ao: apu
at
1
P
(t x 0) + j- -o
(0, x,
at
8
a
ax ae) W
0 < j < p - 1,
0) = cpj (x, 0),
where a
aj u (t x 0) = 0,
a___
a
Pi (ax' '90V (X, 0) _
(3.9)
A
ax° a9Q
(x, 0)a,,o
are right-hand differential operators.
Theorem 3.4. The spaces of test and generalized functions and E'(CA'm) are classes of well-posedness of the Cauchy problem (3.8), (3.9).
Chapter II. Generalized Functions
78
Proof. We note, first of all, that ao
l
aLU
ao
Let us consider a Cauchy problem in the space of generalized functions
E'(C"'). The Fourier transformation F' turns this Cauchy problem into the Cauchy problem for an ordinary differential equation in the Frechet space apii
O-V ii
E a (t, y, e) Pi (y, 0) = 0, at (t, y, 0) +P_' =o
(3.10)
pp
a (0, y,
(y,
ati
0 < j < p - 1,
(3.11)
where
Pj (y, ) aQ
are symbols for right differential operators P3 (az , B) . A solution of this Cauchy problem exists and is unique (as usual, the Cauchy problem (3.10), (3.11) is written in matrix form, and it remains to use the fact that the exponent of a matrix with entire S-analytic coefficients is an entire S-analytic function). The continuous dependence on the initial conditions follows from estimate (2.1). Applying the operator we get the statement of the theorem for E'(Cn'm). The Cauchy problem in the space of test functions can be considered by analogy.
Corollary 3.1. In the space of generalized functions E'(CA'm) there exists a unique fundamental solution of the Cauchy problem (3.8), (3.9).
In the same way we can prove the well-posedness of the Cauchy problem and the existence of a fundamental solution for left-differential operators: a
a
P, (-,) p(x, B) _ 190
a,.0
a°+A
ax°aep (x, B)
3. Fourier Transformation
79
In this case, Eq. (3.8) in the space E'(Cpm) becomes an equation in the space A(C" m): av f,
p
(t, y, 0 + P_' E P, (y, 0 a (t, y,
(3.12)
0,
i=o
where PP (y, e) _ E i cxp
are symbols for left-differential operators Pi (ax , ae) We can see from the proof of Theorem 3.4 that the spaces E(Cn'm)
and E'(CAm) are classes of well-posedness of the Cauchy problem (3.8), (3.9) for infinite order differential operators as well, the symbols P; (y, e) for which belong to the class A(CA m) .
3.4. Poisson formula for heat conduction equations. We denote by E = Eo ® E1 the real part of the complex CSA A = AO ® Al
(A = E (D iE). Let the matrix T consist of elements of a CSA E and have a block structure
T=
Too
Tol
Tol
T11
where the matrices Too = (n x n) and T11 = (2k x 2k) consist of even elements, the matrix T01 consists of odd elements (i.e., the matrix T Rr2k); let the matrix Too is associated with the operator T: Rr'2k be symmetric and the matrix T11 be skew-symmetric. We consider the superanalog of the Laplace operator: a2
n
AT= ,1 E T°°o ax;axi + 2
Tt'
L
01 axiae;
a2
2k
a2
n,2k
+
T t'
L
11 ae;aei
and the Cauchy problem for the heat conduction equation on the superspace:
at
(t, x,
e)
= 2 ATU(t, x, e),
u(0, x, B) = O (x, B).
(3.13)
Chapter II. Generalized Functions
80
The Fourier transform F of the fundamental solution is a quadratic exponent, E(t, Y) = exp{- (TY, Y) }, where we denote by Y the supervector (y, ).
z
We shall show that for t > 0 the fundamental solution is associated with the Gaussian "measure" on the superspace RE,2k (i.e., the action of the fundamental solution on the test function is equal to the Gaussian integral with respect to this function). To this end, we must impose a number of constraints on the matrix T which ensure the existence of a Gaussian integral (see Chap. I, Secs. 3.5, 4.5). Let the matrices Too = too+'yoo, T11 = t11+'711, where too and t11 are
numerical matrices and 'yoo and 'yi, are matrices with quasinilpotent elements. For instance, if E = G1 00 is Rogers' algebra, then any matrix can be represented in this form. In addition, we shall assume that the matrices too and t11 are invertible (it follows from the condition of quasinilpotency that the matrices Too and T11 are invertible as well) and also that the matrix too is positive definite, too > 0. Lemma 3.1. The matrix T-1 has a block structure A C T-1 =
-C* D
where A = (n x n), D = (2k x 2k) are matrices with even elements, C is a matrix with odd elements; the matrix A is symmetric and the matrix D is antisymmetric. Proof. In order not to cram the calculations by the indices of the matrix T, we shall carry out the calculations for the matrix M = T-1. By multiplying the matrices we verify that
M-1 -
CD-1C')-1 -A-1C(D + (A + D-1C`(A + CD-1C*)-1 (D +
C'A-1C)-1
C'A-1C)-1
The matrices Too = (A+CD-1C`)-1 and T11 = (D+C`A-1C)-1 consist of even elements and the matrices T01 = -A-1C(D + C`A-1C)-1
and T10 = D-1C'(A+CD-1C')-1 consist of odd elements. Furthermore, we note that if U and V are matrices with odd elements, then (UV)* = -V * U` Consequently, .
Too = ((A +
CD-1C*)-1)* = (A'
-
C(CD-1).)-1
3. Fourier Transformation
= (A -
81 CD'-1C')-1 = Too,
and we can verify by analogy that Tll = -T11, and, finally,
T01 = -(A-1C(D + C'A-1C)-1)'
_ -(D' +
(C'A-1C)')-1C'A'-1
= -(-D -
Tlo.
We shall now make use of the results of Chap. I; the quadratic exponent, namely, the Fourier transform of the fundamental solution, is the Fourier transform of the Gaussian "measure" on a superspace. We have thus proved that the fundamental solution of the Cauchy problem (3.13) for real t > 0 is the Gaussian "measure" considered in Chap. I. A solution of Cauchy problem (3.13) can be obtained by means of the Gaussian convolution
u(t,X) _
dY
/'
J R^ xEzk
(27r)- sdet T
exp{- 1 (Y,T-1Y)} x cp(X - Y), 2t
i
(3.14)
where X = (x, 0), Y = (y, C).
Formula (3.14) is an analog of Poisson formula in a supercase. The Poisson formula defines the solution of Cauchy problem (3.13) for real time, t > 0, for the initial conditions of the class E(CA'm). Using the Poisson formula, we can essentially extend the class of initial conditinos for which the Cauchy problem (3.13) is solvable. We can
show that the Cauchy problem (3.13) is solvable for the functions cp E S2,°°. Note that since the integral with respect to anticommuting variables is defined only for polynomials, it follows that when using the Poisson formula, we cannot reject the condition of analyticity with respect to anticommuting variables. There also exist constraints concerning commuting variables, and these constraints are more severe than in the standard theory of the heat conduction equation. As the following example shows, the class of the initial conditions cannot be
extended to So,'.
Example 3.1. Let E = G' and b: G100 -+ R be a canonical projector onto R; W(x) = b(x). Then the function u(t, x) defined
Chapter II. Generalized Functions
82
by (3.14) is not S-differentiable. There is no solution of the Cauchy problem (3.13) with the initial condition cp(x). Although the heat conduction equation on a superspace is one of the simplest partial differential equations with commuting and anticommuting variables, there still remain many open questions concerning this equation and Poisson formula (3.14). For instance, can we weaken the condition of double S-differentiability with respect to even variables in (3.14)? Is there a solution of problem (3.13) for initial conditions which are not S-analytic with respect to anticommuting variables?
3.5. Inversion formulas. Let us derive inversion formulas for Fourier transforms F and P. These are formulas according to which the function f E A(CA'm) can be used for finding the generalized
function µ E E'(C"), ,E'(µ) = f, and the function f c E(C"m) can
f
be used for finding the generalized function µ E
.
By virtue of propositions 2.6 and 2.7, any generalized function from the spaces and A'(Cn'm) can be represented as a series of generalized derivatives of the b-function on CA m. This series corresponds to Taylor's series for the Fourier transform f of the generalized function µ. Every entire S-analytic function can be expanded in the left Taylor series
f (y, o _
aGa Y
(3.15)
(0)
ap
Using the formula B) = the right inverse operator F':
we obtain the formula for
av
ay
CIO
(°).
(3.16)
Every entire S-analytic function f can also be expanded in the right Taylor series Q+p
f (y, ) _
a1 !
ay°a
f
(3.17)
3. Fourier Transformation
83
e) =
we obtain the formula for the
Using the formula left inverse operator .P:
(FI)-1(f) _
IAI
(Z)'C"
a
ay
ap
a Q (°)
ax aepe)
(3.18)
Using the formula F() = (-i9j).F(v), with the aid of relation (3.17) we obtain the formula for the left inverse operator F: (
f)
(i)k-H+IAI 4+0f 3°'(x, 0)
-
a!
Qp
ayQW ax' a0'
() 3 19
Using the formula F(a ) = .F(v)(i0j), with the aid of relation (3.15) we obtain the formula for the right inverse operator F: aR+Aa(x, 0) aL+Q f
-F-1(f) _
a!
00
axa'a0A
ayaacA (0).
(3.20)
3.6. Operational calculus on a superspace. For partial differential equations on a superspace we propose an operational calculus of the type of the classical Heaviside operational calculus. In the framework of this calculus, the fundamental solution of the Cauchy problem (3.8), (3.9) for the right differential operators can be found according to the following scheme.
Consider the Cauchy problem (3.11), (3.12) for ordinary differential equations with coefficients dependent on the superparameter Y = (y, ) with the initial conditions ilij = 6j(p-1), 8jk is Kronecker delta. The solution E (t, Y) of this Cauchy problem is an entire Sanalytic function of the superparameter Y. From the inversion formula (3.16) we obtain E(t, x, 0) =
is
is
iaR
iaR
t, axl , ..., ..., axn , aa,n' aBl
6(x, 9).
(3.21)
We can also use the right formula (3.18). Then .6 (t, x, 0) _ e Ct'
is axl
, ...,
0
-iaL
axn' ae,n
...,
-iaL ,, 0
6(x, 0).
(3.22)
Chapter II. Generalized Functions
84
in Example 3.2. Consider the equation at (t, 9) = e the momentum space aG (t, ) = G(t, ) (-*e), and, consequently, ) = e-it ' = 1 - itea. From relation (3.21) we obtain
(t, 0) _
C t,
aR ) aRj(B) a. ae) (e) = 1 + ta (e) _ (9) + t
a
OR
a9
C
We shall use this example to illustrate the importance of the order in which the fundamental solution and the test function enter into the convolution. Using formula (3.5), we calculate the solution of the Cauchy problem for the odd initial function W (O) = 0: u(t, 0) = cp(9) * E(t, 0) = f (9 - e)e(t)(de)
[9_t()a] = [0+t(e)c] =9+ta. If we take the convolution in a different order,
a
f e(t)(<)(9 - ) = a9 - t
a9 - ta(a),
then we get an incorrect answer.
In order to calculate the fundamental solution, we can also use formula (3.22): E(t, e) = 1 - to,
(t, 0) _
(t,
a9L
6(9) = 1- to, (a)a9)
_ 8(9) - tc(a)aLa(e) a9
_ J(9) - taL6(0) a9
Example 3.3. Consider the equation z
(t, B1, e2) = a91a92 (t, e1, 02),3;
5(9)
3. Fourier Transformation
85
in momentum variables:
ac ( t, at
1 ,
) = c(t
1, 6) (-C2 1Q),
i.e., E(t, 1i 2) = 1 - t&10; from relation (3.21) we have E(t,
OR OR 01, 02)
a02 , ael
_
(i+t_
ae ae2
0)
o(01, 02)
x(01, B2).
Similarly, from relation (3.22) we have (6(01,02)
ael
1902
2
)(0102).
atae'9L
(here E(t, 1, e2) = 1 + tQe1e2).
Example 3.4. Consider the more general equation a(t,01 ...,em) _
aRu 190, ...aem
(t,e1i...,em)0;
in momentum variables (t, 1, ..., Sm)
-
C-' (t,
1+
t(t, 1, ..., m) = from relation (3.21) we have
(t, B1,
..., em) _ (1
+
am
tael. aem
Q
em),
Chapter II. Generalized Functions
86
and from relation (3.22) we have 1 mam C(t, 01i ..., 0,n) = I 1 + taBm)..aR R 6(01, ..., Bm).
In order to see that the two expressions for the fundamental solution coincide, we must use the formula aR b aOl...aem
(-1)m
=
aL a0m...a01
For the left differential operators, the operational calculus on a superspace can be realized according to the same scheme (only instead of Eq. (3.10), we must consider Eq. (3.12)). The fundamental solution can be found from formulas (3.21), (3.22).
a
Example 3.5. Consider Cauchy problems for the equations a = and at = - a These problems have the same fundamental solu-
tion E(t, 0) _ (1 + t aB)6(0). However, the solutions of these problems for odd initial conditions are different (if cp(0) = 0, then u(t, 0) = 0 + t for the right equation and u(t, 0) = 0 - t for the left equation).
3.7. Fundamental solutions of Cauchy problems for heat conduction equation, Schrodinger, d'Alembert, and Laplace equations. With the aid of operational calculus it is easy to find the fundamental solutions of Cauchy problems for the most important equations with constant coefficients on a superspace. Let a C A0. The fundamental solution of the Cauchy problem
at
(t, x, 0) = aLTU(t, x, 0),
u(0, x, 0) = cp(x, 0)
can be written as the formula E(t, x, 0) = exp{atLT}b(x, 0).
For a = 1, we obtain a heat conduction equation. As has been shown, in this case the infinite-order differential operator of the bfunction can be realized with the aid of the Gaussian integral over a
3. Fourier Transformation
87
superspace. For a = -1, we get Schrodinger equation. Restricting the class of initial functions to functions rapidly decreasing along the real axis (see Sec. 4), we obtain a representation for an infinite-order differential operator in terms of an oscillating integral over a superspace. Let us now consider a Cauchy problem for the d'Alembert-Laplace equation 3z (t, x, 9) = aLTU(t, x, 9). 8t2
The fundamental solution of this problem can be expressed by the relation
E(t, x, 9) _
sink a7 T
3.8. Pure anticommutative case. If there are no commuting variables, then the whole above-described theory becomes simpler and
is of a pure algebraic character. Here the space of entire S-analytic functions coincides with the space of polynomials. Generalized functions are A-linear functionals on the space of polynomials.
Proposition 3.1. Every generalized function L E P'(A', A) is defined by the polynomial g E P(Am, A): L(f) = f dOg(9) f (9), with L(9-) = (_1)P(-)oIc1I(gl-,,), where m
(1,...,1),
p(a) = mlal - Ial(Ial - 1)/2->jai. j=1
Proof. Using the right A-linearity, we obtain l
0- f,,)
_
L(9-)ff` = fdog(O)f(O)
fdOm...OiO1_1...O1_cxmgz_aOa1...9mfc,
_
f d9 91-a9aa1 `1(gl-c`) f..
It remains to point out that 9"9' = (-1)P(°)91.
Chapter II. Generalized Functions
88
For instance, let L = 8 be Dirac 8-function concentrated at the point 1; E Am. Then it follows from this proposition that Le = b(B-l;), where b(O) = 0. Here the Fourier transformation operator.F: P -+ P has the form
NO = .F(g)(e) = fdOg(9)exp{i(O,e)}.
Proposition 3.2. If g(0) = E Oaga, then a
a
m
where
ga = (-1)r(a)tlal91-a, r(a) = m1 al - 17aj j_1
Proof. Indeed, alai
aa exp{i(9, )}le=o
a
_
it = E
it al Lam ... a
or
It remains to apply Proposition 3.1.
Theorem 3.5 (inversion formula). The inverse Fourier transformation operator is defined by the relation .F-1(f)(B) =
1m 2
f
(l;)
8)}.
(3.23)
Proof. Let f = g. Then Proposition 3.2 gives
f df
exp{iB)} _
9a21a1(-1)r(a)f!-a
Furthermore, fA-a =
im-lal(-1)r(1-a)9a,
m
r(l - a) = m(m - jaJ - Ej(1 - a,)). j=1
Consequently, .F(f) (B)
_ > Baim(-1)r(a)+r('-a)g(a). Q
3. Fourier Transformation
89
Corollary 3.2 (Fourier formula). The inequality
g(x) = i_m fderndem...deidoig(o)exp{i(e,x_o)} holds for every polynomial g E P.
If the space is even-dimensional m = 2k, then .F-1 = F, and if the space is not even-dimensional m = 2k + 1, then .F-1 = i-1.F. In the even-dimensional space m = 2k it is convenient to write the variables as (01 i B1, ..., 9k, Bk) and the dual variable as ( 1, 1, ..., k, Sk) k
We set dO dO = rl dOp dOp. In this notation the Fourier transform p=1
J(e) = f dB d9 f (9, B) exp{i(O, e)
Proposition 3.3. For every element a from the CSA A we have
f dl; d exp{a(l;, ) +
B) + i(, B)} = [J (-a+ Op#p).
(3.24)
P=1
Thus the Fourier transform of a quadratic exponent may not be a quadratic exponent even if the number of variables is even.
Proposition 3.4. The relation (9,
n!
B)k_n
(k
- n)i
=
f
de <
e)" exp{i(C, 9) + i(C, B)}
(3.25)
holds true.
Proof. Using relation (3.24), we have
I(B,B) = Jdedexp{a() +i() +i(,O)} _ ll(a+9jdj). j=1
Furthermore, k
II(a+bj) =ak+ak-1(bl+...+bk)+...+a°bl...bk j=1
Chapter II. Generalized Functions
90
for even elements b; . If b,2 = 0, then
(bl + ... + bk)n = ( E b;l...bin)n! 91 G..
Therefore
1 (0, 0)
k
(eve)n
a k-n
_
an(99)k-n
k
However, on the other hand,
fdd
E n=
n exp{i((, 0) +
B)}
n1
whence follows formula (3.25).
Proposition 3.5. The relation Aa(9, 9)n = (k - n + 1)n(B, 9)n-1,
n = 1, 2, ..., k,
(3.26)
holds for the Laplace operator with respect to anticommuting variables Aa = E a000j, a2 j=1
Proof. We shall carry out the proof by induction. For k = 1, relation (3.26) is valid; let us assume that it is valid for k = m - 1. Then m-1
Da(B, 9)n
_
m-1
a2
a2
ae; a9; + An a9m _
(m-1
OjO,)
j=1
j=1
/m-1
\ n-1
=(m-n)nI\ 9;9;I /
#j
ll
n-1
/m-1
+nI E9;9;) /m\-1
+n9rn9m(m-n+1)(n-1)I T 9;9; \j=1
n-1
n-2
3. Fourier Transformation
91
Proposition 3.6. For the Dirac 6-junction 6(0, 0) we have a representation 6(0, 0) = k(0, 0) k. ,
Proof. As has been pointed out, if the even elements bj are such
that b = 0, then (b1 + ... + bk)n = n! (
bj1...
n)'
71<...<3n
and therefore (9, 0)k /k! = 6101...9k0k = b(9, 0).
Proposition 3.7. In the space P(Aik, A) there exists a fundamental solution of the iterated operator (Da + b)m, where b is an invertible element of A. The fundamental solution is defined by the relation em(0,e)
= k (0,9)p(m+k-p-1)! j=1
bk-p+m(m - 1)!p!
(3.27)
Proof. Applying the Fourier transformation, we obtain
(b -
e) = 1.
However, k
(b-(, ))-m=V n=0
()
)n
(m + n - 1)!
n!bn+m(m - 1)!
It remains to use Proposition 3.4. Note that in the space of polynomials P(Aik, A) the equations oae(0, 9) = 6(0, 9), a E(0, 9) = b(0, 9)
j
have no solutions. In conclusion, note the useful formula for integration by parts for anticommuting variables
f d9 j (9)9(0) = f d9 f (0) f-(0).
(3.28)
92
4.
Chapter II. Generalized Functions
Superanalog of the Theory of Schwartz Distributions
We give here the theory of generalized functions on a superspace which can be regarded as a supergeneralization of the theory of Schwartz distributions. We use a new parameter of the theory of generalized functions, a type of superalgebra, for constructing a theory which is simpler and more convenient for applications. A superspace over Banach CSA is, evidently, not very suitable for constructing Schwartz supertheory. Spaces of test functions of the type and D can be very naturally introduced on a superspace over superalgebras in which elements with zero numerical component are nilpotent. In these CSA, it is convenient to introduce not a topology but a pseudotopology or a convergence of an inductive limit with respect to the order of nilpotency of the elements.
4.1. Body and soul projectors. De Witt introduced very useful projectors, body and soul projectors, on Grassmann algebras. The following definition is the generalization of the De Witt definition to the case of an arbitrary Banach CSA B = Bo G B1. Definition 4.1. The body projector onto the CSA B is a (nonzero) continuous homomorphism b: B -+ Ke; the map c = (1 - b) (where 1 is a unit element into B) is known as soul projector. It is often convenient to identify Ke with the field K and regard the body projector b as a K-valued functional. A body projector is a linear continuous multiplicative functional on a Banach algebra (a character). As usual, we show that the norm of this functional is equal to unity, and be = e, and that the kernel Ker b is the maximal ideal in the algebra B (see, e.g., [38, p. 521]).
Theorem 4.1. We can use any maximal ideal M in a complex Banach CSA B in order to construct a linear continuous multiplicative functional b such that M = Ker b.
Proof. Here everything is similar to a commutative case (see, e.g., the book by Kolmogorov and Fomin [38, p. 522]). We must only verify that just as in a commutative case the quotient algebra B/M is
4. Schwartz Distributions
93
isomorphic to C. Note, first of all, that B1 C M. Indeed, we assume that there exists an element a E B1 and a ¢ M. Let us consider the principal ideal Ta = aB and the sum of ideals R = Ta + M. Then R is a proper ideal containing M. Indeed, let us assume that R = B. Then e = Aa + m, A E B, m E M, i.e., a= .\a2 + ma E M. Furthermore, the ideal M can be represented as a direct sum M = M° ® B1, where M° is the maximal ideal in Bo. The quotient algebras B/M and Bo/M° are algebraically isomorphic. Let us verify whether they are isometric. Let j be a canonical homomorphism of the algebras into quotient algebras (we denote it by one symbol since B1 C Ker j). Then
z°EMo f1EB1(Ila° + z°II + Ilzlll)
= inf Il.°+z°ll = IIj(A°)IIBo/MozoEMo
Thus the quotient algebra B/M = B°/M° is isomorphic to the field C (here we use the well-known property of maximal ideal in commutative Banach algebras). Consequently, the maximal ideal M is of codimension unity, and the identity e of the algebra B does not belong to the ideal. It remains
to set be=eandblM=O. By virtue of this theorem, there exists a one-to-one correspondence between the body projectors in CSA and the maximal ideals in CSA. This is a generalization of the well-known theorem for commutative Banach algebras (commutative Banach algebras are also CSA in which the odd part B1 consists of zero element).
Remark 4.1. Body projectors play an important part in physical applications (see De Witt [27, p. 239], the definition and the properties of physical observables in superquantum theories). Superalgebra is an ideal object which can be used for describing a physical phenomenon. Receiving in calculations an answer in the form of an element of CSA, we must extract from this answer certain numerical values. Body projectors are usually used precisely for this purpose.
Chapter II. Generalized Functions
94
Let G be a subset of the CSA B. The set b(G) is called a body of G and the set c(G) is called a soul of G. Body and soul projectors can be extended coordinatewise to the superspace KB'm. The body and soul for the subset G of the superspace KB'm can be introduced by analogy. The body of the superspace can be identified with K". All odd elements get in the soul of the superspace.
Theorem 4.2. Suppose that in the Banach CSA B = Bo ® Bl regarded as a K-linear Banach space there exists a topological basis {ej }'?_o, where eo = e; {ej }, j j4 0, are nilpotent. Then there exists in the CSA B a unique body projector b1 > cj ej I = co,
c3 E K.
Proof. Let e,' = 0. Then bed = (bej)k = 0, i.e., be3 = 0.
Corollary 4.1. In a finite-dimensional Grassmann algebra there is a unique body projector.
Corollary 4.2. In the Banach CSA G100 there is a unique body projector.
Using the theory of commutative Banach algebras, we can give numerous examples where a body projector is not unique. In what follows, of particular interest to us are body projectors consistent with the norm of the CSA B IJxMM = 11bx + cxJJ = JbxJ + JIcxJJ.
4.2. Pseudotopological algebra with a nilpotent soul. As before, A is either a commutative Banach algebra or a Banach CSA
with a trivial A1-annihilator. Let b: A -- Ke be a body projector consistent with the norm in A (e.g., the body projector onto G1 is of this kind). We set C = c(Ao). Then AO = Ke ® C, C is the maximal 00 ideal and, in particular, a subalgebra of the algebra Ao. We denote by N a nilradical (a set of nilpotent elements) of the algebra A0. By Cm we denote a subset of the algebra C consisting of
4. Schwartz Distributions
95
nilpotent elements of order m: c E Cm b cm = 0. Then the algebra N can be represented as a union of an increasing sequence of subsets {Cm}: N = M=2 U Cm Lemma 4.1. The subsets Cm of the algebra N possess the following properties: Cn + Ck C C..+k,
C Cp,
p = min(n, k).
(4.1)
(4.2)
These properties can be immediately verified. On the algebra N = UCm we can introduce a very natural inductive pseudotopology TN using the sequence of subsets {Cm}.
Remark 4.2. The readers whose interests are far from the problems of topology and pseudotopology can everywhere in what follows consider simply the convergence of sequences in the algebra N. The sequence xn -* x in N if xn -+ x in A0, i.e., Ilxn - xII -+ 0, and there exists a set Cm containing a sequence {xn}. Recall the definition of pseudotopology. A filter on the set P is every nonempty set of parts of P which satisfies the following conditions: (1) 0 V z/', (2) if B1, B2 E z/', then
B1 n B2 E ', (3) if B E z/i, C C P, and B C C, then C E ?/i. Furthermore, if 0 is a filter in the set P, then the basis of -% is every subset 'fib of the set V) satisfying the condition VA E V 3B E Yob B C A.
Thus, ifb is the basis of the filter 'Y, then A E a 3B E Ib, B C A so that the filter can be uniquely restored from its basis. The simplest example of a filter is a system of neighborhoods of a point in a topological space. The basis of this system is the basis of the filter. We denote by (P) the set of all filters on the set P and by T((D(P)) the set of all parts of the set c(P). The pseudotopology in P is the map T: P -+ T(O(P)) satisfying the following conditions: (1) Vx E P the filter [x] which consists of all subsets containing x belongs to r(x), (2) Vx E P (W1, W2 E T(x)) = ((cP1 n W2) E T(x)),
Chapter II. Generalized Functions
96
(3)VxEP(oET(x),'c'E'(P),'tbjg)=(z/)ET(x)) The filter go is said to be convergent at the point x in the pseudotopology T if cp E 7-(X)-
Actually, to define a pseudotopology is to define, for every point, a collection of filters converging to this point which satisfies the natural constraints. We shall use the symbol cp . x for the filter cp converging to the point x. The pseudotopology TN in the algebra of nilpotent even souls A is defined as follows: cp . x in A b go x in the Banach algebra AO and there exists a set Cm E W.
Theorem 4.3. (N, -r) is a pseudotopological algebra.
Proof. Let the filters cp c rN(a), 0 E TN(b). We shall show that cp + V) E TN (a + b) and got E TN (ab) . Here we denote by cp + b a filter generated by the basis consisting of various sets of the form A + B, A E cp, B E and by go a filter with the basis AB, A E cp, B E 0. Using (4.1) and (4.2), we have the following: let Cm E cp, Ck E zb, and then Cm + Ck C Cm+k Cm+k E cp + ib, CmCk C C s = min(m, k), = C, E cpib.
Remark 4.3. It is not clear whether N is an alebra relative to an inductive topology or relative to a pseudotopology. Note that the sets Cm are not linear spaces, they are not even convex or bounded. Let us now construct a new CSA U = Uo ® U1 setting Uo = Ke
N, U1 = A1. This is a pseudotopological CSA in which the soul is nilpotent and the annihilator of the odd part is trivial if 'A1 = 0. Recall that the set G in the pseudotopological linear space (X, T) is bounded if OG .. 0, where 0 is a filter in the field K whose basis consists of the neighborhoods of zero in the field K. The following proposition immediately follows from the definition of the pseudotopology TN and Theorem 4.3.
Proposition 4.1. Let the set G be bounded in the pseudotopological space (Af,TN). Then there exists a set Cm containing G.
4. Schwartz Distributions
97
4.3. Differential calculus over a supertopological superalgebra. We introduce a superspace Rum over the CSA U = UO ® U1. We assume here that the algebras A and U are real. The complexification of the algebra A is denoted by Ac, Ac = A ® iA. The definition of S-differentiability for mappings from the superspace into the Banach A°-module M repeats Definition 1.1 from Rum
1. The remainder o(h) satisfies the condition o(th)/t -+ 0 in the Banach space M uniformly on any bounded subset of G of the
Sec.
pseudotopological superspace RuIM As usual, we introduce classes of S-differentiability of the maps Sn,m S°o
Proposition 4.2. For every function f E C°°(R", A°) there exists a unique extension of the class S°°(Uo , A`).
Proof. The extension of f to UO is defined by Taylor's formula f (X)
f
_ k=0
(kk
bx) (cx)k
(4.3)
for x E UO, cx E Cm. By virtue of Proposition 4.1, function (4.3) f : Uo
Ac belongs to the class S°°(Uo, A°).
Proposition 4.3. Let -A1 = 0, the set f (Uo) being bounded in Ac. Then f is a constant. It follows from the nilpotency of elements h E Al and the boundedness of the set f (xo +Al), x0 E R" that f'(xo)N = 0, and therefore f'(xo)Ul = 0. It remains to use the triviality of the U1-annihilator.
4.4. Spaces of the types C and D on a superspace. We introduce superanalogs of the space of infinitely differentiable rapidly decreasing functions G(R") and the space of infinitely differentiable functions with compact supports D(R") on a superspace over a CSA with a nilpotent soul. By virtue of Proposition 4.3, we cannot impose the condition of boundedness even with respect to even variables on infinitely S-differentiable functions on Rum. Therefore we shall use conditions of boundedness and compactness of supports with respect
to the real subspace R"
R"e of the superspace Rum. Thus, we
Chapter H. Generalized Functions
98
set G(RR'm, Ac) = { f E S°°(Rum, Ac): for any multiindices Q and (t) II < oo, where (x, 9) E ry we have IIf IIQ,7 = E sup (1 + ItI)II az tERT
Ru'm}. The topology in c' the space G(Ru'm, Ac) is defined by a system This topology is equivalent to a (formally of subnorms {II stronger) topology defined by the system of subnorms -
IIA,7}.
IIf IIA,'r,G = E sup sup (1 + ItI) a hEG tERn
a7+°` f
h) ax7aea (t +
where {G} are bounded subsets in (N, TN).
Proposition 4.4. G(Rum, A°) is a Frechet space. Let M be a Banach CSM over the CSA Ac; we introduce a space of infinitely differentiable rapidly decreasing functions from R" into M, setting
G(Rn, M) = If E C°°(Rn, M) : IIf IIA,7 = sup(1+Itlo)IIf(7)(t)II < oo}. t
Then the space 9(R/'m, Ac) is isomorphic to G(Rn, P(Am, Ac)). The space G (Rum, A°) is known as a space of test functions which
rapidly decrease along the real axis. We introduce a space of test functions which are finite along the real axis D(RR'm, Ac) = If E Soo(Rum, Ac): f IRn has a compact support}.
As usual, in the space D(RR'm, Ac) we introduce a topology of an inductive limit. The Fourier transform of the function f E G (Rum, Ac) is defined by the relation
.F(f)(y, ) = fR' fm dx d9 f (x, 0) exp{i(y, x) + i(0,
Theorem 4.4. The Fourier transform F: 9 (Rum, Ac)
9 (Ru'm,
Ac)
e)}.
(4.4)
4. Schwartz Distributions
99
is the isomorphism of the Frechet CSM, and F-1(g) (x, 9) = cn,m fR^ fi" dy
cn.,m =
For commuting variables the proof does not differ from that for numerical functions (with the replacement of the module I I by the norm I I I I) and for anticommuting variables we must use relation (3.23).
The following result shows that if we consider infinitely S-differenti-
able functions rapidly decreasing along the real axis on a superspace over the Banach CSA A (and not over U in which the soul is nilpotent), then pathologies are possible, namely, the Fourier transform of the function f E A`) may not be defined on the whole superspace RA'm
Proposition 4.5. Let A = G. Then, for any e > 0 there exists an element h E AO such that sup
IIeitnlle-tl-E
= oo.
(4.5)
tER+
Proof. We set h = > an/n1+E, where a1 = gig2, a2 = q3q4, n=1
Note that
00 g(t) = Ileitnll = H II exp{itan/n1+E}II =00H 1 + t2/n2+2E
n=1
n=1
Consequently, 00
2ing(t) = > ln(1 + n=1
t
n2+2E )
t2 > i In (1 + n2+2e) n>tl/(l+E)
It remains to point out that 00
L/l+)
t 2 ln(1 + (x1+E) )dx < oo.
It is not known either whether there exist infinitely S-differentiable 0 whose restriction to Rn is functions on the superspace finite.
Chapter II. Generalized Functions
100
5.
Theorem of Existence of a Fundamental Solution
The polynomial p: Ru'm ate if the polynomial bp(x) If the polynomial
A' is said to be a scalarly nondegenerE b(pa)xa 0 0. a
b(pa)xa
+ E papxaep,
p(x) = > a
o9 191#0
then bp(x)
b(pa)xa, and cp(x)
C(pa)xa
a
+
papxa0$. ap
The map bp(x) is called the body of the polynomial p(x, 0), and cp(x, 0) is called its soul. The soul of a polynomial is said to be nilpotent if it assumes only nilpotent values. Lemma 5.1. The coefficients of a polynomial of commuting variables which assumes only nilpotent values are nilpotent. Proof. It suffices to consider only the case of a polynomial on Rn. m Let g(s) = E anSn and let s E R be a polynomial assuming nilpotent n=0 values. Then the element ao = g(0) is nilpotent, and, consequently, the polynomial gl (s) - ao assumes nilpotent values. Hence, we find that the polynomial gl(s) = s-1(g(s) - ao) assumes nilpotent values and, hence, the element al = gl (0) is nilpotent. We can prove by induction that all coefficients ak, k = 0, ..., m, are nilpotent. It remains to carry out induction on the number of variables.
Lemma 5.2 (on the superextension of generalized functions). Let u E G'(Rn) (u E D'(Rn)). Then there exists a superextension of ua E 9'(U on, A0) (u5 E D'(U0 , Ac)).
Proof. Since A = Re®C®A1, the space of test functions G(Rn) is embedded into the space G(uo , Ac). Thus, any functional u E G'(Rn) can be regarded as a functional defined on the subspace G(Rn) of the
5. Existence of a Fundamental Solution
101
space G(U , A`). The problem arises of the extension of the functional u: G(RT) -+ C to the Ac-linear continuous functional u,: G(uo , A`) A`. Let us use the well-known representation for generalized functions u(co)
= f f (t). (c) (t) dt,
(5.1)
where f (t) is a continuous function of a medium growth on Rn. Since A` is a Banach space, relation (5.1) defines the Ac-linear continuous functional on G(U0, Ac).
Similarly, using the structure of the generalized functions u E D'(Rn), we obtain the superextension of u, E D'(Uo , Ac). Generally speaking, the superextension is not unique. We shall call superextension (5.1) canonical.
Proposition 5.1. Let A = G. Then the superextension of the generalized function u E G'(Rn) is unique.
Proof. Every test function cp E G(Rn, A') can be expanded in a series cp(t) = E cpa(t)ga which is convergent in the space a
G(Rn, Ac),
qa = ql l...qn,,..., a = (all ..., an, ...), laI < 00.
Note that the system of subnorms {II IIp,7} is equivalent to the -
system of subnorms I I W I I a,7 = f (l + I t l p) I l f (7) (t) I I dt.
Thus, it suffices to show that (An = (a: aj = 0, j > n)):
paVAn
W - E cpaga ' aEAn
= f (1 + Itlp)
II(p(7)(t)II dt - 0, n - oo.
However, this follows from the Lebesgue theorem of the majorizing convergence.
Thus, if the functional u, E G'(Rn, A`), then we have u9(co) = E u3(W.) q'. a
Theorem 5.1. In the space G'(Run", Ac) of generalized functions there exists a fundamental solution of every linear differential operator
Chapter II. Generalized Functions
102
with constant coefficients whose symbol is a scalarly nondegenerate polynomial with nilpotent soul. Proof. Applying a Fourier transformation, we reduce, as usual, the problem on a fundamental solution to problem of division by the polynomial p(x, 0)E(x, 0) = 1; p(x, 0) = bp(x) + cp(x, 0). By virtue of Lemma 5.1, all coefficients of the soul c(p,,) are nilpotent. Let j be such that (cp(x, 0))'+1 0. The generalized function E(x, 0) is defined by the relation E(x, 0) =
[(_1)k(cp(x,
0))k(bp(x))'-k I Regs(1/(bp)'+1)(x),
(5.2)
k-0
where Reg(1/(bp)j+1)(t) is a regularization of the function 1/(bp)i+1(t) in the space of generalized functions G'(R") and Reg,(1/(bp)j+1)(x) is its standard superextension (5.1). Indeed, (p(x, 0)e(x, 0), cp(x, 0)) 0))k(bp(x))'+1-k
C [E(_1)k(cp(x, k=0 9
+ L(
1)k(CP(x,0))k+1(bp(x))'-k]
k=0
J
xReg,(1/(bp)'+1)(x), W (x, 0)
_ ((bp(x))'+1) Reg,(1/(bp)'+1)(x), p(x, 0)). It remains to prove that (bp(x))i+1Reg,(1/(bp)'+1)(x) = 1 in the space G'(Ru'"`, Ac), i.e., the relation ((bp(x))'+1) Regs(1/(bp)'+1)(x), W(x)) = f cp(t) dt
holds for any test function cp E G (Uc , A`) . Let y: Ac -+ C be a continuous C-linear functional and f (t), t E R" be a continuous function of a medium growth such that f (O) (t) _ Reg(1/(bp)3+1)(t). Then ('y,
((bp(x))'+1) Reg,(1/(bp)'+1)(x),
w(x)))
R^
5. Existence of a Fundamental Solution (Reg3(l/(bp)'+1)(x),
_ (7,
_ (_1)°
fRn
103
(bp(x))'+1co(t)) dt)
f(t)a°((bp(t))'+1('Y, (p(t))) dt
=f Reg(l/(bp))'+'(t)(bp(t))'+1('Y, w(t)) dt = ('Y, fR(t) dt). It remains to use the Hahn-Banach theorem. As is shown by the examples in Sec. 3.7, a scalarly degenerate linear differential operator may not have a fundamental solution in G'(Rum, Ac) and D'(Rum, Ac). Moreover, we have the following general result. Theorem 5.2. Let p be a linear differential operator with constant coefficients whose symbol is scalarly degenerate. Then, in the space of generalized functions G'(Rum, Ac) and D'(Rum, Ac), there is no fundamental solution of this operator.
Proof. The symbol p(x, 0) has no body but only has a soul. Assume that there exists a solution of the equation p(x, 9)E(x, 0) = 1. Then (p(x, 9)E(x, 9), cp(x, 0)) = f cp(t, 9) dt d9.
We set cp(t, 0) =
g(t)
f
g(t) dt,
but
(p(x, 0)E(x, 0), cp(x, 0)) = f cp(t, 9) dt d9
belongs to the soul on the CSA. Proposition 3.7 provides an example of a differential operator whose
soul is not nilpotent and which has a fundamental solution.
Example 5.1 (a fundamental solution of a Laplace operator on a superspace). On the superspace Ru,2k we consider a Laplace operator n
OSuper =
9=1
a2
ax,
+
k
192
7=1
199;a9;
Chapter II. Generalized Functions
104
This operator can be represented as the sum of the body opeator Ab = bAsuper = E a and the soul operator Ac = CASuper = E aeaae 7=1
7=1
3
>
>
(= A°). Using formulas (5.2) and (3.25), we find that the fundamental solution of the Laplace operator has the form e(x, 9, 9)
( = (u P=O
pIp)
(0, #)PA)ek+1(x),
where Ek+1(x) is a superextension (5.1) of the fundamental solution Ek+1(t) of the iterated Laplace operator Ak+1 on R". An elliptic operator on a superspace is a differential operator whose body is an elliptic operator on R. In order to obtain a fundamental solution of an elliptic operator on a superspace, it suffices to use the well-known formulas for fundamental solutions of elliptic operators on R" and formula (5.2).
Example 5.2. A fundamental solution of the heat conduction operator on a superspace. Using formulas (5.2) and (3.25), we find that the fundamental solution of the heat conduction operator -'Super) has the form
E(t' x'
O, 8) =
1. (-1)k-p (k - p)! (0, P=O
p!
#)Pgk-P+1(t, x),
where em(t, x) is a superextension of the fundamental solution of the iterated heat conduction operator (a A)m on R". Note that
-17
tm 1 1 x) = (t) m(t' (m - 1)! (41rt)n/2 exp
(
x, x ) 4t
where i7(t), t E Uo, is the superextension (5.1) of the Heaviside function. Therefore we obtain E(t' x, 9, 9) = 77(t)
X (-1)k
(47rt)n/2
-
expf (4t )
P(9, 9)p(-1)p] !tP
5. Existence of a Fundamental Solution (2k-n)/2 t(47r)n/2
=
(
j eXp
105 ) + 4(0) 0)l x, x4t
l-f
J
L
If n = 2k, then 77 (t) (4-7r) )/2
E(t, x, 0, 0) =
expl - [ (x' x)
4t4(e) 0)11-
Example 5.3 (a fundamental solution of the Schrodinger operator on a superspace). Calculations similar to those carried out in the preceding example show that the fundamental solution of the Schrodinger operator (i A + Super) has the form E(t, x, 0,
(2k-n)/2
-ar(t)
(4x)n/2
exp{2 [ (x'
x)
44(e' B)
- 2 (2k + n)]
If n = 2k, then E(t, X, B, B)
1
- -ir7(t) (47r)n/2 exp{-
X)
D-
4t
`
Example 5.4 (a fundamental solution of the d'Alembert operator on a superspace). Formulas (5.2) and (3.25) lead to the following expression for the fundamental solution of the d'Alembert operator, []Super = (aatz - ASuper)
(-1)k-P(9p9)P(k
e(t, x, 0, 0) =
- p)! Of ]Ek+1(t, X), J
P=O
where b = -&S - Ob is the body of the d'Alembert operator and Ek+1(t, x) is the superextension of the fundamental solution of the iterated d'Alembert operator k+l on R. Using the formula for this fundamental solution on Rn (see, e.g., Vladimirov [15, p. 188]), we obtain k
e(t, x, 0, 0) = Cn 2
P=O
(_1)k-p(epe)p(k
_ P),
b
X),
X(t,
where X(t, x) is the superextension of the characteristic function on the ) light cone of the future and C, , is the constant 2 + 1 x nn+1 r n+1 (nn+1 2
Chapter II. Generalized Functions
106
6.
Unsolved Problems and Possible Generalizations
1. Theories of generalized functions over Banach CSA with nontrivial annihilators of odd subspaces and, in particular, over finitedimensional Grassmann algebras.
2. Analogs of the spaces G and D for the superspace R"'. 3. Theorem of the existence of a fundamental solution for linear differential operators with constant coefficients whose souls are not nilpotent, for instance, for operators with a quasinilpotent soul. 4. The problem on the division by an S-entire function (Loyasevich supertheorem).
5. Classification of quasilinear second-order differential equations.
6. Equations in convolutions.
7. Theory of a potential. 8. Superwaves.
9. Integral equations.
Remarks All the results from this chapter were obtained by the author [65, 68, Other superanalogs of the spaces g and D were introduced by Nagamashi and Kobayashi [103]. The methods of investigation of the Cauchy problem in the spaces of S-analytic test and generalized functions are similar to those used by Dubinskii [31] for functions of a complex variable (inversion formulas, existence of a fundamental solution of the Cauchy problem, classes of well-posedness of the Cauchy problem). 146, 153].
Note that when defining infinitely differentiable functions of the elements of the CSA U, in which all nonnumerical elements are nilpotent, we use a very old idea which was sufficiently used in the definition of functions of matrices (see, e.g., Gantmacher [23, p. 98]). Suppose that the p x p-matrix A = AI + H, where I is an identity matrix, and H =
6. Unsolved Problems 0
1
0
0
107
0 .
.
Then
f (A) = f (A) +
P (A)
(PP_ 1)
+ ... +
lei)
Hp-i
In Chap. 5, when proving the Lyapunov theorem for probability distributions on a superspace, we use the logarithm of an element of a superalgebra, In u; its definition is essentially the same s the well-known definition of the logarithm of the matrix In A = In(.\I + H) (see [23, p. 206]).
Chapter III
Distribution Theory on an Infinite-Dimensional Superspace In this chapter we consider the general theory of a superspace, a superspace over commutative supermodules [65, 68]. As distinct from the superspace KK'm, this superspace may contain an infinite number of commuting and anticommuting coordinates and serves as a natural mathematical foundation for Fermi quantum theory and the quantum theory of superfields, boson fields and strings with Faddeev-Popov anticommuting ghosts, superstrings, supergravitation, and quantum chromodynamics [7, 27, 28, 40, 42, 43, 52, 53, 75-86, 95]. In the definition of S-differentiability on a superspace over commutative supermodules, my idea (proposed in [67, 144]) was realized that superanalysis was an analysis of nonlinear mappings on a superspace admitting a A-linear approximation on a commutative supermodule covering a superspace. Thus, superanalysis is an analysis on a pair (superspace and a covering commutative supermodule). With this approach, superanalysis is a natural development of the classical mathematical analysis. In the classical mathematical analysis, a superspace (=space) K" coincides with a covering module K", and therefore the approximations by K-linear maps on the module K" and the space K" coincide. Superanalysis is a more complicated structure in which a superspace and a covering module do not coincide. In particular, Vladimirov-Volovich superanalysis is an analysis on the pair (KK'm, An+m) (see Theorem 2.3 in Chap. 1).
Chapter III. Distribution Theory
110
This interpretation of superanalysis differs essentially from the interpretation of superanalysis as a law of signs accepted in the algebraic approach [43].
There is no reason to restrict the consideration of linearity only over CSA. Any (generally speaking, nonassociative and noncommutative) algebra A is associated with the analysis of nonlinear mappings
that admit A-linear approximation on an A-module covering an Asuperspace. Algebras which are not commutative superalgebras can be found, for instance, in the theory of strings, they are Witten noncommutative associative algebras [86] and nonassociative commutative superalgebras of Aref'eva-Volovich [76, 77]; see also Connes noncommutative geometry [105].
In this chapter we consider a very fine mathematical structure; it is impossible not only to study a superspace without a covering module but also construct an analysis on a supermodule without a superspace. In a noncommutative case, the S-differential calculus on a supermodule degenerates into a linear calculus.
In accordance with the theory proposed by Fomin [134], in the infinite-dimensional case generalized functions are functionals on spaces of Q-additive measures on an infinite-dimensional space and functionals on functional spaces are generalized measures or distributions. In
this book, we use the term "distribution."
Polylinear Algebra over Commutative Super-
1.
modules If Vk, k = 1, ..., n, and U are linear spaces, then we denote by Ln ( IZ V', U) the space of n-linear mappings from H Vk into U. If vk
k=1
k=1
= V k ®V k and U = Uo ®U1 are Z2-graded spaces, then in the space n
Ln (11 V k, U) we can distinguish subsets L° and Ln which consist of k=1 even and odd maps respectively: (1) a map bjal, E L° if b(xi 1, ..., xn^) E Uo for even I a I and b (x", E U1 for odd
x`-)
1. Polylinear Algebra
111
(2) a map b E L,l, if b(xi 1 , ..., xnn) E U1 for even IaI and b(xi 1 , ...,
x°`n )
E U° for odd IaI.
Proposition 1.1. If Vk, k = 1, ..., n, and U are Z2-graded spaces, then
n
n
n
Ln(11 Vk, U) = Ln(H Vk, U) ® Ln(ll Vk, U). k=1
k=1
k=1
Let Mk, k = 1, ..., n, and N be CSM over the CSA A. In the space n Ln (ji Mk, N) we distinguish subspaces Ln,r (right A-linearity) and k=1
Ln,1 (left A-linearity). The map b E Ln,r if we have b(x1i..., xka°, ..., xn) = b(xi, ..., xn)a°, b(x1 , ..., xkkal, ..., xan) = (-1)BTkb(x1
,
...,
(1.1)
xan)a',
(1.2)
where Brk = > aj for xj E M, and a = a° ® al E A. The map j=k+1
b E Ln,, if condition (1.1) and the condition b(x11, ..., xkkal, ..., xnn) = (-1)B'kalb(xi 1, ...)
xan),
(1.3)
k-1
where Bik = > aj, are satisfied. j=1
Proposition 1.2. Let the map b be right (left) A-linear. Then its even part b° and odd part bl are also right (left) A-linear.
Proof. Suppose, for instance, that b E Ln,r. We introduce a function W: N -+ {0, 1} W(2n) = 1, cp(2n + 1) = 0. Then we have
b'(x"x3 ,...,
x3 0, ..., x1n))3 ,..., xan) n = cp(1 + IaD)(b(x°1 1 ..., n
+(1 - (p(1 + IaI))(b(x11, ..., xj'0, ...,
xnn))1-J
[W(1 + IaI)(b(xil, ..., x° `n)0))'
+(1 - 7'(1 + IaI))(b(x"1, ...,
xnn)0)1-',(-1)B=J
Chapter III. Distribution Theory
112
for 0 E A1, s = 0,1. Note that for any 0 E Al and b E M, where M is a CSM, the relation (b0)' = b'-'O, s = 0, 1 holds true. Therefore
1 ,...,x-0,...,x--) Uj b8(xl 1) B, [(b(xl',
+(b(xi,
..., x"))1-57'(1 + jal)
..., x1^))'(1
- '(1 + l al
))]0.
It follows from Propositions 1.1 and 1.2 that Ln,,. = L°, ® LV,r, where Ln T = Ln,r n Ln, a = 0, 1, and Ln,1 = L° I ® Ln,1, where Ln i =
L,1 fl L, a = 0, 1.
Proposition 1.3. Let Mk = Mo ® Mj , k = 1, ..., n, and U = Uo ® U1 be CSM. Then n
n
k=1
= Lon z(ll Mk,U). k=1
L°r(ll Mk, U)
Proof. Let b c L°,,, 0 E A1. Then (see the proof of Proposition 1.2)
0xC'j,...,x1^) _
+(1 - cp(1 + Iaj))(b(xi',
..., x1"))o)0
IaI)(b(xi', ..., x1^)0))1
(1 + IaD))(b(xi', ..., xn"))0;
if
2k, then
b(xi', ...,
x1") _ (-1)BTj+°j0(b(xi', ..., x1"))o
and if Ial = 2k + 1, then
9x', ..., x1n) _ (-1)Brj }1+cj8(b(x", ..., x1"))1 _ (-1)BliOb(x1c', ..., xn
1. Polylinear Algebra
113
Proposition 1.4. Let Mk = Mo ® Mi , k = 1, ..., n, and U = Uo ® U1 be CSM, b E Ln r (b E Ln 1). Then, for any multiindex a = (al, ..., an), aj = 0,1, there exists a map b,,, E Ln,1 (b,, E Ln r) such that b
bc, IMal X...XMan
M 1 X...XMan
Proof. We set ba(xl, ..., xn) = (-1)I`Ib(o(x1), ..., o (xn)). Note that a(9x') = -9o(xO) for 0 E A1. Therefore b,,, x1 , ..., 9xk , ..., xn
_
(-1)1+IaIb(Q(xAl), ..., 9Q(xkk), ..., U(Xlno
o(xnn))0
(-1)1+Ick1+Brk+13
(-1)1+IaI+Brk+Ak0W(1
-(1 - (p (1 +
+
IYI)b(a(21),
QI))b(o(xQ1),
..., o, (xnn))o
..., or (xnn)))1;
if 101 = 2m, then b,,
(xQl 1 , ..., 9xkok , ..., Qxnon
_
(-1)IaI+Brk+Ak0(b(o(41), ..., a(xnn)))1
_ (-1)B'k0b,,(41, ..., xnn) and if 101 = 2m + 1, then 01 b,,( x1 ,...,9xkQk ,...,QxnQn) _
_
(-1)IaI+Brk+1+Ak0(b(a(xA1), ..., or (xnn)))0
_
(-1)B'k0b,, (x11, ..., xnn).
Thus, condition (1.2) is satisfied, and condition (1.1) can be verified by analogy.
In the linear spaces Ln,r(j-j Vk, U) and Ln,1(fl Vk, U), where Vk = k=1
k=1
V k ®V k, k = 1, ..., n, and U = Uo ®U1 are CSM, we introduce a struc-
ture of modules setting
Chapter III. Distribution Theory
114
(1) (7b)(x1, ..., xn) = Ab(xl,..., xn),
xn) = b(Axl,..., xn)
(b\)(x1,...,
forbELn,r,AEA; (2) (fib) (x1, ..., xn) = b(x1i ..., xnA), (b)) (x1, ..., xn) = b(xl, ..., xn)A
forbELn,1,.AEA. Proposition 1.5. The spaces n
V k' u)
Ln,r (II
n
n
= Ln,r (IT V k' u) ® Ln,1(II V k, u) ,
k=1
k=1
n
n
k=1
Ln,1(11 V k' U) = Ln 1(II V k' U) ® Ln 1(II V k' U), k=1
k=1
k=1
where Vk = Vk ® Vk, k = 1, ..., n, and U = U° ® U1 are CSM, are CSM.
Proof. Let us verify, for instance, that Bbl = -b10 for 9 E Al and bl E Ln r. Indeed, (bl0) (xl, ..., xn) = b' (9x1, ..., xn) = b' (a(xl), ..., a(xn))0 b (x1al, ..., xnan )9 1
-
1
b(xal,
..., xnan )9
lal=2k+1
ja1=2k
=-O E bl(xal,,xa") n lal=2k
-9E
bl(xal
,
xa^) n
lal=2k+1
_ -Bbl (xl, ..., xn). There exists a natural isomorphism of the CSM Ln,r and Ln,1. For b = b° (D b', ba E Ln r, a = 0,1, we set I(b) = I(b°) ®I(bl), I(b°) = b°,
I(bl) = bl(® Q). k=1
Proposition 1.6. Let Vk = Vk®Vk, k = 1, ..., n, and U = U°®Ul be CSM. Then the mapping n
I: Ln,r (11 V k, k=1
is an isomorphism of the CSM.
u)
n
-+ Ln,1 (11 V k' U) k=1
1. Polylinear Algebra
Proof.
115 E)
E L,,,r and A _ A° ®A1 E A. Then
I(b)t) = b°)° + b'A1 + I(b°A1) + I(b)O). It remains to note that (b°)i°)(xl,..., x,a) = b°(a°x1,..., x,,) = (I(b°)a°)(x1,..., xn); (bl)l) (x1, ..., xn) _ (I (bl)A1) (x1, ..., xn); I (b°A1) (x1, ..., x,i) _ (b°)1) (a(xl), ..., a(xn)) = b°(A1a(x1), ..., o (x,.)) = b°(x1, ..., xn)A1 = (I (b°)A')(x1, ..., x'+);
I (bl)t°)(x1, ..., x,a) _ (b1a°)(a(x1), ..., a(xn)) _ (I (bl)A°)(xl, ..., xn).
The verification of the fact that I(Ab) _ AI(b) is similar. By virute of Proposition 1.4, I(Ln,r) C L,a,1. We introduce a mapping T: Ln,1 -4 Ln,r, T(b°) = b°, T(bl) = b1(®o.). Then T o I(b°) = b°, k=1
T o I(bl) = b1(®a2) = bl; T = I-1. k=1
In the sequel, we identify, in most cases, the spaces Ll,r(M, A) and L1,1(M, A) and denote them by M* = MO* ® Ml , where MM '=' Li,r
Lii, a=0,1. The CSM M* is known as an algebraic conjugate of M and the elements of the space M* are called A-linear functionals on M. We introduce the form of duality between M and M* by setting (m*, m) = Ir(m*)(m),
where Ir: M* -+ Ln,r and I,: M*
IoIr=I,,ToI,=Ir.
(m*, m) = I1(m*)(m), Ln,1 are canonical isomorphisms;
Proposition 1.7. The forms of duality between M and M* possess the following properties:
(mm) -
m,
(m, m*) = 0
(1.4)
for homogeneous m* E M* and m E M, and (Am*Q, ma) = .1 < m*,
Qm > a
(1.5)
for any A, Q, a E A and m E M, m* E M* (i.e., LO,, Proof. Suppose, for instance, that Im* _ m = 1. Then (m*, m) _
Ir(m*)(m) _ -Ir(m*)(a(m)) _ -I(Ir(m*))(m) _ -Ii(m)(m*) _ -(m*, m), Ir(.Am*,Q)(ma) = .\Ir(m*/3)(m)a = AIr(m*)(Qm)a.
Chapter III. Distribution Theory
116
2.
Banach Supermodules
2.1. Z2-graded norms. A CSM M over a CSA A is called a Banach CSM if M is a Banach module over the Banach algebra A (i.e., IlAmll < IILII Ilmll, A E A, m E M) and the direct sum M = Mo ® M1 is topological (i.e., the projectors 7r,,, are continuous). The norm on the CSM M is said to be Z2-graded if Ilxll = Ilx0 II + IIx1lI. In this case, Il-7r II = Ilir°II +
Ilirlll for x E M.
We shall only consider Z2-graded norms on Banach CSM. In this section, the symbols M = MO ® Ml and N = No ® N1 are used to denote Banach CSM over the Banach CSA A = AO ® A1.
2.2. Supermodules of A-sequences. The CSM introduced below constitute a natural generalization of Banach spaces of sequences of real (or complex) numbers 1P and co. We introduce Banach CSM consisting of A-sequences x = (x1) ... xn, ...), xj E A,
co(A) = {x :m xn = 01; 00
1
1 (A)={x:
I17raxnllP)1'P<00},
IIxIIP=EM
1
a=0 n=1 1
l,,. (A) _ {x
We consider the norm II the space co(A).
:
Ilxll... = E sup ll7r,,xnll < oo}. ck=0
IIP
n
in the space 1P(A) and the norm II III in
2.3. Conjugate supermodule. For the Banach CSM M and N we use the sumbols G1,,. (M, N) and Ll,i (M, N) to denote the submodules of the CSM L1,r (M, N) and L1,1(M, N), respectively, which consist of continuous mappings. If M = N, then these CSM will be denoted by L1,, (M) and L1,1(M). For the space of K-linear continuous operators, we use the standard notation £(M, N) (G(M)).
The space L(M, N) is a Banach space with respect to the ordinary uniform norm IIILIII = sup{IIL(m)II: IImll < 1}. This norm induces the structure of the Banach spaces on the CSM Ll,r (M, N) and
117
2. Banach Supermodules
L1,1 (M, N). The norm III' I I is consistent with the Z2-graduation since I
7r°(L) = 7r°L7ro+7r1L7r1, 7r1(L) = 7rIL7r°+7r°L7r1i i.e.,
We introduce a norm I I L I I= I I I11° (L) I I I+ I
I
III-raIII < 2IIILIII.
I II1(L) I I I This norm is
Z2-graded.
Note that the restriction of the canonical isomorphism I to the CSM £1,,.(M, A) is an isomorphism of the CSM £1,,.(M, A) and G1,1(M, A)
Theorem 2.1. The isomorphism I is an isometry of the Banach CSM L1,r(M, A) and L1,i(M, A).
Proof. Let the operator L E LI,r (M, A). Then I (L) = L° + L1 o a, and, consequently, III (L) II = IIL°II + IIL1 o all, sup
IIL10 all =
IILI(x° - xl)II
11x011+I1x1ll
sup
IIL1(x°+x1)11 = IILIII.
11x011+Ilxlll<1
By virtue of this theorem, we can identify the Banach CSM Gl,,.(M, A) and G1,1(M, A): M' = MO' ® M1' = GI,r(M, A) G1,1(M, A), where
M.' = Gi,r (M, A) '" Gi i (M, A), a = 0, 1.
The CSM M' = MO' ® M1' is a conjugate of the Banach CSM M = M° ® M1. By analogy with the ordinary theory of Banach spaces, the elements of the CSM M' will be called functionals. Here is an example of a Banach CSM M such that M' = {0}. Example 2.1. We set A = C[O, 1] (a space of continuous functions on the interval [0,1]), M = LI[0,1] (a space of summable functions on the interval [0, 1]). Then M is a Banach commutative module for the commutative Banach algebra A.
Let the functional L belong to M'. Note that A C M, and for g E A we have L(g) = L(1)g, where L(1) E A. However, the set A is everywhere dense in M, and therefore, for any g E M, there exists a sequence g,, E A, gn -4 g in M. However, the sequence L(1)gn converges to L(1)g in M, and, consequently, L(1)g = L(g) E A, but for any continuous function L(1) A 0 we can choose a summable function g such that L(1)g V A.
Chapter III. Distribution Theory
118
Here is an example of a Banach CSM M such that the even part in M is trivial (there are no even elements except for the zero element) and in the conjugate CSM M' the odd part is trivial (there are no odd elements except for the zero element).
Example 2.2. Let A = G1 and M = G1,1 = {x E G1: x = uq, u E K}. There are no odd functionals (except for the zero one). All functionals are even and have the form L(x) = Ax, A E K, M' = K.
Example 2.3. Let M = A. Then M' is isometric to A, and we have a relation
IIILIII = sup II(L,x)II = IIIL°III + IIILIIII = IILII
(2.1)
Ilxll<1
Example 2.4. Let A = G2 and Mo = {x E G2: x = tQ1g2i t E K},
M1={xEG2: x=Bg2iBEK}. Then MM=K, and Ml={xEG2: x = tQ1, t E K}. In this case we have relation (2.1) and the supremum is attained on the odd part of a unit ball.
Proposition 2.1. We have an inclusion lq(A) C 1,(A), 1 < p < oo, 1/p+ 1/q = 1; 11(A) C co(A).
Proof. Every vector u E lq(A) is associated with a A-linear functional L E 1,(A) defined by the relation (L, x) = > unxn, and 1
IILIIlp(A)
- E sup 11E unxn
IIuIIq(EIIxnIIP)'
IIUIIgIIxIIP.
a=0 11416<1
The case u c 11(A) can be considered by analogy.
Open questions 1. Are the inclusions lq(A) c l' (A), 1 < p < oo, 1/p + 1/q = 1; 11(A) C c'0(A) isometric?
2. Are the spaces lq(A) and 1'(A), 1 < p < oo, 1/p + 1/q = 1, and 11(A) and c' (A) isomorphic?
For what classes of Banach CSM does the equality of norms (2.1) hold true? Let us consider the norms II - IIP, 1 < p < oo on the CSM M = An. Then the CSM M' is algebraically isomorphic to the CSM M for any 3.
2. Banach Supermodules
119
one of these equivalent norms. It is not known, however, whether the Banach spaces (A', II . II ,)' and (An, II ' II9), 1 < p < oo, 1/p + 1/q = 1 are isometric.
Proposition 2.2. The Banach CSM 1', (A) and l,,,, (A) are isometrically isomorphic.
Proof. The series x = i enxn, where en = (0, ...,1, 0, ...) is a 00 n=1
canonical basis in 11(A), converges in the Banach space 11(A). Consequently, we have (L, x) = E Lnxn for the functional L E 1'(A). Let ILO = > sup I I Ln I Then VE > 0 3N, a = 0,1, such that a
n
I
sup I I Ln < I I LN I I+ E It remains to note that n
IILIIl1(A) >-
11 (L", eNQ)II >- IILII00 - 2E.
We find, as a consequence, that the Banach spaces (An, II II1)' and (An, 11.11 I) are isometric.
The situation realized in Example 2.1 is pathological. In order to exclude cases of this kind from the consideration, we give the following definition.
Definition 2.1. The CSM M and M' are dual if M' separates the points of M. If M and M' are dual, then M is embedded into M": x '-* lx E M",
1x(y) = y(x) for y E M'. It suffices to verify that the operator is M -+ M", x H lx belongs to the class LO,, (M, M") = C',, (M, M"). Indeed, lax(y) = (lax, y) = (\x, y) = )(x, y) = Aix(y), i.e., lax = )lx and i E G1,i(M,M"). The parity of the operator can be immediately verified.
Definition 2.2. A Banach CSM M is semireflexive if is M -4 M" is an algebraic isomorphism, i(M) = M'. Definition 2.3. A Banach CSM M is reflexive if M is semireflexive and the canonical isomorphism i is an isometry. In the theory of linear Banach spaces, the semireflexivity implies reflexivity since the canonical inclusion of E into E' is an isometry (the
Chapter III. Distribution Theory
120
concepts of reflexivity and semireflexivity differ only for topological linear spaces) .
Theorem 2.2. Suppose that for any vector x E M there exists a vector y E M' such that IIyII < 1 and IIxII = IIy(x)II. Then the canonical inclusion of M into M" is an isometry.
Proof. Let x E M. Then there exist ya E M', a = 0, 1 such that II(xa,ya)II
= IIxa!I and IIyall < 1. Consequently, IIl2II = sup 11W"011 Ilvll<1
> IIxalI, i.e., 111 "11 = IIxII and 111.11 = IIxII
Open question Is the Banach CSM (An, II II1) reflexive?
It has been shown that (An, II II1)' is isometric to (An, II . II00). It is not known, however, whether the CSM (An, II - II,)' is isometric to (An, II 111) for the arbitrary Banach CSA A.
2.4. E-superalgebras. The following definition play an important part in duality theory for Banach CSM. Definition 2.4. The Banach CSA A is called a E-algebra if, for any elements a1, ..., an E B, we have n
n
> IIajlI = sup 11Eajajll,
j=1
I10,11-51 i=1
where a = (a1, ..., an) are homogeneous vectors from An. The algebra K is the simplest example of the E-algebra.
Theorem 2.3. The CSA G1 00 (T), where T is an arbitrary commutative Banach algebra, is the E-algebra.
Proof. Let 00
a7 =
E 1:
aj-,q-,, ...gryn
a77 E T,
j = 1, ..., m.
n=0 71 G..
For any E > 0 there exists a finite set of indices r such that Ilai -air II < E
m,
where air =
yEr
2. Banach Supermodules
121
We set I3 = qK3, j = 1, ..., m, where qK, # q.y, for all indices 'y E IF and kj j4 k; when j i. Then m
m
IIE Qjajrl = > IlajrM. j=1
j=1
Furthermore, m
M
m
IEajajl <EIlajli <E+EIlajrll
sup
j=1
I1c11o051 j=1 M
j=1
m
=E + 11 E Ojaj
E+
j=1
m
Qj (air - aj) I + 11 E Qjaj j=1 j=1 M
< 2E + sup
11E
ajaj
.
Ila11oo<1 j=1
Open questions 1. Are finite-dimensional Grassmann algebras E-algebras? 2. Are there finite-dimensional CSA which are E-algebras?
2.5. Duality of Banach supermodules over E-superalgebras. Theorem 2.4. Let A be a E-algebra. Then (1) the CSM l,(A) is isometrically isomorphic to the CSM lq(A),
1
Proof. Let the functional L E l,(A). Note that the series x = 00
n
Therefore (L,x) _
> xnen converges in the Banach space lP(A).
n=1
Lnxn, Ln = (L, en). We introduce vectors 1
U. - (n=1IILn
Ii°)-1P
L
en0enjILli°-1'
n
n=1
where i = 0, 1; a = (al, ..., aN), IIa1I < 1, and a is a homogeneous vector. Then IIuN.IIP =
(E IILnII9)-VP(E n=1
n=1
IILnIIP(9-1))1/P'
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122
note that p(q - 1) = q, and, consequently, Ilur,,,IIP < 1. Thus the vectors uN. lie in a unit ball of the Banach CSM lp(A). Furthermore, IILII = IIL°II + IILIII =
>
i=o
sup II(Lx)II I1x16<_1
sup II (L uNJII i=o IIaHI.<1
E sup i=0 11 IIoo_
1
IE Lnan
IILnIIq-ll (r
n=1 1
IILn!Iq)-1/P
n=1
N
_ E(Y
IILnII9)1iq
= IILIiq-
i=0 n=1
It remains to use Proposition 2.1.
Corollary 2.1. Let A be a E-algebra. Then the Banach CSM lp(A), 1 < p < oo are reflexive. Corollary 2.2. Let A be a E-algebra. Then the CSM l2(A) is isometrically isomorphic to the CSM 12(A).
Corollary 2.3. Let A be a E-algebra. Then the CSM (An, II IIP), and (An, II l 1q), 1 < p < oo, 1/p+1/q = 1 are isometrically isomorphic. 2.6. Topological bases in Banach supermodules. A basis (topological basis) in a CSM M is a system of vectors {ap} such that any vector x E M can be uniquely represented as the sum x = > xpap,
xp E A.
We assume, as usual, that no more than a countable set of coefficients in the sum is nonzero.
Example 2.5. M = A', en = (0, ...,1, 0, ..., 0). Example 2.6. M = lp(A), 1 < p < oo, e = ( 0 , Example 2.7. M = P(Ak, A), e,3 (0) = OP.
...,
1,
0,
. . .)
.
2. Banach Supermodules
123
In these examples, the bases are homogeneous; in Examples 2.5, 2.6 the bases are even and in Example 2.7 lepl = 1,81 (mod 2). There exist Banach CSM without bases, these CSM may even be finite-dimensional K-linear spaces. Example 2.8. Let p be a fixed element of the CSA A. We denote by M the CSM pA, i.e., Mo = pAo, M1 = pA1. If p2 = 0, then there is no basis in the CSM M. Indeed, assume that these is a basis {ap} in M. Then x = E xpap, where xp E A. However, ap = pbp, by E A, p
and, consequently, x =
ypap, where yp = xp + p.
Theorem 2.5. Suppose that there exists a body projector in the CSA A. Then the number of vectors in the basis of the CSM M over A is an invariant relative to the CSM M.
Proof. Let {aa} and {ap} be two bases in the CSM M. Then a. _ >p &pap, ap = E. Kp,,a,,,. The matrices K = {K,,,p} and K = {&p} are associated with the operators K, K: M -+ M, and KK = 1 and KK = 1. It follows from these relations that KbKb = 1 and KbKb = 1. The statement of the theorem immediately follows from these equalities.
Open question Is the dimension of the CSM over a CSA in which there is no body projector an invariant?
Theorem 2.6. In every finite-dimensional CSM M over a CSA A with a quasinilpotent soul there exists a basis consisting of homogeneous elements.
The proof of this theorem (which is long enough) repeats verbatim the proof of a similar statement for a CSA with a nilpotent soul which can be found in the book by De Witt [27, pp. 21-23]). Of the most interest for applications are modules over the super-
algebra G. Open question Is there a homogeneous basis in the finite-dimensional CSM M over a CSA A whose soul is not quasinilpotent?
Chapter III. Distribution Theory
124
In his book, De Witt advanced a conjecture that infinite-dimensional CSM with a basis may not contain a homogeneous basis even if the soul of the CSA A is nilpotent. This conjecture has not yet been proved.
2.7. Operator matrix. Let M and N be CSM with bases {aa} and {ba} and let L: M -4 N be a left A-linear operator. Then
Lx = > xaLaa = > xa > Lapbp a
a
p
xaLap, bp, p
a
i.e., the operator L is realized by the matrix L = {Lap}. In this case, when the operator acts on the vector from the CSM, the vector row is postmultiplied by the operator matrix. Right A-linear operators can be considered by analogy.
Let now M be a CSM with a homogeneous basis {ice}, where a = 0, 1 and liael = a. A left-handed coordinate system is defined for every vector x E M: x = E > xaiiae = xaiiae (we use the stipulation a i
concerning the summation over the repeating indices). We set eai = iae and introduce a right-handed coordinate system:
x = E Ei eaiiax = eaitax. Q
Note that xai = Let the operator U E £1,1(M). Then, in the left-handed coordinate system, y = Ux = xaiu(iae) = xaiiaU13'13e, i.e., yp7 = xaiiaUp7. Consequently, the operator matrix U has a block structure ora(iax).
u=
(ou°
luo
ou1
lul l
If the operator U is even, then the matrices 0U° and 1u1 consist of even elements and the matrices 0W and 1U° consist of odd elements, i.e., JQU1J = (-1)a+0+IuI; if the operator U is odd, then the matrices oU° and 1u1 consist of odd elements and the matrices 0U' and 1U° consist of even elements, i.e., IaU1l = (-1)a+p+IUI again.
2. Banach Supermodules
125
Suppose now that U E £1,,(M). In this case, it is convenient to use a right-handed coordinate system, y = Ux = eQfl Uaji°x, j13y = j#U,,i"x. The operator matrix again has a block structure
u=
°u1
lu01u1)
Ou°
the relationship between the parities of matrix elements and the parity of the operator is the same as for a left A-linear operator.
If the operator U is even, then it is simultaneously right and left A-linear, with the right and left matrices related as j16uai = (_1)a(Cf+#)i.upj.
2.8. Continuous operators in Banach supermodules 1P(A). The continuous right A-linear operator L: lP (A) -4 1P (A), 1 < p < oo, is defined by its matrix 00
(Lx)i = E Lijxj,
i = 1, 2, ...
(2.2)
j=1
The space l... (A) does not have a topological basis, and therefore not every continuous operator is defined by a matrix. In this case, however,
we shall again restrict the consideration to an operator of the form (2.2).
We shall prove, as usual, that for the continuity of the operator L in the Banach space 1P(A), 1 < p < oo, it is sufficient that
IIL11 = E (I(E
II7II9)P/4)1/P < 00;
cx=0 i=1 j=1
in the space l.. )(A) it is sufficient that 1
IILijII < oo;
sup
IILII00 Q=0
j=1
and in the space 11 (A) it is sufficient that 1
00
a<=0
j=1
oo.
IILIII=
Chapter III. Distribution Theory
126
Theorem 2.7. Let a CSA A be a E-algebra. Then the norms IILII and IILII coincide for the operator L E G1,,.(l... (A)).
Proof. For any e > 0 there exists an iE such that 00
Lij11 < E+ j iiL°jii.
sup =
j=1
j=1
N
R
Let us consider vectors upN = > ej/j, where the vector Q = (01i ..., NN) 7=1
is homogeneous and I1 Q 11 OO < 1. Then n
IILII > sup IIL'uaN11. = sup jjELj'Ojjj i,II0II00<1 j=1 II011051 N
> Consequently,
I
I L" 11 >
00
I
I L° j 1 > I I La 11 I
7=1
Laicjaj
- E.
We have a similar result for left A-linear operators.
2.9. Adjoint operator. Let the operator L E £1,1(M, N). A adjoint operator L' is defined by the relation (Lx, y) = (x, L'y),
y E N'.
x E M,
Proposition 2.3. (LR). Let the operator L E £1,1(M, N). Then the adjoint operator L' E £1,,.(N', M'), with IILII <- IIL11
Proof.
The functional uy(x) _ (Lx, y) is left A-linear and continuous for any y E N', i.e., the operator L': N' -+ M' is defined. 2. The operator L' is right A-linear 1.
(x, L'(yA)) = (Lx, y\) = (Lx, y) A = (x, L'(y) A).
3. Let the operator L be homogeneous. Then 1
1
IIL'yll = E sup II (L'yQ,x)11 < IILII > 11y'II = IILII IlyM1. Q=0 11451
0=0
2. Banach Supermodules
127
It remains to use the following statement.
Proposition 2.4. The relation (L')' = (L°)', a = 0, 1 is valid. It is easy to construct an example of an operator for which IIL'1I < IILII.
Proposition 2.5. Let the CSM M and M' be dual and let the canonical inclusion of M into M" be an isometry. Then IILII = IILII.
Proof. Using the fact that the inclusion into the second conjugate module is isometric, we have 1
1
IILII = E sup 11(y,Lx°)11 < IIL'II E IIx'11= IILII IIxII. a=O
a=O I1y1151
Proposition 2.6. Let the CSA A be a E-algebra. Then the relation I I L I = IILII1 I
holds for any operator L E G1,r(l1(A)).
Proof. Note that co(A) = l1(A) and li(A) = l.. )(A). It follows from the first relation that sup
IIzC'111=
II (u, z°) II
I1ulloo<1,uEco(A)
sup
11(u, zcl) II,
lulloo<1,uE1.(A)
a = 0, 1. Thus we have IILII = IILII, where L' E ,C1,,(l.(A)), for the operator L. It remains to note that I I L' 11 = I I L' I I . = 11 L1 I I As a consequence, we get a similar result for the Banach CSM A. .
Suppose now that the operator L E C,,,(M, N). The adjoint operator L' is defined by the relation (L'y, x) = (y, Lx),
y E N',
x E M.
Proposition 2.3. (RL). Let the operator L E G1,r(M, N). Then the adjoint operator L' E G1,i(N') M'), with IILII < IILII Propositions 2.4, 2.5, and the analog of Proposition 2.6 are valid for right A-linear operators.
2.10. Banach supermodules of type lp with arbitrary graded bases. Banach CSM of A-sequences considered above were defined as
Chapter III. Distribution Theory
128
A-spans of the even canonical basis {en}. This construction admits an obvious generalization to the case where the canonical basis is not even.
Suppose that A and B are finite or countable sets of indices, {noe}nEA are even vectors and {nle}nEB are odd vectors.
We introduce Banach CSM ip,A,B (A)
_
{x = xaiiae. Jjxjjp
1
= a=0 2 (E iEA
11(xoi)aM1p
+
II(x1i)1-0jjP)h'P < oo}, iEB
1
Proposition 2.7. The inclusions lq,A,B(A) C lp,A B (A), 1 < p < oo, 1/p + 1/q = 1 are valid.
Lemma 2.1. Let the elements al, ..., ak, bl,..., bi E G00' (T). Then k
I
EIlanll+EIlbnll n=1
n=1 k
sup
I E anon + E bn/3
UaHIoo,IIAlloo<1 n=1
I,
n=1
where aj, /33 E A, Iaj I = 0, 1/3, I = 1.
The proof of this lemma is similar to the proof of Theorem 2.3 (the elements of the algebra G , (T) can be "moved apart" not only with the aid of generators but also with the aid of pairwise products of generators).
Theorem 2.8. Let the CSA A = G' (T). Then the CSM lq,A,B (A) is isometrically isomorphic to the CSM lq,A,B(A), 1 < p < oo, 1/p + 1/q = 1.
2. Banach Supermodules
129
Proof. Consider the case of countable sets of indices. We introduce vectors N
n
U N = M II(ioL)°`III+ i=1
i=1
N
N
x (EQiioe
II(ioL)aIII-1 +
EQiile
II(i1L)1-allq-1),
i=1
i=1
where a = 0 ,
1/P
II(i1L)1-°Ilq)
IQl J = 0, I/ 1= 1; i.L = (iae, L). < 1 (by virtue of the choice of coefficients 0 and Q, the vectors Ll. are even). Furthermore, 1;
IiBMMOO, IIQII. < 1 ,
Then 11 U.-O ll P
1
IILII
II(U
sup
,L°)II
a=o IIC1II-<1,11I311oo<1
N
1
_E
sup
IE/3 0(L°`)
1I(ioL)'jIq-1
a=o 11011-51,11,611-<_1 i=1
II (i1
L)1-allq-1([
i=1
E II(i1L)1-'IIq)-1/PII, i=1
i=1
a = 0, 1. with io(L) = (ioL)", i1(L") = It remains to use Lemma 2.1. Corollary 2.4. The Banach CSM IP,A,B(A), 1 < p < oo, A = (i1L)1-a,
G , (T) are reflexive.
Corollary 2.5. The Banach CSM 12,A,B(A), 1 < p < oo, A = G ,(T) is isometrically isomorphic to 12,A,B(A).
We can see from the proof of Theorem 2.8 that it remains valid for any Banach CSA A which possesses property (2.3). Requirement (2.3) is stronger than that in the definition of the E-algebra, where we considered homogeneous vectors. We do not yet know anything concerning CSA which possess property (2.3) and are different from G100 (T).
Open question Is Theorem 2.8 valid for an arbitrary E-algebra?
Chapter III. Distribution Theory
130
3.
Hilbert Supermodules
3.1. Hilbert supermodule with an even basis. Let the CSM M be Banach and let CSM M and M' be dual. Then the form of A separates the points of the modules M duality ( , ): M x M' and M', is continuous, and possesses property (1.5) (and, in particular, is even).
Definition 3.1. The continuous bilinear form ( , ): M x M -4 A, where M is a Banach CSM, which separates the points of the CSM M and possesses property (1.5), is known as the scalar product on M.
Definition 3.2. The Banach CSM 12(A) provided with the scalar product (x, y) _ >00xnyn is called a coordinate Hilbert CSM (with an n=1
even basis).
Proposition 3.1. For the coordinate Hilbert CSM 12(A) we have the Cauchy-Bunyakovskii inequality II(x,y)I1
(3.1)
11X11 IIyII
Definition 3.3. The triple (M, I I . , (. , )), where M is a Banach CSM with norm I I I I and scalar product ( , ) is known as a Hilbert II
CSM (with an even basis) if there exists an isomorphism S: M -* l2 (A) which preserves (1) the scalar product (Sm1i Sm2) = (m1i m2), (2) the norm IISmMI = IImII
Note that in the definition of a Hilbert CSM it suffices to require the existence of a R-linear isomorphism of linear spaces M and 12(A) which possesses properties 1 and 2.
Note, in the first place, that if the operator L is left and right A-linear, then it is even. Indeed, L(x)Q = L(x/) = L(crHHH (,3)x) = a1'1(Q)L(x),
3 E A.
Next, we assume that there exists an R-isomorphism S: M -+ 12(A) which possesses property 1. Then (Am1i m2) = (S(Am1), Sm2),
3. Hilbert Supermodules
131
but A(m1, m2) = (.\Sml, Sm2), i.e., (S(.\ml) - .1Sm1i Sm2) = 0. It remains to note that S(M) = l2(A). The right A-linearity can be proved by analogy.
It follows from the definition of Hilbert CSM that the CauchyBunyakovskii inequality (3.1) holds true, the scalar product on M possesses not only property (1.5) but also property (1.4), and there exists an even topological basis in the CSM M.
3.2. Riesz theorem for Hilbert supermodules. Corollary 2.2 for a coordinate Hilbert supermodule immediately implies a superanalog of the Riesz theorem.
Theorem 3.1. Let M be a Hilbert CSM over a E-algebra A. For every continuous A-linear functional L on M there exists a unique element m E M such that L(x) = (m, x),
x E M,
(3.2)
Conversely, if m E M, then relation (3.2) defines a continuous A-linear functional L on M such that IILII = IImII Thus, the CSM M and M' are isometrically isomorphic.
with IILII
=
IImII.
Open question Is there a superanalog of Riesz theorem for Hilbert CSM over an arbitrary Banach CSA?
3.3. Hilbert supermodule with an arbitrarily graded basis. The Banach CSM 12,A,B(A) provided with a scalar product (x) y) = x0:or a(Y'),
(3.3)
is called coordinate Hilbert CSM with a basis of parity (A, B). The Caychy-Bunyakovskii inequality (3.1) also holds for the scalar product (3.3). The scalar product (3.3) no loger possesses property (1.4) (if B j4 0). Property (1.4) is valid only for vectors of the form x = x0ijoe, and for the vectors x = x1';le we have a relation (x, ±) = (_1)1X141+1(±' x).
Chapter III. Distribution Theory
132
(
)) of parity (A, B) is defined with the aid of the isomorphism of the coordinate Hilbert supermodule The Hilbert CSM (M, 1 1
1
1
,
,
12,A,B(A) which preserves the scalar product and the norm. The definition of the Hilbert CSM of parity (A, B) and Corollary 2.5 yield
Theorem 3.2. Let M be a Hilbert CSM of parity (A, B) over the CSA A = G ,(T). Then the CSM M' is isometrically isomorphic to the CSM M.
The definition also implies the Cauchy-Bunyakovskii inequality (3.1).
Definition 3.4. The homogeneous basis {iae} in a Hilbert CSM is said to be orthogonal (ONB) if (jae, ipe) = kap.
Proposition 3.2. A scalar product in coordinate with respect to ONB has the form (3.3).
3.4. Orthogonal operators. The operator U E £1,1(E, F), where E and F are Hilbert CSM, is orthogonal if U maps the CSM
E onto the CSM F (Im U = F) and preserves the scalar product ((Ux,Uy) = (x,y)). Proposition 3.3. Every orthogonal operator is even, and therefore an orthogonal operator is also right A-linear. Proposition 3.4. Every orthogonal operator is invertible, and the inverse operator is also orthogonal. Proof. By virtue of Banach inverse operator theorem, it suffices to show that KerU = {0}. Indeed, let us assume that x E KerU. Then (x, y) = 0, y E E, but the scalar product separates the points of the CSM E and, consequently, x = 0 (note that the relation (x, x) = 0 does not, in general, imply x = 0). Corollary 3.1. Orthogonal operators in a Hilbert CSM M form a group.
3. Hilbert Supermodules
133
A group of orthogonal operators is denoted by Os(M).
Proposition 3.5. The operator U E L01', (E, F), ImU = F is orthogonal if and only if it maps an ONB into an ONB.
Proof. 1. Let {iae} be an ONB and U be an orthogonal operator. We set iaa = uiae. Since ImU = F, any vector y E F can be represented in the form y = xaiiaa, i.e., {iaa} is a topological basis in F; it remains to use the fact that an orthogonal operator preserves a scalar product. 2. Let the operator U: E -+ F be even, continuous, and Im U = F, Then with {iaa = uiae} being an ONB for the ONB {ice}.
(ux,uy) = xai(iaa, ypjjpa) = xaiaa(yai) = (x, Y) -
We can see from the proof that it is sufficient that at least one of the ONB be mapped into an ONB.
Proposition 3.6. The operator U E C0,1 (E, F), ImU = F is orthogonal if and only if its matrix elements for the ONB satisfy the relations iau7kQ7(9pu7k) = 6i,jap.
Proof.
1.
(3.4)
Let the operator U be orthogonal. Then the vectors
{iaa = uiae} are ONB for any ONB {iae}, i.e., 8ijap = (iauryke,jpuµssµe) = iau7k(k7e,aµe)oµ(Apuµ9).
2. Conversely, it follows from condition (3.4) that the operator U preserves the scalar product. Example 3.1. (the orthogonal group 0.(12,{1},{1}(A)). Conditions (3.4) for the matrix 1
u = (OUoo °ul
IOU° I
= Bull = 0,
Iou11 =IiU°I=1,
Chapter III. Distribution Theory
134
have the form (ou°)2
- (°ul)2 = 1,
ou°lu° + oullul = 0,
(lu°)2 + (lul)2 = 1, 1U°0u°
- luloul = 0,
i.e., the orthogonal group Os consists of matrices u =
0 C
b
I
,
where
a2 = b2 = 1, with a and b being even.
3.5. L2-supermodules. For applications we need superanalogs of the Hilbert spaces L2.
We denote by cpa(x) Hermite (normed) functions on R. The Hilbert CSM L2 (R", dx) of functions, cp: R" -+ A, which are square summable with respect to the Lebesgue measure on Rn, is introduced as a completion of the tensor product A ® L2 (R", dx); here L2 (R", dx) is a space of real-valued square summable functions 00
L2 (Rn, dx) = If (x) = E facoa(x): f,, E A, Q=o 1
00
IIf112 = i(i IIfQ7112)1'` <
00
y=o a=O
This is a Hilbert CSM with even basis {cp,,(x) } which is isomorphic to 12(A). The scalar product on the CSM L2 (R", dx) has the form
(g, f) =
JR^
9(x)f (x) dx = E faga.
If the CSA A is a E-algebra, then, by virtue of the superanalog of the Riesz theorem, any A-linear continuous functional L on the Hilbert CSM L2 (R", dx) is defined by the function
9(x) E L2 (R", dx): L(f) =
f g(x)f(x) dx.
Suppose now that (Q, a) is a measurable space and p is a or-finite countably additive measure on (Q, Q). The Hilbert CSM L2 (R", dµ)
3. Hilbert Supermodules
135
can also be introduced as a completion of the tensor product A 0 L2 (1,dµ). Under natural constraints imposed on the measure p, this CSM is also isomorphic to 12(A).
3.6. Nested Hilbert supermodule. Suppose that M is a Hilbert CSM, E is a locally convex CSM which is topologically em-
bedded into a CSM M. We assume that the E-annihilator in M' is trivial:
1E = Ann (E, M) = {L E M': LIE = 0} = {0}. For this annihilator to be trivial, it is not necessary to require that the CSM E be densely embedded in the CSM M.
Example 3.2. Suppose that the set of indices A consists of one element and the set of indices B is empty. Then M = 12,A,B (A) = A (the basis in M is a unit element e from the CSA A). Let us consider the case A = G. Then E = is a (Banach) CSM embedded into M, with --E _ {0} although E is not dense in M. If lE = {0}, then the CSM M' is embedded into the CSM E' and we get a quadruple of embedded CSM
ECMCM'CE'.
(3.5)
Quadruple (3.5) is known as a nested Hilbert CSM. If the CSA A is a E-algebra, then M M', and quadruple (3.5) is replaced (just as in ordinary functional analysis) by a triple of embedded CSM
ECMCE'.
(3.6)
The framing of the CSM L2 (R", dx) is carried out, as usual, with the aid of the CSM of test and generalized A-valued functions on R. For instance, G(R", A) C L2 (R", dx) C (LA(R", dx))' C G'(R", A) or
D(R", A) C L2 (R", dx) C (L2 (R", dx))' C D'(R", A).
Chapter III. Distribution Theory
136
Now if the CSA A is a E-algebra, then
g(R", A) C L2 (R", dx) C g'(Rn, A), D(Rn, A) C L2(Rn, dx) C D'(Rn, A).
Here the triviality of the annihilator follows from the density of embedding of the CSM of test functions into the CSM L2 (Rn, dx).
3.7. Superalgebras with involution. Recall that an algebra D over a field C is called an algebra with involution if the operation f -4 f * possessing the properties (1) (f *) * = f , (2) (Af + µg)* = A f * + µg*, (3) (f g) * = g* f *; f , g E D, A, p E C is defined in D. In a topological algebra D it is assumed that the involution is continuous. If f * = f , then the element f is real and if f * _ - f *, then the element f is imaginary. Any element h is representable in the form h = x + iy, where x, y are real elements, x = 2 (h + h*),
y = 2i (h - h*).
The involution * in the CSA A = A° ® Al is consistent with the Z2-graduation if *: A° ® Al -+ A° ® Al is an even operator. In this book we only consider involutions consistent with the Z2-graduation. Example 3.3. (involution in G1(C)). It suffices to determine the involution on the generators qj; qj* = qj (i.e., the generators are real
elements). Then, by virtue of property 3 of the involution, we have the monomial qjl...qj is real if 2n(n-1) is even (q and imaginary if 2n(n - 1) is odd. By analogy, we can define the involution in the CSA G00' (B), where
B is a commutative algebra with involution. Let A be a banach CSA with involution *. We can introduce in A an equivalent norm I I I- I I satisfying the condition I aI I I= I I Ia* I I (a star norm), for which purpose we must set I I a I I I= max(I I a I I, I I a* I I) Setting IIaII' = IIIa°III + IIIa'III, we obtain an equivalent norm satisfying the star condition I I a I I' = I I a* I I' and a Z2-graduated a I I' = I I a° I I' + I I a' I I'. Everywhere in the sequel, we only consider star Z2-graduated norms on CSA with involution. I
I
I
.
I
3. Hilbert Supermodules
137
Note that the norm on G'00(B) is a star norm if the norm on B is a star one.
3.8. Hilbert supermodule with involution. A module W over the algebra D with involution * is called a module with involution if the operation (denoted by the same symbol ') f H f' possessing the properties (1) (f*)* = f, (2) (.*f) = f*)*, (3) (f + g)' = f ` + g1,
f, g E
W, A E D is defined on W. Just as we did in Sec. 3.7, we define real and imaginary elements and the involutions in the CSM consistent with the Z2-graduation. A Banach CSM 12(A) over a CSA A with involution * endowed with *-scalar product 00
(x, y) = L xnyn n=1
is called a coordinate Hilbert CSM (with an even basis) with involution.
Here are the main properties of a *-scalar product. 1. Hermiticity: (x, y)` = (y, x). 2. `-A-linearity: (Ax, pya) = \(xa*, y)µ'. 3. The Cauchy-Bunyakovskii inequality: (x,y)
x112y2 Similarly, coordinate Hilbert CSM with involution with bases of an arbitrary graduation; *-scalar product in 12,A,B (A) over a CSA A with involution is defined by the relation (x, y) = x' (y°')' In contrast to the scalar product (. , ), the properties of a *-scalar product ( , ) do not depend on the graduation of the basis.
Definition 3.1. (*). The bilinear form ( , ): M x M -+ A which separates the points of the Banach CSM M and possesses properties 1-2 is called a *-scalar product on the CSM M. Definition 3.2. The triple (M, 1 1 , ( , )), where M is a Banach CSM with norm I I II and *-scalar product ( , ) is called a Hilbert CSM with involution if there exists an isomorphism S: M -+ 12,A,B (A) which preserves the *-scalar product and the norm. 11
If a CSA A is a E-algebra, then a superanalog M' ^_' M of the Riesz theorem is valid for Hilbert CSM with involution.
Chapter III. Distribution Theory
138
3.9. Self-adjoint operators in Hilbert supermodules with involution. Let M be a Hilbert CSM with involution. The operator at which is an (Hilbert) adjoint of the operator a E £1,1(M, M) is defined by the relation (ax, y) = (x, a*y). Just as we did for the opeator a', we can verify that a*: M' -+ M' is continuous. However, in contrast to the operator a', the type of A-linearity for the operator a* is the same as for a, at E £1,1(M', M'). Indeed, (x, a* (Ay)) _ (ax, Ay) = (ax, y) A* _ (x, Aa*y)
Let the CSA A be a E-algebra. Then M' ^_' M and the operator a*: M -+ M. The operator a is self-adjoint if a* = a. Unbounded self-adjoint operators are introduced in the same way as in the ordinary functional analysis. As usual, if (1Qa0j) is a matrix in the ONB {i,,e} of the self-adjoint operator a in a Hilbert CSM with involution, then the matrix elements satisfy the condition iaaAi =
(jfa")*
Conversely, any A-linear continuous operator satisfying condition (3.7) is self-adjoint.
3.10. Unitary operators in Hilbert supermodules with involution. An operator U E £1,1(E, F) (G1,,.(E, F)), where E and F are Hilbert CSM with involution, are said to be unitary if U maps the CSM E onto the CSM F (ImU = F) and preserves the *-scalar product ((Ux, Uy) = (x, y)). As distinct from an orthogonal operator (Proposition 3.3), a unitary operator is not necessarily even.
Example 3.4. Let M = lz,{1},{1}(A) _ {x = xooe + x11e: xa E A},
where oe is an even base vector, le is odd.
Consider an operator
U: M -* M, Uoe = 1e, 111e = oe. Then (Ux,Uy) = (x,y), U E L'1'1 (M, M) Proposition 3.4. (*). Every unitary operator is invertible, and the inverse operator is also unitary.
3. Hilbert Supermodules
Corollary 3.1.
139
Unitary left (right) operators in a Hilbert
(*).
space CSM form a group. Groups of unitary left and right operators are denoted by U,,, (M) and U,,,. (M) respectively.
Proposition 3.5. (*). An operator U E G1,,(E, F) (U E G1,r(E, F)), Imu = F, is unitary if and only if its matrix elements for ONB satisfy the relations u7k
u7k *
= joij.
Example 3.5. Let the CSM M be the same as in Example 3.3 (one even vector and one odd vector). Then the group of unitary operators U,,,(M) consists of matrices U=
°U0 0ul ) lu°
lul /
whose elements satisfy the conditions OUOOUO* + OU1OU1*
= 1,
lu°lu°* + lullul* =
1,
= 1. If the CSA A = G ,, then, for instance, any matrix of the form OU01UO* + 0U11U'*
U=
1+gtg3
-qj
qj 1
+ gtg3
satisfies these conditions.
Proposition 3.7. Let the operator a E £1,,(M, M) be self-adjoint. Then the operator U = e" is unitary. The proof repeats the standard one.
3.11. L2-supermodules with involution. Let A be a real CSA and A' = A ® iA be its complexificatoin. We introduce a superspace
Chapter III. Distribution Theory
140
T ,m = Ao x (A')' in which even coordinates are real and odd coordinates are complex. Let us consider a system of functions go :
V,' --) Ac, cp0p (x, 0) = cp,,(x)0P, where {cp,,(x) } are Hermite func-
tions on R"; the parity of the functions W,,# is 101 (mod 2). We introduce a Hilbert CSM with basis {cpa }:
(
L2`(Tn'm, dxdO) = If (x, 0) ffQ E AC, 11f
112 =
_
(x, 0): aQ
E
IIf.10112+
161=0
(mod 2)
191=1
(mod 2)
+(E IIffp112 + > 191=0
IIf.10112)1,2
IIff0112)1/2
< oo}.
161=1
(mod 2)
(mod 2)
This Hilbert CSM is known as a CSM of A`-valued functions which are square summable with respect to the distribution dxd0 on a superspace
TA''. The involution in this CSM can be introduced coordinatewise: f ` _ {f}. pThis involution is consistent with the Z2-graduation: the *-scalar product (f , g) _ ffpg.*,6 can be represented as an integral over a superspace: (f,g) = fSfl,m f (x, 0) (g (x, 0))`dx dB` d9. A
In order to frame the Hilbert CSM L2 ` (TA'm, dxdO), we must do the following. Instead of the superspace TA'm we must consider a narrower superspace Tu'm, where U is a pseudotopological CSA with
nilpotent soul constructed with the use of the CSA A. The framing of the Hilbert CSM L2`(Tn'm, dxd0) is carried out with the aid of the spaces of test and generalized functions on the superspace Tu'm. In the second chapter, we introduced spaces of types G and D on the superspace Ru'm = Uo x Ltl `; by a complete analogy, we can introduce
spaces of types g and V on the superspace Tu'm = U x (Ul )"` (here the anticommuting variables become complex, but the functions with respect to these variables are polynomials). In the case where A is a E-algebra, we have (TT'm, A`) C L2 ` (TA'm, dxd0) c
(Tu'm, A`)
4. Duality of Supermodules
141
or
D(TT'm, A`) C L2`(TA'm, dxdO) C D'(TT,m, Ac).
4.
Duality of Topological Supermodules Everywhere in the sequel we use the symbols M and N to de-
note topological CSM over the topological CSA A. By Ln,r and we denote the submodules Ln,,. and Ln,i of the CSM that consist of continuous mappings; Ln (L - L,) is a space of continuous n-linear mappings over the field K. In addition to continuous polylinear mappings, we shall be interested in mappings which are continuous on compact sets. The spaces of these mappings are denoted by JKn,r and lc,,,, respectively. If the topology on a CSM is metrizable, then the continuity on compact sets implies continuity, i.e., in this case we have Ln,r = Kn,r, Ln,1 = Kn,l.
The analogs of Propositions 1.1-1.8 are valid for the CSM Ln,r, Ln,l, Kn,r, Kn,l
4.1. Topologies of a bounded and compact convergence on a conjugate supermodule. Just as in a Banach case, by identifying the CSM Ln,r(M, A) and Ln,l(M, A) we obtain a CSM M' which is topological conjugate of M. The forms of duality ( , ): M x M' -* A and ( , ): M' x M -+ A possess properties (1.4) and (1.5). As in the theory of topological K-linear spaces, in the theory of topological CSM an important role is played by the topologies of uniform convergence on the conjugate CSM M'. Let us consider the case of locally convex CSM M and CSA A. Every bounded subset B C M and every prenorm II II from the defining system of prenorms IPA on the CSA A are associated with a prenorm on the conjugate CSM M': IIILIIIB = sup
IIL(x)II.
(4.1)
xEB
If {B} is a system of bounded subsets in the CSM M, then the topology of bounded convergence (topology of uniform convergence on bounded subsets) is defined by the system of prenorms (4.1). In
Chapter III. Distribution Theory
142
the same way we can introduce the topology of compact convergence on M' (the topology of uniform convergence on compact subsets). In a standard way, from prenorms (4.1) we can obtain equivalent Z2-graded prenorms JIL11B = UUIL°I11B+1HIL'IIIB. If M is a Banach CSM over Ba-
nach CSA A, then the topology of bounded convergence coincides with the normalized topology on M'.
4.2. Topological bases and Schauder bases. As in the Banach case, we can define the topological basis {ep} in the topological CSM M.
Example 4.1. Consider functional CSM A(Cm) and The monomials ec,p = x°9p form a topological basis of these CSM.
Example 4.2. Consider CSM of generalized functions and E'(C"m). The generalized derivatives e'Q0 = ;i ax-ae A
of the
J-function form a topological basis in these CSM. It follows from the definition of a topological basis that there exists
a family of functionals ep E M* such that x = E (x, eQ)ep, and the families {ep} and {eQ} are biorthogonal. Just as in ordinary functional analysis, we shall call the topological basis {ep} a Schauder basis if the functionals ep are continuous, ep ep E M'. For the Schauder basis we have
x=
(x, eQ)ep.
(4.2)
p
For a wide range of locally convex K-linear space, the pointwise convergence of a sequence of linear continuous operators implies a uniform convergence on compact subsets. Locally convex spaces of this kind are known as barreled space [71]. In particular, metrizable and Banach spaces are barreled spaces. For Banach spaces this is well-
known Banach-Steinhaus theorem. Moreover, every inductive limit of barreled locally convex spaces is barreled. Consequently, inductive limits of Banach and metrizable spaces and, in particular, practically all spaces of test functions that are encountered in analysis are barreled. We can use all these results in superanalysis since modules are
4. Duality of Supermodules
143
K-linear spaces and A-linear operators are, in particular, K-linear. In this case, series (4.2) converges uniformly on compact subsets of these CSM M.
Example 4.3. A CSM A(C"m) is a Frechet space, and, consequently, the series f (x, 0) = i f,,px'013 converges uniformly on any Qp
compact subset of a functional CSM.
Example 4.4. A CSM E(Cn'm) is an inductive limit of Banach spaces, and, consequently, the statement of the preceding example is valid here as well.
4.3. Dual topological supermodules. An essential difference from the theory of locally convex K-linear spaces is that the duality forms do not, in general, separate the points of the CSM M and M' (even in a Banach case, Sec. 2). In order to exclude these pathological cases from consideration, we distinguish a class of "good" CSM with the aid of the following definition.
Definition 4.1. Topological CSM M and N are said to be dual if there exists a bilinear form ( , ): M x N -+ A (form of duality) continuous on compact sets which separates the points of the CSM M and N and satisfies condition (1.5).
In particular, a duality form is even. The duality form ( , ): N x M - A is defined by the relation (1.4). Dual bases are the topological bases {ep} in M and {e'} in N, biorthogonal relative to the duality form, with respect to which the series converge uniformly on compact subsets of the CSM M and N respectively; a duality form in coordinates is written as (x, y) = > xpyp
Example 4.5. Let the CSM M = A(C" m) and the CSM N = A'(Cnm). Then they are dual CSM with dual bases
and {e' p} (it should be pointed out that the space of generalized functions A'(CA'm) is an inductive limit of Banach spaces, and therefore series (4.2) converges uniformly on compact subsets).
Adjoint operator. For topological CSM, an adjoint operator is defined in the same way as for Banach CSM.
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144
Proposition 4.1. Suppose that the conjugate CSM M' and N' are endowed with a topology of compact or bounded convergence; the operator a E G1,,(M, N). Then the adjoint operator a' E C,,r(M', N').
The continuity of the operator a' follows from the fact that for any compact (bounded) subset B C M the set a(B) is compact (bounded) in N.
Example 4.6. Let the CSM M =
(Run", Ac) or D(Ru", AC), Then the operators e E £1,1(M), j = 1, ..., n, and Am), E(Cnm) A(C M' =g'(Rum, Ac), D'(Ru'm, A`), the adjoint operators (B A'(Cn'm), E'(Cn'm) respectively. .
5.
Differential Calculus on a Superspace over Topological Supermodules 5.1. Superspace over topological supermodules.
Definition 5.1 [144]. A K-linear topological space X = MO ® N1, where M = Mo ® M1 and N = No ® N1 are topological CSM, is called a superspace over a pair of CSM M and N.
Definition 5.2. A CSM Lx = M ® N is said to be a covering of the superspace X = MO ® N1.
Example 5.1 (Vladimirov-Volovich superspace). Let M = Am, N = A". Then X = MO ® N1 = KA ", LX = Am+" In particular, if A = G"(K), G', then X is a Rogers superspace. De Witt superspace is the simplest example of a non-Banach space (Chap. I, Sec. 4).
Example 5.2 (superspace of Fermi fields). We denote by M a CSM consisting of some class of functions (perhaps, generalized) cp: Rk -+ Am. Then X = M1 = {cp E M, W: Rk -+ Am} is a superspace of Fermi fields.
Example 5.3 (superspace of a system of intersecting Bose and Fermi fields). We set M = {cp: Rk - Am}, N = {co: Rk -+ A"} (in
5. Differential Calculus
145
specific models, conditions of smoothness and boundary conditions are imposed on functions from the classes M and N); X = MO ®N1 = {cp: KK'"}. Rk ->
Under a new approach to a superanalysis, a boson field assumes values in Ao and not in Km. Example 5.4 (superspace of a boson string with Faddeev-Popov ghosts). We denote by M a CSM consisting of paths q: [0, 7r] -+ AD,
D = 26, satisfying the boundary conditions q'(0) = q'(7r) = 0 and by N a CSM consisting of paths z = (c, c): [0, 7r] -+ A2 satisfying the boundary conditions c'(0) = c'(ir) = 6(0) = 6(7r) = 0. We can impose different conditions of smoothness on the paths and topologize the CSM M and N respectively. The superspace X = MO ® N1 = {W:
cp = (q, z)} is a coordinate space of a boson string with FaddeevPopov ghosts, Mo is the string part of the superspace, N1 is the ghost part of the superspace [25]. Under the new approach to a superspace, a boson string assumes
values in Ao and not in R'. The topological basis (Schauder basis) in the superspace X = Mo® N1 is the topological basis (Schauder basis) in the covering CSM Lx =
M ®N. Let {ej}jEJ and {ai}iEJ be topological bases in the CSM M and N respectively. Then {ej; ai}jEJiEI is a topological basis in the space X:
x = E x°ej + E x1ai, jEJ
1x; l = lej 1,
lxi 1 = 1 - jail.
iEI
Example 5.5 (Hilbert superspace). Let M, N 12(A) be Hilbert CSM and let {ej} and fail be canonical bases in M and N respectively. The superspace H = Mo®N1 is known as a Hilbert superspace, {ej, ail is Schauder basis in the Hilbert space H, 00
00
j=1
i=1
The scalar product 00 (x)
y) = (x°, y°) + (x1, y1)
j=1
00
X59 + > xi yi i=1
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146
assumes values in A0.
Let M and N be Hilbert CSM with involutions `. The involutions in the CSM M and N induce an involution in the Hilbert superspace H (since I' I = 0); the *-scalar product has the form 00
00
(x, y) = (x°, y°) + (x1, yl) _
x°(y9)i + E xi (yi )'. j=1
i=1
By analogy, we can consider a superspace over the Hilbert CSM M, N 12,A,B (A)
5.2. Superdifferentiability. Let us recall, for the beginning, the definition of a differentiable mapping of topological K-linear spaces El and E2. In differential calculus, there are several dozens of definitions of differentiability for topological K-linear spaces. In order to define differentiability, we must, first, fix a certain class of K-linear operators 1-1(E1, E2) to which the derivatives will belong and, second, fix a type
of smallness of the remainder, i.e., define in what sense o(h) E E2,
hEE1. The mapping f : U -+ E2, where U is an open subset of the space
El, is said to be differentiable at the point x E U if, for all h E El such that x + h E U, we have
f(x + h) - f(x) = A(h) + o(h),
(5.1)
where the operator of the derivative A = f'(x) belongs to the class 7 l (El, E2).
The differentiability with respect to systems of bounded and compact subsets is especially widely used. For the differentiability with respect to a system of bounded subsets we take £(E1, E2) as 3l(E1, E2) and define o(h) as o(th)/t -4 0,
t -+ 0,
t E K,
uniformly on any bounded subset of the space El. For the differen-
tiability with respect to a system of compact subsets f(E1, E2) = K(E1, E2), and in the definition of o(h) we use a uniform convergence
5. Differential Calculus
147
on compact subsets. The Gateauz differentiability is also of importance. For this differentiability we also have 3l(El, E2) = G(E1, E2) and o(th)/t -4 0, t -4 0 for any vector h in El. For Banach spaces, the differentiability with respect to a system of bounded subsets coincides with the well-known Frechet differentiability (see Sec. 1.2, Chap. I). Using a pair, namely, a superspace and a covering supermodule, we can extend the definition of differentiability for topological K-linear
spaces to a supercase. We fix a certain class of A-linear operators Hi,r(Lx, Ly) (7i1,1(Lx, Ly)) (it is customary to assume that this class is a CSM) and a certain definition of o(h).
Definition 5.3. The mapping f : U -a Y, where Y is a superspace and U is an open subset of the superspace X, is said to be right (left) Sdiferentiable at a point x E U if, for all h E X such that x+h E U, we have relation (5.1), where the operator of the S-derivative A = 8f (x) belongs to the class 3-ll,r(Lx, Ly) (f1,1(Lx, Ly))
When necessary, the right-hand S-derivative will be denoted by aR f (x) and the left-hand derivative by aL f (x). By a complete analogy with the case of topological K-linear spaces, we can define S-differentiabilities with respect to systems of bounded and compact subsets and the Gateauz S-differentiability. The definitions of o(h) can be extended without changes; for the S-differentiabi-
lity with respect to a system of bounded subsets and the Gateauz differentiability we take the CSM Gl,r(Lx, Ly) (G1,1 (Lx, Ly)) as the class Ni,r(Lx, Ly) (711,1(Lx, Ly)); for the S-differentiability with respect to a system of compact subsets I,r(Lx, Ly) = IC,,,- (Lx, Ly) (' 11,1(Lx, Ly) = K1,1 (Lx, Ly))
Definition 5.3 can be reformulated as follows: the mapping f is differentiable as the mapping of K-linear topological space X and Y, the derivative belonging to the class of A-linear operators R1,r (f1,1)
Remark 5.1. We cannot restrict the consideration to some type of linearity on a superspace and not on a covering supermodule. A superspace is only a A0-module, but the A0-linearity of the derivative is insufficient for S-differentiability (see Example 2.4, Chap. I).
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148
Let M=M°®M1, N=No ®N1, R=RoED R1, S=So ®Sl be topological CSM over a topological CSA A, X = M°®N1 i Y = Ro®S1.
The function f : X -4 Y, f = (f °, f 1), f ° E Ra, f 1 E S1, is right Sdifferentiable. Then
of
of
of
= 19x°, axl
of
aa axe
=
,
aE
where the operators ax E 7{° o l,r (M, R),axr E 1-ll ,r(N R) , 3li r(M, S), a E Vi,r(N, S). If the CSM R and S coincide, Y = R° ® R1 = R, then we can regard the S-derivative a f as an element of the space W1,r (Lx, R): e
of=of°ED af'. Definition 5.4. The mapping f : U -4 N, where U is an open subset of the CSM M, is said to be right (left) S-differentiable at a point x E U if, for all h: x + h E U, we have relation (5.1), where af(x) E Hi,r(M, N) (af (x) E 71,1(M, N)) In contrast to an analysis on a superspace, an analysis on a CSM is very meagre and is of no particular interest.
Example 5.6. Let us consider the mapping f : A -4 A, f (x) _ ax,3, where a, fl E A1. We shall regard the domain of definition of A as a CSM = CSA and not as a superspace KA1 = A° x A1. The mapping f is not (right or left) S-differentiable. The same mapping regarded as a mapping of superspaces is both right and left S-differentiable. By virtue of Remark 5.1 and Example 5.6, it is obvious that an analysis on a pair (a K-linear superspace, an A-linear covering CSM) is a "golden mean" between an analysis on a K-linear space and an analysis on an A-module.
Proposition 5.1. If the operator CSM fl,r = Ll,r and 711,1 = L1,1 or 7-ll,r = 1Cl,r, 7-11,1 = 101,1, then the right and left S-differentiabilities
are equivalent.
This proposition is a direct corollary of Proposition 1.4.
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149
Theorem 5.1 (Leibniz formula). Let M, N, S be Banach CSM; suppose that the functions f : X = MO ® Nl --* S and g: X -+ S' are Frechet S- differentiable at a point a E X, and g(a) is a homogeneous element of the CSM S'. Then the function W(x) = (f (x), g(x)) is Frechet S- differentiable and we have a formula
l
ax° J f (a)
axe (a)
+(-1)°I9(a)I
{o.19a)I
axf
(a)l g(a),
a = 0, 1.
(5.2)
Proof. It suffices to note that P = ` axe (a) V, g(a)) =
(a),
I9(a)I (_(a)h'\"
by virtue of Proposition 1.7 (relation 1.4). If jg(a) I = 0, then P=
((a)f (a)ho /
if jg(a) I = 1, a = 0, then P
f
= Kg(a)' a ('9x° (a)) ho)
if jg(a)j = 1, a = 1, then P
(g(a), a
((a)) hl)
These three equalities yield the right-hand side of relation (5.2). The ordinary theorems of differential calculus (on an inverse function, ...) are valid in superanalysis. They are obtained in Sec. 1, Chap VII in a more general case of analysis on a superspace over an arbitrary noncommutative algebra.
Chapter III. Distribution Theory
150
We introduce a right annihilator of the superspace X = M° ® N1 by setting 1X - Ann (X; ll,r(Lx, R)) = {a E 'Hj,!(Lx, R): Ker a D X}, where R is a CSM. By analogy, we introduce the left annihilator.
Proposition 5.2. If the condition 1A1 - Ann(A1; R) = 0
(5.3)
is satisfied, then Ann (X; Ll,r(Lx, R)) = 0 (and Ann(X; L1,i(Lx, R))= 0)
Proof.
1.
Suppose that the operator a E Ll,r(Lx, R) and the
restriction of a to M is zero. Then we have a(m10) = a(m')O = 0, i.e., a(ml) E 1A1, for all 0 E Al and ml E M1. Consequently, aim = 0. 2. Let a E L1,r(Lx, R) and ajN1 = 0. Then we have a(n°9) _ a(n°)9 = 0, i.e., a(n°) E 'A1i for all 0 E Al and n° E N°. Conse-
quently, a iN = 0.
Thus, by virtue of Proposition 5.2, the triviality of the annihilator of the superspace for any class of S-derivatives Hi,r(Lx, R) follows from the triviality of the A1-annihilator for the module R (see (5.3)). As was pointed out in Chap. I, an S-derivative is, in general, not uniquely defined.
Let the function f : X -4 Y be S-differentiable (with a space of S-derivatives ?'ll,r(Lx, Ly)). By factoring the space of S-derivatives with respect to the annihilator of the superspace, we get a one-to-one mapping aR f : X -+'Hl,r(Lx, Ly)/Ann(X, Lv). Consequently, we can define the second derivative aRf (x) E Ni,r(Lx, W i,r(Lx, Ly)/Ann(X, Ly)) and higher-order derivatives. 5.3. Supersymmetries of higher-order superderivatives. Let us restrict the consideration to the case of single-valued S-derivatives: 1X = 0. By virtue of Proposition 5.2, it suffices to require that 1A1 = 0.
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151
Thus, suppose that 'Al = 0. Proposition 5.3. For the Frechet S-differentiability on a Banach superspace, the restriction of higher-order derivatives to a superspace is symmetric.
Proposition 5.3 is a direct corollary of Frechet differentiability in K-linear Banach spaces [37].
Remark 5.2. When formulating the results concerning S-differentiability, we consider Banach superspaces only for the sake of simplification of the exposition. We can also consider S-differential calculi over topological and pseudotopological superspaces (cf. the K-linear case [38, 541).
Lemma 5.1. Let X = MO ® N1, b E Lp,r (LX, R) and let the restriction of b to XP be symmetric. Then we have the symmetry b(yl, ...) yk, ..., y;, ..., yP) (5.4) ak+j(yj), ..., Qk+j(yk), ..., yp) _ onYP, Y = M1oNo. Proof. Let 03 E Al for j = 1, ..., p. Using relations (1.1) and (1.2) -b(yl,...,
and the symmetry on XP, we obtain b(yl01, ..., yk8k, ..., yjOj, ..., yp9P) = b(y101,...,y3O,,...) ykOk,...,yyOP)
=
j-1 y;, ... , QP-1 yp)e1...Bk...B;...9p yk, ..., Q j-1 P-1 k-1 yp Bl...e;...ek...ep. Q Or yke ..., Or
b (yl, Oyz, ..., 0'
- -b (yl
k-1
y2r Let us set yj = Qi-lyt Then
b(yi, ..., yk, ..., y', ..., y'P)el...ek...e;...ep
_
b(yl, ..., or
k+j yk,
I I ..., a k+j y;, ..., yP)el...ek...B;...ep
In order to prove relation (5.4), it remains to use the triviality of the Al-annihilator.
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152
Lemma 5.1 yields
Proposition 5.4. Let X = Mo ® N1, b E LP,,.(LX, R) and let the restriction of b to XP be symmetric. Then the restriction of b to M? and No is antisymmetric.
Lemma 5.2. Suppose that the conditions of Lemma 5.1 are fulfilled. Then b x1Q1, ..., xkQk, ..., xQi , ... , xPQP) i-1
QkQ;+(Qk+Q;)
Q;
b(x1..., xp', ...)xkk, ..., xQP),
(5.5)
where/3,=0,1,xAEMo for/3=0 and XP ENo for /3=1. Proof. Let 03 E A1, j = 1, ..., p. Then b(x#1
0Q1
, ..., x1 k akk , ..., X13' 01' , ..., xpp 9pp )
xA'9p, ..., xkk ekk, ..., xpp epp)
= b(xp1
=
b(xQ1 1,
9Q1...9Qk ..OOP ..., xak k, ..., xQi k ...B~i 7 P j , ..., XOP) P
Xk = b(xQ11, ..., xQi 7, ..., k, ..., xQ9)0Q1...Bp'...BQk...BQP 7 k P P 1
i-1
X(-1)
Qk1;+(Qk+Q;)+ > Qi :=k+1
It remains to use the triviality of the A1-annihilator. Lemmas 5.1 and 5.2 show that the symmetry of an A1-linear form on a superspace entails many (rather unexpected) supersymmetries on a covering CSM.
Lemma 5.3. Any form b c Lp,,.(LX, R) is uniquely defined by its restriction to the superspace XP.
Proof. Let a, Q = 0, 1. We regard the sum a + Q mod 2. Let zC1Q j = 1 , ..., p belong to M, , for / 3 = 0 and to N, , for 0 = 1 For 0 E Al .
we obtain 0Q1+Q1 Bop+QP b(za1 , ..., z°P 1Q1 1 PQP P )
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153
(-1)6b(zQ' lfl,
,
zOP °P )BQ1+Q1 1
0Qp+Op .. p
where 5 = b(a, Q). Note that the vector z = (z1A1B11+A1,
,
P+Rp)
belongs to XP. Therefore, if blxp = 0, then P
> (Q; +Ai )
b(z) E L (Aj ' l
Theorem 5.2 (on the properties of higher-order S-derivatives). Let the function f : X -+ Y, where X and Y are Banach superspaces, be n times Frechet right S-diferentiable at a point x E X. Then its S-derivative of order n is uniquely defined and belongs to the CSM Gn,r(LX, Ly); the restriction of the S-derivative to the superspace is symmetric and supersymmetries (5.4), (5.5) hold on the covering CSM.
Theorem 5.2 is also valid for other types of S-derivative on locally convex superspaces, for instance, for differentiability with respect to a system of compact sets.
Example 5.7. Let M = A2, N = A2, X = RA2, Lx = A4 and let the function f (x, 0) = ax1x2 + /30102i where a, /i E A, x = (x1, x2) E A2, and 0 = (01 i 92) E A2. Then 8Rf (x, 0) (h, h') = a(v2u1 + v1u2) + Q(e1712-e2711), h = (v, e), h' = (u, 71) E A4. The restriction of 8Rf(x, 9)
to the superspace RA2 is symmetric and the restriction to o ® A2 and to A2 ® o is antisymmetric. Note that 8Rf (x, 0) has many K-linear extensions to the covering CSM A4. For instance, (h, h') -+ v2au1 +vi au2 + bft - e20711. These extensions are not right A-linear if a, /3 V Ao. A higher-order Frechet S-differentiability is defined by the classes
of operators 'Hn,r = Gn,r (Wn,i = Gn,l) to which the S-derivatives belong and by the Frechet differentiability in K-linear Banach spaces. Similarly, every differentiability in topological K-linear spaces and any sequence of classes of forms Wn,r C Ln,r (fn,l C L, ,1) are associated with a higher-order S-differentiability.
5.4. Taylor formula. It follows from Theorem 5.2 that if Taylor formula holds for the differentiability in K-linear spaces [38, 54], then,
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154
for the corresponding S-differentiability, we have Taylor's formula on a superspace
f (x) _
aRf xo) Ti.
n=0
(X l
- x0i ..., x - xo) + Tm(x - x0),
where Tm is the remainder, 8R f (xo) E fn,r, and the restriction of oR f (xo) to Xn is symmetric.
In particular, the Taylor formula holds for an S-diffrentiability with respect to a system of bounded (compact) subsets: rm(th)/tm -4
0, t -+ 0, uniformly on bounded (compact) subsets of the superspace X; in Banach superspaces, this is equivalent to the fact that IITm(h)II/IIhII -+ 0, h + 0.
5.5. Superanalyticity. The Taylor formula on a superspace leads to the following definition of S-analyticity. Definition 5.5. The mapping f : U t -+ Y, where U is the neigh-
borhood of the point x0 E X, is right (left) S-analytic at a point xo if, in a certain neighborhood of the point x0i the mapping can be expanded in power series 00
f(x)=Ebn(x-xo,...,x-x0),
(5.6)
n=0
where A-n-linear forms of bn belong to the supermodules (L'r, Ly) (Ln , Ly)) and the restriction of these forms to X n is symmetric. We obtain various definitions of S-analyticity corresponding to dif-
ferent classes A-n-linear forms ?in,,, fn,t and to different types of convergence of the power series (5.6).
Definition 5.6. The mapping f : U
Y, where Y is a superspace over a locally convex CSM, is said to be compact (bounded) S-analytic at a point xo E U if Kn,r(Ln,r), in,t = 1Cn,i(Ln,i) and if there exists a neighborhood V = V(xo) of the point xo in the covering CSM
LX such that for any compact (bounded) subset B C V and any prenorm II II E FL,, we have 00
Ilf IIB = E Sup Ilbn(x1 - x0, ..., xn - xo)II < 00. n=0 xi E B
(5.7)
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155
It follows from Proposition 1.4 that the left and right compact (bounded) S-analyticities coincide.
It follows from Definition 5.6 that every compact (bounded) Sanalytic function f : U(xo) -* Y can be extended to the neighborhood of the point x0 in the covering CSM L. From estimate (5.7) it follows that every compact (bounded) Sanalytic function is infinitely S-differentiable with respect to a system of compact (bounded) subsets.
If the function f is compact (bounded) S-analytic on the whole superspace X, then it is said to be compact (bounded) S-entire. An S-entire function can be extended to the covering CSM Lx. Consider compact and bounded S-entire functions on a superspace KA'm over a locally convex CSA A with a trivial Al-annihilator.
Proposition 5.5. The spaces of compact and bounded S-entire functions f : KA'm -+ A coincide.
Proof. Let f be a compact S-entire function:
f (y) = 001: 1:
bk(ejl, ...,
ejk)yjl...yjk,
k-0 jl...j
is a canonical basis in KA'm, and for any compact set where B C An+m and prenorm II . II E I A we have 00 IIfI
bk(ejl, ..., ejk)zljl...zkjk 11 < 00.
I B = E sup
k=0 zyCB ji...jk
We set BR = {Rel, ..., Ren+m}, and then IIfMIBR =
E00 sup IIbk(ejl, ...,
ejk)IIRk
< 00.
k=0 it ...7k
Consequently, 00
IIfIIR=
Ilbk(ejl,...,ejk)IIRk < IIfMIBr < 00, k=0 ji...jk
(5.8)
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156
r = R(n + m). Suppose now that If M R < oo for all R > 0. Then, for any bounded set B C An+m: IIf JIB < If IIR, where R = sup up JJxJJ.
Corollary 5.1. Let A = A be a Banach algebra. Then the space of compact (=bounded) S-entire functions f : KAn'm -+ A coincides with a space of S-entire functions in the sense of the Vladimirov-Volovich definition (Chap. I). The corollary follows from estimate (5.8).
Remark 5.3. It should be emphasized that convergence (5.6) in Definition 5.6 holds on compact (bounded) subsets of the covering CSM Lx and not only of the superspace X.
The article [118] is a brilliant illustration to this remark. The author of this work constructed a "counterexample" by virtue of which (in his opinion) "Khrennikov's theory of superanalyticity (1988) is not well-grounded." The author of [118] considers a superspace KA1, where
A = lim ind Gn is a Nagamashi-Kobayasi topological CSA [103]. It is easy to show that the series E Q1...gkxkB converges uniformly on every compact k=0
subset of the superspace KA1 and defines the function f (x, 0) on KA'1 (it follows from the properties of inductive topology [71] that every
compact subset in A is contained in one of the finite-dimensional Grassmann algebras Gn). This function possesses a number of "pathological" properties. In quantum theory, the fields of the function cp(x, 0) are fields on a superspace. Of a considerable importance is the expansion of a field in the powers of 0: cp(x, 0) _ cop(x)0', (5.9) where the coefficients cpp (x) are physical tensor fields. Expansion (5.9) does not hold for the function f (x, 0).
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However, the function f (x, 0) is not S-analytic in the sense of Definition 5.6. In the covering CSM A2 we take a compact set KE consisting of one point {Ee, Ee}, where E > 0 is any arbitrarily small number. Assume that 00
Ilf IIK, =
L
IIg1...gkEk+lII < oo
(5.10)
k=0
for any prenorm I II E I'A. Then the series E Q1...gkek+1 converges to k=0
A, and, consequently, the sequence of partial sums of this series must be contained in one of the finite-dimensional Grassmann algebras G,,. Therefore (5.10) does not hold true.
Corollary 5.2. For every compact (-bounded) S-entire function go: KA'm -+ A expansion (5.9) holds true; the coefficients Wp(x) are compact S-entire functions.
This corollary can be immediately obtained with the aid of estimate (5.8). It follows from the triviality of A1-annihilator that the coefficients are uniquely defined.
If the superspace X is of an infinite dimension (over A), then the S-analytic function f of anticommuting variables can be a nonpolynomial.
Example 5.8. Let X = N1, where N = 12(A); the function f (0) = exp{ w 8;93+1} is bounded S-entire, but is not a polynomial. 7-1
In an infinite-dimensional case, the compact S-analyticity does not imply a bounded S-analyticity. We denote the space of compact S-entire functions f : X -+ A by A(X) (by virtue of Corollary 5.1, for the superspace X = KK'm this
notation is consistent with that used in Chap. II); the topology in the space A(X) is defined by a system of norms (5.7), where II I is a norm on A, x0 = 0.
Chapter III. Distribution Theory
158
6.
Analytic Distributions on a Superspace over Topological Supermodules
The theory of distributions on a superspace over CSM was constructed in [141, 144, 65]. It was pointed out in Chap. II that this theory was based on two ideas, namely, A-linear functionals are considered instead of K-linear functionals and the theory of distributions is developed over a CSA with a trivial A1-annihilator. Only over a CSA A with a trivial A1-annihilator can we consider functions of an infinite number of variables (infinite-dimensional superspaces over CSM is of the main interest for physical applications).
6.1. Dual superspaces. A superspace X' = Mo ® N1', where M' = MO 'E) M1' and N' = NO ED N1' are CSM conjugate to M = M° ®M1
and N = No ® N1, is a conjugate of the superspace X = M° ® N1. The form of duality between the superspaces X and X' is defined by the relation (m° ® nl, u° ® v') = (m°, u°) + (n', v'),
(6.1)
where ( , ) are forms of duality between conjugate CSM. In contrast to the form of duality between CSM, the form of duality between superspaces assumes values in A0.
Generally speaking, the form of duality does not separate the points of the superspaces X and X' even if the forms of duality separate the points of the conjugate CSM M and M', N and N'.
Example 6.1. Let M = Am, N = An. Then M' = Am, and
N'=An,X=KA'm=X'; m
(m° ®nl, u° (D vl)
n
- Ej=1m°uj + j=1 E nj
m °,u° EAo,
n 1,v l E A. Ai
Let the CSA A be a Grassmann algebra Gk, k < oo. If the number of generators k is odd, then the form of duality does not separate the points of the superspaces X = KA'n , X' = KA'n although the form of
6. Analytic Distributions
duality (a, b) _
d
159
a3bj separates the points of the conjugate CSM Ad
and Ad.
Definition 6.1. The superspaces X = M° ® N1 and Y = RO ® S1 over the pairs of dual CSM M and R, N and S are said to be dual. The form of duality is defined by relation (6.1) with the aid of the forms of duality between the CSM.
Definition 6.2. The dual bases in the dual superspaces X and Y are dual bases in the covering CSM LX and Ly. Recall that we have accepted the notation MI (M')'C, a = 0, 1. In precisely this way must we interpret all corresponding symbols in the subsequent examples.
Example 6.2. Let X = M° ® N1 be a Hilbert superspace and Y = X; {e3; a canonical basis of X. The orthogonality of the basis means that it is dual to itself. The basis lies in M° ® N°.
Example 6.3. Let M = A(CA'm), N = A(C; °), X = M° Ni ®= Ao(CA'm) ® A1(CA9), Y = X' = A°(CiA'm) ® Al(Cr'A9). The basis in the superspace X is formed by the monomials onomials e p (x, 0)
= X419, x c Ao, 9 E Am, e7P(z, ) =
z E Ap, S E M;
f(x,z,0, ) = f°(x,0) +fl(z,O E ffpe,p(x, 0) + E f7Pe7P(z, S), ap
where
7P
I f7Pl = 1 - e7Pl'
The basis in the superspace Y is formed by the derivatives if the 5-function a
a
ax' ae) -
1 a°+p6(x, 0) a! axaaop
1 a7+P6(z,e) e7P
a
a
a
P (ax, az' 190'
a 111 9C
7!
az7 a °
a
,
a
a
- P (ax' aB) +P
1
a
( az' al;)
Chapter III. Distribution Theory
160
°
,
a
a
1
P°,6e.,c ax(' ae ) +
P7Pe,P
7P
where 1 P.1# 1 = l eQp I , I PP J = 1 - l e7P 1.
Ca
al
az a
The form of duality between
the superspaces X and Y is written as (f, P) = (f °, P°) + (f 1, P1)
_ E fpPap + E f7PP7P at3
(E Ao).
7P
Example 6.4. X = E0 (Cnm)
(D El (COQ),
X = E°(Cnm) ®
y = X' = Eo (C m) ®Ei Y = X' = EE(CAm)
Example 6.5. X = 9(RR'm, Ac) ®G(Ru4, Ai),
Y = X'= c'(Ru'm, A') ®g'(R f, A1); X = D(Rum, Ac) e D(Ru°, Ai), Y = X' = D'(Ru'm, Ao) ®D'(Ruq, Ai)
Proposition 6.1. Let X = M° ® N1 and Y = R° ® S1 be dual superspaces over CSM over a CSA A with a trivial A1-annihilator. If there exist dual bases in the superspaces X and Y, then the form of duality (6.1) separates the points of X and Y.
Proof. Let {ej, ak} and {e'., ak} be dual bases: ej E M, ak E N, e E R, ak E S (by definition, the bases lie not in superspaces but in covering modules). Then we have x = (x°, ej)ej + E(x', ak)ak j k for x = x® ® x1 E X. Assume that (x, y) = 0 for ally E Y. Then (x°, O c a'.) = 0 for all le = 0 and (xl, ak) = 0 for all jakj = 1. Let A1, and then (x°, e')0 = 0 for all lej1 = 1 and (x', ak)0 = 0 for all IakI = 0. It remains to use the triviality of the A1-annihilator.
6. Analytic Distributions
161
6.2. Fourier transform. Let V = Mo ® Ni and W = R® ® S1 be dual superspaces over pairs of dual CSM (M, R) and (N, S) over the Banach CSA A.
For every vector w E W we introduce on V a function f,,(v) _ e`(","). We denote by 'Y(V) a certain CSM consisting of S-smooth (or S-analytic) functions f : V -+ A and containing all functions f,,, w E W. We choose T (V) as a space of test functions on the superspace V ; W' (V) is a space of distributions on the superspace V.
Definition 6.3. The Fourier transform of the distribution of L E 'k'(V) on the superspace V is the function .F(V) on the dual superspace W defined by the relation F(L)(w) = f L(dv) f,,(v). We denote by D(W) the Fourier-image of the space of distributions W'(V). If Ker F = {0}, then we can define the space of distributions on the dual superspace W:
M(W)=III eV(W):IL =µg, 9E`y(V), fco(y)(dy) = f .F-1(w)(dx)9(x),
co E
ID(W)j.
Thus, every function g from the space of test functions on V is associated with a distribution p9 on W. The funcion g is called a Fourier transform of the distribution µ9 and is denoted by .i'(µ9). A harmonic analysis arises on a pair of dual superspaces V and W :
`y'(V) 4''(W), 11(V) - M(W). By definition, we have a Parseval equality
f (L)(w)µ(dw) = fL(dv)Y()(v). Then we use the notation of cp for
(6.2)
cp E' (W).
Theorem 6.1 (on the kernel of a Fourier transform on a superspace). Let the covering CSM LV be complete and locally convex.
Chapter III. Distribution Theory
162
Suppose that in the superspaces V and M there exist dual topological bases; A = A is a Banach CSA with a trivial A1-annihilator, the space of test functions W(V) = A(V) is a space of compact S-entire functions. Then the kernel of the operator of the Fourier transform .F is zero.
Proof. Let a E Ker.F. Then OO in
> (a, (. n-o n1
(a,
,
w)n) = 0
for all w E W.
It follows that (a, ( , w)n) = 0 for all w E W, n = 0, 1, ... Let {ej; a;} and {ej; ai} be dual bases in the superspaces V and W. Let ej' I = 0 or I aiJ = 0. Then, setting w = e' or w = ai, we obtain (n = 1): (a( , e'j)) = 0 or (a( , e;)) = 0. Let le'1 = 1 or Jail = 0. Then, setting w = or w = ai0 for any value 0 E A1, we obtain (a(. , e'))9 = 0 or (a(. , ai)) = 0. It remains to use the triviality of the .
A1-annihilator.
By analogy, we obtain (n = 2,...): (a,
0
for all jl... jk, il...im If the superspaces V and W are finite-dimensional, then the theorem is proved. Suppose that the dual bases are countable. We introduce projectors 00
7rmkv = E (v°, j =M
00
E(v1, a')aj. j =k
Uniformly on compact sets of the covering CSM Lv, 7rmk + 0,
m,k-+oo.
Let B c Lv be a compact set. For every absolutely convex neighborhood U in the CSM Lv we can construct a finite U-net of the set 00 00 U U 7rmk (B) = C. Indeed, for any neighborhood U in the CSM Lv m=1 k=1 there exist mo, ko such that 7rmk(B) c U for all m > mo, k > ko. However, the sets 7rmk(B), m = 1, ..., mo, k = 1, ..., k° are compact,
6. Analytic Distributions
163
and therefore there exists a finite U-net for them. Since the space Lv is complete, the existence of a finite U-net for any absolutely convex neighborhood U implies that the set C is compact. Let the form b E IC,a,,(L ', A). Then, for any c > 0 there exists a neighborhood U of zero in L such that sup jIb(v1,..., vn)jj < E. V2 E Cf U
Therefore, for any e > 0 there exist mo, ko such that sup Ilb(7rmkv1, ..., 7f,nkvn)II < E v, EB
lim b(®(1 - 7rmk)) in the space for m > mo, k > ko. Thus b = m,k-+oo 1
A(V). It remains to note that for any function f E A(V) there exists m a sequence of forms {bn}, bn E Kn,r(LV, A), f = lim E bn in the m_+oo n=0 space A(V). In particular, Theorem 6.1 is valid for all superspaces considered in Examples 6.2-6.5.
Remark 6.1. One must distinguish between two causes for the noninjectivity of a Fourier transform on an infinite-dimensional superspace. The first cause for the noninjectivity is not connected with infinite-dimensionality. It is the nontriviality of the A1-annihilator (see Chap. II). The second cause is not connected with the superstructure and is due to the infinite-dimensionality of the space. The conditions of injectivity of a Fourier transform on K-linear infinite-dimensional spaces were discussed in the articles [137-141, 67]. It was shown there that for a Fourier transform to be injective, a weaker condition was sufficient, namely, the fulfilment of the approximation property [71]. The approximation property for superspaces can be defined as for Klinear spaces; an identity operator can be approximated uniformly on compact sets by finite-dimensional A-linear operators. Everywhere in what follows, we consider the theory of generalized functions in which the space of test functions %P(V) = A(V); V and W are complete locally convex superspaces with dual bases, and the A1-annihilator is trivial.
Chapter III. Distribution Theory
164
Proposition 6.2. The functions from the space Z(W) are compact S-entire. Proof. Every function f E -1>(W) has the form 00
in
f (w) _n=E ?n1 (L, (,
)n),
where L E A'(V). Let B,, be a compact subset of the CSM Lw. Then 11f
fIIBW =
00
1
SUP II(L, (.,wl))...(.,wn)II.
n=0 n' w, EBW
However, since L is continuous, there exists a compact set B in the CSM Ly such that for any form b E 1Cn r (LV, A) we have II(L,bn)II <_ CL Sup
Ilbn(v1,...,vn)II
viE By
Thus,
Sup Sup IIABv < CL E 1 n=0 n! vj EBo Wj EBW
II(v1,w1)...(vn,wn)II
It remains to use the continuity of the form of duality on compact sets.
Proposition 6.3. Suppose that the superspace W = V' is endowed with a topology of uniform convergence on compact sets. Then the functions from the space (D(W) are of the first order of growth. 6.2.
The proof of this proposition is contained in that of Proposition Indeed, the estimate I f (w) I < C exp{ I I Iw I I }, where I Iw I I I= I
I
I
I
sup II (v, w) II is a continuous prenorm on W, is valid for functions from
vEBv
the space 4)(W).
Proposition 6.4. Suppose that the sequence of vectors {rj},'?_1 is contained in a compact subset B covering the CSM Lv, and let
6. Analytic Distributions
165
{Ljl...;n} be a sequence of elements of the CSA A satisfying the condition 00 00 R-n
CL =
E 1141 ... in I I< o0
jl...in
n=O
for a certain R > 0. Then the function 00
00
E E Lil...in(ril
1
Wl)...(rin, Wn)
n=0 n. 71...in
belongs to the space (D(W).
Proof. We introduce a functional 00
E n=0
00 1 ni
anb(y)
,
jl...in
a;
is a generalized derivative of the s-function in the direction of rj. Let us prove that L E A' (V ) where
II(L,.f)II <
1 sup n! zjE R
Ilanf(0)(zl,...,zn)II
where BR = RB. We have not obtained an explicit description of the space of test functions D(W).
Remark 6.2. The problem of describing Fourier transforms of the spaces of measures and distributions on infinite-dimensional spaces was posed in the well-known work by Fomin [62]. However, no essential advances (even in the K-linear theory) have been obtained.
Remark 6.3. The operators .F and F of the Fourier transformation belong to the spaces LO,, (A',,D) = L0,1 (A', c) and LO,, (M, A) _ L°,i(M, A).
The direct product and the convolution of distributions on the superspaces V and W are defined as usual (see Chap. II). The operations of direct multiplication and convolution in the spaces A' and M are
Chapter III. Distribution Theory
166
correct. The spaces of distributions A' and M are convolution CSA. The operators F and F' of the Fourier transformation transform a direct product and a convolution of distributions into a product of Fourier transforms. Remark 6.4. For the superspace CA'm the space of generalized functions M(CA'm) = E'(CAm) is a topological conjugate of the space of test functions (Cn'm) = E(Cnm). In the general case, the question concerning the topology in the space of test functions -(D(W), in which M (W) = c' (W) , remains open.
7.
Gaussian and Feynman Distributions
Out of numerous of definitions of Feynman integrals over infinitedimensional space (see, e.g., [1, 3, 5, 7, 24, 26, 45, 59]) we choose a definition based on the Parseval equality. This definition (for K-linear spaces corresponding to boson theories) was used by many authors. We can point out the articles by De Witt and Morette [97], Albeverio and Hoegh-Krohn [1], Slavnov [126], Uglanov [133], Smolyanov [127], and Khrennikov [138-141, 145, 67].
This definition is based on a simple but very fruitful idea. Note that (in the sense of the theory of generalized functions) the Fourier transform of a quadratic exponent 2n , x E R, is also a quadratic 71 et22
exponent Let the function cp(x) be a Fourier transform of the function cp(s) (generalized, in general): cp(x) = f d. Then e-in(2/2.
eix2/26
J co(x)
27rib
eix2/26
dx
-
ix2/26
_
0,
F ( e 27rib)) =
Jd(e)e2/2.
(7.1)
It is an infinite-dimensional analog of the Parseval equality (7.1) that is used for defining the Feynman integral. The notation det B 1 w 1
(x)e`(B-lx,x)/2
00 ( dx;
H =1
27ri
7. Gaussian and Feynman Distributions
167
where B is a linear operator in an infinite-dimensional space, is set, by definition, equal to
f
e-i(Bt,t)/2(P(<)
= ((P,
e-=(B(,()/2),
(7.2)
where (p is a distribution on an infinite-dimensional space, (p(x) = (the space of test functions of an infinite-dimensional f argument must contain quadratic exponents). A detailed analysis of this definition, in particular, the choice of spaces of test functions, can be found in [128]. Note that the analog of (7.1) holds for Gaussian measures on R, and in an infinite-dimensional case definitin (7.2) turns into an equality:
f (P(x)vB(dx) =
fe_'2(de),
where vB is a Gaussian measure with a covariant operator B. As was pointed out in Chap. II, Gaussian integrals even over a finitedimensional superspace are integrals with respect to generalized func-
tions and not with respect to measures. Therefore, in a supercase, it is natural to use the Parseval equality for defining both Feynman integrals and Gaussian integrals.
7.1.
Quasi-Gaussian, Gaussian, and Feynman distribu-
tions. Definition 7.1. A quasi-Gaussian distribution on a superspace W is the distribution rya,B E M (W) with the Fourier transform Jr'(7a,B) (v) = exp{- 2 B(v, v) + i(v, a) },
(7.3)
where (mean value) a E W, B (the left-hand covariance functional) belongs to the space K2,1(L2 , A), the restriction of B to the superspace V2 is symmetric. By virtue of Parseval equality (6.2), we have fw
(P(w)'Ya,B(dw) = fv (p(dv) exp{-2B(v, v) + i(v, a)}.
Chapter III. Distribution Theory
168
If there exist projectors b of the body and (c = 1 - b) of the soul in the CSA A, then we can define Feynman and Gaussian distributions on a superspace. Definition 7.2. A Gaussian distribution is a distribution 7a,B for which Re b(B) > 0, Im b(B) = 0.
Definition 7.3. A Feynman distribution is a distribution rya,B for which Re b(B) = 0.
7.2. Feynman distributions in the quantum field theory. Countably additive Gaussian measures on R-linear spaces are Gaussian distributions. These distributions can be extended from a space 4> of test functions to a space of continuous exponentially growing functions.
Feynman "measures" for boson systems (see [7, 26, 50, 53]) are Feynman distributions. Of especial interest are the following examples of Feynman distributions with fermion degrees of freedom.
Example 7.1. (Feynman distributions for a spinor field). We set V = q(R4, A8) and W = 9'(R4, A8). It is convenient to write the elements of the superspaces V and W as v = (r)j, 77j)'-o and w _ ('0j, ,0j);.0. Then the form of duality 3
(v, w) = = j=0
,/,
,/1
(x) j (x) + ; (x) l'j (x)) dx.
R
14
We denote by ryS a Feynman distribution on the superspace W with zero mean and covariance functional
BS(v, v) = -2i I l(x)Sc(x
- y)77(y) d4x d4y,
where
S`(x - y) =
1
(27r)4
+m I P2 -Pm2 + io
ei(P,x-y)d4P
is Green's causal function of the Dirac equation
(i8 - m)Sc(x) = 6(x)
7. Gaussian and Feynman Distributions
169
3
3
µ=0
µ=0
(we use the notation j3 = E yµpµ, a = E 'y-, where -yµ are Dirac µ
matrices). The Feynman integral f U(,0, z,)'ys(d dzb) is defined for the func-
tions U from the class 4>(W). This integral is a strict mathematical definition of a Feynman path integral for a spinor field [7, 50, 53]:
fU(b,ii)exp{ifR
(x)(za - m) V) (x) d4x} II dV) xER4
(in this symbolical notation the determinant of the operator (ia - m) is included in the normalization of the functional differential
II d'(x)di/i(x)) xER4
Example 7.2 (Feynman distribution for a chiral superfield [123]). In the superspace Ru4 we consider a SUSY transformation (cf. Chap. I (1.8)):
9'=0,+EQ.
xµ=xµ+2EyµO,
The generators of the SUSY transformation a i3 S.=89°+2
3
a
(ryµ9)ax
µ=0
µ
where e = C-10, C is a charge conjugation matrix, satisfy the commutation relations IS., Sa} = i ('y C).a aiµ . The operators of the µ=0
covariant differentiation on the superspace D°
3
a
a
E (fµe)° ax = ae - 2 µ=0 µ
are invariant relative to the SUSY transformation; they satisfy the commutation relations 3
[D., Da}
-i E (1'µC).0 8 µ=0
µ
Chapter III. Distribution Theory
170
We !introduce spaces of chiral scalar neutral superfields A): (1 + 75)1)(p± (x, 9) = Q.
9(R4,4
Gt(RA4, A) = {cPf E
,
We set
V=
9+(R4,4
,
w=
A) ®G_
(R4,4
,
A),
A) ®G'
A).
The form of duality between the superspaces V and W is defined by the relation
(v, w) = f (- DD) 2
(v+w+ + v-w_) d4x.
We denote by 'yss the Feynman distribution on the superspace W which has zero mean and covariance functional
f[1(v)2(v + a 2 + M2 + io v-) 1
Bss (v, v)
-(DD) V+
-1/2M -1/2M d4x' + v_ v-)] v a2+M2+io a2+M2+io +
M>0,
a=(9XA a )3 µ=o
For any function Z E O(W), the Feynman integral f Z(w+, w-)'yss(dw+dw-) is defined which is a strict mathematical definition of a Feynman path integral for a chiral neutral superfield [123]:
f Z(w+, w-) exp{i f[D)2(w+w)
-1(DD) (DD) (2M(w+ +)]}
II
dw+(x, e)dw-(x, O).
(z,O)ER '4
Example 7.3 (interaction of a boson and a spinor field). We set V = G(R4, RA8), W = G'(R4, RA8) and denote by rySB the Feynman
7. Gaussian and Feynman Distributions
171
distribution on the superspace W with a covariance functional BSB = Bs + BB, where BS is a covariance functional for a spinor field,
BB(V, V) = i fR8 cp(x)D`(x - y)co(y) d4xd4y
is a covariance functional for a scalar boson field, where D`(x) is Green's causal function for the Klein-Gordon equation D`(p)=p2_m2+io,
f u((P,.,, ) x exp{i fR4
f `
fi(x)
x 11
(
2
2m
)W(x)
+ (x)(ie -
f u((P,'),0)'YSB(dcpdV)
xER4
U E -1)(W).
Remark 7.1. It should be emphasized that with our approach to the superanalysis, a boson field assumes values not in the field of real numbers R but in the even part of the algebra A. The simplest example of the Gaussian distribution on a superspace is a distribution on R`n,,m (see Chap. II). Its infinite-dimensional generalization is given by the following example.
Example 7.4 (Gaussian distribution on a Hilbert superspace). Suppose that M and N are Hilbert CSM, V = Mo ® Ni , W = V. We denote by -y the Gaussian distribution on the superspace W with a covariance functional B(y ® ®t, y ®C ®) _ (y, y) + 2(e, e). Then, by definition, we have
f U(x,9,6)exp{-2(x,x) - (9, 6)}dxdOd9 f U(x, B, 6)'y(dx dO d6).
Chapter III. Distribution Theory
172
Let, for instance, V = 12(A0) ®12(A1) ®12(A1). Then 00
ry(dxdOdB)
=exp{-1 Exj 29=1
00
E0363} 11 00 dxjdOjd#j. 7=1
9=1
If V = L2 ° (Rn, dx) ® L21(Rn, dx) ® L21 (R n, dx), then
ry(dcp dib dpi)
= eXp{
2
1R^ 1P 2(x)dx - fR' i(x)0(x)dx}
x II dcp(x) dz/'(x) di'(x). xER^
7.3. Formulas for integration by parts and for an infinitesimal variation of a covariation. In the theory of Gaussian measures on infinite-dimensional spaces an important part is played by formulas for integration by parts (an extended stochastic integral [26, 6], Malliavin calculus [113, 6]). The simplest formula for integration by parts for the Gaussian measure 'YO,B with a covariance operator B is f (cp'(x), Ba)ryo,B(dx) = f (p(x)(a, x)'YO,B(dx).
(7.4)
For Feynman integrals, formula (7.4) was obtained in [53, 126, 145].
The formula for integration by parts on a superspace differs essentially from formula (7.4). When integrating by parts, in a quasiGaussian integral another integral appears which is not quasi-Gaussian in addition to the standard quasi-Gaussian term (see [151]). Let T be an associative Banach algebra with identity e. For t, s E T we set
exp{t; s} = e +
tksn-k 1)i
n=1 (n + k=0 This function is two-parameter generalization of the exponent:
exp{t; t} = exp{t}.
7. Gaussian and Feynman Distributions
173
For the forms c, b E 1C2,1(L2 , A), we denote by x(c; b) the distribu-
tion belonging to the space M (W) and having a Fourier transform .F'(ic(c; b)) (v) = exp{-2c(v, v); -2b(v, v)}.
For the bilinear form b E 1C2,1(L2 , A), whose restriction to the superspace V2 is symmetric, we introduce a bilinear form b_ (v, v') = b° (m e
n, m'en') -bl(men,m'en'), where v = m®n, v' = m'® n' E Lv. For the restriction of the bilinear form b_ to the superspace V2 we have a relation b_ (v, v) = a(b(v, v)),
v c V.
For instance, suppose that we are given a superspace CA,m and a form
m
m n
b(v, v') = i,j=1
mimjaij + E Dminj + njmi)Qij i=1 i=1 n
+ I (ninj - njni)7ij.
(7.5)
i,j=1
Then the form b_ has coefficients a(aij), a(Qij), a('Yij) We also introduce a diagonal and an antidiagonal form a+(b)(v, v') = b(m, m') + b(n, n'),
a-(b)(v, v') = b(m, n') + b(n, m'). For the bilinear form (7.5) we have a+(b) (v, v')
m
n
= > mimjaij + > (ninj - njni)'Yij; ij=1
i,j=1
m rn
a (b) (v, v') =
njmi) i=1 j=1
In what follows, we shall use the notation ryb to denote a quasiGaussian distribution with zero mean and a covariance functional b and the notation Kb to denote the distribution ic(b_, b).
Chapter III. Distribution Theory
174
Theorem 7.1 (formula for integration by parts). Let the function cp belong to the class 1(W) and let the vector a E Lv. Then f co(y) (a, y)'yb(dy)
f [a+(b°)(a, aR) + a (b1)(a, aR)] (co) (y)-yb(dy)
+ f [a (b°)(a, aR) + a+(b1)(a,
(7.6)
Proof. Note, in the first place, that for arbitrary m E M and n c N the functional p = Ann: A(V) -+ A, P(f) = i
8x°
(0) (m) + 8x1 (0) (n)
is continuous.
Furthermore, (a, y) = F(p)(y), a = m ® n. Thus, we have fw co(y) (a, y)'yb(dy)
- fv
fv
= fc*p(dv)exp{_b(v,v)}
p(dvl) exp{
2b(v + vi, v + vi)}
We set gv(vl) = [b(v + v1i v + v1)]n, and then, for m° E M°, we have 8x°v (0)(m°) =
> 2bk(v, v)b(m°, k=0
v)bn-k-1(v,
v);
l
similarly, for n1 E N1 we have aLgv
8x1
n-1
(0)(n1) = E 2bk(v, v)b(nl,
v)bn-k-1(v, v).
k=0
Furthermore, b(m°, vo ® v1) = A° ®A1 E A° ® A1, where .1° _ b°(m°, v°) + bl(m°, v'), Al = b°(m°, v') + bl(m°, v°); similarly, b(v° ®
7. Gaussian and Feynman Distributions
175
v1, v° ® v1) = a° ®a1 E Ao ® A1i where a° = b°(v°, 0°) + b°(v1, v1) + 2b1(v°, v1) and a' = bl(v°, v°) + bl(vl, v1) + 2b°(v°, v'). Therefore we
have b(v, v)b(m°, v) = (a° ® a')(\° ® A1) = )°b(v, v) + Alb- (v, v). Similarly, for the vector n1 E N, we have b(v, v)b(n', v) = µ°b(v, v) + p'b_(v, v), where µ° = b°(n1, v1) + bl(n1, v°) E Ao, µ' = bl(nl, v1) + b°(n', v°) E A,. Employing these formulas, we obtain q9a 8x°v (0) (mo) = 2n\°[b(v, V)] n-1 n-1
+2)1 >
[b_
(v, v)]k [b(v,
v)]n-k-1 ,
k=0 19
2nµ°[b(v, 8 v (O)(n1) = n-1 +2µl E [b_ (v, v)]k[b(v,
v)]n-1
v)]n-k-1
k=0
Let us calculate the integral with respect to the distribution P= Pmonl E A'(V):
f p(dv,)exp{-2b(v+vl,v+vl)} 1 2
2n 1)n! (n[b(v,
v)]n-1(A° + µ0)
n--1
+(A' + µ1) L. [b-(v, v)]k[b(v,
v)]n-k-1)
k=0
= i(A° + µ°) exp{-2b(v, v)}
+01 + µl) exp{-2b_(v, v); -2b(v, v)} Finally, for the vector a = m° ® n' E V we have
f
w (y) (a, y) -yb(dy)
Chapter III. Distribution Theory
176
= if cp(dv)[b°(,m°, v°) + b'(rn°, v') + b°(nl, vl) +b1(nl,v°)]eXp{-Zb(v,v)}+if cp(dv)[b°(m°, v1) + bl (m°, v°) + bl (n', v') + b° (nl, v°)] x
exp{-2b-(v, v); -2b(v, v)}.
Extending relation (7.6) in the left A-linearity from the superspace V to the covering CSM Lv, we get formula (7.6) for the vectors a c Lv.
Corollary 7.1 (cf. (7.4)). Suppose that the function cp belongs to the class '(W), the vector a E Lv, and the restriction of the covariance functional b to the superspace V2 is even-valued. Then f cp(y) (a, y)'yb(dy)
= f b(a, aR) (cP) (y)'yb(dy)
Example 7.5. Consider the Gaussian distribution on the superspace Ai: 'yb(d9) = exp{-01027 - 0304/3} dO 2p 1,
where ry = 'yo +'y', /3 = 00 + 01, /3,, -y, E Aj, j = 0,1; there exist /30 1, 7o 1; p = 'y/3 + /3ry. Then (see Example 3.5 Chap. I)
('Yb)( ) =
f
exp{-66A -
e)},
where A = 2/3p-1, B = 2ryp-1. Consequently,
exp{-2b-(e,
_b(,)}
= e - 4 [b- (e, e) + b(e, e)] + +b_ (e,
4!
[b2
b(e, ) + b2 (e, )] = e - [e11;2Ao + 3&4Bo]
7. Gaussian and Feynman Distributions
+1
177
r(B)Q(A) + a(B)A + cr(A)B
+AB + BA] = e - 66A0 - U4B0 + exp{-S1C2Ao
- U4Bo}.
Thus, in the example that we are considering 'b is a Gaussian distribution kb(d9) = exp{-9192Ao 1 - 0304Bo 1}d9AoBo.
Note that
A=(7o1-010o1'Y17o2)-y1'Yo2=Ao+A1, B=(001-1'1'Yo1Q1Qo2)-01002=Bo+B1, AO 1
= ('Yo
1
- 0100
1'Y1'Yo 2)
1 = 'Yo + Q1 Q0 17'1,
Bo 1 = (00 1 - 'Y1'Yo 1Q10o 2) 1 = 00 +'Y1'Ya 101
Let us calculate the integrals I1 = Jb0(,aR)()(o)yb(do); 12
_ I1 =
f
= f b1(C,a.)((P)(0)r1b(de),
_ ae2) a ae 2)
aR
We1
+ f Lf aRae4 A3) _
09Rae3 4)
AO exp{-0102ry - 93940}d9 2p Bo exp{-9192ry - 93940}d9 2p-1
= Ill + 112.
Using the formula for integration by parts in the integral with respect to anticommuting variables (Chap. II, formula (3.27)), we obtain 111 = f (o(9) [e1Ao aL a91
19L
exp{-0102'y - 9394Q}
a02
exp{-0102-Y - 93940}] d9 2p-1
Chapter III. Distribution Theory
178
= fw(O)[eioi +e292]Ao('Y -
0304P)d02p-1.
By a complete analogy, we get 112 = JP(0)[e303 + e404]Bo (N - 0102p/2)de 2p
1.
In the same way we can calculate the integral 12: 112 = f (P(e)[e181 + e2e2]A1(Ao 1 - 0304Ao 1Bo 1) dOAoBo,
122 = fw(9)[3O3 + e484]B1(BO 1 - 0102Ao 1BO 1) dO AoBo.
Consequently, Il + 12 = f W(e)[e1e1 + e202]
X [(Ao7 +
A1Bop)
+ f cP(0)[6 03 + G O4] [(Bo1 +
2p-1
- 9304(Ao + A1) 2] dO B12 Ao
- 0102(Bo + B1) 2]
d92p-1
Furthermore, Ao'y + Al Bop 2
-11"Yo 2/30
= ('Yo 1
- 010o 1'Yl'Yo 2) ('Yi +
1(1 _ 'Yl'Yo 1Ql/3o 1)(/3o'Yo + 01'Yo +'Y1Qo) = 1; AP 2
Similarly, Bo/3 + B12 ° = 1,
=
2QP-1P = Q. 2
= y. Thus we have
I = f(9)[ei9i + e202](1 - 00384)dO +
YO)
f
W(9)[6O3+5484](1-'YB182)dO
2p-1
2p-1
f (P(e) [ 01 + e282] (1 - 01027 - 0304Q + 101e2B3e4P) dO
2p-1
7. Gaussian and Feynman Distributions
179
+ f cP(9)[683 +e404](1 - 0102'Y - 0304/ + 1 01 02
0304P)d02p-1
= fco(0)(, 0)yb(d9).
It is easy to give an example showing that for the distribution 'yb(dO) considered above the standard formula for integration by parts (7.7) is not valid.
Example 7.6. Let the function cp(0) = 029394. Then
f w (0) (e, 0)yb(d9) = f b(e, aR) ((P) (0)yb(d9)
f ei01 02 93 04 7b(d9)
= 2j1P-1i
= f e1A91029394y d9 2P 1 =
(7.8) (7.9)
If (7.8) = (7.9) for all 1 E A1i then Ay = 1 since the A1-annihilator
is trivial. Let y = 1 + yl, Q = 1 +,31. Then Ay = 1 - 01y1 j4 1 if Q1-y1 j4 0.
For countably additive Gaussian measures on R-linear spaces, the following formula of infinitesimal variation of covariance is valid (see, e.g., [24]): dt f co(y)'Yb(t) (dy) =
2
Jb(t)(a,o)(2)(Y)-Yb()(dY),
(7.10)
where yb(t) is a family of Gaussian measures with covariance function-
als b(t), t is a parameter, b(t) - dtt Just as the formula for integration by parts, formula (7.10) for an infinitesimal variation of covariance cannot be directly extended to a supercase. The following theorem can be proved by analogy with Theorem 7.1 (see [151]).
Theorem 7.2 (infinitesimal variation of a covariance). Suppose that the superspaces V and W are Banach and b: AO -+ £2,1(L,, A) is a continuously S-diferentiable function. Then, for any function cp from the class '(W), we have a fcQ(Y)7b(t)(dY) =
2
[f (bo (t) (acx, a.0)
180
Chapter III. Distribution Theory +b° (t) (a', aR) + 2b' (t) (aR, aR))co(y)7b(t) (dy)
+ f(bl (t) (aR, a°R) + bl (a', aR) + 2b' (t) (aRl, aR))'(y)nb(t) (dy)]
.
Corollary 7.2. Suppose that the conditions of Theorem 5.2 are satisfied and the restriction of the covariance functionals b(t) to the superspace V2 is even-valued. Then formula (7.10) holds true.
8.
Unsolved Problems and Possible Generalizations
In this chapter we outlined the main directions of development of analysis on a superspace over CSM. We hope that this analysis will be successfully developed. In general, an infinite-dimensional analy-
sis has much in future. We think that with the aid of an infinitedimensional analysis and, in particular, the theory of distributions on infinite-dimensional spaces, we shall be able to expose, on the mathematical level of strictness, the quantum theory of a field and a string outside of the framework of the perturbation theory; probably, some other infinite-dimensional objects will appear in physics. We observe a standard situation where, along with infinite-dimensional bosons commuting coordinates (boson fields, strings, string fields, membranes) there are also infinite-dimensional fermion coordinates. The infinitedimensional superspace X = M°®Nl over a pair of CSM M = M°®M1 and N = No ® Nl arises in practically all quantum models. The ordinary infinite-dimensional analysis developed during the last hundred years, beginning from the works by Volterra, Frechet, Danielle, Wiener, Levy, Gateaux, Hadamard (see [22]) and following to the works by Gross, Fomin, Smolyanov, Berezanskii, Daletskii, Hida, Uglanov, Khrennikov, Shavgulidze, Bogachev [2, 6, 26, 54-55, 62, 64, 65-68, 96, 127-130, 133, 134]. It is natural that in the framework of this book we cannot propose as well developed infinite-dimensional superanalysis, the more so that a considerable part of the book is devoted to finite-dimensional super-
8. Unsolved Problems
181
analysis. Therefore a wide range of problems remain unsolved (many problems may give rise to whole theories). Topological supermodules. 1. Theorems of the type of Hahn-Banach and Krein-Mil'man theorems.
2. Topologies on conjugate CSM. 3. Superanalog of Mackey topology. 4. Reflexivity theory for locally convex CSM. 5. Weak topology Q(M, M'). Weak compactness. 6. Topological properties of spaces of test and generalized functions on a superspace.
7. Unbounded operators in Hilbert CSM: self-adjoint operators, unitary groups, Stone theorem. 8. Theory of semigroups of operators in Banach and locally convex CSM.
9. Operators of trace class and Hilbert-Schmidt operators in Hilbert CSM. 10.
Nuclear locally convex CSM. Superanalog of Grothendieck
theory. S-differential calculus. 1. Successive exposition of differential calculus on topological and pseudotopological superspaces.
Distribution theory. 1. Theories of nonanalytical superdistributions. 2. Existence theorem of a fundamental solution for a linear differential operator with constant coefficients on an infinite-dimensional superspace. 3. Cauchy problem for linear differential equations with variable coefficients on an infinite-dimensional superspace. Quasi-Gaussian distributions. 1. Extension of the class of integrable functions of an infinitedimensional superargument. 2. Formulas for integration by parts and an extended stochastic integral.
182
Chapter III. Distribution Theory
Remarks Sec. 2. These results were published in [65, 166]. Sec. 3. These results were published in [166]. Hilbert modules over C`-algebras were introduced by Paschke [116]; in connection with the applications to the theory of pseudodifferential operators they were studied by Mishchenko [48]. The main differences between the theory of C'-modules
and the theory of supermodules are generated by the differences in the properties of Banach algebras over which these modules are considered. All proofs of the theory of C'-modules are based on positive linear functionals, in superanalysis these methods are inapplicable. Sec. 5. A superspace over a pair of CSM was introduced in article [144]. The S-differential calculus on these superspaces was developed in [65, 68, 148].
Sec. 6. Here wide use was made of the methods of infinite-dimensional analysis. Actually, the results of the works [136-141, 145, 67] were extended
to the supercase. In turn, these works were based on the investigations of Fomin, Smolyanov, and Uglanov concerning the theory of distributions on infinite-dimensional spaces.
Sec. 7. Uglanov was the first to define Feynman's "measure" as a distribution on an infinite-dimensional space. I have done this for a supercase.
Chapter IV
Pseudodifferential Operators in Superanalysis In this chapter we expose the theory of PDO on a superspace over topological CSM. These superspaces can have a finite as well as infinite number of supercoordinates. Thus, the proposed PDO calculus serves as a mathematical basis for the quantization of physical supersystems with a finite as well as infinite number degrees of freedom. In a finite-
dimensional case, we obtain quantum mechanics on the superspace and in an infinite-dimensional case we obtain a quantum theory of a superfield, in particular, that of a superstring and superstring field, and fermion theories and boson theories with anticommuting Faddeev-Popov ghosts. Only the first steps have been made in the PDO theory on a superspace. In [65, 68, 153] I constructed a PDO calculus (composition formulas,...), proved the correspondence principle, investigated evolutionary pseudodifferential equations. No considerable results concerning a superspace (even for a finite-dimensional one) are available in many important branches of the PDO theory (such as, for instance, a parametrix, spectral properties).
1.
Pseudo differential Operators Calculus Let us begin with considering PDO on a space R". Recall (see,
e.g., [73, p. 178]) that a PDO a in a space of functions on R' with
Chapter IV. Pseudodifferential Operators
184
the T-symbol a(q,p) is an integral operator
a*')(q) = f a((1 - T)q
+Tq',p)(P(q')e`(q-q',P')
(2)
For T = 0, 1, 1/2 we obtain qp, pq, and Weyl symbol respectively. Let us now consider the case of an infinite-dimensional K-linear space (Hilbert, Banach, locally convex). In an infinite-dimensional
case, the Lebesgue measure dq dp is absent, and therefore PDO are introduced either as limits of finite-dimensional PDO [87] or by proceeding from polynomial operators [3], or, else, with the use of the distribution theory on infinite-dimensional spaces [74, 127, 129, 136, 139, 141] (on the mathematical level of strictness, the first variant of PDO calculus in the framework of the distribution theory on infinitedimensional spaces was proposed in [127]). In my works [139, 141] I introduced a Feynman integral on a phase space and defined an infinite-dimensional PDO by the relation
a(W)(q) = f a((1 - T)q +Tq',P)W
(q')e'(q-q'P'1dq'dp',
(1.2)
where the symbol ei(q-q'P')dq'dp' was used to denote Feynman distribution on a phase space (the same formulas as (1.1), but the normalizing factor 1/(27r)°° was "driven in" the Feynman distribution).
I have constructed spaces of functions of an infinite-dimensional argument which possess a remarkable property, namely, every formula of the PDO theory in R" is also valid for infinite-dimensional
PDO in these spaces with a replacement of the complex measure e`(9-q',P'-P)dq'dp',
a E C, by a Feynman distribution on a phase space. In [144, 146] I introduced a Feynman integral over a phase super-
space and defined the PDO a with the T-symbol T E Ao, a(q,p) E O(Q X P) by relation (1.2). This definition of the PDO is used in the sequel.
Let P and Q be dual superspaces satisfying the constraints imposed
in the process of construction of the distribution theory (see Sec. 4, Chap. III). The superspace Q x P is known as a phase superspace. The superspaces Q x P and P x Q are dual.
1. Pseudodifferential Operators Calculus
185
We set b.(p®q,C(D 77) = 2Z [(p°,77°)+(q°,C°)]
aEAo,p,eEP,q,r7EQ. The form b,, is A-linear both on the right and on the left, continuous on compact sets, and symmetric. We shall denote by i
f co(q',p) exp {--(q' - q, p' - p)} dq'dp an integral with respect to Feynman distribution on a phase superspace with mean a = q ®p and covariance functional (-2b,,). It is this symbol that is used in definition (1.2) of PDO. Theorem 1.1. Every PDO a with a 7--symbol a E (D(Q x P) is a right A-linear operator in the CSM (D(Q).
Proof. For every q E Q, we introduce a A-linear continuous operator Sq: A(P x Q) -- A(P2 x Q) by setting Sq(f)(P1,P2, q') =
e'(1-'r) (Pi,q)f
(7-pi + p2, q').
We can present the integrand Vq(q', p') = a(7-q' + (1 - 7-)q, p')cp(q') in (1.2) (q plays the part of a parameter) in the form z/)q(q', p') = T ((d 0 () o Sq)(q', p'). Consequently, this function belongs to the space 4)(Q x P). Thus, the operator a is defined on the whole space
(Q)
-
We shall show now that a(,p) E -ID(Q) for any cp E O(Q). Using the
Parseval equality (5.3) from Chap. III, we obtain
a(co) (q) = I a 0
cp(dpdq'dp")
x exp{iT(p', q') + i(p', q) + i(p", q' + q)}.
(1.3)
We introduce a A-linear continuous operator S: A(P) -+ A(P2 X Q) setting S(f) (p', q', p") = eir(P',q')+i(P",q) f(p' + p"). Then we can represent the function &(W) (q) as
a(W)(q) = F((a 0 ) o S)(q) E (D (Q)
Chapter IV. Pseudodifferential Operators
186
Theorem 1.2. Let a be a PDO with a 7--symbol a E -1>(Q x P). Then we have a representation dgi)e'(Pl,a+'rq1 )W(q
f a(dpi
+
qi).
(1.4)
Proof. Using relation (1.3), we obtain et ((p) (q)
(P(dp2)e''2,q+qi))
= f a(dpi dgi) (f xe=(pl,q)+iT(pi,gi) = (1.4).
Let {ej, ai}jEJ,iEI and {e'j, a=}jEJ,iEI be dual bases in the superspaces P = Po ® Pl and Q = Qo ® Q1 consisting of even elements
p = Ep°ej +>piai, jEJ
q=
jEJ
iEI
q°ej +Eq'ai iEI
We introduce the (left) operators of the coordinate and momentum corresponding to the resolution with respect to the bases P°, 4°, j E J; jii, 4ii, i E I. These operators satisfy the canonical commutation relations on a superspace (which coincides with (1.7), Chap. I in a finite-dimensional case).
Theorem 1.3. Suppose that a is a PDO with a r-symbol a E c(Q x P) and the superspaces P and Q have even dual bases. Then, for any function cp E 1(Q) we have relations a(cp) (q) = fa(duciv)
x exp{i >(Tu°v° + u°4°) + i (TUkv1 + uk4Lk) } jEJ
kEI
x exp{i > v°p° + i jEJ
vk7Lk}w(q); kEI
a(cp)(q) = fa(dudv)
(1.5)
1. Pseudodifferential Operators Calculus
187
x exp{i E((7- - 1)v°u° + vjp°) + i E((T - 1)ukvk + vkpLk) } jEJ
kEl
x exp{i E(u°4° + i E(ukgLk}W (q); jEJ
(1.6)
kEI
&(W) (q) = fa(dudv)
x exp{i E((T - 1/2)u°vjo + v°pj + u°4jo) jEJ
+i E((T - 1/2)2Gkvk + ukgLk +
vk11
PLk)}co(q)
(1.7)
kEI
Theorem 1.3 is a direct corollary of representation (1.4). Formula (1.5) is considerably simpler for the qp-symbol, formula (1.6) is simpler for the pq-symbol, and formula (1.7) is simpler for Weyl's symbol.
Example 1.1. Let P = Q = Al, and then 4i(cp)(q) = qw(q), pL(c')(q) = -i5LCo(q). Let the function a(q,p) = qp, and then a = 4p for the qp-symbol, a = -p4 for the pq-symbol, and a = z (4p - p4) for Weyl's symbol (i.e., Weyl's symbol with respect to anticommuting variables leads to antisymmetrization). Theorem 1.4 (on the relationship between the symbols for PDO). Let at and a, be, respectively, t- and s-symbols of the class -D(Q x P) of PDO a and let t, s E Ao. Then
at(P,q)= f a,(q',p)eXp{t
(q'-q,P -P)}dq'dp.
Proof. Using formula (1.4), we obtain h(cp)(q) = f a,(dp'dq') exp{i(s - t) (p', q')} x exp{i(P', q) + it(p', q')}W(q + q'), i.e., at (q, p)
= fas(d7idq')
(1.8)
Chapter IV. Pseudodifferential Operators
188
x exp{i(s - t)(p', q') + i(p', q) + i(q', p)} _ (1.8).
Theorem 1.5 (composition formula). If a, al, a2 are PDO with T-symbols (T E Ao E {0,1}), a, al, a2 E (D(Q x P), and a = al o a2, then a(q,p) = fai(q',p')a2(qh',p") x exp{
1
T
(q - q", p - p) + T (q - q', p - p') }dq'dp"dq"dp'. (1.9)
Proof. Using formula (1.4), we obtain et (W) (q)
=
f a1 ®a2 (dp dq'dp"dq")
x exp{i(p',q+Tq') +i(p",q+q'+Tq'")}cp(q+q'+q") =
f(ai ® a2) o B(dpdq')e=(P'q+Tq')W(q + q'),
where B is a A-linear continuous operator defined by the relation
B(f)(p,q,p",q") = exp{i(1 -r) (p",q) xf(p'+p',q'+q"),
- iT(p',q")}
B:A(PxQ)-+A(P2xQ2).
Using formula (1.4) once again, we have a(q,p) = f (al ® a2) o B(dpdq')e'(P',q)+i(q,P)
= f al 0
a2(dp'dq'dp"dq")
x exp{i(1 - T)(p", q') - iT(p , q") + i(q' + q",p) + i(p' + p", q)} _ (1.9).
Passing to the limit in relation (1.9) as T -3 0 (T
1), we obtain
composition formulas for the qp (pq) symbols:
a(q, p) = f ai (q, p)a2(q',
p)e-'(q'-q,P'-P)dq
dp,
(1.10)
1. Pseudodifferential Operators Calculus
189 p)e'(q'-q,P'-P)d9
a(q,p) = f a 1 (q', p) a2 (q,
(1.11)
dp'.
Theorem 1.6. Let a be a PDO with a qp-symbol a E 4)(Q x P). Then, for any function cp E 4)(Q) we have (dp).
a(W)(q) = f a(q,
(1.12)
Proof. We set bq(p) = a(q, p) (where q plays the part of a parameter). Then a(W)(q) = f bq(P)ip(9
)_i('
f bq(P)(f ei(P",q')c(dp"))e-i(q'-q,P')dgdp
ff
=
bq(p')e-i(q'-q,P')+i(p",q')dq'dp'(
(dp")
(we have used the supercommutativity of the operation of direct multiplication of distributions and the fact that a Feynman distribution on a phase superspace is even). Consequently, et (W) (q) = f (f bq ®(SPii (dq'dp') x e'(P'q')+i(P',q)) gdP 1)
= f (f bq(dq')e'(P",q'))ei(P",q)( (dp")
=
f (f
bq(d4)e'(q',0r(P")))ei(P",q)c(dp
)
From methodological point of view, it is useful to consider formula (1.12) by way of a simple example.
Example 1.2. Let P = Q = A1i a(p) = p, W(q) = aq, a E A. Then a(cp) (q) = a(a). Recall that every functional u E A' is associated with two functionals ur = Ir (u) E Gl,r (A, A) and ui = Ii (u) E G1,j(A, A) and that ui = I (ur) = u° (Dur o a. For the functional gyp, the right A-linear realization has the form cpr = 228W Indeed, .
esPl)
=- (a Z
app)' eiP9)
= a((S(p), aa g) _ W(q)
Chapter IV. Pseudodifferential Operators
190
Note that the generalized function
al OR6(p)
0
Pr Therefore
2
ap
i
I
cpt = (APT) =
app
is odd. Consequently,
ao M(P)
1
cp, = i
,
ap
al ORb(P) ® ao 19R6(P) o Q.
ap
i
ap
i
Thus we have
f 1) =
al i
aRJ(P) C
ap
f) +
ao 'M(P) (
ap
a0 d (o) + iaO ap
,
a(f))
(o)
i ap
Using the formula for the transformation from the left-hand derivatives to the right-hand ones (Chap. I, formula (1.6)), we have
(f7 (A) = - (al
a ap(f)
ao
(0) +
a f (0)) _ -
(o) te(a)
Consequently, aLb(P) cr(a) ap
i
,
in this case,
W(4) = -(&
a (P) o (a) ). ,
P
Furthermore, f a(o(p))e'P9c (dp) =
(pe`P4,
a a(P) a(a)) _ _(a) P
Theorem 1.7. Let the function a E A(Q x P). Then the PDO a defined by relation (1.12) maps the space -(D(Q) into A(Q).
Proof. Consider an arbitrary function b E A(Q x P) and a p c A'(P). Let us estimate the norm of the function
functional
1. Pseudodifferential Operators Calculus
191
g(q) = f b(q, p)p(dp) in the space A(Q). For the arbitrary compact subset BQ in the CSM LQ we have 11
ro(0, p) (hl, ..., hn)P(dp)II
sup
I19I1BQ = 00
f
n=0 n. h, EBQ
q
1
00
00
GCPLn n=0
m=0 7n !
'
an+'nbb
x sup sup II f L agnapm hj EBQ u, EBp
0) (hl, ..., hn, ul, ..., um) I,
where Bp is a compact subset in the CSM Lp which exists by virtue of the continuity of the functional J. Using this estimate, we have k
00
I1911BQ
CP E k=0
x
sup V3 EBQxBp
IIaL9(0)(v1,
kl
lE Cn n=0
vk)II <- CPI
2(BQXBp)
By analogy with the PDO calculus on Rn (see, e.g., [5, 32, 44, 73, 74]), we can rewrite formula (1.8), which connects various types of symbols and the composition formulas (1.9), (1.10), (1.11), with the aid of infinite-order differential operators. We define a differential operator with constant coefficients as a PDO with a polynomial symbol (of class A(P); it should be emphasized that in an infinite-dimensional case there exist polynomials which are not compactly S-entire) which depend only on momenta:
R(-iaL)(co)(q) =
f
R(o,(p))e=(n,a)c(dp)
(1.13)
Proposition 1.1. Let the form b E 1Cn,,.(Lp A) and suppose that the restriction of b to the superspace Pn is symmetric. Then we have a relation b(-i8L)(co)(q)
Chapter IV. Pseudodifferential Operators
192 n
k=0
"n
b(e71, ..., ejk, ail, jl...jkEJ il...ikEI 1 0 1 xpjl0 ...pjkpLil ...pLin_k
(1.14)
Proof. It follows from formula (1.13) that n
b(-iaL)(co)(q) _
(-1)n-kCn
k=0
x f b((p0)k,
(p1)n-k)ei(p°,q°)+i(p1,g1)gdpodpl)
n
_ E(-1)n-kCC k =O
E b(e71, ..., elk, ail, .-., ain_k jl...jk il...in_k
x f p1
(1.14).
Proposition 1.2. Let a be a PDO with a symbol depending only on momenta, a E A(P). Then the operator ea: -1>(Q) --* -1>(Q) is defined which is a PDO with symbol e°().
The proof is a direct consequence of (1.12) for PDO with an Sentire symbol. In particular, an exponent is defined for any differential operator with constant coefficients.
Theorem 1.8. If at, a, E -1)(Q x P) are t- and s-symbols of the PDO a, then aL aL l j - s)Caq' ap)}a,(q, p). at(q,p) = exp{i(i
(1.15)
Proof. Using formula (1.8), we have at (q, p) = f a,(dp dq') exp{i(p', q) + i(q', p) + i(s - t) (p', q')}.
1. Pseudodifferential Operators Calculus
193
We set X = P x Q, Y = Q x P, and then a,(dx)ei(x,Y)ei(s-c)b(x)
at(y) = f
where b(x) = (p', q'). Note that JbI = 0 and b(a(x)) = b(x). Hence, we obtain at(y) = f ei(,-t)b(a(x))ei(x,y)a,(dx) = (1.15). Theorem 1.9. Let a, a1, a2 E D(Q x P) be 7--symbols of the PDO a, a1i dz and a = al o a2. Then aaL
aL
a(q,p) = exp1zr( g1, apt) + z(T - 1)
aL 41L
aqz apl (1.16)
xal(g1,p1)a2(g2,p2)I91=q2=q P1=P2=P
Proof. Using formulas (1.9)-(1.11), we obtain
a(q, p) = f a1
®a2(dp'dq'dp"dq")e(1-7-)i(p q')-Ti(P',q")
x ei(P',q)+i(P",q)+i(q',P)+i(q",n)
We set X = Pz X Qz, Y = Q2 X Pz, and then
a(q,p) = f a1 ®
a2(dx)e(1-T)ib1(x)-rib2(x)ei(x,v)
Y=(q,P,q,P)
where bi (x) = (p", q'), b2 (x) = (p', q"). Note that I b1 I _ I bz = 0 and bi(o(x)) = bi(x), bz(o(x)) = b2(x). Therefore
a(q,p) = f
e(1-T)ib1(a(x))-i-rb2(a(x))
x ei(x,+J)a1 0 az (dx) l y=(q,r,q,P) = (1.16).
Let the function cp belong to the class A(P). Then it follows from formula (1.14) that
_ P(-21L) _
n,aRW(0)((-iaL)n).
Chapter IV. Pseudodifferential Operators
194
Theorem 1.10. Let a, al, a2 E 'ID(Q x P) be 7--symbols of the PDO a, al, a2 and a = a1 o a2. Then
a(q,p) = al(q+iT
aL
api
,p+i(T - 1)
x a2 (q + ql, p + a (pl))
'9L
)
9ql (1.17)
P1=0.
91=0
Proof. Using formula (1.16), we obtain a(q,p) = f (al o'yr) ® a2(dpi dgldp2dg2) x et(P2,g1)+1(p1,g2)+i(P2,q)+i(g2,P)
where ryr is a A-linear continuous operator, 'yr(f) (pl, ql) = ei(P',q)+i(g1,P) f (-Tpl, (1 - T)ql),
-yr: A(P x Q)
A(P x Q).
Furthermore,
(a1 o'yr)(dpi dgl)a2(q+ql,p+a(p1))
a(q,p)
Let us calculate the Fourier transform of the distribution a1 o ryr
.P(a1 o'yr)(q',p) =
f al
07r(dpldgl)et(p1,q')+i(g1,P')
= al (q - Tq', p + (1 - T)p') In conclusion, we shall use formula (1.4) for a PDO with a symbol depending only on momenta. We set X = P x Q and Y = Q x P, and then a(q,p) = bgp(fgp) (0) _ (1.17), where bqp is a PDO with a al(q-Tq',p+(1-T)p'), fqp (x) = a2(q+q',p+a(p')), symbol bgp(x) = x = (p', q'). Let a be a PDO with a qp-symbol a(q, p). Then, by virtue of Theorem 1.3, we can symbolically write this operator as 2
2
1
1
0 1 0 1 a = a(40,43,P3,P31),
1- 1
4 = 4L,
1- 1
P = PL
1. Pseudodifferential Operators Calculus
195
Here we use Maslov's notation [44]. The digits over the operators denote the order in which the operators of the coordinate and momentum act. For instance, for the pq-symbol a(q,p) we have 1
2
1
2
0 0 -1) = a(Qj, 4j,pj,jj 1
The PDO calculus that we constructed above constitutes rules for dealing with ordered resolutions in terms of operators of (left) coordinates and (left) momenta satisfying the commutation relations pk }
= [qj
qk }
l0j , qk } _
= 0,
(1.18)
In many applications there appear resolutions of operators a in terms of the operators u and v that satisfy commutation relations different from (1.18). Let C: P -4 P be a fixed even A-linear operator which is diagonal in the basis {ej; ai}; Cej = c°ej, Cai = c;a;, Ick = 0. To every
operator C there corresponds its own PDO calculus with S-analytic symbols. The PDO ac can be determined from the relation (1.19)
ac(cp)(u) = f a(u,
where the symbol a(u, v) E A(QxP). The PDO ac can be symbolically written as 2
2
1
1
ac = a(u°, u;, v°, where the operators ucl ,
[v
,
vjcl
, a = 0, 1, satisfy the commutation relations
vk} _ [zt , uk} = 0,
[v uk } = -ihCk J.pbjk. ,
(1.20)
The calculus for PDO (1.19) can be constructed by a complete analogy with the PDO calculus with qp-sumbols. If C = I is an identity operator, then formula (1.19) coincides with formula (1.12) and the commutation relations (1.20) coincide with (1.18). The most important part in applications to the quantum field theory is played by Wick (normal) symbols. We can obtain these symbols
Chapter IV. Pseudodifferential Operators
196
if we choose the operator of multiplication by i in P as the operator C. In this case, the operators f i9 denoted by a, and the operators In the field theory, the operators a°' are the operators of vjc' by a,". creation of bosons, the operators a'* are the operators of creation of fermions, the operators a° are operators of annihilation of bosons, and aj are operators of annihilation of fermions. These operators satisfy the canonical commutation relations
[a
[aJ ak } = 0,
[ai , ak
ak.
} = hbapbjk.
,
(1.21)
The PDO f with the Wick symbol f (a', a) E A(Q x P) can be symbolically written as 2
2
1
1
Here the operators of annihilation are the first to act, and then their action is followed by the action of the creation operators. The relativistic PDO calculus also plays an important part in the quantum field theory. Let T and R be dual superspaces P = TD, Q = RD, p = (p")D o , q = (q")D o , and suppose that {gµ"} is Minkowski metric (-goo = 911 = = 9D-1D-1 = 1). Let the operator C act according to the rule Cp" = g""p", v = 0,..., D - 1. Then the PDO a defined by relation (1.19) can be symbolically written as 2
0 v,
2
1 (qk)",(pk)
1 (Pk)"),
a = a((qk
where the operators of the coordinate and momentum satisfy the relativistic commutation relations [(pj )", (Pk)µ} =
[(qj)", (qk)µ} = 0,
[(P7 Y, (qk )µ} = -2h6.#6jk9µ".
This formalism can be generalized in many directions, for instance, we can choose an operator C which is nondiagonalizable or odd. The generalization of the calculus of pq-symbols can be constructed by
197
2. The Correspondence Principle
analogy. Of special interest here are anti-Wick symbols. PDO with anti-Wick symbols can be symbolically written as 1
f=
1
f(a;.,a,
z
z
In contrast to Wick PDO, here the operators of the creation of bosons and fermions are the first to act, and their action is followed by that of the annihilation of bosons and fermions. The PDO calculus developed in this section is the calculus of operators which are right A-linear. In particular, for 7--symbols the operators can be resolved in terms of left operators of the supercoordinate and supermomentum. By analogy, we construct the calculus of PDO which are left A-linear. PDO with the T-symbol can be resolved in terms of the right operators of supercoordinate and supermomentum. Mixed calculi are also possible in which we use both left and right operators of supercoordinate and supermomentum. These PDO are also very important for applications and a successive exposition of the theory of left-right PDO would be of a considerable interest. We can
obtain another interesting generalization of the PDO calculus on a superspace by considering the T-symbols for the parameter T E A.
2.
The Correspondence Principle With an operator approach, the procedure of quantization of a
physical system which has boson and fermion degrees of freedom can
be realized as follows. The functions a(q,p) on a phase superspace (finite-dimensional or infinite-dimensional) are put into a correspondence with a PDO (see formula (1.2)): et (W) (q)
= f a((1
- 10q + rq', p)co(q') eXp{ h (q - q', p') }d9 dP
,
where h is a Planck constant. In this case, the operation of pointwise multiplication of functions becomes an operation of quantum multiplication * (see composition formula (1.16)): al (q, p) * ai (q, p)
Chapter IV. Pseudodifferential Operators
198
= exp{ihr( aL , aL) + 2h(, - 1)( aL , aL )} 9q2 api aqi ape x al (qi, pi )a2 (g2, p2) I q1=92=q D1=P2=P
[al, a2}. = al * a2 -
1
1
)IaiIIa2l a*2a1.
The principle of correspondence between classical and quantum
theories consists in the fact that in the approximation h = 0 the quantum theory turns into the classical theory: lim ai (g, p) * a2 (g, p) = ai (g, p) a2 (q, p),
o
(2.1)
=: {al, a2}(q,p),
u o h[ai, a2}.(q,p) where I. , .1 are Poisson brackets on a phase superspace.
Remark 2.1. The Poisson brackets on a superspace were introduced by Martin in his work [114].
We introduce Poisson brackets on the phase space Q x P setting aL agz
If, 9}
aL
api)
-
x f (qi, pi)9(g2, p2)
aL
aL
ape C age
)]
91=92=q
D1=p2=P
Using the formula that connects the right-hand and left-hand derivatives (Chap. I, formula (1.6)), we obtain 09f
{f,9}= [Ca p°'a4 )-\ 19g
af
ag
,apo)]
+[/aRf 9L9) + /aRf aL9 \ 9ql , apl \ '9P1 aq l ,
where Ip°l = jq' I = a, a = 0, 1.
2. The Correspondence Principle
199
Definition 2.1. The Lie superalgebra is a Z2-graded linear space G = Go ® G1 in which the bilinear operation [ , ] (called a commutator) is defined, and the identities I[x,y]I = 1x( + IyI; (_1)lxl lyl+1[y,
[x, y] = [x, [y,
z]](-1)1x1IZI + [z,
x];
[x, y]](-1)IZI IYI + [y, [z, x]](-1)IYI I=I = 0
are valid for homogeneous elements. The last identity is a superanalog of Jacobi identity (the second and
the third term in it result from a cyclic permutation of the elements x, y, z in the first term).
Definition 2.2. The Lie superalgebra G = Go ® Gl which is a CSM over a CSA A and in which the commutator is A-bilinear, [)xa, y/3] = \[x, ay]Q,
A, a, Q E A,
x,yEG,
is a Lie superalgebra over the CSA A.
Proposition 2.1. A CSM
x P) with a commutator equal to a Poisson bracket is a Lie superalgebra over a CSA A, and
if, g} = - f x ((l /, q') - (',
f
®g(dp'dq'dp'dq")
q"))e'(P'+p",q)+i(q'+q",p)
Proof. Since there are dual topological bases in the superspaces Q abd P, we can restrict the consideration to the case p E C"1, q E We set u(e, 77) = e'(f,q)+'(+r,p), where q and p are fixed. Then f f ® 9(dpi dg1dp2dg2) x(paq + pagi - gzpi + g2pi)u(pl + P2, q1 + q2)
= f f (dpldgl) (q° f 9(dp2dg2)pzu(p2, q2)
Chapter IV. Pseudo differential Operators
200
-P'l f 9(dp2dg2)gzu(p2, q2) + f 9(dP2dg2)pau(P2, g2)gi
+ f 9(dp2dg2)g2'u(p2, g2)p1)u(pi, qi) =
f
f (dpldgl)4°u(p1, q1) f(dp2dq2)pu(p2,q2)
- f f (dpi dg1)p°u(p1, qi) f
.
(dp2dg2)gzu(p2, q2)
+(-1)1§1+1[f f (dpldgi)giu(pi, q1) f.(dp2dq2)pu(p2,q2)
+ f f (dpldgl)piu(pl, q1) f 9(dpd2)g2 (p2, q2)]
of ag
of ag
- [ap0 aq° - aq° app
+ (_1)Isl+1
aRf aRg 0R! aRg apt aql + aql ap1) _
-1f,
g}.
The commutator [ , }, on the space of symbols -1 (Q x P) is known as a quantum commutator. The results of Sec. 1 yield [
,
Proposition 2.2. A CSM D(Q x P) with a quantum commutator }, is a Lie superalgebra over the CSA A.
Theorem 2.1. (correspondence principle). Relations (2.1) and (2.2) are valid for any functions al, a2 E c(Q x P).
Proof. Formula (1.16) immediately yields (2.1). From the same formula we get
l o h [al, a2 }. (q, p) =-[TCaL aL )+(T-1)\aL
aL aq2' apl
aqi ape
x a,(g1,pl)a2(g2,p2)
91=92=9
Pl =P2=P
+(-1)1alI1a21 [T(
aL
aL ,
age
19P1
+ (T - 1) \
aL
aL
aql 'ape
J
2. The Correspondence Principle
201
xa2(g2,p2)a1(g1,pi)I q1=q2=q = {al, a2}. P1=P2=D
Thus, the Lie quantum superalgebra over the CSA A, namely, (41(Q x P), [ , },), is a continuous deformation (cf. [46]) of the clasx P), { , }). sical Lie superalgebra over the CSA A, i.e., Apparently, the concept of Lie superalgebra is not adequate to the correspondence principle in the quantum field theory. In a finitedimensional case (supersymmetric quantum mechanics), the Lie su-
peralgebra 1(Q x P) (a space of entire S-analytic functions of the first order of growth) contains symbols of all real Hamiltonians. In an infinite-dimensional case, the Lie superalgebra D(Q x P) only contains symbols of model Hamiltonians. Indeed, the Lie superalgebra D(Q x P) contains only polynomials which satisfy constraints of the type of nuclearity (see Proposition 4.4, Chap. III). For instance, the Lie superalgebra (D(Q x P) contains the Hamiltonian function H(q, p) = (Ap, p) + (Bq, q),
where A and B are nuclear operators (boson system), whereas real quantum-field Hamiltonian functions have the form (for free theories)
H(q,p) =
1(p,
p) +
(B-'q,
q),
where B is a nuclear operator.
Definition 2.3. A Lie supermodule over the Lie superalgebra G = Go ® Gl is a Z2-graded linear space M = MO ® M1 in which the bilinear commutators [ , ]: G x M -* M, [ , ]: M x G -+ M are defined, and the identities I[9,m]I=191+Iml; [g, m] = (_ 1)1911-1+1 [m, 9] [91, [92,
+ [m, [91,
m]](_1)'91I I-1 + [92, 92]](_1)1-L 1921
= 0,
are valid for homogeneous elements.
[m,
9, 91, 92 E G,
91]](_1)19211911
mEM
Chapter IV. Pseudodifferential Operators
202
If M and G are CSM over the CSA A and the commutator is A-bilinear, [,\ga, m/3] = ,\[g, am]/3,
a, /3, .1 E A,
g E G, m E M,
then M = Mo ® Ml is a Lie supermodule (over the Lie algebra G = Go ® Gl) over the CSA A.
Let f be a PDO with a qp-symbol f E A(Q x P) and let g be a PDO with a qp-symbol g E -1)(Q x P). Then g maps the space 4) into 0 and f maps the space
fog: D(Q) -4 A(Q) is defined.
Furthermore, relation (1.4) makes it possible to extend the domain of definition of the PDO g from the space c to A. Thus, the operator
g o f : (Q)
A(Q)
is defined.
We denote by 00 and OA the spaces of PDO with symbols from the classes 1 and A respectively. We introduce a commutator
[f g} = fog ,
(-1)111 M§01.
Proposition 2.3. The space OA is a Lie supermodule over the Lie superalgebra O, (over the CSA A). Proof. It suffices to prove that the space OA is a two-sided module over the algebra O0 relative to the operation of composition of PDO. We set u = f o g. When proving Theorem 1.1, we established that g(W) _ .P((g ®,p) o S), where S is an even operator. Using (1.12), we obtain
u(co)(g) = f f (q, ha(p))e",9)g((P)(dp).
(2.4)
Note that the direct product g ®cp is a left direct product, i.e., first the functional cp acts from the left and then its action is followed by
2. The Correspondence Principle
203
that of functional g. Therefore we must transfer the functional g(cp) in relation (2.4) from right to left: '
(co)(q) = (-1)I9(v)I Ifl f 9(()(dp)f (9,
ha(p))e'(P,q).
Next, using the definition of the operator S, we have u(W)(q) _ (-1)I9(W)I
X f (q, hu(P +
_
III
f 9(dP dq% (dp) p))e'(p+P',q)+i(P,q')
(_1)I9(W)I IfI+Igl(lvl+Ifl)
f(
x (f f (9, ha(p'
(dp)e'(P,q)
+p))e'(P'q)+i(P,q)§(dp
dql )
= f u(9, ha(p))e'(P,q)(P(dp), u(q, p) = Jf(g,hcrQi) +
p)e'(P',q)+'(q',P)9(dp'dq')
E A(Q x P).
We set v = g o j. From relations (1.4) and (1.12) we have
v(W)(q) = f
9(dp'dgl)e'(P'q)
x f f (q + hq', hu(p))e=(P,q+hq)(Pldp)
= f v(q, ha(p))e'(v,q)c(dp), v(q, p) = f 9(dP dq')f (q + hq',
p)e'(P',q)+=(q',P)
The commutator [. , } on the operator Lie supermodule A(Q x P) induces a commutator on the space of symbols:
[f, 9}. = f f (9, ha(p) + p)e=(P'q)+i(q'P)9(dP d9 ) -(-1)III I9l r 9(dP dq')f (q + hq',
p)e'(q',P)+i(P',q>
(2.5)
Chapter IV. Pseudodifferential Operators
204
The Lie supermodule A(Q x P) over the Lie superalgebra (D(Q x P) with supercommutator (2.5) is known as a quantum Lie supermodule. Commutator (2.5) is an extension of the commutator which can be defined on the space 4D(Q x P) with the aid of the * operation.
Proposition 2.4. The space of S-entire symbols A(Q x P) with a Poisson bracket as a commutator is a Lie superalgebra (over the CSA A)
In particular, the space of symbols A(Q x P) is a Lie supermodule over the Lie superalgebra (D(Q x P). This Lie supermodule is said to be classical.
Theorem 2.2. The classical Lie supermodule A(Q x P) is a limit as h --- 0 of the quantum Lie supermodule A(Q x P), i.e., (2.2) holds
for any f Proof. Transforming relation (2.5) for a quantum commutator, we have
r (f (q, hu(p') + p) - f (q + hq', p)) xe'(q',p)+i(p',q)g(dp dq')
=
i(aa f (q, p), f o,
-i(af a (q, p), f aRf aLg ,
,9p
aq)
(p')e`(q'p)+=(q,a(p'))9(dp
q'e'(q',P)+i(p',q)9(dp dq'))
aRf (aq
d4 ))
+ o(h)
aL (ap)g) + o(h)
= { f, g} + o(h).
Note that commutator (2.5) can be written as
[f, g}* = exp{ -ih(
-, aL
aL
aq2 aP1
) } (f (qi, pi)g(g2, p2)
-(-1)1h1 i9ig(q,,Pl)f(g2,P2))I q1=q2=q P1 =P2 =P
3. Feynman-Kac Formula
205
Relation (2.6) allows us to extend the domain of definition of the commutator [ , }. with A(Q x P) x ''(Q x P) to wider classes of symbols. If the function [1 , g}.(q, p), which can be calculated from formula (2.6), belongs to the class A(Q x P), then, by definition, we
set [f,.} = [f,-g}. (i.e., the PDO [f,.} is defined as a PDO with an S-analytic symbol [f, g}.). Note that this PDO can even be defined in the cases where both operators f o g and g o f are not defined. This fact is of a special interest in an infinite-dimensional case (even for ordinary numerical coordinates). All results obtained in this section can be immediately generalized to the PDO a defined by formula (1.19). In particular, the correspondence principle for the quantum Lie supermodule A(Q x P) also holds for Wick symbols.
It should be emphasized that qp-symbols (as well as all generalizations of these symbols (1.19)) play a special part in an infinitedimensional case. For other symbols (including pq-symbols and Weyl symbols), we cannot construct anything similar to the quantum supermodule A(Q x P). For these symbols, we can only consider the quantum Lie superalgebra 4)(Q x P) which, as has been noted, is insufficient for applications to the quantum field theory.
3.
The Feynman-Kac Formula for the Symbol of the Evolution Operator
The new functional approach to superanalysis makes it possible to consider continual integrals over spaces of paths in a superspace. We
interpret a path as a real mapping of the range of variation of the evolution parameter into a superspace. When we used an algebraic approach to superanalysis, we used the symbol of an integral along paths but no paths (maps) really existed, and the continual integral with respect to anticommuting variables constituted a purely algebraic
construction and was rather an object of algebra than that of the functional analysis. In this section, we shall consider a Feynman path integral over a space of paths in a phase suprespace which appears in the Feynman-
Chapter IV. Pseudodifferential Operators
206
Kac formula for the symbol of the evolution operator of a PDO. Use is made of the PDO calculus on a superspace developed in the preceding sections. We restrict our consideration to a finite-dimensional (dimension over A) case.
3.1. Composition formulas on CA''. For the superspace coincides with the space the functional space 4)(CAm) = of entire S-analytic functions of the first order of growth E(CA'm). The PDO a with a qp-symbol a(q, p) E E(C" '2m) is extended, with the use The of formula (1.4), to a continuous operator in the space
PDO a with a qp-symbol a(q,p) E A(Cl '2m) is defined by relation into A(Cn'm). This operator is (1.12) and maps the space also continuous.
In what follows, when deriving the Feynman-Kac formula on a phase superspace, we shall use the composition formula for the PDO qp-symbols. Formula (1.10) can be rewritten in terms of Fourier transforms a and b of the symbols a, b E E(CA '2m), a * b(q, p) = f a(dp'dq')b(dp'dq'")
x exp{i(p", q) + i(p' + p", q) + i(q' + q", p)}.
(3.1)
Now if a is a PDO with a qp-symbol of the class E(C2 '2m) and b is a PDO with a qp-symbol of the class A(C2 '2m), then we shall use the composition formula a * b(q, p) = f a(dpdq')b(q + q', p) exp{i(p', q) + i(q', p)}.
(3.2)
3.2. Representing the qp-symbol of the evolution operator as a quantum chronological exponent. In the space of S-entire symbols of A(C2 '2m) we consider the Cauchy problem 8a(t, q, p) = h (q, p) of
*
a( t , q, p) ,
a(0, q, p) = 1.
(3 3) .
(3.4)
3. Feynman-Kac Formula
207
This Cauchy problem is equivalent to the Cauchy problem for the operator-valued functions
dta(t) = ha(t),
a(0) = I.
Thus, the solution of problem (3.3), (3.4) is the symbol of the evolution
operator for the PDO h. In the following theorem, the evolution parameter t belongs to the commutative Banach algebra A0.
Theorem 3.1. Let the PDO symbol h have the form h(gx,g9,px,po) _
h.Q(gx,px)geqe) as
q = (qx, qo),
p = (px, po)
E Cp m, h,Q(dpdq')e'(q'e,P=)+i(P'e,q-)
hcip(gx,px) = fR2,.
where hc,p are A-valued measures with compact supports on R2i. Then there exists a unique, S-analytic with respect to t E A0, solution of the
Cauchy problem (3.3), (3.4), a: Ao -4 A(C2 '2m). The solution is defined by the relation tk
a(t, q, p) _ (*) exp{th(q, p) } -
h*
.k.
*h(q, p)
k=0 oo tk
=
k=0 k!
f
h(dpidgi)...
k
fii(dpdq) k
x exp{i > (pj, qi) + i(y:pj, 4) + i(E qj, p)}. 1<1<j
j=1
(3.5)
j=1
Proof. The symbol h belongs to the class E(C2 '2m) and, consequently, the expression h * ... * h has sense and the expression h * a has sense for any function a of the class It suffices to prove the convergence of series (3.5). Since all functions are polynomials with respect to Qei pei it suffices to prove the
Chapter IV. Pseudodifferential Operators
208
theorem for an even case. We restrict the consideration to the case
n=1,m=0. In the first place, we have
Ila(t)IIR <00:
IIk
IIh * ... * hIIR.
k=0
We denote by IhI the variation of the measure h and by var h the total
variation, varh = IhI(R), where K is the support of the measure h, and by K = sup{Ikj: k E K}. Using the composition formula (3.1), we obtain
00
00 Rn+m
IIh*...*hIIR<w
n=0 m=0 n!m! k
xJ 00
k
Qjm
KIhl(dpidq')...fxlhi(dpkdgk)IEPiInI, 7=1
7=1
00 Rn+m
<EE
n=0 m=0 n !m!
(var h)krn+mkn+"` = (var h)ke2Rk
Consequently,
IIa(t)IIR < exp{IItII varhe21}.
We denote by M(K) the space of A-valued measures whose supports are in the compact set K. The space M(K) with norm var p is a Banach space. We set
M(Rn) = lim ind M(K), K
In the following theorem the evolution parameter t is real.
Theorem 3.2. Let h(t) be a family of PDO which depends on the parameter t E R, 0 < t < a, with symbols of the form h(t, 4x, q0, px, pe)
hap (t, Qx, px)4B pe
1
crp
h,,,6 (t, 4x, px)
= f.2n
h,,p(t, d4 dp)e`(ge,p:)+i(P'e,9s)
3. Feynman-Kac Formula
209
[0, a] -+ M(R2n) are continuous functions. Then there
where
exists a solution of the Cauchy problem for the symbol of the evolution operator 8a(t, q, p) = h(t, q, p) * a(t, q, p), at a(0, q, p) = 1.
The solution belongs to the class C1((0, a), A(C2 '2m)) and is defined by the quantum chronological exponent: t
a(t, q,p) = T(*) exp{I h(s, q,p) ds} 00
k=O 00
=
k
t
f 0dtlh(tl, q, p) * ... * fo
tk-1
dtkh(tk, q, p)
t
O
f dtl... ft'-' dtk f h(tl, dpi, dqi)... f h(tk, dpk, dqk 0
a
k
k
x exp{i E (pj, qj) + i(E pj, q)+ i(E qj, p)}. 1
)
j=1
(3.6)
j=1
Proof. We can again restrict the consideration to the case n = 1, m = 0. We shall show that series (3.6) converges uniformly on the interval [0, a]. We set A = max var h(t), r. = sup lc(t), lc(t) tc [O,a]
tc [O,a]
sup{IrI: r E supp h(t)}, and then we have 00
00
00 Rn+m
IIa(t)IIR << k=0 n=0 m=0 X
f
c,
dt1... f
tk
1 dtk f I h(tl)I (dpidgi) In
X J I h(tk)I (dpkdgk)
00 (ya)k 00 00 Rn+m
1:
k!
f
n=O
m=0
n m'
k
pj j=1
k=O
n!m!
j=1
qjlm
Chapter IV. Pseudodifferential Operators
210
By analogy, we can prove the convergence of series (3.6) which we have differentiated with respect to t. It is natural to call the first part of (3.6) a quantum chronological exponent.
Everywhere in this book we use an additive representation of a chronological exponent.
Suppose that R is an arbitrary topological algebra, h(t) is a function acting from the interval [a, b] into the algebra R (satisfying a number of constraints that depend on the algebra). Then the additive representation of the chronological exponent T exp{ fa h(t)dt} is the series
00
r
k=oJa
6
dtlh(tl)
r
Ja
tk
tl
dt2h(t2)... f dtkh(tk) a
(for whose convergence we impose the constraints on the function h(t)). If the algebra R is Banach, then the continuity of the function h: [a, b] -* R is sufficient for this series to converge. A chronological exponent defines the solution of the Cauchy problem for the evolu-
tionalry equation u(t) = h(t)u(t) in the algebra R. If R is a matrix algebra, then the additive representation for a chronological exponent is called a matrizant (see Gantmacher [23, p. 405], series (40)). A more complicated situation is considered in Theorem 3.2, namely, the operation * of the composition of PDO is chosen as a multiplication operation (therefore this chronological exponent is called a quantum exponent). The function h(t) maps the interval into the algebra E(C2 '2m). An additive representation series can be written for this function. However, this series does not converge in the topological algebra E(C2 '2i`). The series converges only in a wider space A(Cl '2m) which is not an algebra relative to the operation *. Along with an additive representation of a chronological exponent, use is often made of a multiplicative representation (a multiplicative chronological integral). However, we do not use this representation
in this book. A detailed exposition of the theory of multiplicative integrals for operator-valued functions can be found in [26] (see [23, p. 407] for the description of a multiplicative integral for matrix-valued functions).
211
3. Feynman-Kac Formula
3.3. Feynman path integral in the framework of perturbation theory. The class of functions of an infinite-dimensional argument for which the Feynman distribution integral is defined (see Chap. III) is too narrow to be applied to the Feynman-Kac formula. This class can be extended considerably with the use of perturbation theory on an infinite-dimensional superspace. Let Y be an infinite-dimensional superspace and let the function f (y) belong to the class A(Y) with its S-derivatives belonging to the class 1(Y). If the series 00
1
E n1 f f(n) (O) (y, ..., Y)'Ya,B(dy) converges, then the function f (y) is said to be summable in the framework of the perturbation theory with respect to the Feynman distri-
bution 'Ya,B and the sum of series (3.7) is denoted by the integral fy .f (Y)'Ya,B (dy)
3.4. Representation of the symbol of an evolution operator as a Feynman path integral over a space of paths in a phase superspace. We denote by B([O, t], Cam) the Banach space of Borel bounded paths q: [0, t] -+ Cn'm (i.e., the mapping q of the interval [0, t]
into the Banach space C m is measurable relative to the a-algebras of Borel subsets and IIgII00
= sup
Ijq(s) jI < oo).
s E [O,t]
We set Q([0, t], CA'm) _ {q E B([0, t], CAm): q(t) = 01; P([0, t], Cn'm) = {p E B([0, t], Cn'm): p(0) = 0};
P([O, t], CA'-) = Q'([O, t], CA'm);
Q([0, t], Cn'm) =
Pl([O,
t], CA'm);
Y([0, t], Cn '2m) = Q([0, t], Cn'm) X P([0, t], Cn'm). X ([O, t], Cn'2m) = P([0, t], CAm) X Q([0, t], CA'm);
Chapter IV. Pseudodifferential Operators
212
On an infinite-dimensional superspace X([0, t], C2 '2'") we consider a quadratic form t
b(x, x) = f f 0(s - T) (p(dT), q(ds)) 0 0 x = p ®q,
(P
where B(s) is an ordinary Heaviside function, and 0(0) = 0 (the last condition is very important and is actualy equivalent to the choice of a qp-symbol for a PDO). On the dual superspace Y([0, t], C2,,2,) we introduce a Feynman distribution v with zero mean and correlative functional (-2ib).
Theorem 3.3. Suppose that the conditions of Theorem 3.2 are fulfilled. Then the symbol of the evolution operator can be represented as a Feynman path integral (in the framework of perturbation theory) over the space of paths in a phase superspace t
a(t, q, p) = fY([0,t],C2m` T exp I h(T, q + q(T ),
p+P(T)) dT} v(dq(.)dp(.))
(3.8)
Proof. Note, first of all, that functions on the paths in a phase superspace
uk(q(.), P(.)) = f0t dt1...
ftk-1
dtk h(tl, q + q(tl), p + p(tl))
...h(tk, q + q(tk), P + p(tk)) = F(Lq,p,k) (q(.), p(.)) E ID(Y),
where the distributions Lq,p,k E A' (X) are defined by the relation ft
(Lq,p,k, f) =
1
dt1... f tk dtk f h(t1, dpi, dqi)... f h(tk, dpi) dqk) k
k
x exp{i(E pj, q) + i(E qq, p)} j=1
i=1
3. Feynman-Kac Formula
213
k
x f\
k
pj6(T - tj), j=11
q,6 (-r - ti)) j=1
E A(X), where b(T - tj) are Dirac for the test functions f measures concentrated at the points tj. Using relation (5.3) from Chap. III, we find that
a(t,q,p) = >00f F(Lq,,,,k)(q(-),p(.))v(dq(.)dp(.)) Y k=0
00
a fx Lq,p,k(dp(.)dg(.)) exp{i f t f t 0(s - T)(p(dr), q(ds))} dtk f h(tl, dpi, dqi)... f h(tk, dpk, dqk) f0 dt1... f 00
tk
1
0
k=0
k
k
exp{i(>Pj, q) + i(> qq,p)}
x
j=1
x exp{i
j=1
Jot 1 0(s - T)J(T - tj)
(p,, q j=1 1=1
00
- ti)drds} _ E
xb(s
k=0
ft
dt1...
f 4-1 dtk 0
0
x f h(t1, dpi, dqi)... fil(tk,dp,dq) k
k
k
x exp{i(E p q) + i(E qq, p) + i ,
j=1
j=1
00
j=1 1=1
tk
k=0 fo
t
1 dtk f h(t1, dpi, dqi)... f h(tk, dpk, dqk)
dt1... fo
k
k
x
k
E(pj, q)0(tt - tj) }
exp{i(E p q) + i(E qq, p)+ i E ,
j=1
j=1
(pj,qi)}.
1<1<j
3.5. Generalized solutions. Recall that any entire S-analytic function f (x, 0) on a superspace C"' is a polynomial with respect to
Chapter IV. Pseudodifferential Operators
214
anticommuting variables 0 and is therefore representable in the form
f (x, 0) = i fp(x)A, where fg(x) are entire S-analytic functions of commuting variables x E Ao A. We shall use the fact that the topology in the functional space A(C" m) is equivalent to the topology defined by the system of norms Ilf I1x = max
maxEC
IIb(zle,
..., z,le)
The Cauchy problem (3.3), (3.4) in the functional superspace A(Cl '2m) is of a very general nature and includes, in particular, all partial differential equations with entire coefficients of the first order of growth if we consider the symbols h E E(C2 '2m) as well as systems of equations of this kind. The existence of a solution of problem (3.3), (3.4) which would be classical for t is very problematic even in the absence of superstructures. The coefficients from the superalgebra make the matter even more complicated. In what follows, we construct a solution of this problem that is generalized with respect to t. We can write Cauchy problem (3.3), (3.4) as a Cauchy problem for an ordinary differential equation in the Frechet CSM A(CA ,2m),
a(t) = h.a(t),
a(0) = 1,
t c A0,
(3.9)
where h.: A(C2 '2m) A(C2 '2m), a H h * a. As has been pointed out, this operator is continuous if the symbol h(q, p) belongs to the class E(CA '2m).
Let us consider, in an arbitrary locally convex CSM M, the Cauchy problem (3.10) r(t) = Ax(t), x(0) = xo, t E A0,
where A is a A-linear operator in M. The formal solution (3.10) of this problem can be written as an exponential series 00 tnAnx0 x(t) = -
ni
(3.11)
n=0
If series (3.11) converges in the CSM M (at least for sufficiently small
t), then it defines the mapping from the range of variation of the
3. Feynman-Kac Formula
215
evolution parameter t (a ball in the Banach space Ao) into the CSM M which is a classical solution of the Cauchy problem (3.11). However, the qualitative estimates of series (3.11) for the general symbols from the class E(C2 '2m) show that the series diverges for arbitrarily small
time moments t (i.e., the solution is "chucked out" from the space of analytic functions in an arbitrarily small time moment). In this situation we can consider a generalized solution with respect to t. For the linear differential equation from (3.10), the generalized solution with respect to t can be defined as a solution of the equation
fx(t)c(t)dt+Afx(t)co(t)dt = 0,
(3.12)
where cp(t) belongs to some space of test functions U(Ao) and the generalized solution x(t) belongs to the space of (vector-valued) generalized functions U'(A0, M), namely, the space of A-linear continuous operators from the space of test functions It (Ao) in the CSM M.
Remark 3.1. It stands to reason that the classical solution of the Cauchy problem (3.10) may exist even if series (3.11) diverges. For instance, for an unbounded self-adjoint operator in a Hilbert space a continuous solution of the Cauchy problem ±(t) = iAx(t), x(0) = xo exists for all initial conditions xo from this space although series (3.11) may diverge (series (3.11) converges for analytic vectors). Since the solution a(t) is "chucked out" from the functional space A in an arbitrarily small time moment, it is natural to choose as test functions the functions W(t) which affect the solution only dur-
ing infinitely small time intervals, i.e., functions with a support at zero. However, these are generalized functions. Thus, we propose a new conception concerning a generalized solution, namely, spaces U containing generalized functions can be chosen as spaces of test functions.
Remark 3.2. The ordinary classical solution fits in this scheme. The classical solution is a generalized solution with a space of test functions U(R) that contains generalized functions (measures) W, (t) =
b(t - s). If the space U(R) = D(R) or G(R), then we obtain an ordinary definition of a generalized solution.
Chapter IV. Pseudo differential Operators
216
Remark 3.3. Spaces of o -additive differentiable measures on infinite-dimensional spaces are widely used as test spaces in the theory of infinite-dimensional distributions. Let x(t) be a classical solution of the Cauchy problem (3.10) from the class A(Ao, M). This solution can be regarded as a linear continuous functional on the space of test generalized functions A'(Ao):
f
x(n (0)
dt =
n!
n
(t , P(t)),
n=0
and the value of the classical solution at a point is defined by the relation x(s) = (x(t), 5(t - s)). Therefore it is natural to choose as a space of test functions U(Ao) a certain subspace of the space of generalized functions A'(Ao). We set
W(A0) = { E A'(Ao): IIWil,,
= E n! II(tn,
,)Ile,n2
< oo
n=O
va > 0}.
The topology in the space W (Ao) is defined by a system of norms Il,,}, a > 0, which is a countably normed space (a Frechet space). Lemma 3.1. If W(t) E W(Ao), then the generalized derivatives of all orders cp(k) (t) E W (Ao) .
Proof. Indeed, =
i(p
W(k))Ileanz
ll(tn' nE n!
00
= n=k E 1,n II(tn, Lemma 3.2.
co)Ile'(n+k)2 <
CII(PIIa
a' > a.
The space W(A0) is isomorphic to the space of
infinite-order differential operators 00
{P(a) &
=
E 0)
n=0
Pn6(n)(t): IIPIIP
= E IlPnlln!pn < C)0' p > o}. n=0
3. Feynman-Kac Formula
217
Theorem 3.4. Let the symbol h(q, p) belong to the class E(CA ,2m). Then there exists a unique generalized solution of the Cauchy problem (3.3), (3.4), the solution of the problem
f a(t, q, p)cp(t) dt + h(q, p) *
a(t, q, p)cp(t) dt = 0,
J
fa(t,q,p)o(t)dt= 1. The solution is defined by relation (3.5). Proof. It suffices to show that series (3.5) converges in the space of generalized functions W'(A0, A(C2 '2m)) if the symbol of the PDO belongs to the class E(C2 '2m). Since the topology in the space A(C2 ,2m) is a topology of coefficientwise convergence for an expansion in terms of anticommuting variables, we can restrict the consideration to a purely
commutative case. We can assume, without loss of generality, that n = 1. Since the functional h E A'(A2), there exists R > 0 such that 11
f h(dpdq)f (p, q) I < Ch max
Iz1I,Iz21
11f (zle, zee) jI, zj E C.
Consequently,
,17-21 Irilma
I f h(dpldgl)... f h(dpkdgk)
k
k
j=1
j=1
xexp{iTl>pj+iT2>qj+i 1<1<j
pjgllll
k
max
max
IriI,IT2I
exp{ZT1 E zlj j=1
k
+2T21: z2j+i E j=1
zljz21 }I
1<1<j
< Ch exp{2RR1k + R2k(k - 1)/2}.
Chapter IV. Pseudodifferential Operators
218
Let cp c W(A0), and then 11f a(t, q, p)W(t) dt IRl
< 000k (tk, W) 11 exp{2RRlk + R2k(k - 1)/2} k=0
< Bh,R 11WIlp,
p > R2/2.
Representing a generalized solution as a Feynman path integral over a space of paths in a phase superspace. 3.6.
Our aim is to obtain formula (3.8) for a generalized solution of the Cauchy problem (3.3), (3.4). We shall consider here symbols which do not depend on the time t but belong to the class E(C2 '2m). We shall consider the time t from the commutative Banach algebra A0, i.e., the Feynman path integral will be an integral along the paths from A0 into a phase superspace. The Feynman path integral in the Feynman-Kac formula (3.8) was considered in the framework of perturbation theory. We shall also use here the definition in the framework of perturbation theory, but the series of perturbation theory converges in the space of generalized functions W'(A0, A(CA '2m)) A closed interval in A0 is the set
[0, t] = {s E A0: s = at, a E [0, 1] C R}. Recall (see Chap. I) that an integral over a closed interval
f f (s)ds = t f 0t
I
f (at) da.
We denote by B([0,t],C"m) a space of bounded Borel paths q: [0, t] --+ CAm. This is a Banach superspace with norm s
Just as we did in Sec. 3.4, we introduce superspaces X([0, t], Cn'Zm), 1'([0, t], c'2) and define the Feynman distribution v on the infinitedimensional phase superspace Y([0, t], C2 '2m). Theorem 3.4 and calculations similar to those used in the proof of Theorem 3.3 give a Feynman-Kac formula in a phase superspace for generalized solutions.
3. Feynman-Kac Formula
219
Theorem 3.5. Let the symbol h(q, p) belong to the class E(CA '2m). Then the symbol of the evolution operator a(t, q, p) can be represented as a Feynman path integral over a phase superspace
f a(t, q, p)cp(t) dt f
=/ J
r
[
J
t
T exp{f h(q + q(T),
Y([O,t],cn '2m)
p + p(T))
cp(t) dt,
where cp E W(Ao).
3.7. Paths in a phase superspace on which the Feynman distribution is concentrated. Here we discuss a problem concerning the choice of phase superspace Q([0, t], Cnm)P([0, t], CA 'm) the continual integration over which gives the symbol of the evolution operator. By virtue of relation (5.3), Chap. III, we have fy
V(q('),p(.))v(dq(.)dp(.))
= fx dcP(P(.)q(.)) eXp{i f Jot 9(s - T) (p(dT), q(ds)) }, where cp is a linear continuous functional on the space of entire Sanalytic functions on a dual phase superspace X. Consequently, as a space of the trajectories of coordinates Q([0, t], CA") and those of momenta P([0, t], Cn'm) we can choose any superspaces of paths for which the covariance functional of the Feynman distribution is continuous. We shall call an infinite-dimensional phase superspace which possesses this property a superspace of paths of the Cauchy problem (3.3), (3.4).
It stands to reason that a phase superspace constructed on the basis of a superspace of bounded Borel paths is a superspace of paths of the Cauchy problem (3.3), (3.4), but this superspace is too large and it is interesting to find narrower superspaces of paths.
Chapter IV. Pseudodifferential Operators
220
We shall restrict the consideration to a real evolution parameter t. We denote by VQ0, t], CA ") the superspace of functions of the bounded variation v: [0, t] -+ CA "`; j jv M v = var v l o which is the total variation on the interval [0, t]. We denote by R([0, t], Cn'm) the superspace of functions without discontinuities of the second kind r: [0, t] -+ CA'"`, Ilrlloo _ sup 1jr(T)II. TE[o,t}
Theorem 3.6. Let Q([0, t], Cam) = {q E R([0, t], Cn'm): the coordinate q(T) is right continuous, q(T + 0) = q(T), and q(t) = 01, P([0, t], Cn'm) = {p E V([0, t], Cn'm): p(0) = 0}. If the CSA A is a Ealgebra, then the infinite-dimensional phase superspace Q([0, t], CA'"`) X
P([0, t], CA'"') is a superspace of paths of the Cauchy problem (3.3), (3.4).
Proof. Let us consider a family of functions q,(T) = 9(s - T),
0 < s < t.
These functions have no discontinuities of the second kind, are right continuous, and q, (t) = 0, 0 < s < t. Consequently, and the function q, E
1(s) =
fq(r)(dr) == (gs(T),P(r)), t
1 < j < n + m,
is well defined. We shall show that this function has a bounded variation (cf. the proof of Riesz theorem [37]). Let 0 < so < sl < ... < sn = t be partitioning of the interval [0, t]. Then we have nn
,,.,7
> M11(Sk) -P'(Sk-1)II k=1
n
= Sup IE CYk(Pi(Sk) - P(sk-1)) II0kII!S1 k=1 n
sup I\I ak(q', - /y Y
IkakII!5l
k=1
n
< 1011 sup I ak(gk - k_1)1100 11001<-1 k=1
221
4. Unsolved Problems <-11911,
P E P([0, t], Cn'm) = Q'([0, t], Cn m).
Thus, IIpMIv < IIPII We note that qo(T) = 0, and therefore p(O)
_
(qo,P) = 0. Thus, p(t) E P([0, t], Cam), and the quadratic form
x = P ® q H (p, q)
is well defined, with II (p, q) II < Consequently, the covariance functional of the IIpIIvMIgMI < IIPII I. Feynman distribution v is continuous.
Theorem 3.7. Let Q([0, t], Cn'm) _ {q E V ([O' t], CA'-): q(t) = 0},
P([0, t], CA'm) _ {p E R([0, t], CA 'm): momentum p(s) be left continu-
ous, As - 0) = p(s) and p(O) = 0}. If the CSA A is a E-algebra, then the superspace Q([0, t], C m) x P([0, t], CA m) is the space of paths of the Cauchy problem (3.3), (3.4).
Proof. Let us consider the family of functions pr(s) = 9(s - T), 0 < T < t. These functions have no discontinuities of the second kind, are left continuous, and pT (0) = 0, 0 < T < t. Consequently, pT E P([0, t], Cnm) and the function q3 (T)
qi) is well defined.
By analogy with the proof of the preceding theorem, we find that varglo < IIgII Then we have pt(s) - 0 and, consequently, q(t) _ (pt, q) = 0. The next part of the proof repeats that of the preceding theorem.
Theorems 3.6 and 3.7 reflect the uncertainty principle. If the trajectories of coordinates have a bounded variation, then we can only require that the moments have no discontinuities of the second kind, and, conversely, if the trajectories of momenta have a bounded variation, then we can only require that the coordinates have no discontinuities of the second kind.
4.
Unsolved Problems and Possible Generalizations
As was pointed out in the introduction to this chapter, only the first steps have been made in the PDO theory on a superspace. It
Chapter IV. Pseudodifferential Operators
222
should be emphasized that the PDO calculus is constructed on an infinite-dimensional superspace. Thus two problems have been simultaneously solved here. The first problem is connected with an infinite dimensionality of a phase space and the second problem is connected with a superstructure. It has turned out that these (seemingly quite different) problems have much in common, namely, both in an infinitedimensional case and in a case of a superspace we have to solve the same problem, the problem of construction of a PDO theory on a
phase space on which the dx measure which is shift invariant (the Lebesgue measure, the Haar measure) will be absent. The reasons for the absence of dx in an infinite-dimensional case and in a supercase are different but the problem is the same. We have first overcome this difficulty in an infinite-dimensional case and then applied the methods employed in an infinite-dimensional case to superanalysis. Therefore we have practically not used the techniques of the standard PDO the-
ory on R. We think that in further investigations it is expedient to split the PDO theory developed in Sec. 1 into three separate theories, namely, infinite-dimensional PDO over the field of real numbers, PDO on a finite-dimensional superspace (dim over A), and PDO on an infinitedimensional superspace. The third theory is the most general and the most difficult. Below we formulate certain problems which can be solved by the
methods developed in this book. We have only to overcome some technical difficulties (though, perhaps, rather essential).
4.1. PDO calculus on a finite-dimensional superspace. A superanalog of the ordinary PDO theory on R" is the PDO theory on the superspace R"' over the pseudotopological CSA B in which the Bi-annihilator is trivial and all even souls are nilpotent. The definition of the PDO a of order rn can be directly extended to this case (see, e.g., [32, p. 57]). Here use is made of the theory of generalized functions (D(RBm), D'(RBm)). For these PDO, we can try to construct a systematic theory which would include (1) PDO on supermanifolds, (2) a parametrix for superelliptic differential operators,
4. Unsolved Problems
223
(3) canonical transformations, (4) Maslov's canonical operator, (5) Fourier integral operators, (6) spectral PDO theory on supermanifolds, (7) boundary value problems, (8) wave fronts.
4.2. Infinite-dimensional PDO. The most promising is the construction of a PDO theory on a locally convex space with a fixed Gaussian measure. Here it is easy to formulate a spectral problem on infinite-dimensional PDO. It would be natural to consider, as the first step, PDO whose symbols satisfy constraints of the type of nuclearity. Also of interest is a PDO calculus on a locally convex space with a fixed diferetiable measure. This calculus can also be successively used to study the spectral properties of infinite-dimensional PDO. In the framework of PDO calculus with analytical symbols, Sec. 1, we can obtain relations of the type of the Feynman-Kac formula, similar to relations from Sec. 3. The construction of a PDO theory on infinite-dimensional manifolds is a technically very complicated problem. Here we can evidently use the calculus discussed in Sec. 1.
4.3. PDO on an infinite-dimensional superspace. We have to construct a superanalog of the space L2 with the use of Gaussian measure. This will be a Hilbert CSM. It is natural to develop this theory over E-superalgebras. In this Hilbert CSM we can pose a problem concerning the spectral properties of PDO. However, this is a very complicated problem. At present, we do not have any results concerning the spectrum of the operators in Hilbert CSM. It is evidently easy enough to obtain Feynman-Kac formulas, similar to the formulas from Sec. 3, for an infinite-dimensional superspace. We can hardly say anything definite about PDO on infinite-dimensional supermanifolds. We do not yet have a theory of infinite-dimensional supermanifolds.
224
Chapter IV. Pseudodifferential Operators
Remarks The fundamentals of the PDO theory on a superspace were laid in the works [65, 68, 144, 153]. I managed to use here the methods of the theory of infinite-dimensional PDO [67, 136, 139, 141]. For the first time, infinitedimensional PDO were considered in [127]. The main part in my constructions is played by the article [139] in which I introduced functional spaces that were used in all further considerations. Sec. 1. The results given in this section were published in [65, 68, 144, 153, 148]. For the infinite-dimensional PDO theory over the field R see [127, 129, 136, 139]. For infinite-dimensional PDO in spaces of functions square summable with respect to u-additive Gaussian measure on a locally convex space see [141, 67, 149, 150].
Sec. 2. The results considered in this section were published in [148, 153, 149]. Here the main difficulties are connected with infinite dimensionality of a phase space. The introduction of Lie supermodules is due precisely to infinite dimensionality. Sec. 3. The most systematic exposition of the Feynman-Kac formulas for PDO symbols can be found in Berezin's review [5] (see also [3, 53, 26]). In these works, a sequential approach to the definition of the Feynman integral was used (the continual integral was defined as a limit of infinitedimensional integrals). We regard the Feynman integral as an integral with respect to the distribution on an infinite-dimensional space. For the first time, this idea was realized on the mathematical level of strictness by Uglanov [133].
In the framework of algebraic superanalysis, the Feynman-Kac formulas for PDO symbols (a sequential Feynman integral) were obtained in [3]. The Feynman-Kac formula for PDO symbols on an infinite-dimensional superspace in the framework of the new functional approach to superanalysis was obtained by Khrennikov [144, 68]. As has been pointed out, the main advantage of this functional integral is that it is actually a functional integral and not an integral with respect to the infinite-dimensional Grassmann algebra (which is a purely algebraic object). This is more consistent with Feynman's concept [59] of continual integral as an integral along paths or over classical fields. The Feynman-Kac formulas for the heat conduction equation and for the Schrodinger equation with a potential on a finite-dimensional superspace were obtained in [146, 147]. In these works, the Feynman-Kac for-
4. Unsolved Problems
225
mulas connected with the Wiener process were also obtained. In [153] I considered the Feynman-Kac formulas for the heat conduction equation and the Schrodinger equation with a potential on an infinite-dimensional Hilbert superspace. Interesting formulas of the type of the Feynman-Kac formulas on a phase superspace were obtained by Rogers and Ktitarev [122, 109, 110].
Generalized solutions (for t) of the Cauchy problem to the symbol of an evolutionalry PDO were studied in [68, 148] and generalized solutions (for t) of the Cauchy problem for the Liouville infinite-dimensional equations were obtained in [170]. I was apparently the first to consider the Feynman path integral which converges in the sense of the theory of generalized functions (see [68]). Sec. 4. A boundedless sphere of action opens here indeed. It is
beyond doubt that a superextension of the results obtained by Egorov, Maslov, Treves, Hormander, Shubin [32, 44-46, 58, 73] and by other authors
in the PDO theory on R" and on manifold over R" can be obtained on
the superspace RBm over the pseudotopological CSA B with a trivial Blannihilator and a nilpotent soul. In an infinite-dimensional case, of especial interest is the spectral PDO theory on a locally convex space with a fixed Gaussian measure. For these PDO, we can try to get results similar to those from the theory of infinitedimensional differential operators on a space with a fixed Gaussian measure (see the systematic exposition of this theory in the monograph by Berezan-
skii [2]). An important part can also be played by the Hida calculus of generalized Brownian functionals [64]. The theory of differentiable measures on infinite-dimensional spaces was proposed by Fomin [62] in 1968, the modern exposition of this theory can be found in [6].
Chapter V
Fundamentals of the Probability Theory on a Superspace
In this chapter we consider probability models in which probabilities as well as expectations and variances are elements of Banach CSA and more general noncommutative algebras. Limit theorems of probability theory on a superspace are obtained, a Wiener superprocess is introduced, and a representation of a solution of heat conduction equation is obtained as a probability mean with respect to Wiener process. A spectral interpretation of A-valued probabilities is given and the connection with the multivalued probability theory is considered.
1.
Limit Theorems on a Superspace
Limit theorems play an exceptionally important part in the ordinary probability theory. We also begin the exposition of the fundamentals of the probability theory on a superspace from limit theorems, namely, the central limit theorem (CLT), the law of large numbers (LLN), and a supergeneralization of Lyapunov theorem. We begin with considering a general case where a superspace may be infinite-dimensional. In the ordinary probability theory, limit theorems on infinite-dimensional spaces are no less important than finitedimensional theorems.
Chapter V. Probability Theory
228
The methods of the theory of analytical distributions that we use
here allow us to obtain as limit distributions in the CLT not only Gaussian but also quasi-Gaussian distributions. In particular, we have obtained a CLT for Feynman distribution. This CLT is of interest by itself, without being connected with superanalysis.
1.1. Mean value and covariance functional. Definition 1.1. A momemt of order m of the distribution p E M(W) is the form bm(v) = f (v, w)mp(dw). The first-order moment bl is called a mean value. The form a2 = b2 - bl ® bl is the covariance functional of the distribution M.
Lemma 1.1. Let the distribution p E M(W). Then the relation F'(µ) (v) = f et("'W)p(dw)
(1.1)
holds for the Fourier transformation P. Proof. Assume that the kernel of the Fourier transformation .F': .M (W) -4 A(V) is a function g, c -D(W ), v E V. Then
F'(µ) (v) = f
f.v(th.")F'(l2)(V'),
i.e., g is a Dirac 5-measure concentrated at a point v E V. Therefore f(9v)(w) = e'(' ).
Lemma 1.2. Let the distribution µ E M(W). Then bm (v)
(_j)maT .F'(p)(0)(vm).
This lemma follows from relation (1.1). It follows from this lemma that Definition 1.1 and Definition 7.1 from Chap. III are consistent.
Definition 1.2. The distribution p E M (W) is normalized if
µ(W)= fp(dw)=1.
We use the symbol µc, where c E C, to denote the image of the distribution p under the homothety c and the symbol ('µ)" to denote the convolution of n copies of the distribution A.
1. Limit Theorems on a Superspace
229
1.2. Central limit theorem on a superspace. Theorem 1.1. Let µ E M (W) be a normalized distribution with zero mean and covariance functional b c £2,1(L ,, A). Then fco(w)'m(dw) = lim f W(w)(*µl/,/n)n(dw)
for any function W E c(W) (-yb
(1.2)
rya,b)
Proof. It suffices to show that the Fourier transformations of the distributions (`µl/,/n)n converge to the Fourier transformation of the quasi-Gaussian distribution ryb in the functional space A(V). We set f (v) = F'(p)(v), and then we have
)=1-
f (v/
2nb(v,
v)
00
+ni+1/2
k
k1n(k-3)/2f (0)(v ) k=3 b(v,
v) )
+ 9(n, v)
2n
n1+1/2
We introduce functions fk,n(A, u), k = 0, ..., n, A, p E A by means of the relation n
(A + lpl p)n =
L. fk,n(A, 1 )IPk,
P E C.
k=O
These functions are homogeneous over the field C. With respect to the variable A the order of homogeneity is equal to (n - k) and with respect to the variable µ it is equal to k. It should also be pointed out that fo,n(A, µ) = An, and we have an estimate (1.3) Ilfk,n(A,/.L) < CnIJAIIn-kllllk. Using the function fn,k(A, µ), we can write [f (v//) ]n in the form [f (vl V'L)]n _ (1
-
b(v, )
)n
Chapter V. Probability Theory
230
+ kE
1
nk(1+1/2) fk,n
Furthermore,
-
( n
k=1
(1
- n
b(2)
((1
)
)n - exp{- b(2 y) }
b(y, y) k Cn
1
) (nk
2
- b(v,2nv) ), g(n, v ).
°O
Vii) +
(- b(y,2 y) ) k
1
k=n+1
.
!
k
Having verified that (C - k,) -+ 0, n -+ oo, we find that (1
- b(v,
))n
exp{-b(2
v) }
in the space A(V). Using inequality (1.3), we find that fn,k ((1 - b(2n v) ), 9(n, v)) I B
< Cn(1+ II2n
)n-k11g(n,v)Ila
for any compact set B from the covering CSM Lv. By virtue of this estimate, the functional sequence [f (v/V qn]n converges to exp{-° zv } in the space A(V). The convergence in Theorem 1.1 differs from the weak convergence of probability measures even when there is no superstructure. Infinite dimensionality does not play a decisive part here either.
Let us consider Theorem 1.1 in a finite-dimensional case for numerical coordinates. In this case, the space 0 coincides with the space E(Cn) of entire functions of the first order of growth. Thus, in the CLT we consider a weak convergence not over a space of continuous bounded functions on Rn but over a space of entire functions of the first order of growth. For ordinary Gaussian measures, on Rn, the standard CLT does not yield Theorem 1.1 and Theorem 1.1 does not yield the CLT.
1. Limit Theorems on a Superspace
231
What is new that Theorem 1.1 gives for Gaussian measures on R"? In this case, the main difference from the standard CLT is that we can choose as µ not only a probability distribution on R" but also a signed measure µ on Rn as well as a complex-valued measure or even a generalized function. The following example shows that the standard CLT is no longer valid for signed measures.
Example 1.1 (signed initial measure µ and a standard Gaussian distribution on R). Consider a discrete signed measure on the real line 3
1
2Jo-2b1+2J3, where 8t is a Dirac b-measure concentrated at a point t E R. We can immediately verify that this measure is normalized, its mean is +00 zero, and f t2p(dt) = 3 > 0. By virtue of Theorem 1.1, the limiting -00
process (1.2) takes place for any entire function of the first order of growth on C. Suppose now that co(t) is a continuous unbounded nonnegative function (not identically zero) whose support lies on the interval (-oo, 0) (i.e., does not intersect the support of the measure µ). Then we have 0000
1
0
+00
for any n whereas f to(t)e-t2/6dt > 0. -00
This example shows that if we consider even the simplest probability supermodel with a-additive measures on the real line, which assume values in a Banach CSA, then there is no weak convergence on the space of continuous bounded functions on the real line with values in the Banach CSA A. In connection with example 1.1, we would like to discuss one more classical theorem of the probability theory. It is known that in the ordinary probability theory the convergence of characteristic functions implies a weak convergence of distributions. Example 1.1 shows that
Chapter V. Probability Theory
232
this is not true even for signed measures and all the more so for measures with values in a Banach CSA. A wider choice of initial distributions p leads to a wider class of limit laws. In an infinite-dimensional case, the CLT for the Feynman distribution is especially important for applications. As the following example shows, in the absence of a superstructure, we can choose a countably additive complex-valued measure on an infinite-dimensional space as an initial approximation of p for the Feynman distribution. Then the limiting process (1.2) can be regarded as a new method for calculating Feynman path integral. Under this approach, the function W(w) on an infinite-dimensional space is said to be Feynman integrable if there exists a limit (1.2) which is called a Feynman path integral of the function co(w). In the framework of this approach, the class of functions 4) integrable (by virtue of Theorem 1.1) can be essentially extended.
Example 1.2. Suppose that H is a separable real Hilbert space, vl and v2 are Gaussian countably additive measures on the a-algebra of Borel subsets of H with zero mean and covariation operators B and 2B. It follows from the theory of measure on a Hilbert space that the operator B is a nuclear operator and the measures vl and v2 are mutually singular. Consider a countably additive complex-valued
measure on the a-algebra of Borel subsets v = (2v1 - v2) + i(v2 v1). This measure is normalized, has zero mean, and its covariance functional is equal to i(Bv, v). By virtue of Theorem 1.1, the limiting process
1Hco(w)exp5-2(B-1W,W)Jdw=L c(w)7iB(
= lim
r
corwi +... +Wn
Iv(dwl)...v(
Hn
)
n)
is valid for any function co from the class 4)(W).
1.3. The law of large numbers on a superspace. By analogy with Theorem 1.1 we can prove
1. Limit Theorems on a Superspace
233
Theorem 1.2. Suppose that the distribution µ E M (W) is normalized, and its mean value µ is zero. Then the limiting process
. f (P(w)(*A'/f)n(dw)
0(0) = n,lim
is valid for any function cP E c(W).
1.4. The central limit theorem for exponential distributions. Quasi-Gaussian distributions were introduced as distributions whose Fourier transform has the form of a quadratic exponent. A natural generalization is distributions whose Fourier transform has the form of an exponent of a polynomial on a superspace over a CSM. Definition 1.3. An exponential distribution on a superspace W is the distribution yp E .M (W) whose Fourier transform is .F'('yp) (v) _ eP(V), where P(v) = k,bk(vk) is a polynom on a superspace V. Here k=0
the forms bk E Xk,l(Lkv, A), and their restrictions to a superspace are symmetric. An exponential distribution is a fundamental solution of the Cauchy problem on a superspace of (t, w)
at
= P-0) f (t w) , ,
f(0,w) = fo(w). Let b E 1Cn,i(Ln, A). We shall denote the exponential distribution with a Fourier transform F(ry) (v) = exp{ n, b(vn) } by ryb (if n = 2, then this is a quasi-Gaussian distribution with zero mean and covariance functional b). It follows from Lemma 1.2 that the moments of order 1, ..., n - 1 of the distribution ryb are zero and the moment of order n is equal to b.
Theorem 1.3. Suppose that the distribution µ E M (W) is normalized, moments of order 1, ..., p - 1 are zero, and the moment of order p is equal to b, with b E Lp,j (L A). Then the limiting process W(w)(*µn-1/v)n(dw)) f W(w)'Yb(dw) = nlim J
Chapter V. Probability Theory
234
is valid for any function cp E -(D (W).
For C-linear locally convex spaces, the complete proof of this theorem can be found in [140, p. 123]. We can obtain a proof for a supercase by combining the proof of Theorem 1.1 with that given in [140]. If p = 2, then Theorems 1.1 and 1.3 coincide. Theorem 1.3 is a typical limit theorem for a fundamental solution of the Cauchy problem for a linear differential equation with constant coefficients. In particular, the heat conduction equation is associated with the ordinary CLT. If we consider the Cauchy problem for the equation 2q
then we obtain Krylov's theory [106]. Thus, Theorem 1.3 is an ordinary limit law which includes laws that are already known. This law was obtained on a superspace, the superspace being infinite-dimensional.
It should be pointed out that in the limit theorem for Eq. (1.4) Krylov also considered a limiting process for integrals with respect to analytic functions of an infinite-dimensional argument. Our class (D is considerably wider than that of Krylov. Here is an example of a countably additive measure on the aalgebra of Borel subsets of a Hilbert space satisfying the conditions of Theorem 1.3. 00
Example 1.3. Let the numbers )j E C, E )j = a# 0, the series j=1
converging absolutely. On a Hilbert space H we consider a p-linear form 00 .1j (ej, v) P, v E H, i-1 where {ej} is an orthonormal basis in H. Let the function f (t), t E R, be summable, and let
b(vP) _
+00
+00
ff(t) dt = a1, ftf(t) dt = 0
-00
-00
(n=1,...,p-1),
1. Limit Theorems on a Superspace
235
+00
ftPf(t) dt = 1. -00
We denote by pj the measure concentrated on the one-dimensional subspace Hj = {Aej} which is absolutely continuous relative to the Lebesgue measure corresponding to the scalar product on H and having a density f (xj), where xj = (ej, x), x E H. We set µ = E00Ajpj, and then this is a countably additive normalj=1
ized measure, and
f (v, w)mp(dw) = (E00 .1j (ej, v) m) H
E Aj(ej,
tm f(t) dt
-00
7=1
1000
f
+00
M = 1, ..., p - 1,
P, m = p.
j=1
In addition, +00
JH Ilxllmlµl(dx)
j=1
Aj)
f Itim If (t) I dt <
00
-00
+00
if the integral f I t I m 1f (t) I dt is finite. The fulfilment of the last condition for all m is sufficient for the measure µ to be associated with the distribution from the space M(W). By analogy, we can introduce countably additive A-valued measures on the covering CSM Lye which satisfy the conditions of Theorem 1.3.
1.5. Superanalog of the Lyapunov theorem. Here, as before, the main difficulties arise already for signed measures. In particular, already for signed measures it is necessary to introduce an additional condition into the formulation of the Lyapunov theorem which was automatically fulfilled for probability measures.
Chapter V. Probability Theory
236
Definition 1.4. The distribution S E A'(V) is regular if there exists a compact subset B of the covering CSM Lv such that II(S, f) 11 < Cs maBx Ilf(v)II.
The space of regular distributions on the superspace V is denoted by AR(V); TR(W) = F'(AR(V)).
Proposition 1.1. If A = C, then A'(V) - AR(V). In order to prove this proposition, it suffices to use the Cauchy integral formula for analytic mappings of complex locally convex spaces.
Proposition 1.2. If A = C, then the space A'(V) consists of countably additive complex-valued measures with compact supports on the o -algebra of Borel subsets of the locally convex space V.
Proof. The space A(V) is embedded into the space CC(V) of functions which are continuous on compact subsets of the locally convex space V. By the Hahn-Banach theorem, any distribution S E AI(V)
can be extended to the functional S E CC(V). Using proposition 5 from [56, p. 18] and [22], we find that S = p, where the measure p satisfies the conditions of the proposition. The uniqueness of the measure p follows from the triviality of the kernel of the operator of the Fourier transformation. It is not clear as yet what the situation is in a supercase. Furthermore, when proving the Lyapunov theorem, we shall use a logarithm on the CSA A. Suppose that in the CSA A there exists a body projector b consistent with the norm on A (see Chap. II). The logarithm of the element A of the CSA A is defined by the relation
lnA=InbA+ln(1+
c'O
ba
n-1
(-1) i-1(c-` Tt
This function is defined for all A satisfying the condition Ilcall < Ibal,
bA
0.
ba )
n
1. Limit Theorems on a Superspace
237
Note that if aj, j = 1, ..., n, are even elements, then n
n
exp{E In (1 + aj)} _ 11 (1 + aj), j=1
JJcaj JJ < 11 + bad l;
j=1
in this case In( f (1 + aj)) may be undetermined. 7=1
Theorem 1.4. Suppose that in the CSA A there exists a body projector consistent with the norm. Let µnk, k = 1, ..., n, n = 1, ..., be normalized even-valued distributions belonging to M(W) and possessing covariance functionals ank and zero means. If there exists a form a E L2,k A) such that Orn
(1)
n = k ank -4 a; k=1
n
(2) E I1ank112 -4 0; k=1
n (3)r
0 Sup I J pn = k=1 0
f O(W)'Ya(dw)
=1n
f W(W)µn1 * ... * Ann(dW)
is valid for any function cp E TR(W).
Proof. Using Taylor's formula, we obtain V, V)
unk (v) = .F(µ'nk) (v) = 1 - ank (2
+ rnk (v),
where the estimate Ilrnk(v)II < Sup I f
(v)W)3eia(v,ur)Unk(d41)
O
holds for the remainder rnk. Here we used Lemma 1.2. Consequently, n the sum E IIrnkll tends to zero uniformly on the compact sets of the k=1
CSM Lv, and, in particular, max I l rnk l l -4 0. 1
Chapter V. Probability Theory
238
It follows from condition 2 that max I I ank I I -+ 0. 1
Furthermore, since Cunk = -2-1cank + Crnk, it follows that max I I conk l 10,
1
and since bunk = 1 - 2-'bank + Crnk, it follows that I bunk l> 1 - 2-1 max I bank l- max I brnk l 1
Thus, for any E > 0, there exists a number n such that 1min I bunk l> 1 - E.
lmka n I l CUnk l l< E;
Therefore, for sufficiently large n we have an inequality (
Max IICUnkII)/( min Ibunkl) <
1
1
E
1-E
< 1.
Consequently, the function n
n
gn(v) = 1: In unk = k=1 k=1
n
oo
+E
k=1 m=1
In bunk
(-1)m-1 Cunk) M ( m bunk J
is defined. The article [140, p. 128] contains a complete proof of the fact that 00
the sum of complex-valued functions k In bunk converges to (-ba/2) k=1
uniformly on compact sets. Let us prove that Cgn -+ (-ca/2) as n -4 oo uniformly on compact sets of the CSM L. Note that n
CU
n
1 m-1
o0
nk + Cgn = k=1 bunk k=1 m=2
(
)
m
( CU nk) M = An + (nbunk
1. Limit Theorems on a Superspace
239
Next, we have n
n
1
Crnk
E bunk
`-
k=1
E I I Crnk I I - 0; 1- E k=1
n
Cank
- cank) <
1 1 E
(lmax
IlCankll
k=1(bunk n Ilbankll2)1/2)
XEIlbrnkll+ 2 k=1
1l<
-4 0;
k=1 IlCankll' k=1 2
1 11 CUnk 11m-2 I CUnk n k=1 I I bunk I I2 m=2 717 b7dnk I l
bn
<
I
? E) 2
(1
E(4-'IlCankll'
00
X E 1( E m=2 7n
+
IICrnkll')
k=1
1-E
m-2
--3. 0,
n --4 oo.
If µnk are countably additive A-valued measures, then Condition 3 in Theorem 1.4 can be replaced by a more common condition as compared with that from the ordinary probability theory, namely, n
Ef
(dw) -+ 0,
k=1
where Iµnkl are variations of vector-valued measures. In the ordinary probability theory, there is no analog of condition 2. If condition 2 is not fulfilled, then the limiting process (1.5) is, in general, absent even for signed measures on the real line. If the measures µnk are not even-valued, then the limiting process (1.5) is also, in general, absent even for A-valued measures on the real line.
Chapter V. Probability Theory
240
2.
Random Processes on a Superspace
In this section, we propose a formalism of the theory of random processes on a (finite-dimensional) superspace in the framework of which a Wiener process was constructed and a representation of a solution of the heat conduction equation on a superspace was obtained as a probability mean (a probability version of the Feynman-Kac formula).
2.1. Cylindrical distributions. Suppose that we are given a consisting of sequence of locally convex superspaces un,m = A. We assume that these CSM satisfy the followfunctions cp: ing condition of consistency: for any operator B E Gi,i(An+m, Ak+l) which maps the superspace C m into C and for any function W E l tk,1 the function 0 = cp o B belongs to the CSM Un,m 1
Let M=Mo®M1and R=Ra®R1,N=No®N1and S=So®S1 be two pairs of dual topological CSM, V = MO ® N1 and W = Ro ® S1 being dual superspaces over these CSM. We denote by II(W) a collection of finite-dimensional A-linear projectors defined by the elements of the Ao-module MO ® No, i.e., maps
7r: W -> Cn",," of the form 0 0 0 0 7rw = ((w, ml), ..., (w, mn,r); (w, nl), ..., (w, MOM,
where m?eMO) j=1,...,n,,;no ENo,i=1,...,m,,. We introduce a functional CSM consisting of cylindrical functions
U(W) = U { f (w) = cp o 7r(w): c' E aE11(W)
For any projector 7r: W --> Cn,,m" we introduce a mapping j,:Un,f,m, U(W), co y wo7r. The CSM U(W) is endowed with a topology of an inductive limit of the family j.r}, 7r E 11 (W). By j,., 7r E II (W), we denote an operator from U'(W) into Un',r ,n,r which is an adjoint of 3a.
Definition 2.1. A cylindrical distribution on a superspace W is a A-linear continuous functional on the CSM of cylindrical functions U(W) (i.e., the element µ E U'(W)).
2. Random Processes on a Superspace
241
Definition 2.2. Finite-dimensional distributions of a cylindrical distribution p is a collection of distributions {A r = jrp}, 7r E II(W) on finite-dimensional superspaces.
Theorem 2.1 (the condition of consistency of finite-dimensional distributions). Let µ be a cylindrical distribution on a superspace W, the projector 7r E II(W), and let A be a A-linear operator from. An-+m-into A'+' which maps the superspace Cnw,m r
into C. Then, for any
function cp E U1,3, we have
f
cp o A(w)p.,,(dw) = f cp(w)pAO,(dw).
Cnir.mx
(2.1)
CA A
A
Conversely, let the family 7r E II(W), p E Un,,,nw satisfy the condition of consistency (2.1). Then {µ,} is a family of finitedimensional distributions of the cylindrical distribution p defined by the relation
ff(w)p()= f co(w)µff(dw),
f= W o rr.
Cner.m, A
Remark 2.1. If the algebra A = R, then condition (2.1) coincides with the condition of consistency of finite-dimensional disributions for cylindrical measures on locally convex spaces. 2.2. Cylindrical random processes with continuous tra-
jectory. We set SZ = {x E C([O,1], Cp,°): x(O) = 0}. This is a Banach superspace with a norm JJxJJ. = max 11x(t)II. We set o
in the superspace Cr'. This is a Banach superspace with a norm IlvlIv = Varvlo.
These are dual superspaces and the form of their A0-duality is defined by the integral
(x, v) = E f x° (t) dv° (t) + > f x (t) dvi (t), 1
1
j=1
j=1
Chapter V. Probability Theory
242
where x(t) = x°(t) ® x'(t), v(t) = v°(t) ® v'(t).
The finite-dimensional projectors 7r: Q -* 7rx = (
have the form
f l x° (t) d ,k (t); j f x (t) da°t (t)), j
where k = 1 ,
...,n,
l = 1, ..., 7)Z,,,
vjOk, ail E V(l0, 1], A°), vjk(0) =
a°l (0) = 0.
Definition 2.3. A cylindrical random process x(t) = xµ(t) with with continuous trajectories is a normalvalues in the superspace ized cylindrical distribution p on a superspace Il (it is convenient for us to consider random processes emanating from zero).
2.3. Quasi-Gaussian cylindrical random processes. We set Un,,.n = E(C"m) which are spaces of S-entire functions of the first The cylindrical random process order of growth on superspaces x(t) all of whose finite-dimensional distributions are quasi-Gaussian distributions on Cn"'"`x, is called a quasi-Gaussian cylindrical process.
Gaussian and Feynman random processes can be introduced by analogy.
Remark 2.2. The random processes introduced in this section are not random processes in the standard sense even if the algebra A = R. In this case, we deal with the generalization of the concept of random process under which generalized functions on C "` can be taken as finite-dimensional distributions. Random processes of this kind are encountered, for instance, when we consider Schrodinger equations. The distribution on a space of trajectories that corresponds to these processes may not be a probability measure.
2.4. Random processes. Let W(1) be a topological CSM consisting of functions G: I - A and containing a space of cylindrical functions U(1). The continuous A-linear functional µ: 'I(S2) -4A is a IV-continuation of the cylindrical distribution p if µ4u(n) = A. A random 'Y-process x(t) = xµ(t) with continuous trajectories is a distribution µ. E W' (1) which is a 'Y-continuation of the normalized cylindrical distribution p.
2. Random Processes on a Superspace
243
For any cylindrical quasi-Gaussian random process whose covariance functional is continuous on E x E there exists a continuation to
the 0-process, 0 = 4)(0). We shall use the symbol of the mean value M to denote the integral (in the generalized sense) with respect to the distribution µ.
Remark 2.3. If the algebra A = R, then the standard theory of random processes results if we take a space of bounded Borel functions on the space of trajectories SZ as the space W(Q).
2.5. Wiener process. We consider a finite-dimensional superspace C,,2k. On the corresponding superspace E we consider a quadratic form
c(v, v) = f f
min(t, s)(Mdv(t), dv3)),
where the covariation matrix M satisfies the constraints indicated in Sec. 3.5 of Chap. I. A random quasi-Gaussian process with a covariance functional c(v, v), v E E, is called a quasi-Wiener random process. Suppose that there exists a body projector in the CSA A. A quasiWiener random process for which bA > 0 is called a Wiener process on the superspace C" m. If the soul of the CSA A is quasi-nilpotent, then the results of Secs. 3.5 and 4.5 from Chap. I are applicable and we can write out finite-dimensional distributions of a Wiener random process.
Suppose that a finite-dimensional projector 7r on the superspace Q is defined by points, i.e., 0 = to < t1 < ... < ti, 7r = 7rtl...t,, 7rx = (x(t1), ..., x(t1)) E CA'21k. Then we have 1-1
µa(dx) = [((27r)n sdet M)1 fl (tj+1 - tj) n-2k F-
12
J
j=0
x exp{-9
(xj+1 - xi,
M-1(xj+1
- xj))/(tj+1 - tj)}dx.
j=0
Note that if n = 2k, then the time disappears from the normalizing factor.
Chapter V. Probability Theory
244
2.6.
Representing the solution of the heat conduction
equation on a superspace as a probability mean. We consider a Cauchy problem 09U
at n
+2
2k
1
(t, x, B)
=2
a2
n
2
[il Ai' ax ax; a2
2k
i=1=1 c=' axiae;
i,7=1
+Bi'
aeiae
u t, x, 0)
+v(x) 0)u(t, x, 0),
u(0, x, 0) = cp(x, 0),
t E R+.
(2.2) (2.3)
Theorem 2.2 (Feynman-Kac formula). Let the potential v(x, 0) = > ga(x)0', where ga are Fourier transforms of A-valued measures with a compact supports on Rn and let the initial condition cp(x, 0) belong to the class Then the probability mean t
u(t, x, 0) = MT exp{J v(x + w(T), 0 + e(T)) d7-} xcp(x + w(t), 0 + l;(t)),
where w(t) = (w(t),e(t)) is a quasi-Wiener random process, defines the solution of the Cauchy problem (2.2), (2.3). In order to prove this theorem, we must pass to Fourier transformations for the potential and the initial condition, write the integral with respect to the quasi-Wiener distribution also in terms of the Fourier transformation (on an infinite-dimensional superspace).
3.
Axiomatics of the Probability Theory over Superalgebras
We propose here the generalization of Kolmogorov's axiomatics to the case of probabilities with values in a CSA A and, in general, in an arbitrary Banach algebra.
3. Axiomatics of the Probability Theory
245
3.1. Measures with values in a Banach space. Let (SZ, a) be a measurable space and E be a Banach space. The mapping p of the a-algebra a into the Banach space E is called a vector measure if, for any linear continuous functional l on E, the set functions is a bounded (signed, in general) measure (a-additive) on (SZ, a) (i.e., a charge). The a-additivity of compositions of vector measure with all linear
continuous functionals on E (a weak a-additivity) entails a strong o-additivity, namely, p(U Aj) _ E µ(A3) in the sense of a normed i=1
i=1
topology on E for any contable family of pairwise nonintersecting sets.
3.2. Generalization of Kolmogorov's axiomatics to probabilities with values in a Banach algebra. Suppose now that E is an arbitrary Banach algebra over R with a unit element e. We want to obtain a generalization of Kolmogorov's axiomatics [38] to the case of probabilities with values in E. In accordance with Kolmogorov's axiomatics, probabilities are (a-additive) measures on a measurable space (1, a) which assume values in the interval [0, 1]; P(Q) = 1. It is obvious that the generalization of the concept of measure is a vector measure with values in the Banach algebra E. It remains to find out what serves here as an analog of the concept of a probability measure, i.e., a measure assuming values in the interval [0, 1]; P(11) = 1.
We propose the following approach to this problem.
Definition 3.1. A probability measure with values in the Banach algebra E is a vector measure p satisfying the following conditions. 1. For any set A E or the spectrum of the element µ(A) lies in the interval [0, 1], the spectrum being nonempty. 2. p(Q) = e, where e is a unit element of the algebra. Everywhere in what follows, we denote by Spec (A) the spectrum of the element A E E. Definition 3.2. The collection of objects (1, or, P), where (a) 0 is the set of points w;
Chapter V. Probability Theory
246
(b) a is the Q-algebra of the subsets of S2; (c) P is a probability measure with values in E, is called a E-probability model. We have thus proposed a generalization of a standard probability model. If E = R, then Definition 3.2 coincides with Kolmogorov's definition. In the E-probability model, the probability of the event A E a is an element of the Banach algebra E.
Example 3.1. We consider as E the algebra of continuous functions on the compact set T: E = C(T). Then Spec (p(A)) consists of the set of values of the function µ(A) (t), t E T, and the probability measures are vector measures p such that for all t E T we have the following: (1) 0 < µ(A)(t) < 1 for any t E T, (2) µ(S2) - 1. As usual, discrete measures are the simplest examples of probability measures. The general scheme for constructing C[a, b]-valued probabilities is as follows.
Let us consider an ordinary R-probability model (S2, a, P) and a random process x(t, w), a < t < b, with values in R and with continuous trajectories satisfying the following conditions: (1) for almost all w E S2 we have 0 < x(t, w) < 1 for all t E [a, b], (2) Mx (t, w) - 1. Then the vector measure PE(A) fA x( , w)P(dw) is a probability measure.
This example can be immediately generalized to the case of the algebra C(T) of continuous functions on the compact set T, the C(T)probabilities are constructed with the use of continuous random functions x(t, w), t E T.
In particular, if the set T is finite, then we obtain probabilities with values in R" corresponding to random vectors.
Remark 3.1. Example 3.1 will be important in the frequency interpretation of E-probability models (see Sec. 3.8).
Example 3.2. Let us consider an algebra Mats (n x n) of complex matrices n x n as the algebra E. Then the probability measures are the vector measures p for which the eigenvalues A (A) of the matrices µ(A), A E a, lie on the interval [0, 1] and µ(S2) is an identity matrix.
3. Axiomatics of the Probability Theory
247
We can again construct numerous examples of discrete Matc (n x n)-probabilities; the general construction is based on random matrices. Let a(w) be an ordinary random matrix, Ma(w) = e. Then, under certain constraints on the random matrix a(w), the vector measure PMatc(nxn)(A) = fA a(w)P(dw) is a probability. Into this example we can also include probabilities with values in finite-dimensional Grassmann algebras. Consider, for instance, an algebra G2 and its regular representa-
tion in Matc(4 x 4). Let ej(w), j = 0, ..., 3, be ordinary random variables satisfying the following conditions: (1) 0 < eo(w) < 1 a.e., (2) Meo(w) = 1, MCj(w) = 0, j i4 0. Then
Pcz (A) _
MAeO
0
0
0
MA1
MAeO
0
0
MAC2
0
MAeO
0
MAS3 -MAC2 MAe1 MAeO where MAej (W) = fA j (w)P(dw) is a G2-valued probability. The following example is an infinite-dimensional generalization of Example 3.2.
Example 3.3. Let X be a Banach space. Consider an algebra G(X) as the algebra E. The probability measures are those G(X)valued measures p for which the spectrum of the operators µ(A), A E a, lies on the interval [0, 1] and p(Q) is an identity operator. We can use discrete measures in order to construct these probabilities; the general construction is based on random linear operators. Into this example we can include probabilities with values in the CSA G1 00 by considering the regular representation of this algebra.
3.3. A spectrum of en event and the multivalued probability theory. The set Spec (P(A)) constitutes the real probabilities of an event A E a. We actually deal with the multivalued probability theory. Every event A E a is associated with a whole set of probabilities Spec (P(A)). If the spectrum of the element P(A) consists of one point, then the set of real probabilities reduces to one number, namely, the probability
248
Chapter V. Probability Theory
of the event A. Conversely, we can regard every classical probability as an element of a Banach algebra. It is not necessary to take the field of
real numbers as this algebra, we can, for instance, regard this probability as a diagonal matrix and, in general, as any element of a Banach algebra which has a one-point spectrum equal to this probability.
3.4. Splitting real probabilities. Another essential peculiarity of the new probability theory is that by extending the field R to the algebra E we can "split" the real probabilities of an event. We can demonstrate this in the most visual way by considering a zero probability. In the ordinary probability theory an event A of zero probability is not at all impossible, it is only "very rarely" realized. Thus, the zero probability comprises a large class of events which are "very rarely" realized. However, the frequency of realization of events of zero probability may differ rather considerably. It would be expedient to sort out somehow events of zero probability. We can do this by extending the number field R to the Banach algebra E. In contrast to the number field, a Banach algebra may contain arbitrarily many elements with a zero spectrum, i.e., the zero of the number field extends somehow to form a Banach algebra. We can now add various elements of the Banach algebra E which have a zero spectrum in the E-probability model to the two events A and B which have zero probability in the R-probability model. In the same way we can split any other real probability. Thus, from this point of view, the new probability theory makes it possible to study finer properties of a probability model which disappear under real approximation. Here we can draw an analogy with the quantum field theory. We begin with real probability (an analog of the free theory) and then consider the perturbation of this initial approximation by operator probabilities. This perturbation describes finer effects which were not taken into account under the first approximation.
3.5. The soul and the body of probability. We choose an algebra G' as a Banach CSA. Any element of this algebra can be represented as) = b) + c), where cA is a quasinilpotent element. Consequently, Spec) = {b)}.
3. Axiomatics of the Probability Theory
249
This means that all G.-probability models are single-valued. Any event A is associated with a unique real probability, namely, the body of the element P(A). Any G'-probability assumes values in the set [0, 1]e x c(G ,). Any event A E or whose probability belongs to the soul c(G'00) has a zero real probability, i.e., an element of the number field R is extended to the soul c(G100 ). In general, G1-probability models can be regarded as a splitting of ordinary R-probability models with a layer c(G'). For any G1-probability P we can distinguish a body of the probability Pb(A) = bP(A) and a soul of the probability PE(A) _ cP(A): P = Pb + PP. The body of the probability Pb is an ordinary probability and the soul Pc is the c(G100 )-probability. The real probabilities for the soul PP are zero.
Thus, we can represent the G'-probability model as follows. We have an ordinary probability model (S'l, or, Pb). Then we extend
the zero element of the field R to the algebra c(G100), i.e., we ascribe new values P(A) = Pc(A) to the events of zero probability, Pb(A) = 0; in the same way we extend the other probabilities. If the CSA A # G', then the spectrum of the elements of the soul may be nonzero and the real probabilities may be multivalued.
3.6. Conditional probabilities. Independent events. The majority of concepts and constructions for R-probability models can be easily extended to E-probability models.
Suppose that (SI, a, P) is a E-probability model, A, B E a, and P(B) is an invertible element of the algebra E. The right (left) conditional probability of the event A relative to the event B is an element
P,.(A/B) = P-1(B)P(A n B) (P,(A/B) = P(A n B)P-1(B)). Note the formula that connects the right and left conditional probabilities, P,-(A/B) = P-1(B)P1(A/B)P(B).
Proposition 3.1. Let E be a subalgebra of the algebra of continuous functions C(T) or E = G. Then the conditional probabilities are E-valued probabilities.
Chapter V. Probability Theory
250
Proof. Consider the case E = G. We have Pr(A/B) = (Pb(B) + PC(B))-1(Pb(A n B) + Pc,(A n P)), and, consequently,
Spec (P,.(A/B)) = {Pe 1(B)Pb(A n B)} E [0, 1].
An event A is said to be right (left) independent of an event B if
P(A n B) = P(B)P(A) (P(A n B) = P(A)P(B)). If the event A is right independent of the event B, then the event B is left independent of the event A and vice versa. If the event A is both right and left independent of the event B, then the events A and B are independent. The probabilities of independent events commute. If the probability of the event B is invertible, then the event A is right (left) independent of the event B if and only if
Pr(A/B) = P(A) (P1(A/B) = P(A)). We must also note the formula for total probability of E-probability models. Let {A1, ..., An} be a complete group of incompatible events and P(Ai) > 0, i = 1, ..., n. Then n
P(B) _
P(Ai)Pr(B/Ai)
P(B n Ai) _ i=1
i-1
n
Pt(B/Ai)P(Ai) i=1
As usual, this gives the Bayes theorem (we give the right version)
PP(Ai/B) = P-1(B)P(A n B) n
_ [E P(Ai)Pr(B/Ai)]-1P(Ai)Pr(B/Ai) i=1
Let us generalize the concept of independence to n events.
Let or = (jl, ..., jn) be a permutation of the subscripts (1, ..., n). The events A;, j = 1, ..., n are said to be or-independent if P(nA;) =
3. Axiomatics of the Probability Theory
251
The events A3, j = 1, ..., n which are a-independent for any permutation a are said to be independent.
3.7. Random variables. The spectrum of expectation. A random variable is any measurable map C: (S2, a) --4 (E, p), where Q is a a-algebra of Borel subsets of the algebra E. The expectation of the random variable C(w) is the integral
M = fC(w)P(dw).
(3.1)
We shall not discuss here the mathematical definition of integral (3.1). Note that this integral can be embedded into the general theory of a bilinear integral on locally convex spaces (a bilinear form is defined by the multiplication operation in algebra). The spectrum Spec (MC) constitutes the real expectations of the random variable C(w). Here again we encounter multivalued quantities. However, if E = G1, then the real expectation (just as the probability) is a single-valued variable since Spec (MC) = bMC. In
this case, the random variable can be decomposed into the sum of the body Cb(w) = bC(w) (of an ordinary real random variable) and the soul C ,(w) = ca(w) (of c(G')-valued random variable), and we have relations bMC = MbCb
=
f
Cb(w)Pb(dw),
cMC = fCc(41() + f Cb(w)Pc(dw) We can represent the random variable in G' as the result of a perturbation of the ordinary random variable Cb(w) by a certain random variable Cc(w) which is nonzero although the spectrum of this variable is zero, i.e., C ,(w) "assumes a zero real value."
3.8. Frequency interpretation. As is known (see Kolmogorov [38]), the axiomatic probability theory was preceded by the frequency probability theory. All Kolmogorov's axioms reflect some properties of relative frequencies. The most systematic exposition of the frequency probability theory was proposed by von Mises [47]. His frequency theory is based on the concept of a collective.
Chapter V. Probability Theory
252
Let S be an experiment with a set of outcomes H. For simplicity, we shall assume that the set II is finite, II = {irl, ..., 7rn}. If we repeat the experiment N times and record the outcome after each experiment, then we get a finite sample x = (X1, ..., xn) in which we can calculate the relative frequencies v' = n2 /N, where n1 is the number of realizations of the outcome 7rr in the first N trials. A collective is a mathematical abstraction of a finite sample, it is an infinite sequence x = (x1; x21 ...) xm, ...),
(3.2)
where xj E II, for which there exists a limit of the sequence of the relative frequencies
P = lim v3, N-oo
v3 = n3 IN,
for each outcome irk (there occurs a statistical stabilization of relative frequencies). This limit is called a probability of the outcome 7rj. In the frequency probability theory, a collective is regarded as a
fundamental object and the whole frequency probability theory reduces to various operations performed on the collectives. Consider now an infinite sequence of outcomes (3.2) for which there is no statistical stabilization of relative frequencies for some characteristics Irk. Such a sequence is not a collection and, consequently, cannot be regarded as an object of the frequency probability theory. However, a sample of this kind also carries some information concerning the event which is investigated in the experiment S, and it would not be wise to reject all events in which there is no statistical stabilization. We propose the following formalism (which is, naturally, only the first step in this direction). For every characteristic Irk we denote by Spec (vk) the set of limit points of the sequence of relative frequencies {vk}. The probability of the characteristic -7rk is an element of a Banach algebra E whose spectrum coincides with Spec (vk). The algebra E and the rule according to which every characteristic is associated with an element of the algebra are defined in accordance with the properties of the probability model in question.
3. Axiomatics of the Probability Theory
253
This formalism does not cover the case of an empty set Spec (vk). However, we can include this case into the formalism considering elements with an empty spectrum in the definition of the E-probability model, and then the characteristic Irk for which Spec (vk) = 0 is associated with an element Ak E E with an empty spectrum. In the following model we can also arrive at multivalued probabilities.
Suppose that we have a sequence of instances of time t E T and every instance is associated with a collective It = (xlt, ..., X t) ...). For each instance t we calculate relative frequencies vk (t) = nk (t) /N
and probabilities Pk (t) = N-too lira vk (t). We obtain a discrete vector probability on the set II of characteristics. If, for any one of the characteristics, the function Pk(t), t E T is continuous, then it is a C(T)-valued probability (see Example 3.1). In Secs. 1 and 2 we considered more general probability models on a superspace. Distributions that we came across in these models
are not a-additive A-valued measures. Even if the algebra A = C, they are complex-valued distributions of, in general, an unbounded variation. These distributions and random processes are similar, in many respects, to constructions from the theory of quantum random processes [36].
With respect to the degree of complication of the mathematical formalism, the results obtained in this chapter should be arranged as follows: the frequency probability theory, the analog of Kolmogorov's axiomatics, the generalized probabilities from Sec. 1, and the generalized random processes from Sec. 2. We have exposed the results in the reverse order because the most interesting results have been obtained precisely for generalized Gaussian-type probabilities. It is also important that quasi-Gaussian distributions play a significant part in applications to the quantum field theory. No interesting models have been obtained as yet for or-additive A-valued probabilities. However, we can choose or-additive A-valued probabilities considered in this section as initial distributions p in the limit theorems from
Chapter V. Probability Theory
254
Sec. 1. The frequency interpretation for Gaussian distributions on a superspace can be realized in the same way as in the ordinary probability theory. Using the central limit theorem, we can represent a Gaussian distribution on a superspace as the distribution of a sum of infinite number of discrete A-valued random variables. The frequency interpretation is valid for these variables, and then the Gaussian distribution on a superspace is approximately considered to be equal to the approximating distribution with a sufficiently large number.
4.
Unsolved Problems and Possible Generalizations
We can see from the content of this chapter that only the first strokes with a paint-brush have been put in the probability superstructure. I can formulate some problems which we are of the most interest.
Limit theorems. 1. The superanalog of Lyapunov's theorem for the distributions µnk with noncommuting values. 2. Limit theorems, in which a limiting process takes place for the functions cp, which are continuous and bounded with respect to commuting variables and which are polynomials with respect to anticom-
muting variables. The most interesting case here is that of a "purely Gaussian" distribution on a superspace. Nothing is known even in a finite-dimensional case, even for the simplest Gaussian distribution 'YB,
0
0
0
0 -1
0
1
0
1
B=
3. Limit theorems for dependent random variables. 4. Infinitely divisible distributions on a superspace.
4. Unsolved Problems
255
Random processes. 1. Stochastic differential equations and diffusion processes (at least on a finite-dimensional superspace). This seems to be the most interesting problem. A Wiener process already exists. The stochastic differential equation can be written in the space of (D(1l)-processes de(t) = a(e(t))dt + b(e(t))dw(t).
Now we have a problem of the existence and uniqueness of a solution in the space of 4) (1)-processes. 2. The Ornstein-Uhlenbeck process on a superspace and Malliavin calculus on a superspace connected with it. 3. Investigation of parabolic equations on a superspace with the aid of the theory of random processes. 4. Poisson superprocess. 5. Relationship between the theory of quantum random processes and the theory of random processes on a superspace. It seems to me that we can obtain here something like a correspondence principle.
Frequency interpretation. 1.
Of a considerable interest is a systematic exposition of the
frequency supertheory on a mathematical level of strictness (a superanalog of Mises theorem). 2. Construction of specific examples of the use of E-probability model in natural sciences. Here the ideas can be realized of an extension of the number field R, in which the standard probabilities assume values, to a Banach (or even topological) algebra.
Remarks Sec. 1. The results of this section were published in [154, 161]. The central limit theorem for Feynman distributions on real locally convex spaces was formulated in the article by Smolyanov and the author [128]. A different version of the central limit theorem for Feynmann distributions was obtained by Ktitarev [108]. Krylov's articles [106] seem to be the first in which the central limit theorem was obtained for noncountably additive distributions.
256
Chapter V. Probability Theory
Sec. 2. The results of this section were published in [161]. The Brownian motion on an infinite-dimensional superspace was introduced in [144], the Feynman-Kac formula was obtained in the same article. Independently, the Brownian motion on a finite-dimensional phase superspace was constructed by Rogers [122]. Sec. 3. Here the ideas were realized and used which I proposed when constructing a p-adic probability theory [169]. In general, the Mises frequency theory of probability (almost forgotten by now) is the most powerful method for the development of new probability formalisms. In the p-adic probability theory the reasoning was carried out according to the same scheme. A general principle of statistical stabilization of relative frequencies was advanced. By virtue of this principle, the convergence of sequences of relative frequencies can be considered not only in a real topology on the field of rational numbers (and all relative frequencies are rational) but also in any other topology. A random computer modeling was carried out in the p-adic probability theory as a result of which random samples were obtained for which statistical stabilization of relative frequencies does not exist in the field of real numbers but exists in the field of p-adic numbers. Here p is chosen in accordance with the properties of the probability model in question. The choice of algebra E plays a similar part in the formalism from Sec. 3. In the p-adic probability theory, the probability of an event may be a negative number, an imaginary unity, a natural number, exceeding unity, so that
the p-adic probability theory is, essentially, an intermediate step on the way from R-probability models to E-probability models.
Chapter VI
Non-Archimedean Superanalysis
Traditionally, all constructions of mathematical physics were carried out over the field of real numbers R. However, a different point of view is also possible according to which on fantastically small distances (of order 10-33) the space-time has non-Archimedean structure and, consequently, cannot be described by real numbers. The philosophy and ideology of non-Archimedean physics were laid as a foundation by I. V. Volovich (1987), he also advanced an invariance principle (which got the name of the Volovich invariance principle). By virtue of this principle, rational numbers formed the experimental basis for any physical formalism, and physical formalism must be invariant with respect to the choice of the completion of the field of rational numbers. Thus, along with real physical theories, certain theories were worked out over other number fields, in particular, over fields of p-adic numbers (see [21]).
In order to describe non-Archimedean fermions and non-Archimedean superfields as well as non-Archimedean (and, in particular, padic) superstrings, we need a non-Archimedean generalization of superanalysis. The first work in non-Archimedean superanalysis was the article by Vladimirov and Volovich [19] in which the authors considered a superspace over an arbitrary locally compact field. In this chapter I expose my version of the theory of generalized functions, partial differential equations and Gaussian (continual inclusive) integrals on a non-Archimedean superspace (both finite-dimen-
258
Chapter VI. Non-Archimedean Superanalysis
sional and infinite-dimensional).
1.
Differentiable and Analytic Functions Recall that the absolute value (valuation) on a field K is said to be
non-Archimedean if, instead of the inequality Ix+ y J K< I x J K+ I y J K, a stronger inequality, namely, Ix + Y K < max(IxIK, I yI K), is satisfied. An absolute value is non-Archimedean if and only if I n I K <_ 1 for all
elements n from the ring generated in K by its unit element. If JXJK =
1 for all x E K*, then the absolute value is said to be trivial. The absolute value of the field K is a homomorphism of the multiplicative group K* of the field K into a multiplicative group R' of the field R. We denote the image of this homomorphism by F. Everywhere below we denote by K a complete non-Archimedean field (Char K) = 0 with a nontrivial absolute value.
Example 1.1 (fields of p-adic numbers). Any rational number x 0 can be uniquely represented as x = p" m, where (p, n) = (p, m) = 1. Here p is a fixed prime number, p = 2,3,5... Each p is associated with its own field of p-adic numbers. The absolute value for the rational
number x is defined by the relations jx 1P = p-", x # 0, 101p = 0. This absolute value is non-Archimedean. The completion of the field of rational numbers relative to the metric p(x, y) = Ix - ylP is called a field of p-adic numbers and is denoted by Q,,. This field is locally compact. It was pointed out that InIK < 1 in a non-Archimedeand field, and therefore 1 / ln! I K does not decrease but increases with the growth of n. For what follows, we must know the order of growth of 1 / I n! I K. It is known (see, e.g., Borevich and Shafarevich [11]) that the growth is exponential for K = QP 1/In!IP < pn/(P-1).
(1.1)
It should also be pointed out that there are not very many possibilities for the construction of non-Archimedean absolute values on the field of rational numbers Q. By Ostrovskii's theorem (see, e.g.,
1. Differentiable and Analytic Functions
259
[11, 49, 72]), any absolute value of the field of rational numbers Q is equivalent either to one of the p-adic absolute values or to a real absolute value. Consequently, there are no other completions of the field of rational numbers except for the field of real numbers and fields
of p-adic numbers (if we consider only completions with respect to metrics defined by absolute values). It stands to reason that there exist non-Archimedean fields which are finite and infinite extensions of the fields of p-adic numbers. Using inequality (1.1) and Ostrovskii's theorem, we find that 1 / n! I K grows exponentially in any K, 1/In!IK = 1/In!IP < pInl(p-1)
Suppose that the quadratic equation x2 - T = 0, T E K, has no solutions in the field K. We denote the quadratic extension K(f) by Z. The elements of Z can be represented as z = x + fry, x, y E K; the conjugation operation in Z is defined by the relation z = x - Vf'ry; the absolute value on Z will also be denoted by I IK, it is defined by the relation JzI K = Iz21K = 11x2 7-y2I K. We shall use the symbol z 12 to denote the square of length of the element z E Z: I z 12 = zz (it belongs to the field K).
-
Example 1.2. Let K = Qp. Then there exist seven different quadratic extensions of the field Qp if p = 2 and three if p # 2. As distinct from an Archimedean case where the quadratic extension R(i), which is a field of complex numbers, is algebraically closed, the quadratic extensions of field of p-adic numbers are not algebraically closed. Extensions of any finite order are not algebraically closed either. A non-Archimedean norm on a K-linear space E is a map II . E -+ R+ satisfying the following conditions: (1) IIPxII = IAIKIIxII, A E K, x E E, (2) Ilx + yll < max(Ilxll, Ilyli), II
(3) IIxiI
=0s.x=0.
A complete linear normed space whose norm is a non-Archimedean is known as a non-Archimedean Banach space. A non-Archimedean
Chapter VI. Non-Archimedean Superanalysis
260
Banach algebra and a non-Archimedean Banach commutative superalgebra (CSA) are defined in the same way, cf. Chap. I. A non-Archimedean Banach CSA is denoted by A = AO ® A1. In order to construct a rich non-Archimedean superanalysis on a CSA A, we must impose the following natural constraints 1. IIx®Oil =max(IIxII,II0II),xEAo,0EA1. 2. Either the algebra A is commutative (i.e., A = Ao and Al = 0) or the A1-annihilator in the algebra A is trivial. 3. The set rno = { I I A I I: A E Ao } coincides with r.
The removal of these constraints complicates considerably the development of non-Archimedean superanalysis.
Example 1.3. Let B be a commutative non-Archimedean Banach algebra with a unit element and a norm 11 The CSA G,,.(B) = A°°B can be introduced by analogy with an Archimedean case (see Chap. I). Different subalgebras of the CSA GA(B) are the main model examples of a CSA which are used in non-Archimedean superanalysis. In Archimedean superanalysis, a significant role was played by the algebras G00' (B). However, the 11-norms do not satisfy the strong triangle inequality and are not Archimedean. We must consider oo-
I
I
.
norms of the type of a supremum or maximum. We denote by GOO (B) the subalgebra of the CSA Goo(B): GOO 00
(B) = if c Go.(B): If Il,,. = Sup If Ili < oo}.
The CSA GOO (B) is a non-Archimedean Banach CSA satisfying conditions 1 and 200for any algebra B. Condition 3 for the CSA G00" (B) is satisfied if rB = r and the algebra B possesses the following property: for any bounded sequence {b,,} of elements of this algebra sup IlbnIl E FB n
Let, for instance, B = K = Qp. Then sup,, IIbnIIP = supnp "n E r = {p": v = 0, ±1, ±2,...}, i.e., the CSA G'(Qp) satisfies condition 00 3. Now if B = K = Qp is an algebraic closure of the field Qp, then r = {pr: r E Q} and, consequently, supra IIbnIIQ; = Supra p-r^ does not, in general, belong to r, i.e., the CSA GOO 00 (QP) does not satisfy condition 3. Similar arguments hold for the field K = CP = Qp which is a completion of the algebraic closure.
1. Differentiable and Analytic Functions
261
We denote by G° (B) the subalgebra of the algebra G'00 (B): G°00 (B) = {f E Gm(B): lim Ilfill = 0, IiI = i1 + ... + in}. Let us verify, for instance, that Go00 (B) is an algebra. Let f, g E Go 00 (B) and cp = f g, and then
f
21 < ... <
Zn, 2s
= I'm U at;
where a7Q is a signum function. Using the fact that B is a nonArchimedean Banach algebra, we obtain Il
i1...in ll < Max 11f71...7k II yua=i
ll'
Furthermore, since f , g E Go (B), there exists N = NE : I I f7 , I I g,, I I < El max(llf II., ll MIA) if I'yl > n and lal > N. Suppose now that lil > 2N. Since Iil = Iyl + Ial, it follows that either I-yI > N, or a > N and, consequently, IIcjII < e, i.e., cv E G°00 (B), with IIcIl < If ll00IIgMI00. It is obvious that the CSA Go (B) satisfies conditions 1-3 for any Banach algebra B for which rB = F. The monomials {gi1...gin } form a topological basis in the K-linear space Go00 (K). I
I
Remark 1.1. A series in a non-Archimedean normed space converges if and only if its general term tends to zero. This follows from the strong triangle inequality. Just as in Chap. I, we introduce a non-Archimedean Banach superspace KK,m, the norm IIull = max Ilnj II being non-Archimedean, 1<j
Chapter VI. Non-Archimedean Superanalysis
262
We denote by Ap = A(UP x Am, A) the space of functions which are S-analytic at zero and for which Taylor's series
f (x, 0) = E
1
ack+vf
a! axaaOA (O)x'O
converges on Up x Am, i.e., lim
1a1-+00
f
PI«1
axa800 (0)
la!lx
=
0.
In the space Ap we introduce a norm ac,+'O f
llfllp = max axctaeo (0) This is a non-Archimedean Banach space.
In the same way, we introduce spaces of S-analytic functions A(Up', x ... x Upn x Am, A') with norms aa1+...+ak+fl f
llf Ilpi...pk = ma x
P1
...Pk°kf
axl1...8xkkao (0) lal!...ak!lK
We use the symbol Ao - AO(KA'm, A) to denote the space of all functions which are S-analytic at zero, the space being endowed with an inductive topology Ao (KA'm, A) = lim ind A(U x Am, A). p-,0
By the symbol A - A(KA'm, A) we denote the space of S-entire functions f : KA'm - A which is endowed with a projective topology: A(KA'm, A) = lim proj A(UP x A', A). A non-Archimedean prenorm is a function ll ll satisfying conditions 1 and 2 from the definition of a non-Archimedean norm. A topological
1. Differentiable and Analytic Functions
263
K-linear space in which the topology is defined by a system of nonArchimedean prenorms is called a non-Archimedean locally convex space. A metrizable locally convex space is known as a Frechet space (the topology on a Frechet space is defined by a countable system of prenorms).
Non-Archimedean Banach, topological and locally convex nonArchimedean CSM are defined as in an Archimedean case.
Proposition 1.1. The functional spaces A0 and A are complete non-Archimedean locally convex CSM; A is a Frechet CSM.
Proposition 1.2. The differentiation operators az , Ap are continuous, and the inequalities
akf axe
< I!IKIIfIIP, p
of
aoj
a:
AP -+
< Ilfllp p
hold true.
Proposition 1.3. The space Ap is a non-Archimedean Banach CSA, and the inequality IIf911P <_ llf llPll9llp
(1.4)
holds true.
The proof of these propositions is based on the strong triangle inequality for a non-Archimedean norm and on conditions 1-3 imposed on the norms on the non-Archimedean CSA A (when considering higher-order derivatives, we use the fact that l Cn l K <_ 1). The reader can find complete proofs in [160].
Corollary 1.1. Differentiation operators are continuous in the functional spaces AO and A.
Corollary 1.2. Functional spaces AO and A are topological algebras.
Example 1.4. (an infinitely S-differentiable function of anticommuting variables which is not a polynomial). Let K = Qp and let f (t) be a Dieudonne function. It is only important for us that this function
Chapter VI. Non-Archimedean Superanalysis
264
is differentiable on it E Qp: ItIp < 1} with values in Qp and its deriva-
tive is zero (and the function is not a constant). Let A = Go (Qp). Then 0 = 01Q1 + ... + 0123g1g2q3 + ..., Oil... ,I E Qp. We set W(0) = f (01),
and then this function is infinitely S-differentiable (it is Frechet differentiable, and the derivative is determined by the multiplication of the CSA A by zero), but this function is not a polynomial.
2.
Generalized Functions
We choose functional spaces AP, A0, and A as spaces of test functions on a non-Archimedean superspace. The conjugate spaces A'P, Ao, and A' obtained in the same way as in an Archimedean case by means of the identification of the spaces of A-linear functionals which are right and left continuous are spaces of generalized functions on a superspace KA''" For an infinite-order differential operator
P
_
-
P°x,
p
0,3
9
ax-aoa
P,,E A '
Q
where 6(x, 0) is a Dirac b-function on KA'm, we set IIPIIP = sup IIPaA11 Ia!JxpH0I, a$
P E F.
We introduce a space of infinite-order differential operators DP = {P: IIPIIP < 00}Proposition 2.1. The equalities A' = lim ind DP, Ao = lim proj DP hold true.
Proposition 2.2. The equalities A" = A, Ao = Ao hold true. The proof of these propositions is based on the fact that Taylor's series for S-analytic functions converge in the corresponding spaces of test functions. The reader can find the complete proof in [160, 163]. As in an Archimedean case, the differentiation operators in the space of generalized functions are defined by the relations ag aLSP aR9 W)
(g,
W)
= (g, a0j
2. Generalized Functions
265
Corollary 1.1 implies
Proposition 2.3. Differentiation operators are continuous in spaces of generalized functions.
Corollary 1.2 implies
Proposition 2.4. The operation of multiplication by a test function is continuous in spaces of generalized functions.
In what follows, we shall use an integral notation for the action of a generalized function on a test function. As usual (see Chap. II), we introduce the convolution operation and the operation of direct multiplication of generalized functions as well as the operation of convolution of a generalized and a test function.
Proposition 2.5. The operations of direct multiplication and convolution of generalized functions are continuous and supercommutative.
Proof. Let us verify the continuity of a direct product. Let gg E A(UP' x Am', A), j = 1, 2, f E A(U,,1+n2 x Am' +M2 A), and then f (xi, x2) 0l, 02) aa1+a2+Q1+Q2 f
1
a10201,62
Ofl!GY2! axllax22ae1
al a2 Vl qq
22
qq
(0)x1 x2 Bl e2
,
with 0')a1+a2+p1+p2 f
PI a11+1a21
llm 1a1!IKIa2!IK
axl 1 ax22 ae21 ae22
(0)
= 0.
We shall show that the function (92, f)(xl, 01) belongs to the space A(U,, l x Amt, A). Note that 92 = g2 E) 921, where Ig2I = 0, Ig2I = 1. Furthermore, al+a2+r1+Q2 f
PIa11
Ial!IK 1a 1: a2! aa1+a2+/31+r2
+aI
axi1ax22a013qq1ae22
f
axl 1 ax22 ae101 ae22
(0)
0 xa2 eAz )
(g2,
2
2
(0))(gl,x22e22)](-1)101111621
Chapter VI. Non-Archimedean Superanalysis
266
'U1+a2+Q1+#2f
Phil+Ik2I
!5 sup A la1! Kla2! K axi1ax22aepla0 2 11
x sup 11(92, x22e2
(O)M
IIP-I-012
°2132
(we have used the fact that the parity operator a is an isometry of the non-Archimedean Banach CSA A, see property 1 for the norm on A). Note that 119211, = sup 1192, x'01) IIP-I'l Next, we have V E > 0 3 N aQ
IalI+Ia21>N a01+02+fl1+A2 f
PIQ11+1021
<E
lal! Kla2!IK axi1ax22aep1ae22 (0) II
Let 1 a1 I > N. Then I a1 I + 1 a2 I > N for all a2 and, consequently, aOl+02+01+132 f
P101I+I02I
sup
A la1!IKI&2!IK axilax22agA1ae22 11
(0) 11
< E.
Thus we have lim
PIQ
a0l+rl (92, f)
ll
Iall-+00 Ia1!IK
11
axilaoAl
(0)
0,
with II (91®92,f)11
sup
a°1+Q2+p1+02 f
1
(0) < 01a A A2 Ia1!IKIa2! K axilax2zae21ae02 2 I
x11(91,x°10")II II(92,x22e22)II < II91IIPII92IIPIIfIIP
A similar proposition is valid for the convolution of a test and a generalized function.
3. Laplace Transformation
267
Laplace Transformation
3.
In a non-Archimedean case, it is more convenient to use a Laplace transformation rather than a Fourier transformation. Here, in contrast to an Archimedean case, the quadratic extension is not unique and it is more convenient not to employ quadratic extensions at all.
Definition 3.1. The Laplace transform (two-sided) of the generalized function g E AD(KK'm, A) is defined by the relation
L(g)(y, ) = J
g(dxd9)e(x'y)+(B,E)
Theorem 3.1. The Laplace transformation L: .A' (Kn'm, A) -+ A(Kn'm, A) is an isomorphism of a Frechet CSM. Proof. In the proof we shall use estimate (1.2). Let g c A' (Kn'm). Then L(g)(y,0 = E
1
C,p
Y.
and
IIL(g)M1 = scup Ia
11(g,XQ9 )IIP1'1 <
Let f c A(Kn'm). We set (g, x`Bp) = 8
,90
(0). Then
a°+p f
f
1
1P-'a, = Ilf HP p Ia1!IK ax0aep (°) The proof of the fact that Ker L = {0} repeats the proof for an 11
1
Archimedean case. By virtue of this theorem, we have a harmonic analysis
AD(Kn'm, A) - A(Kn'm, A), A0(KK'm) A) 4- A' (Kn'm, A).
(3.1)
Chapter VI. Non-Archimedean Superanalysis
268
By definition, the Parseval equality holds (see Chap. II). The Laplace transformation possesses the usual properties, namely, it transforms a convolution and a direct product into a product and derivatives into a multiplication by variables. belongs to the space Note that the function fx®B(y, l;) = Ap and A' = U A'',. Ap if 1I'M < (ppl/(p-1))-1. Furthermore, A = e(x,y)+(e,0
pE
Let M E A' and let p E r be such that p E A',. By virtue of the Parseval equality, we have L'(p)(x, 0) = (bxeo, L'(p)) = (L(5x®B), p) = (fx®e, p)
=
r e(X'Y)+(e,0p(dydC)
(3.2)
for any point x ® 0 E KA'm, IIxII < (pp'/(p We introduce one more functional space aa+Of
G6 = {f E Ao: Illflll6 = su Note that G6 C
n
ax aae
Ibici (O)
< °O}.
AP and, in particular, G6 C Ao. (p<6p'/(1-P))
Theorem 3.2. The adjoint Laplace transformation L': A'' -4 G6i b = p-' is an isomorphism of Banach CSM. Note that the space Ap is not reflexive if the absolute value I IK is discrete (for instance, for p-adic numbers). Therefore the operator L" transforms the space G'P_1 not into Ap but into a wider space A'P. Then we have a Laplace calculus
A'AA'' GP-1 CAa, APCA'P G''-1 Ao.
(3.1a)
By virtue of the standard properties of adjoint operators, we have L"(g) = L(g) for the element g E Ao fl Gp_1. The Parseval equality is
f g(dxd9)L'(p)(x, B) = f L"(g)(x, 0)p(dxd6),
4. Gaussian Distributions
269
g c G'-1,
p c A'*
(3.3)
Note that L" (g) (x, 0) is not simply a symbolical notation. The elements of the space A'P can be realized as functions on an open-andp, and then 6.,®o E A'P, closed subset of a superspace. Indeed, let x and, by relation (3.3), L"(g)(x, 9) = (L" (g), Sx®e) = (g, L'(8x®e)) =
f
g(dy<)e(y,x)+(E,B)
Thus, the function L"(g) is defined on the set UP x Am. It is easy to verify that this function os S-analytic on the set (Up )" x Am, where Up = {x c Ao; JIxjj < p}. Remark 3.1. Concerning the reflexivity theory for non-Archimedean locally convex spaces see, for instance, [72].
4.
Gaussian Distributions Suppose that the matrix B =
Boo
Blo
B°
,
E A, satisfies
the same constraints as those used in the definition of a Gaussian distribution on an Archimedean superspace, the vector a E A. The Gaussian distribution on K""' is defined as a generalized function from the space A'(KK'm, A) for which the Laplace transform L'('ya,B)(w) _ exp{ (Bw, w) + (a, w) }. 2 Under certain constraints on the covariation matrix B and the
mean a, the Gaussian distribution 'ya,B can be extended from the space of S-entire functions A(K;,'m, A) to wider spaces A(U' x Am, A). Let us use the Laplace calculus (3.1a). If the Laplace transform L'(ya,B) of the Gaussian distribution ya,B belongs to the class GP-1 C A0i then ya,B can be extended to the space AP. A detailed description of these
extensions and the discussion of the problem of normalization of a Gaussian integral can be found in [163] (see also [160]).
Chapter VI. Non-Archimedean Superanalysis
270
Duhamel non-Archimedean Integral. Chronological Exponent
5.
Everywhere in this section we denote by E a Banach module over a (commutative) Banach algebra A0; the symbol LA, ,(E) is used for the space of A0-linear continuous operators in E. As usual, we define S-differentiable and S-analytic mappings from Ao into E or LAD(E). The spaces of S-analytic functions on a ball Up, p E r, are denoted by A(UP,E) and A(UP,Lno(E)). The integral fa f (t) dt for the analytic function f : U,, - E is defined by the relation 16 a
tndt = (bn+1 -
an+1)/(n
+ 1),
a = aP,
b = bp.
Lemma 5.1. Let the function cp(t, s) belong to the class A(U,,, E). Then the function u(t) = f0 W(t, s)ds belongs to the class A(U6, E),
b < p, and IIUIIo < C(p, b, K)IIkIIP.
Proof. We shall prove that in the space A(U6, E) the series u(t) n,m=0
a
(0)/n!(m + 1)! _ E Unm(t) n,m=0
converges, for which purpose we shall estimate the general term of the series an+m W11 Ilunmllb = 6n+m+l &tnasm
(0)II/In!(rn + 1)!IK
I)n+m/Im+1IK.
It remains to note that 1/InIK < ne, a = a(K). Recall that an evolution operator of a linear differential equation in the Banach A0-module E (with evolution parameter t E A0) Wi(t) = a(t)x(t),
a(t) E LAD(E),
(5.2)
5. Chronological Exponent
271
is a family of operators V (t, T) E LA, ,(E) which is a solution of the Cauchy problem
V(t,T) = a(t)V(t,T),
V(T,T) = I.
Theorem 5.1 (Duhamel integral). Let the functions f (t) E,A(U",E), a(t) E A(UO, Lno (E)); V (t, T) constitutes an evolution family of the class A(UP, LA,,(E)) of Eq. (5.2). Then there exists a unique solution of the class A(U6i E), b < p, of the Cauchy problem for the nonhomogeneous equation
x(t) = a(t)x(t) + f (t),
x(O) = xo,
(5.3)
x(t) = V (t' 0)xo + f V (t, r) f (T)dT.
(5.4)
defined by the Duhamel integral t
Theorem 5.1 is a direct corollary of Lemma 5.1.
Corollary 5.1 (a linear homogeneous equation with constant coefficients). Let the operator b E Lno(E); the function f (t) E A(Up, E), where p < (IIbIIp1/(p-1))-'. Then there exists a unique solution of the class A(U6i E), 5 < p, of the Cauchy problem 1(t) = bx(t) + f (t),
1(0) = xo,
(5.5)
defined by the Duhamel integral x(t) = ebtxo + f t e(t-s)b f(s)ds.
Theorem 5.2 (chronological exponent). Let the operator-valued function a(t) belong to the class A(Up, Lno(E)). Then there exists a unique evolution family V (t, T) of the class A(u6 , Lna (E)), where
Chapter VI. Non-Archimedean Superanalysis
272
b<
pi/(P-1)
min(p, IIaIIp 1), of Eq. (5.1) defined by the chronological
exponent V (t, T) = T exp (
f
00
_ Ef
t
a(s)ds}
'n-1 a(s1)ds1...
n=0 Tit
(sn)dsn.
JrT
(5.7)
Proof. We shall show that the function V (t, T) defined by the relation (5.7) belongs to the class A(UU, Gno (E)). Note that V (t, T) 00
E Vn (t, T) n=0 00
00
J1=0
in=0
Vn(t,T) = E ... E ra(il)(T)...a(in)(T)/jl!...in!] X( t - T) )1+...+in+n /(in + 1) (in +jn-1 + 2)...(jn + ... +
1*1 + n)
Vnj (t,
i Using inequalities (1.3) and (1.4), we obtain I I VnjI I <
IIa(j1)
I
I 6 . . . II
a(in)
II6II (t
7-)j1+...+in+n
II61
Ij1!...jn!IK I (in + 1)...(jn + ... + jl + n)I K < IlaIIPPP
1+...+7n+np-(31+...+in)/I(jn + 1)...(j1 + ... + in + n)I K
<
(U \ )1+...+Jn -J1 (IIaIIPb)nII(j1+...+jn+n)!IK
Using estimate (1.2), we get (ap`/(P-1)
)il+...+7n(IIaIIP5p`/(P-1))
II Vnj 116 <
1
n.
P
Thus, lim I I Vn; I I6 = 0. Consequently, the series > Vni converges in Ii1+00 i the space .A(U2, LAO(E)), and we have the estimate for the sum of the series IIVnII6 <
(6p'1(P-1)IIaIIP)n.
6. Cauchy Problem
273
This estimate implies the convergence of the series E Vn in the space n
A(1
,
LA. (E) )
Theorem 5.3. Suppose that the operator-valued function a(t) belongs to the class .A(U'1, ICA,(E)), and the vector-function f (t) belongs to the class A(UPZ, E). Then there exists a unique solution of the Cauchy problem (5.3) of the class A(U , E), where
6 < min(p2, defined by the Duhamel integral
x(t) = T exp{ fo t a(s)ds}xo +
p1pLl(1-P), IIaIIp1p!1(1-P)),
f T exp{ f t a(s)ds t
} f (7-) d7-.
In order to prove this theorem, it suffices to note that the function f (t) and the evolution family V (t, T) = T exp{ f1t a(s)ds} are Sanalytic on a ball of radius J. We have thus constructed an analog of the chronological exponent in the additive representation (5.7) of the Peano chronological calculus [117]. There is, evidently, no non-Archimedean analog of a multiplicative chronological integral (Volterra chronological calculus [94]) in a non-Archimedean case.
6.
Cauchy Problem for Partial Differential Equations with Variable Coefficients It should be pointed out, in the first place, that we can regard
the functional spaces A(UU x Am, A) as modules over a commutative Banach algebra A0. Thus, the results of the preceding section are applicable to spaces of operators LAo (A(UP x A', A)).
Lemma 6.1. Suppose that the function a(x, t, 0) belongs to the class A(U61 x U x Am, A). Then the operator-valued function t -p a(t) E £A0(A(U x Am, A)), a(t)(co)(x, B) = a(t, x, 9)lp(x, 8), belongs to the class A(u6 , LAo (A(U x A', A)) )
Chapter VI. Non-Archimedean Superanalysis
274
Proof. Note that 00 to
0)
an+a+Aa
- E n! L ff
a(t' x,
00 to
00
(0) a!
J=
- un(x' e).
By virtue of estimate (1.4), the operators of multiplication by the functions un (x, B) are continuous, and we have an estimate for the norm of the multiplication operator IlunllGAO(A(uP xA-,A)) < IlUnllp
It remains to note that 6nllunllp/In!IK
li
< lim
(0)11
bn
In!IK
max
la!IK
QQ
=
a1
II acnazaae
0,
PI
and we have an inequality Ilall A(u6,rAO(A(uP (Ar,A))) <_ Ilallip
It should next be pointed out that by virtue of estimate (1.3) the differentiation operator as aA
axe a o
: A(U x Am, A) - A(L[P x Am, A)
is continuous, and as ap 11
ax, ao8
I.cAO(A(UP xAr,A))
< P-I -I l a!l K.
(6.2)
Estimates (6.1) and (6.2) imply Proposition 6.1. Suppose that the coefficients acO (t, x, 0) of the infinite-order differential operator P (t, x, 0,
a
a
ax ae =
00
E a,,p (t, x, e) iai=o,A
as ao ax° aBp
6. Cauchy Problem
275
belong to the class A(U X LAP x Am, A) and satisfy the condition 0.
(6.3)
Then the operator-valued function t -+ P(t, x, 8, az, B) belongs to the class A(U6, LA,(A(UP x Am, A))), and 8
IP(t,x,0' a'
8x aB )I A(u6GAO(A(uv XAm,A)))
< up(P) -
jp.
Theorem 6.1. Suppose that the elements of the matrix differential operator P = (P3(t, x, 0, ax, ae)) satisfy condition (6.3) and the function cp(x, 8) belongs to the class A(U x ZAP x Am, Ak). Then there exists a unique solution of the Cauchy problem
zl(t, x, 0) = P(t, x, 8,
8x'
a8)u(t, x, 8) + f (t, x, 0),
(6.5) (6.6)
u(0, x, 0) = (P (X, 0),
of the class A(U; X Up x Am, A), where s; < min(a, Sph/(1-n), o.(P)-lph/(1-p)), a(P) = maxcr(P1j) =,i
In order to prove this theorem, it suffices to use Proposition 6.1 (estimate (6.4)) and Theorem 5.3. The solution of the Cauchy problem (6.5), (6.6) is defined by the
relation
r
t
u(t, x, 0) = T exp{ f P(s, x) 0,
+f
c
T exp l f
c
8x'
l P (s, x, 0,
aa ,
a8)ds}co(x, 0)
l ) ds } f (T, x, 8) dT.
In particular, condition (6.3) is satisfied by all finite-order differential operators.
Chapter VI. Non-Archimedean Superanalysis
276
In particular, Theorem 6.1 implies the solvability of the Cauchy problem for the diffusion equation on a non-Archimedean superspace
au at
B) _ E
(t
i,j=1
82z aij (t, x, 9) axiaxj (t, x, 9)
+ E bij (t, x, 9) 1
a2 U
a9iaoj
(t, x, 9).
Theorem 6.1 also ensures the existence of a solution for the equation of a non-Archimedean superdifferentiation with a locally S-analytic potential V(t, x, 9). Suppose that T is a finite extension of the field K and E is a nonArchimedean Banach CSA over the field T constructed on the basis of the CSA A. The theory developed in this section can be immediately generalized to the functions f : U x Am -+ E.
In particular, let T = K(J) be a quadratic extension of the field K. The analog of Theorem 6.1 for maps with values in E implies a local solvability of the Cauchy problem for the Schrodinger equation on a non-Archimedean superspace for locally S-analytic coefficients and potentials
at
(t, x, 9)
h2
- 2 [i
32
n
l aij (t, x, B) axiaxj
2
+
bij (t, x, 9) 1
a9i a97
+ V (t' x, 0)1,0(t, x, 9). J
Non-Archimedean Supersymmetrical Quantum Mechanics
7.
We can apply the results of the preceding section to the Schrodinger
equation for the non-Archimedean supersymmetrical quantum mechanics on a Riemannian surface. The formalism of this quantum mechanics develops according to the same scheme as in an Archimedean case.
7. Quantum Mechanics
277
On a flat space in Cartesian coordinates a supersymmetric Hamiltonian is defined by means of conjugate supercharges Q and Q': H = {Q', Q}/h, where
f VG
[Pa
+
1
Q=-
T
VG
1dPd [Pd + v
adVJ
the operators W'° and Wa are, respectively, fermion operators of creation and annihilation which satisfy the commutation relations {XY`°, fib} = hbab,
Pa is the operator of the momentum -!-a,, and 171d is a plane metric. For a Riemannian surface with metric gµ"(x), x E Ao we have
Q' _
[ica
+
1T
Q=-V '
DaV
[co
1T + -oaV
where r,µ = Pµ - fF 'I' pxF,, are covariant derivatives. We have commutation relations Jai,
[P., x°] _ X01
-
'o] _
[P., 4``Q] = 0,
h
*3
rJA0qfµ,
_
[P., 41,6] = 0,
h
A
*k&
[tea, #0] = hR6«o'p'o`yµ
(for detailed computations see [163]). It should be emphasized that every quadratic extension Z = K(V 9-r is associated with its own supersymmetrical quantum mechanics. In addition, we can consider higher-order extensions. Of the prime interest are Galois extensions. Every element of a Galois group is associated with its own superchange. A supersymmetrical quantum mechanics with superchanges arises where n may be odd. For infinite Galois groups we obtain a non-Archimedean supersymmetrical quantum mechanics with an infinite number of superchanges.
Chapter VI. Non-Archimedean Superanalysis
278
The formalism described above can be generalized to number fields without an absolute value. The corresponding operators are realized in the space of formal power series over the CSA over this field. We can choose the non-Archimedean analog of the De Witt CSA A as the CSA G,,.K.
The Hamiltonian H = {Q*, Q}/h of the supersymmetrical quantum mechanics (h E K) is an ordinary differential operator on a nonArchimedean superspace. To this operator we can apply Theorem 6.1 which implies a solvability (local) of the Cauchy problem for the corresponding Schrodinger equation.
8.
Trotter Formula for non-Archimedean Banach Algebras
Different versions of Trotter's formula play an important part in the investigations and construction of evolution families of operators. Here we propose a non-Archimedean version of the Trotter formula. Everywhere in this section we use the symbol B to denote a commutative non-Archimedean Banach algebra over a field K. Since B is non-Archimedean, the exponent eA for the arbitrary element A E B does not exist (eA exists if IIAII < pl/(1-P)). Therefore the ordinary notation of the Trotter formula eA+C
= llm (eA/neC/n)n n-+oo
is meaningless (even if IIAII, IIAII < p1/(1-P))
Theorem 8.1 (Trotter formula). Suppose that A, C E B, IIAII, IIAII < Pl/(1-P), and the sequence cn E K, cn j4 0, is such that liM I cnI K = n+oo oo. Then we have a non-Archim.edean version of Trotter's formula eA+C
= lim (eA/C"eC/`^)`°.
(8.1)
A complete (rather cumbersome) proof of relation (8.1) can be found in [160].
9. Volkenborn Distribution
279
As a consequence of relation (8.1) for a wide class of differential operators (see Sec. 6) we obtain
exp{t(P(x,0,
= limmo( p{ tP(x,
,
B) +V(x,9))}
w0'ax, A -1 8B)
}exp{tVM}).
Consider the Schrodinger operator in a scalar case, A = K = QP. We set cn = p-n, and then we have
exp{-t(2 =L + -V) MX) h
=lim n- oo
pn
n
exp{t pn } exp{-t
hn
V }co(x)
= n-4oo lira o (x)
The function cpn (x) gives an approximate solution of the Schrodinger
equation with an accuracy to within e = nn (see the estimate of the rate of convergence for (8.1) in [160]). It is not difficult to calculate the operators a ,pn6 and eOp°v. Thus, everything reduces to the extraction of a root of degree pn. This algorithm can be used for solving Schrodinger equation over a field of p-adic numbers by numerical methods.
9.
Volkenborn Distribution on a non-Archimedean Superspace
Let K = Qp be a field of p-adic numbers. A unit ball U1 = ZP = < 11 of this field is known as a set of integer p-adic numbers (it is a ring). Any integer p-adic number can be uniquely represented as { x lp
a series convergent in Qp: X = xo +xlp+x272 +... +xnpn +...,
xj=0,1,...,p-1.
Chapter VI. Non-Archimedean Superanalysis
280
The Volkenborn integral over a ring of integer p-adic numbers is defined by the relation p^-1
f
f (s)v(ds)
nlim
p-n V' f (1)
Generally speaking, this limit does not exist for continuous functions. For this limit to exist, the function f (s) must be sufficiently smooth (see Schikhov [72, p. 167]). For our consideration, it is sufficient that this limit exists for entire analytic function on Qp and that the linear functional defined by this integral is continuous. Thus, the Volkenborn integral is associated with a generalized function v E A'(Qp, Qp) In order to define the superanalog of the Volkenborn integral, we shall do as we did in the case of a Gaussian integral. A p-adic Gaussian distribution was introduced as a generalized function whose Fourier transform is a quadratic exponent. We shall introduce a Volkenborn distribution on a superspace as a generalized function from the space A'(KK''n, A) whose Laplace transform can be obtained by means of a superextension of the Laplace transform for the Volkenborn distribution on Qp. This Laplace transform is known (see [72, p. 172]), it is
V (t) = fzPet'v(ds) =
to
°O
t
t _ = n=o E Bn n e 1
where Bn are Bernoulli numbers. Note that IBnlp < p and, consequently, I Bn I K < pt. By this estimate, the function V (t) can be con-
tinued to the S-analytic function on the ball LIP, where p < p and, in particular, V (t) E Ao (Ao, A). Furthermore, the function n
V
(t1i ..,.tn)
= V (t)...V (tn) _ 11
et' t'
is an S-analytic function on the ball U n' p < pt/(1-p), and, in particular, V(t1i ..., tn) E Ao(Ao, A).
9. Volkenborn Distribution
281
Let us consider the continuation of the function V (t) to the anticommuting variables
f e°8(9+2)dB. Definition 9.1. A Volkenborn distribution on a non-Archimedean superspace KA'm is a generalized function v c A'(KK''n, A) whose Laplace transform is defined by the relation ((t, E KK''n): V(t,
[n
et'tj
exp{-2(e1+...+lm}.
11
Proposition 9.1. The Volkenborn distribution can be extended to the space A(U x Am, A). In order to prove this statement, it suffices to note that sup IBnIK < 00n
It follows immediately from the definition that f f (x, O)v(dxdO) =
f (ff(x,o)v(do))v(dx) = ff(x,_)v(dx).
Everywhere in what follows we shall use the notation u for the generalized function L-1(u), where u E A. The properties of the Volkenborn integral over a superspace (cf. Schikhov [72]) :
Proposition 9.2. Let the function f E A(KA 1, A). Then
f f (-x, -O)v(dxdO) = f f (x + 1, 0 + 1)v(dxd9).
Proof. Note that
f (-x, -0) =
ff
(dyde)e-y"a
=
f
A'(f)(dydC)eyx+fe,
Chapter VI. Non-Archimedean Superanalysis
282
where a: Ao(KA"', A) -+ Ao(KA"', A), 9(y, e) ' g(-y, -e), is a Alinear continuous operator. Using the Parseval equality, we obtain
f f (-x, -9)v(dxd9) = fA'(f)(dyde)L'(v)(y,e) =
f f (dyde) [e_y y 11 el{ = fic'(f)(dYde)L'(v)(y,e),
where rc: Ao(KA"', A) -+ Ao(KA"', A), 9(y, C) H ey+Eg(y, C), is a A-
linear continuous operator. It remains to note that
L(rc'(f))(x,0) = f(x+1,9+1). Similarly, for the function f E A(K;,'m, A) we have
f f (-xl) ..., -xn, -01,..., -9m)v(dxd9) = ff(xi + 1,...,x+ 1,0k + 1,...,Om + 1)v(dxd0). This relation is also valid for the function f from the class A(ul x Am, A).
Proposition 9.3. The relation
f f (X,,..., xj + S + 1, ..., xn, 9) -f (xl, ,
af
fa
xj + S, ..., xn, 0))v(dxd9)
(xl, ..., Si ...)xn, 9)v(dxl...dxj-ldxj+l...dxnd9).
holds for any function f E A(KAn'm, A) and s E Ao.
Proof. We shall restrict the consideration to a one-dimensional case (n = 1, m = 0), namely f (f(x + s + 1) - f(x + s))v(dx)
10. Infinite-Dimensional Superanalysis
=
ff
(dy)(eyls+1)
- ey')
y ey
1=
283
f f (dy)yey' = of (s).
The relation that we have proved is also valid for the function f c A(U x Am, A) for shifts by vectors s from a unit ball. Proposition 9.4. For any function f E A(KA'm, A) and for 77 E Al we have a relation
f(f(X,9i,...,Oj+ii+1,...,0m) -f (x, 01, ..., Oj + 77, ...,
fR a0j
(x, 01 i ..., ii, ..., Om)v(dxdOi ...dOj-1d0j+1...dO,,,).
The proof is similar to the proof of the preceding proposition.
Remark 9.1. We have determined the Volkenborn integral only for S-analytic functions. This integral can apparently be extended to some classes of S-smooth functions.
10.
Infinite-Dimensional non-Archimedean Superanalysis
The analysis on an infinite-dimensional non-Archimedean superspace is developed along with an analysis on an infinite-dimensional Archimedean superspace. The main idea is the same, namely, we consider superspaces over a pair of topological CSM and develop the theory of S-differentiable and S-analytic functions on a pair (a superspace and a covering supermodule). Here we expose an infinite-dimensional non-Archimedean superanalysis over specific CSM, namely, spaces A(KA'm, A) or A'(KAn'm, A).
We choose these spaces because of their good topological properties. It stands to reason that we can develop an infinite-dimensional superanalysis over arbitrary locally convex non-Archimedean CSM. All general constructions can be generalized to this case without changes. However, it is necessary to use here the fine results from the theory of
Chapter VI. Non-Archimedean Superanalysis
284
non-Archimedean locally convex spaces, but in the framework of this book we cannot do this. We set ecp = x°OA = 41 ...Xnn 0"1...00- . Note that these monomials form a topological basis in the CSM A(KA'm, A), with the parity These Ie,vl = I/0I (mod 2). Furthermore, we set e' O = monomials form a topological basis in the CSM A'(KA'm, A), with the parity Ieapl = IQI Proposition 10.1. 1. Let b E Gn,r (An, A). Then there exists azaae,0
REF, IIbIIR =ai,.. sup ,an
IIba1A1...".OnIIR-I-I
< oo,
(10.1)
$11 ,...9n R q where ba1Q1...anpn
b(ea1#1I
eanOn)
2. For any sequence b = (ba1#1...'InQn) of elements of the CSA A that satisfies condition (10.1) for a certain R E IF, the form b=Eba1Q1...QnOnel
®...®e'
(10.2)
belongs to the CSM Gn,r(An, A).
In the CSM Gn,r (An, A) we introduce an inductive topology Gn,r (An A) = l irn ind Cn,r (AP, A).
This is a strict inductive limit of non-Archimedean Banach CSM. We set IbIR= sup IIb(f,...,f)II, IIfIIR<1
and then c = 2p'AP-1). Consequently, the topologies defined by the norms I IR and II - IIR are IbIR < IIbIIR < cnIbIR,
equivalent.
Proposition 10.2. 1. Let b c Gn,r((A')n, A). Then, for all R E F, we have
IIbIIR =al,.. sup ,an 01 ... ,Sn
RI'I/Ia!IK < 00,
(10.3)
10. Infinite-Dimensional Superanalysis where
qq b,,1131...cfnpp
_b
285
eon Qn
e'IP1)
2. For any sequence b = {b,,,1#1...'In On} of elements of the CSA A that satisfies condition (10.3) for all R E F, the form b
®... ® e
b01A1...OnAn
(10.4)
-nQn
a.
belongs to the CSM £,((A')", A). In the CSM Cn,r((A')", A) we introduce a projective topology Ln,r((A,)", A) = lim proj Ln,,, ((AR)' , A).
This is a non-Archimedean Frechet CSM. The explicit description of spaces of continuous polylinear map-
pings on the CSM A and A' (relations (10.2) and (10.3)) makes it possible to obtain an explicit description of spaces of S-analytic functions on A and A'. Let us now construct an infinite-dimensional non-Archimedean superanalysis. We set M = A(KK q, A), N = A(K;;S, A) and introduce an infinitedimensional superspace over a pair of non-Archimedean Frechet CSM
(M, N): X = Mo ® Ni =
A(KAP,q, Ao)
® A(Kns) Ai)
The covering CSM Lx = M ® N = A(KAq, A) ® A(K;;S, A). We shall use the symbol UR,P to denote a ball of radius p E F with respect to the norm II IIR, R E F in the superspace X:
UR,P={f EX: f0 IIf 11R = maX[sup 11
axi1...ax°Ta0Q1...aeg9 (O) I ai!...a,,!IK
fl
ap ax11...ax°*ae1 ...aep 11
O 0
11
ial!...ar!I K]
-
By UR,P we denote the corresponding ball in the covering CSM Lx: UR,P = {f E Lx: If IIR < P} (with UR,P = UR,P n X).
Chapter VI. Non-Archimedean Superanalysis
286
Just as in an Archimedean case (see Chap. III) we define the S-differentiability and S-analyticity on an infinite-dimensional superspace.
We denote by A(UR,P, A) a space of maps S-analytic on the ball UR,P. Every map of this kind can be expanded in a power series 00
(10.5)
F(f) = > bn(f, ..., f ), n=0
where the polylinear forms bn c Gn,,.(L jr, A), the restriction of bn to the superspace X being symmetrical and series (10.5) converging uniformly on the ball UR,P of the covering CSM LX (i.e., sup IIbn(f, ..., f) II f EURL,v
Note that pnlbnlR = sup
Ilbn(f,..., f)II
f EUR.v
In the space A(UR,P, A) we introduce norms
FR,=sup Sup n
f EUR.v
Ilbn(f,...,f)Il=sup pnlbnlR,
R, p E r.
n
We have inequalities IIFIIR,P/C< IFIR,P< IIFIIR,P'
(10.6)
Let us assume that AP (X, A) consists of functions which are S-analytic
on balls of a fixed radius p with respect to all norms that define the topology in X. Note that UR2,P 3 UR1iP1 R2 > Rl and A(UR2,P, A) 3 A(UR1,P) A).
We set Ao(X, A) = lim ind Ap(X, A) (note that AP2 (X, A) 3 API (X, A), p2 < pi). The functional space Ao(X, A) is an infinitedimensional analog of the space of functions on KA''n which are Sanalytic at zero. Theorem 10.1 (on the approximation by cylindrical polynomials). A set of polynomials which depend on a finite number of variables is everywhere dense in the space .A0(X, A).
10. Infinite-Dimensional Superanalysis
287
We introduce functional spaces BR,P = {F E Ao: IIFIIR,P < oo},
BP = lim ind BR,P.
By virtue of inequality (10.6), we have
Ao = l
o ind lim ind A(UR,P, A) = l m ind R m ind BR,P
We introduce spaces of sequences of elements of the CSA A:
IIs(A) = {7r = {7rn} 0: 7rn = {7r 3}, a = (al, ..., an), Q = (Ql) ..., On)) where 7r,,p E A, V R, p E r:
1
17fI
I R,P = Sllpn p n I I7rn I I R< o, I I7rn I I
R=
sup II7rc,0II RI-1 < oo}. QQ
Theorem 10.2. The space A'(X, A) of generalized functions of an infinite-dimensional non-Archimedean superargument is isomorphic to the space of A-sequences IIs(A). Let us introduce a dual superspace Y = Mo ® N1' = A'(KK'q, Ao) ® A'(Kns, A1).
The covering CSM
Ly=M'®N'=A'(KAq, A)eA'(KKS,A). We set YR = MR,O ® NR,1, where
MR =A(URxA', A),
NR=A(URXAs, A).
Note that Y = lim ind YR.
The spaces of test and generalized functions on the dual superspace Y can be defined in the same way as on the space X. The only difference is of a topological character, namely, the superspace X is a projective limit of non-Archimedean Banach spaces whereas the superspace Y is inductive. The map F: YR -4 A is said to be S-entire if series (10.5) converges
uniformly on a ball of radius p E r in the covering CSM LyR (here
Chapter VI. Non-Archimedean Superanalysis
288
the polylinear forms b. E Gf,r(Ly , A)). The space of S-entire maps on the superspace YR will be denoted by A(YR, A). The map F: Y --> A is said to be S-entire if the restrictions of F to the spaces YR are S-entire maps for all R E F. The space of S-entire maps on the superspace Y will be denoted by A(Y, A). We denote a ball of radius p E F in the superspace YR by WR,P, WL R,p being the corresponding ball in the covering CSM LyR. The topology in the CSM A(YR, A) is defined by a system of norms IFIR,P = supra sup Ilbn(f, ..., f)II In the CSM A(Y, A) we introduce f EWR,,
a projective topology A(Y, A) = lim proj A(YR, A) which, in the CSM
A(Y, A), is equivalent to the topology defined by the system of norms I I F I I R,P = sup. pn I I bra I I R (by virtue of inequalities (10.6)). The space of S-entire functions on the superspace Y is a non-Archimedean Frechet CSM.
We introduce a space of A-sequences IIs, (A) = {7r = {7rn}°O-o: 7rn = {7raf}, Trap E A, I R, p c F: II7r
IIR,P = SUPP nMI7rIIR < oo, n
II7rnIIR = Sup II70AII I Y. KR-'a1 < oo}. aQ
Theorem 10.3. The space A'(Y, A) of generalized functions of an infinite-dimensional non-Archimedean superargument is isomorphic to the space of A-sequences Us1(A).
Now the Laplace calculus over a pair of dual superspaces X and Y is developed according to the usual scheme (see Chap. III). Theorems 10.1-10.3 give a non-Archimedean analog of a theorem of the Paley-Wiener type for analytic generalized functions.
Theorem 10.4. The Laplace transformation L: A'(Y, A) -p Ao(X, A)
is an isomorphism of a CSM.
11. Unsolved Problems
289
Now we introduce a Gaussian distribution on an infinite-dimensional superspace Y = MO' ® N. This distribution serves as a basis for constructing a theory of continual Gaussian integrals in a non-Archimedean case.
In [167] we gave the proofs of the results contained in this section in the commutative scalar case A = K. The proofs for a supercase are similar.
11.
Unsolved Problems and Possible Generalizations
1. A substantive analysis on non-Archimedean supermanifolds. 2. Theory of differential equations on non-Archimedean supermanifolds.
3. Pseudodifferential operators on a non-Archimedean superspace. 4. Formulas of the type of Feynman-Kac. 5. Gaussian and Feynman integrals over infinite-dimensional nonArchimedean superspaces. 6. Theory of differential and pseudodifferential equations on infinitedimensional non-Archimedean superspaces. 7. Superconformal structures corresponding to Galois groups. 8. Non-Archimedean Hilbert superspace. 9. Non-Archimedean infinite-dimensional superdiffusion. 10. Hida calculus on a non-Archimedean superspace. 11. Formulas for integration by parts for non-Archimedean Gaussian distributions and Malliavin calculi. 12. In their pioneer work, Vladimirov and Volovich discuss a num-
ber of problems of superanalysis over an arbitrary locally compact field. The authors of [1] suppose that the majority of results which they obtained in this work for a field R can be generalized to a nonArchimedean case. However, this has not yet been done anywhere sufficiently accurately. 13.
Fundamental solutions for linear differential operators with
constant coefficients.
Chapter VI. Non-Archimedean Superanalysis
290
14. We have only used spaces of S-analytic generalized functions. It would be interesting to generalize to a supercase the theory of generalized functions over the spaces Ck which is widely used in nonArchimedean analysis (see, e.g., [72]).
Remarks The foundations of non-Archimedean physics were laid by I. V. Volovich
(1987); the p-adic quantum mechanics (with complex-valued wave functions) was constructed by Vladimirov, Volovich, Zelenov, Alacoque, Ruelle, Thiran, Verstegen, Weyers (see [21] and the bibliography therein). The padic quantum mechanics with p-adic-valued wave functions was constructed by Khrennikov [66, 156-159]. Vladimirov and Volovich also considered padic quantum field theory with comlpex-valued fields. Models with p-adic fields were studied by the author [66, 162]. The monograph [21] contains practically everything that was done by now in the p-adic physics. For p-adic physics with p-adic-valued functions see (66].
The most simple, complete, and reasoned exposition of non-Archimedean analysis can be found in Schikhov's book [72]. More subtle problems of the number theory are exposed in the monograph by Borevich and Shafarevich [11].
Sec.1. The theory of S-differentiable maps on a superspace over an arbitrary locally compact field was given in [19]. In [153] the author considered a different version of a non-Archimedean superspace, namely, a nonstandard superspace.
Secs. 2-7. The results of these sections are given in [156-159, 162164]. A p-adic Gaussian integral over a superspace is a natural generalization of a p-adic Gaussian integral proposed by the author in [157]. For the theory of p-adic Gaussian integration see also [66, 167]. M. Endo proved that a p-adic Gaussian integral cannot be extended to a linear continuous functional on a space of continuous functions of a p-adic argument, i.e., as distinct from a real case, a p-adic Gaussian distribution is not a measure. Sec. 8. Trotter's formula over non-Archimedean fields was obtained in [66], its complete proofs can be found in [160]. Of course, it is only the simplest version and wide generalizations of this formula are possible (cf. [26]).
Sec. 9. Volkenborn's integral plays a significant role in non-Archimedean
11. Unsolved Problems
291
analysis. It is possible that its superanalog can also be used for integral representations of special functions on a non-Archimedean superspace (cf. [72]).
Sec. 10. An infinite-dimensional non-Archimedean analysis was presented in [66, 167]. Here we only outlined its supergeneralization. Sec. 11. 1. We mean investigations similar to the investigations carried out by De Witt, Volovich, Rogers, Buzzo, Cianci (see [27, 19, 20, 52, 88-93, 80, 81, 111, 112]). 2. Compare, for instance, with the work by Cianci [112]. 3. The PDO theory on K" was proposed in [156]; the results of this article can apparently be combined with those from Chap. 3 from [156]. 4. It is interesting to try to generalize the results of this chapter to a non-Archimedean case. 5. Gaussian and Feynman integrals on infinite-dimensional K-linear spaces were considered in [66, 167]. Of particular interest is the quantization of a non-Archimedean spinor field and of graded fields in the formalism of a continual integral. 6. For the secondary quantization over K" see [158]. 7. A number of models with a conformal structure over a Galois group were considered in [159] and [162]. 8. Nothing has been done here, but it is clear that the theory from
Chap. III must be generalized to a non-Archimedean case. 9, 10. The non-Archimedean white noise and non-Archimedean Hida calculus (nonArchimedean Brownian functionals) were introduced in the report that I made at the conference concerning Gaussian random fields, Nagoya, 1990. The p-adic theory of probabilities was proposed in connection with a probability interpretation of p-adic quantum mechanics with p-adic-valued wave functions (see also [169]). 11. Nothing has been done here. 12. Nothing has been done here either. 13. The reader should take the article by Vladimirov and Volovich [19] and Schikhov's book [72] and try to combine them. Although the first steps of this theory will be a trivial generalization, essential advances can be obtained in this direction.
Chapter VII
Noncommutative Analysis
When the main text of the book was ready, I got some ideas that allowed me to construct a noncommutative generalization of the supercommutative analysis exposed in the book. Here I again use the scheme which I used when passing from the K-linear ordinary mathematical analysis to A-linear superanalysis. In order to construct analysis over an arbitrary noncommutative algebra A (or on an A-module), it is necessary to define, "in a natural way," the concept of A-linearity which will be used in the noncommutative differential calculus. In the first place, we can use here a further generalization of the methods from Chap. III and consider analysis on a pair (A-superspace, A-module) defining a superspace as a K-linear subspace of the Amodule. M. I constructed this theory as early as in my first works in superanalysis. It does not constitute an essential advance as compared to superanalysis (see Sec. 1). In this theory, the approximating Alinear maps are, as before, right A-linear or left A-linear maps on a covering A-module. Precisely these maps are used as classes of Sderivatives in the analysis on a pair (A-superspace, A-module). A new essential progress in the development of noncommutative analysis can be obtained with the aid of a new class of A-linear maps (noncommutatively linear maps). New nontrivial algebraically topological constructions arise here such as an ordered projective tensor product of noncommutative Banach algebras (one-dimensional noncommutative differential calculus), a projective tensor product of non-
Chapter VII. Noncommutative Analysis
294
commutative Banach algebras which is ordered with respect to two indices (multidimensional noncommutative differential calculus) and similar constructions for A-modules (infinite-dimensional noncommutative differential calculus). Apparently it will later be possible to generalize all main parts of this book to arbitrary noncommutative Banach (or topological) algebras and modules. It should be pointed out that the analysis on a pair (A-superspace, A-module) given in Sec. 1 is contained in the more general noncommutative analysis considered in Secs. 2 and 3. Choosing different K-linear subspaces in an A-module and regarding them as A-superspaces, we obtain S-derivatives (Sec. 1) as restrictions of the noncommutative derivatives from Sec. 2 to the A-superspace.
As in the superanalysis, two equivalent approaches to the construction of noncommutative analysis are possible here, namely, an algebraic approach and a functional one. In this chapter we construct a noncommutative functional analysis, i.e., a theory of functions of noncommutative variables (of maps of sets with noncommuting coordinates). The Connes noncommutative geometry [105], the Wess and Zumino quantum differential calculus [85], the Soni quantum superanalysis [132] are versions of algebraic noncommutative analysis. It should be pointed out that the functional approach to the theory of quantum groups was used by Aref'eva and Volovich [78].
1.
Differential Calculus on a Superspace over a Noncommutative Banach Algebra
Everywhere in this section we denote by A an algebra over a field K which, in general, is nonassociative and noncommutative. All modules are modules over A.
Let Mk, k = 1, ..., n, and N be right modules. In the space Ln (fl Mk, N) we distinguish a subspace Ln,r: the right A-linearity. k=1
The map b c Ln,r if, for any xj E M3 and a E A, we have relations b(xl, ..., xka, xk+1, ..., xn) = b(xl,..., xk, axk+1, ...) xn);
1. Differential Calculus on a Superspace
295
b(xl, ..., xna) = b(xl, ..., xn)a.
By analogy we can introduce a space Ln,1 of maps which are left A-linear on the left modules Mk, N. As in Sec. 1 of Chap. III, we can introduce the structure of modules in the spaces Ln,, and Ln,1, but there is no canonical isomorphism between Ln,,, and Ln,1 for two-sided modules in the general case.
For topological CSM of modules we denote by Ln,r(Kn,r) and Ln,1(Icn,i) the subspaces of the spaces Ln,r and Ln,1 consisting of continuous (continuous on compact sets) maps.
Definition 1.1. A K-linear topological subspace of a topological module M is a superspace over the module M. The module M is said to be covering for the superspace X.
Definition 1.2. The map f : X -+ Y, where X and Y are superspaces over the modules M and N, is said to be (right) S-differentiable
at a point x if f is differentiable (in a certain sense) as a map of topological K-linear spaces X and Y and there exists an operator aR f (x) E Ll,r (M, N) such that aR f (x) Ix = f (x).
As in the case of a superspace over a CSM, the S-differentiability
on a superspace over an arbitrary A-module is defined by a class fll,r (M, N) of operators, which are right A-linear, to which the Sderivative belongs and by the convergence which defines the ordinary derivative in topological K-linear spaces. In the sequel we assume that the classes fl,r (M, N) are submodules of Ll,r (M, N). The left S-differentiability is defined by analogy. The derivatives are not uniquely defined.
The S-differentiabilities on a superspace X over a CSM and on a superspace coincident with the A-module are special cases of Sdifferentiability. We shall formulate the fundamental theorems of S-differential cal-
culus for Banach modules and for Frechet differentiability: fl,r = Li,r and f : X -4 Y is Frechet differentiable as a map of Banach spaces. The generalization of these theorems to the case of S-differential calculus in topological superspaces can be carried out by analogy with [54].
Chapter VII. Noncommutative Analysis
296
In the following theorems, we denote by U, V, 0 the neighborhoods of the points xo E X, yo c Y, and zo E Z, where X, Y, Z are superspaces over the Banach A-modules M, N, R.
Theorem 1.1 (chain rule). If the functions f : U -+ Y, g: V -+ Z are S- differentiable at points xo and yo = f (xo), then the composite
function cp = g o f : U -4 Z is S-diferentiable at a point xo and aRw(xo) = ORg(yo) ° ORf
(xo).
Theorem 1.2 (the differentiability of an implicit function). Let the function F: U x V -+ Z be continuous at a point (xo, yo) and let F(xo, yo) = 0. If there exist partial S-derivatives a and A , which are continuous at a point (xo, yo), and the operator 8y (xo, yo) has bounded inverse, then there exists an implicit function y = f (x) which is S-diferentiable at a point yo = f (xo) and -(aaF(xo,yo))-1
ORf(YO) _
0 (aaF(xo'yo))
y
Theorem 1.3 (differentiability of an inverse function). Let the function f : U -a Y be continuously S-diferentiable and let the operator aR f (xo) have a bounded inverse. Then there exists an inverse function cp = f -1 which is S-diferentiable at a point yo = f (xo) and 1RW(xo)
_
(ORf (xo))-1
As in Sec. 5 of Chap. III, we introduce an annihilator 1X = Ann (X, £1,,.(M, N)).
If the map f : X -+ Y, where X and Y are superspaces over the modules M and N, is S-differentiable, then the derivative aR f is a single valued map from the superspace X into the right module £1,r(M, N)/Ann (X; £1,r(M, N)). Consequently, we can define the second derivative and higher-order derivatives: a2Rf: X -> Gl,r(M, £1,r (M, N) /Ann (X; Gl,r(M, N)))
We shall restrict the further consideration to associative algebras A and superspaces with a trivial annihilator: 1X = 0.
1. Differential Calculus on a Superspace
297
Theorem 1.4. Let the function f : X -+ Y be n-times S-differentiable in the neighborhood of the point xo. Then the nth-order S-derivative 8Rf (xo) E Gn,r (Mn, N) and the restriction of eR f (xo) to the superspace Xn is symmetric.
Proposition I.I. Let the polylinear form bn E Gn,r(Mn, N) and the restriction of bn to a superspace Xn is symmetric. Then the map f : X -+ N, f (x) = bn(x, ..., x), is n times S-differentiable and 8Rf = n!bn.
Theorem 1.5 (Taylor's formula). Let the function f : X --> Y be n times S- differentiable at a point xo E X. Then f (x)
E 8R n!f x0) (x - xo, ..., x - xo) + rn(x - x0),
M=0
where the S-derivatives 8Rf (xo) E Cn,r and their restriction to the superspaces Xn are symmetric;
0, h --+ 0.
The proof of theorems of S-differential calculus on superspaces over Banach modules repeats the proofs for the corresponding theorems of differential calculus in Banach spaces. We must only replace in these proofs K-linear operators by A-linear ones. Taylor's formula for S-differentiable maps on superspaces over Amodules leads to the following definition of S-analyticity.
Definition 1.3. The function f : X -+ Y is right S-analytic at a point xo c X if f can be expanded in a power series in some neighborhood of the point x0, i.e., W
f (x) = > bn(x - xo, ..., x - xo), n=0
where bn E Gn,r and the restriction of bn to the superspace X' is symmetric.
The type of S-analyticity is defined by the choice of modules to which the coefficients of the power series belong and by the choice of the type of convergence of the power series.
Chapter VII. Noncommutative Analysis
298
The construction of a theory of distributions, pseudodifferential operators, and evolutionary differential equations on a superspace over a module over an arbitrary noncommutative algebra is an interesting unsolved problem.
Differential Calculus on Noncommutative Banach Algebras and Modules
2.
Everywhere in this section, we denote by A an associative Banach algebra over a field K of real or complex numbers in which there exists a topological basis (as in a K-linear Banach space) {en}n 1; {rynmk} 00 are structural constants of the algebra: enem = k -Ynmkek k=1
We also assume that {en ® em} is a topological basis in the completion of a projective tensor product (we shall denote this completion by the symbol for an ordinary tensor product, i.e., A ® A). We introduce an operation of multiplication (a1 0 b1) x (a2 ® b2) = a1a2 0 b2b1 relative to which A 0 A is an associative Banach algebra. As before, we denote by G(A) the space of K-linear continuous operators U: A -+ A. We introduce a canonical map j: A®A -4 G(A) by setting j (a 0 b) (x) = axb, x c A.
Proposition 2.1. The map j is a continuous homomorphism of Banach algebras.
In order to prove this proposition, it suffices to use the representation j (> unmen ® em) (x) = E unmenxem and the implicit forms of n,m
n,m
norms in a projective tensor product (see, e.g., Schaefer [71]) and in G(A).
We denote the image of the tensor product A ® A under the homomorphism j: LA(A) = Im j ^_' A® A/Ker J by LA(A) and the Banach algebra A ® A/Ker j by 11(2) (A).
Proposition 2.2. The element u = E unmen®em of the projective n,m
tensor product A ® A belongs to the kernel of the homomorphism j if
2. Differential Calculus on Algebras
299
and only if we have a relation E unm E rynksYsmi = 0 n,m
(2.1)
s
for any k, i.
Proof. Let uh = 0 for any element h E A. Then we have Unmenhem = i unmhkenekem = E Unmhk i 7'nkj7'jmiei = 0, n,m
n,m,k
n,m,k
j,i
i.e., the relation E Unm"fnkjfjmiei = 0 n,m,j,i
holds for any k. Thus relation (2.1) is valid for any k and i.
Example 2.1. Let A = Gn be a Grassman algebra. Then all elements of the form u = g31...q,n® gil...gim, where j, = it for certain subscripts s and t, belong to Kerj.
Proposition 2.3. Let A = MatK(n x n) be an algebra of n x n matrices. Then Ker j = {0} and SZ(2) (A) = A ® A.
This proposition is a direct corollary of (2.1).
Definition 2.1. The map f : G -+ A, where G is an open subset of the algebra A, is noncommutatively differentiable (NC-differentiable) at a point xo E G if f (xo + h) = f (xo) + V f (xo)h + o(h), where the operator V f (xo) E LA(A) ' Sl(2)(A) and
0,
h -+0. Definition 2.1 can be reformulated as follows: the map f is NCdifferentiable if it is Frechet differentiable as a map of a Banach space
and the Frechet derivative f'(xo) E L(A) belongs to the operator algebra LA (A).
Example 2.2. Let A be a unital algebra and e be a unit element. The function f (x) = xn. Then f is NC-differentiable and V f (x) _ e®xn-1+x®xn-1+...+xn-1®e.
Chapter VII. Noncommutative Analysis
300
Example 2.3. Let f (x) = alxa2x...xan+l, where aj c A. Then f is NC-differentiable and V f (x) = al®a2x...xan+l+alxa2®a3...xan+l+ alx...an ®an+1. If A is a commutative unital algebra, then V f (x) coincides with an ordinary derivative in the commutative Banach algebra. For instance,
for Examples 2.2 and 2.3 we obtain V f (x) = nxn-1 and V f (x) _ nal Example 2.4. Let the operator U E G(A) \ICA(A). Then the map f (x) = U(x) is not NC-differentiable although it is Frechet diferen...anxn-1
tiable as a map of a Banach space.
The derivative of the NC-differentiable map f : G -+ A, f (x) _ E fn (x) en, to be more precise, its representative in the algebra A ® A, n
can be represented in terms of base vectors {en ® em}: V f (x) =
JJ Vfnm(x)en ®em,
n,m
where V fnm(x) are numerically valued functions on the set G.
Theorem 2.1. Let f and g be NC- differentiable maps on the set G. Then (1) the map cp = of /3 + Agp, a, /3, A, M E A, is NC- differentiable, with
V (x) = aV f (x)/.3 + )Vg(x)p and
V psp(x) = E 1'knsrymrp(ak/rVfnm(x) + )1kµ'rVgnm(x)), n,m,k,r
(2) the map cp(x) = f (x)g(x) is NC -differentiable; here the Leibniz formula
V (x) = V f (x)g(x) + f (x)og(x) holds true and Wnp(x) = >(7'mkpV fnm(x)gk(x) +'Ymknfk(x)V9kp(x)) m,k
2. Differential Calculus on Algebras
301
Theorem 2.2 (noncommutative chain rule). Let the maps f : G -+ A and g: W -- A be NC- differentiable at points x E G and y = f (x) E W respectively. Then the composite function cp = g o f : G -4 A is NC- differentiable at a point x, with VV(x) = Vg(y) V f (x) and 7'nkp7'lmgVgnm(y)Vfkl(x)
V ppq(x) = n,m,k,l
Theorem 2.3 (on the noncommutative differentiability of an inverse function). Suppose that the function f : G -+ A is NC-differentiable in a certain neighborhood 0(x) of a point x E G, the derivative V f :
0(x)
Q(2) (A) is continuous and maps the neighborhood
0(x) into a subgroup of inversible elements of the Banach algebra 1(2) (A). Then, in a certain neighborhood 0'(x) of the point y = f (x), there exists an NC-differentiable inverse function g(y) = f -1(y), and Vg(y) = (VAX))-,IX=g(Y)-
We introduce higher-order noncommutative differentiability with
the use of Taylor's formula. As before, we denote by £(A, A) the space K of n-linear maps from An = A x ... x A into A. Next, we -10
e
n
shall need a new algebraic construction, namely, an ordered projective tensor product. Let or = (il, -,in) be a permutation of indices (1, 2, ..., n). Then we set
®A=A®...®A O tl
in
(everywhere we use the symbol of tensor product to denote a projective tensor product which is a completion of an algebraic tensor product in a projective norm), i.e., ®A is an ordinary projective tensor product A ®... ® A with ordered symbols of tensor products. Next, we introduce a direct sum of ordered tensor products with respect to all permutations or from the permutation group Sn: F(n+I) (A)
_ oESn ® (®A) o
Chapter VII. Noncommutative Analysis
302
_
{x = E E a.1 ®a-2 ®... ®a.n+1: IIxIIA®...®A Goo}. to it
oESn al...an+1
t2
Example 2.5. E(3) (A) =A®A®A®A®A®A, i.e., 2
1
2
1
x = Ea., ®aa2 ®a Q3 +Eb#1®bp2 p
2
1
of
2
®b031
1
where the elements a ® b ® c and a ® b ® c are not identified. 1
2
2
1
Let us consider the canonical map j: E(n+1) (A) - £, (An, A), j (a ®b ®c (9 ... ®d) (hi, ..., hn) = ahi1 bhi2 c...hind. i1
i2
i3
in
We set Gn,A(A', A) = Im j ' 1(n+1) (A) = E(n+1) (A)/Ker j (the map j is linear and continuous).
Definition 2.2. The map f : G -+ A is n times NC-differentiable at a point xo E G if n
Vkf (xo) (h, ..., h) + o(hn),
f (xo + h) =
(2.2)
k=0
where the K-polylinear forms Vk f (x0) belong to the classes Q(k+1) (A), Ilo(hn)II/IIhMI" -+ 0, h -4 0.
This definition can be reformulated as follows: the map f is n times
Frechet differentiable as a map of the Banach space A and Frechet derivatives belong to the classes cl(k+1) (A)
Example 2.6. Consider a map f (x) = axbxc, where a, b, c E A. Then
Vf(x)=a®bxc+axb®c, 1
1
V2f(x) = a®b®c+a®b®c. 1
2
2
1
Let us now consider a map f (x) = aixa2...xan+1. Then
Onf =
a1®... ®an+1 OESn
it
in
2. Differential Calculus on Algebras
303
Definition 2.2 of higher-order noncommutative differentiability leads to the following definition of noncommutative analyticity.
The map f : G -+ A is NC-analytic at a point x0 E G if, in a certain neighborhood 0(xo) of this point, f can be expanded in a power series
f(x) =
C'O
Ebn(x-xo,...,x-xo),
(2.3)
n=0
where the coefficients of bn E 11(n+1) (A) and the series converges in the sense 00
Ilf IIP = > PnllbnJJn(n+1)(A) < oo
(2.4)
n=0 for a certain p > 0.
We can reformulate this definition as follows: the map f is Frechet analytic as a map of the Banach space A and the coefficients of the Taylor series belong to the classes SZ(n+1)(A).
Proposition 2.4. Every NC-analytic map f is infinitely NCdifferentiable.
NC-analytic maps are series of the form
f (x) =
Ean,ai (x - xo)an,,Z...(x - xo)an,an+1 n=0 a
with coefficients that satisfy condition (2.4). In order to fulfil this condition, it is sufficient that 00
E Pn E
n=0
Ilan,a1II...Iian,an+1 II < 00.
a
Let us pass to further generalizations of the construction described above. Consider maps f : G -+ M, where G is an open subset of A and M is a Banach A-module (two-sided). A new algebraic construction arises here which is an ordered projective tensor product of n copies of the algebra A and one copy of the module M. An ordered tensor
Chapter VII. Noncommutative Analysis
304
product in which the module is at the (k + 1)th place is defined by the relation k+1
®(A, M = O
O M O ®...®A. i1
22
ik
in
ik+1
In particular, 1
®(A) M)=M®A®...®A, i2 in i1 or
n+1
ii in-1 in Next, we define the direct sum of ordered tensor products with respect or
to all permutations of the indices a E Sn and the numbers of places occupied by the module M: k E(n+l)(A, M) _ ®1 ® ®(A, M)
k=1 oESn o
® (M®A®...®A®... (DA®A®...®M).
oESn
11
1
in
12
i2
in
Example 2.7.
E(3)(A,M) =M®A®A®A®M®A®A®A 1
2
1
2
1
®M ®M®A®A®A®M®A®A®A®M. 2 2
2
1
2
1
1
As before, we introduce a canonical map j: E(n+1) (A, M)
-+ £ (A'1, M); q(n+1) (A, M) = E(n+1) (A, M) /Ker j.
Definition 2.3. The map f : G -+ M, where G is an open subset of the algebra A and M is a Banach A-module, is said to be n times NCdifferentiable at a point xo E G if, in the module M, the relation (2.2) holds true, where Vk f (xo) E I (k+l)(A, M), and 11o(hn)JjM1jjhjIn _+ 0,
h-+ 0. Example 2.8. Consider a map f (x) = axbxm, where a, b E A, m E M. Then
Vf(x) =a®bxm+axb®m;
2. Differential Calculus on Algebras
305
V2f(x) =a®b®m+a®b®m. 2
2
1
1
And now let us consider a map f (x) = mxaxb. Then
Vf(x) =m®axb+mxa®b;
V2f(x) =m®a®b+m®a®b, 2 2
1
1
and, finally, for the map f (x) = axmxb we have
Vf(x)=a®mxb+axm®b;
V2f(x)=a0m®b+a®m®b. 2 2 1
1
Let us now consider functions of several noncommuting variables. We shall use the same scheme as above. The map f : G -+ M, where G is an open subset of Am = A x ... x A, is said to be NC-differentiable if it is Frechet differentiable as a map of the Banach space Am into M and the Frechet derivative (gradient) belongs to the class [SZ(2) (A, M)]m: m
V f (x) (hl, ..., h.) _ E Vj f (x) h;,
Vif (x) E
c(2) (A,
M).
Example 2.9. Consider a map f (x, y) = axbym, where a, b E A,
m E M. Then V f= a ®bym, V f= axb ®m. The higher-order differentiability for functions of several noncommuting variables will also be defined with the use of Taylor's formula. A new algebraic construction arises here which is a generalization of the ordered tensor products introduced earlier. In order not to com-
plicate the consideration, we shall begin with studying the case of A-valued maps. We introduce a projective tensor product which is ordered with respect to two indices. In this product, every symbol of a tensor product has two indices, an upper index and a lower index.
Chapter VII. Noncommutative Analysis
306
Suppose that we have a permutation or = (i1i ..., in) E S. and (with repetitions) from the set of indices a sample (1, ..., m).
The permutation or and the sample , are associated with a projective tensor product ordered with respect to two indices, namely, ®A=A®A®32
ar,K
_
it
32
31
{z =
...®A
12
in
7n
aQl ®aU2 ®... ®aan+1 Q1 ...Qn+1
12
11
IIZIIA®...®A <
in
oo}
Then, as before, we introduce a direct sum of ordered tensor products (n+1)
E(m)
(A) = ®(®A)
.
or, r
Let us consider a canonical map
j: ,( )1)(A) 31
32
7n
ii
t2
in
,Cn((Am)n, A),
j (a ® b ® ... 0 c) (hi, ..., hn) = ah7til bh72 ;2 ...h7inn c, 1
h; E Am
.
We set Gn,A((Am)n, A) = Im j ' S2( )1) (A) = E( )1) (A)/Ker j.
The map f : G -- A, G c Am is said to be n times NC-differentiable if it is n times Frechet differentiable as a map of the Banach space Am into A and the derivatives f (k) (x) belong to the classes SZ( )1 (A).
Example 2.10. Considre a function of two noncommuting variables
f (x, y) = axbxcyd,
a, b, c, d E A.
Then we have 1
1
1
1
Oxf = a®bxcyd+axb0cyd,
Dixf = a®b®cyd+a®b®cyd, 1
2
2
1
2. Differential Calculus on Algebras
f=
V
307
b®c®d.
a®b®c®d+a®1
1
2
2
1
1
1
By analogy, we can consider a more general case of functions of several noncommuting variables with values in the Banach A-module. We shall begin with introducing an ordered tensor product of n copies of the algebra A and one copy of the module M. As before, we assign two indices, an upper index and a lower one, to every tensor product and, in addition, assign an upper index to the module M. Thus we have k+1
®(A, M) O,K
71
72
7k
2k+1
jn
A®A®...®M ®...®A. it ik in ik+1
i2
Next, we again introduce a direct sum of ordered tensor products with respect to or, rc, and with respect to the number k of the position of the module M: n+1
k
®(A, M). E 1) (A, M) = k=1 ® o,w o,K
Then we consider a canonical map j: E( 1)(A, M) -4 Ln ((Am) n' M). We set
Gn A((Am)", M) = IM j 1(n+1) E(m)1) (A, M)/Ker j. The map f : G -p M, where G is an open subset of Am, is said to be n times NC-differentiable if it is n times Frechet differentiable as a map of the Banach spaces Am and M into M and its derivatives belong to the classes SZ( 1) (A, M). By analogy, we can define NC-analytic functions of several noncommuting variables with values in the module M.
Suppose that there exists a topological basis {En} in the module M (as in a Banach space); r m, and rnm, are the right and the left structural constants:
en Em = > rnRnsES, 9
Enem =
rnmsE.'.
9
The function f (x) with values in M can be expressed in terms of the basis and its NC-derivative Vf(x) = E(VfnmL(X)En 0 em + VfnmR(x)en ® Em) n,m
Chapter VII. Noncommutative Analysis
308
Theorem 2.4. Let the functions g, f : G -+ M be NC-differentiable. Then the A-linear combination cp = a f Q + Agp, a, 0, A, p E A is also NC-differentiable, with Vco = aV f /3 + \Vgµ and R
V cslL = > 1'mrlrkns(akV fnmLNr + \kV gnmLpr)) VcSIR =
L
/3r + )kVgnmRµr) E7knsrmrl(akVfnmR/
Theorem 2.5. Let the functions f : G --> M and g: G -+ A be NC-differentiable. Then the product co(x) = f (x)g(x) is also NCdifferentiable, and the Leibniz formula V W = V f g + f Vg is valid and we have R
VcOnpL = >(1'mkpVfnmLgk + rkmn {kVgmpL),
VcnpR = >('ykmnfkVgmpR + rmkpVfnmRgk)
Theorem 2.6. Suppose that f : G -+ Am, where G is an open subset of An, and g: W -> Ak, where W is an open subset of Am, are NC-differentiable at points x E G and y = f (x) E W respecAk is also tively. Then the composite function cp = g o f : G NC-differentiable at a point x and Vcp = VgV f . For the module M = Am we can identify M ®A ®A ®M and (A ® A)m and realize the NC-derivatives as matrices with elements from
1Z(2)(A). The functions f (x) and g(y) are vector-functions: f (x) = (f 1(x), ..., fm(x)) and g(y) = (gl(y),..., gk(y)). As in the ordinary analysis, we obtain matrices of the derivatives V f (x) = (V., f' (x) ), Vg(y) = (V g2(x)), V ;fi(x), V :g'(y) E Q(2) (A). The matrix U = (U;3 = a13 ®b;3) acts on the vector h = (hl, ..., hn) according to the law Uh = (n a;jhjb;j). The matrices U = (U, = a13 0 b13), V = (Vj = -1
czj (9 d13) are multiplied according to the law UV = ( a;,cjk 0 d3kb;3).
3. Generalized Functions
309
Example 2.11. Consider the Clifford algebra A2 with two generators Q1 and Q2: o1a2 + o2u1 = 0, tr = o2 = 1. This is a fourdimensional K-linear space with basis e1 = 1, e2 = Q1, e3 = a2, e4 = 5102 Using (2.1), we find that the kernel of the canonical map j: A2 ®A2 -+ C(A2i A2) is zero and the algebras A2 ®A2 and 1(2) (A2) coincide. Let f (x) = a1xo1 and g(y) = or,ya1. Then co(x) = g(f (x)) = x,
VW = 10 1. Using the chain rule, we get the same answer, namely, Ocp = VgVf = (Q1 0 a,) x (Q1 ® a1) = of ® o
3.
.
Generalized Functions of Noncommuting Variables The function f : G -+ M, where G is an open subset of A'", is
said to be p times continuously NC-differentiable if f is n times NC(A, M) is continudifferentiable and the derivative V" f : G -ous. The space of p times continuously NC-differentiable functions f : G -p M will be denoted by NP(G, M). We use the symbol N"(G, M) to denote the space of NC-analytic functions f : G -+ M. In the space NP(G, M) we introduce a topology of uniform convergence on compact subsets D C G together with all NC-derivatives. This topology is defined by a system of prenorms 11f II Dj =SUP J V3f (x) Ijn(i+1)(A M), xED
= 0, 1'...) p.
(m)
The topology in the space N°° (G, M) of infinitely NC-differentiable functions is defined by a system of prenorms {11 I lD,j };°_o. The space NW(G, M) of NC-analytic functions can be topologized by means of a system of prenorms (2.4). The spaces N°° (G, M) and NW (G, M) are taken as spaces of test functions of noncommuting variables. It is natural to introduce generalized functions of noncommuting variables as linear continuous functionals of these spaces. Here we have a rather difficult problem of defining the concept of linearity which would correspond to noncommutative analysis.
Chapter VII. Noncommutative Analysis
310
In the projective tensor product A ® M we introduce a structure
of the right A ® A-module by setting (a ® m) (b 0 c) = ab 0 cm, a, b, c E A, m E M. Next, we consider the space Gr (A 0 M, A ® A) of
right A 0 A-linear continuous functionals TR: A 0 M -+ A 0 A. The map SR from 4(A 0 M, A 0 A) into the space £(M, A) of K-linear continuous functionals is defined by the relation SR(TR) (m) = trA TR(e ®m),
where trA, which is a trace on the projective tensor product A 0 A, is defined by the relation
trA X unmen ® em) = > unmenem n,m
n,m
We denote the image of this map by GA(M, A). This space is precisely the space of functionals which are linear in the noncommutative sense, i.e., NC-linear functionals. We have obtained a right realization of these functionals. By analogy, we can obtain their left realization proceeding from the A ® Amodule M 0 A and the space Gi (M 0 A, A 0 A) of left A 0 A-linear continuous functionals. Both the right and the left construction lead to the same space GA(M, A) of NC-linear functionals. We shall call the space GA(M, A) a topological conjugate of the module M and denote it by M'. The spaces (N°°(G, M))' and (N" (G, M))' are spaces of generalized functions of noncommuting variables. In conclusion, we shall outline a scheme of constructing a theory of noncommuting manifolds. It is natural to regard as a noncommuting manifold a Banach manifold with a model Banach space A'n in which the functions of transitions from a chart to a chart are NCdifferentiable (a finite or infinite number of times) or are NC-analytic. It is possible to consider noncommutative manifolds over Clifford algebras, algebras of matrices and operators, algebras of pseudodifferential operators.
It is obvious that the exposed formalism can be generalized to locally convex algebras and to other types of tensor products (for instance, to inductive products). When the concept of NC-linearity is
3. Generalized Functions
311
defined, it is not difficult to construct an infinite-dimensional noncommutative analysis, i.e., the theory of mapping A-modules (the theory of generalized functions of an infinite number of noncommuting variables and, in particular, Feynman and Gauss continual integrals over noncommuting spaces).
Remarks The results of Sec. 1 were published in [65, 68]. The results of Secs. 2 and 3 are announced in [165].
Apparently, the theory of analytic functions of several matrices constructed by Lappo-Danilevskii [41] was the first version of noncommutative
functional analysis. It should be pointed out that Lappo-Danilevskii also considered analytic functions of a countable number of matrices, i.e., constructed a version of an infinite-dimensional noncommutative functional analysis.
Chapter VIII
Applications in Physics
In this book we tried to explain a new approach to superanalysis. We hope that this approach will be widely used in applications to physics (especially in quantum field theory, quantum string theory, theory of gravitation), the more so as the majority of physicists have intuitively used functional rather than algebraic approach to superanalysis. Speaking about a superspace, physicists usually mean a set of points endowed with a superstructure and not a ringed space. The language of structural bundles is marvellous, but it is too powerful a tool for studying such a simple structure as superanalysis. In any event, studying the works by Salam, Strathdee, Wess, Zumino, Schwinger's pioneer work, I have realized that in these works the functional approach to superanalysis was used at the physical level of strictness. It was not my intent to study serious physical supermodels in the framework of functional superanalysis. This book is a monograph in mathematics, and the main goal was to expose the mathematical apparatus. Any physicist who reads this monograph will be able to use the apparatus of functional superanalysis in the investigations in which there arise fermion degrees of freedom. In this chapter we propose two new physical formalisms. In Sec. 1 we consider quantization in Hilbert supermodules. The main difference of this quantization from the standard quantization in a Hilbert space consists in the application of the theory of A-valued probabilities (Chap. V). However, we can restrict the consideration to the
314
Chapter VIII. Applications in Physics
De Witt formalism and regard as physical only those states which are associated with probabilities belonging to the interval [0, 1] C A. In Sec. 2 we try to give a correct mathematical definition of the amplitudes of the quantum field theory with real interactions of the type of (w4)4 with the aid of the distribution theory on infinite-dimensional spaces. When we consider transition amplitudes for quantum fermion fields, superfields, and gauge fields with the Faddeev-Popov ghosts, we have an infinite-dimensional superspace over a pair of CSM. We prove the convergence of a series from the perturbation theory in a space of distributions on an infinite-dimensional superspace.
1.
Quantization in Hilbert Supermodules
We propose to carry out quantization of systems containing boson and fermion degrees of freedom in Hilbert CSM (or in Hilbert superspaces). This approach to quantization differs from the quantization in Hilbert spaces that was used before (or, in particular, from the quantization of boson-fermion systems that was considered by Berezin in the framework of an algebraic approach to superanalysis [3]). When quantization is carried out in a Hilbert space, the appearance of fermion degrees of freedom does not lead to a serious change in the procedure of quantization. Just as in a pure boson case, we consider a complex Hilbert space H and self-adjoint operators in H. The quantum states have the form
f = E .fneni
(1.1)
n
where fn E C, {en} is an orthonormal basis in H. These states admit an ordinary probabilistic interpretation, namely, { Ifn l2 } are frequencies of realization of pure states {en}. Everything is much more complicated and interesting in the case of quantization in a Hilbert supermodule M. De Witt was the first to consider this kind of quantization at a physical level of strictness [27]. The quantum states have the form (1.1), but the coefficients fn belong to the CSA A rather than to the number field C. However, De Witt regards as physical states only states with numerical coefficients.
2. Transition Amplitudes
315
A more general approach is possible under which all vectors of the Hilbert CSM M (defined at the mathematical level in Chap. III) are regarded as physical states. For the probabilistic interpretation of the
state f = (1.1), f,, E A, use is made of the spectrally probabilistic formalism exposed in Chap. V. The main advantage of the quantization in the Hilbert CSM M is the availability of a structure of a module over A in M. This structure makes it possible to consider the transformations of the state space with parameters from A and, in particular, with anticommuting parameters (infinite-dimensional analogs of SUSY transformations). The main drawback is the absence of a spectral theory of self-adjoint operators in Hilbert CSM. However, this is a mathematical rather than physical problem.
2.
Transition Amplitudes and Distributions on the Space of Schwinger Sources
One of the main problems of mathematical physics is a strict mathematical definition of a continual integral for the amplitude of transition from vacuum to vacuum in the presence of a source fi(x): Z(S)
f exp{ 2 J (aµ(o(x)aµW(x) - m2cp2(x)) dx
-i f V (cp(x)) dx + i f W(x)e(x) dx} 11 dcp(x).
(2.1)
X
By now, the only correct definition of symbol (2.1) is Slavnov's definition in the framework of the perturbation theory (see [126]). Apparently, no other definitions of the Feynman path integrals can be applied in the quantum field theory for real interactions. The class of functionals c(W) constructed in Chap. III (as well as all classes of Feynman integrable functionals known to me, see [3, 5, 24, 26, 45, 50, 53, 55, 59, 65-68, 133]) cover only model interactions of the quantum field theory. The main difficulties in the definition of symbol (2.1) are, evidently, of a computational rather than of ideological nature.
Chapter VIII. Applications in Physics
316
The definition in the framework of perturbation theory gives an answer in the form of a formal series about whose convergence nothing
is known. This cannot be considered to be a satisfactory solution of the problem either. I suggest the definition of integral (2.1) based on the theory of infinite-dimensional distributions. Let us first consider a one-dimensional example +00
ZW =
J
2
exp{i (2 - V (cp) +
}dV.
-00
The integrand function g(W) = f (co)e" is not summable, and the integral is understood as a Fourier transform in the sense of generalized functions of the function f (c,). Recall (see, e.g., [15]) that a generalized function Z(C) is defined by the Parseval equality (Z, u) = (Z, u),
Z(AP) = f M.
(2.2)
The situation is the same in an infinite-dimensional case. We shall make meaningful not the values of Z at fixed points but the distribution Z(dC) on an infinite-dimensional space. Thus, we suggest that the continual integral (2.1) be realized not as a function on an infinite-dimensional space of sources but as an infinite-dimensional distribution.
2.1. Boson fields. Let V and W be infinite-dimensional dual modules over a commutative Banach algebra AO (in particular, linear spaces). For any S-entire function f (cp) on the space V the symbol f f (cp)e'('P,{>dcp is correctly defined as the distribution
Z(0de on the dual space W. By virtue of the Parseval equality (4.2), Chap. III, the expression
(v, Z) = f u(e)Z(de) is defined as
r
(u, Z) = f
u(di7)2(?7),
2. Transition Amplitudes
317
where 2(77) = P(Z)(r7).
In order to make this notation consistent with the notation from physics, it is convenient to change the sign in the exponent when defining the Fourier transform, i.e., f µ(d77) exp{-i(r7,
and then we have Z(77)
=
f e-i(1'0zw)
f e-i('' (f f
(cP)e'(w,t)dw)
de) dco = f f ((p)S((p
= f f (cP) (f
d1
-
17)dW
= f (rl).
Thus, the distribution Z(<) E M (W) is correctly defined by the relation
fu(e)1) = Ju(dii)f(ri),
uE
(W)
For a scalar boson field (see (2.1)) we set V =!9(R'), W = G'(R4). The function f (co) has the form fB(co) = exp{ 2 f (aµcP(x)8µcp(x)
- m2cp2(x))dx - i f V (w(x))dx}.
The function f B (cp) is S-entire on the infinite-dimensional space V for any polynomial potential V(x). Consequently, the distribution for a scalar boson field Z(dC) = Z(e)de, Z(C) = (2.1) belonging to the space M(C'(R4)) is correctly defined. The variational derivatives of the distribution Z(dC) (derivatives in the sense of the distribution theory)
U(xl, ..., xn) (<)
- i 6C(xl) ... i 6C(xn) Z('K)
are also correctly defined. This distribution belongs to the space M(C'(R4)) and acts on the test function according to the law
f u(e)U(xl, ..., X.)(de) 4'(R4)
Chapter VIII. Applications in Physics
318
f
u(dW)co(xl)...(p(xn) fB((p).
g(R4))
Note that the function co H cp(xl)...cp(xn) fB(cp) belongs to the space .A(9(R4)) 2.2. Fermion fields. We set V = G(R4, A8), W = 9'(R4, A8) and
write the elements of the superspace V as a(x) (x), Vij (x))'=o; the elements of the superspace W are Q(x) = (e3 (x), tj The function
fF(i. V)) = exp{z f (Y'(x)i7µ
,
(x)
-M (x)0(x)) + V (fi(x), V) (x))dx},
where V is a polynomial, is S-entire on the superspace V. As in a boson case, the distribution
Z(<<) = Cf fF(
dz)(x)dib(x))
)e'(E,+G)+i(E,+V)
x
x [J de(x)de(x)
is correctly defined on the superspace of sources W. This distribution belongs to the space M(G'(R4, A8))
Ju(e,)Z(ded) = fu(ddib)fF(b). By analogy, we can consider the case of interaction of a fermion and a boson field. Here the distribution Z(dcpdede) is defined on the superspace of sources G'(R4, RA8).
In the theory of gauge fields, a generating functional can be realized as a distribution on a superspace of sources Ja(x), ea(x), ea(x), where Ja are commuting sources of the gauge fields Aa; a and Ea are anticommuting sources of the Faddeev-Popov ghosts ca and c (see [53]).
2.3. Superfields. We set V
g+(RU4,4,
Ao) x g-(Ru4, Ao),
2. Transition Amplitudes
319
W = 9+(Ru4, Ao x G' (Ru4, Ao)
(see Example 5.2 in Chap. III). The distribution Z(dJ+dJ_)
(r fs(w+,
w_)e'(j+,w+)+i(J-,w-)
\J
11
dw+(x, O)dw_(x, 9))
(x,e)ER. 4
x
II
dJ_(x, 9)dJ+(x, B),
(x,B)ER44
where
fs(w+, w-) = exp{ i f (g (DD)2(w+w_)
1M DD (2 (W+ +W2)) - V (w+, w_)) dx } ,
2
V is a polynomial, is correctly defined on the superspace of sources W.
2.4. Some physical consequences. The formalism exposed above makes it possible to assign a mathematical meaning to source functionals that appear in the quantum field theory. The part played by the theory of distributions on a space of sources in mathematical physics is similar, in many respects, to the part played by the theory of generalized functions on R". A strict mathematical meaning is assigned to expressions whose definition as functions of a point and an (infinite-dimensional) space of sources did not met with success (compare with the definition of the Dirac 6-function). In the framework of the distribution theory on a space of sources, the quantum field theory is not a "pathological" theory from a point of view of mathematics. It is an ordinary theory of generalized functions, but only in an infinite-dimensional space. For fermion fields, superfields, and for boson fields with anticommuting Faddeev-Popov ghosts, an infinite-dimensional space is endowed with a superstructure. As an unexpected physical consequence we find that the quantum field theory is a statistical theory with respect to source functions.
Chapter VIII. Applications in Physics
320
For instance, for an electromagnetic field with a polynomial selfaction the amplitude of transition from vacuum to vacuum Z(J) is not defined when the external electromagnetic current J' (x) is fixed (and, in particular, when there is no current at all). However, the averagings of the amplitude over the space of external currents are correctly defined. Thus, under the proposed approach to the quantization of an electromagnetic field, we have nothing to say about the amplitudes for a fixed external current, only the mean amplitudes with respect to the fluctuations of external currents are correctly defined. In particular, there is no state without external currents here. In any event, we cannot get any physical quantities for such a state. We can only compute the means with respect to the fluctuations about the state without external currents.
2.5. The relationship with the Schwinger theory of sources. The proposed formalism is a further development of the Schwinger theory of sources [70]. In the Schwinger formalism, a particle is described by means of a switching in vacuum of a source K2(x) which occupies a finite space-
time domain followed by switching of a stoke Kl (x) which registers the effect of the action of K2 (x) on vacuum (Kl (x) and K2 (x) are combined to form a single source K(x) = Ki(x) + K2(x)) The action of the source K(x) on vacuum is of a probabilistic nature, and therefore we must perform a large number of switchings of the source K(x) in order to obtain the probability characteristics of a particle. Schwinger supposes that in the process of a large number of experiments it is possible to obtain completely similar sources K(x). However, since the number of experiments is very large, the sources fluctuate, and these fluctuations have not been taken into account in Schwinger's formalism.
We propose to complement Schwinger's formalism with assumptions concerning the probabilistic nature of the source. The physical answers in this formalism are means with respect to the fluctuations of the sources.
2.6. Gaussian packets of sources. We restrict our considera-
2. Transition Amplitudes
321
tion to a boson field: V = G(R4) and W = g'(R4). Let us consider the scheme, given in Chap. III, for constructing a space of functions of an infinite-dimensional argument which are integrable with respect to the Feynman distribution. Everywhere in this book, we took the space A(V) as the space 1(V). Restricting the space 'P(V) and providing it with a stronger topology, we can achieve a situation where the space oP (W) = .F(xF'(V)) contains the Fourier transform of all countably additive measures on V [140] and, in particular, the Fourier transforms of Gaussian measures, i.e., all Gaussian packets of the form p(C) = exp{-2B(e, )},
where B is a continuous quadratic form on the space W (the space W is nuclear, and therefore the continuity is sufficient for the countable additivity, see, e.g., [26]). For the Gaussian mean of the amplitude of transition from vacuum to vacuum we have (Z)B = f Z(e)p(e) dC
= f exp{-2B(C,e)}Z(de) = ffB(co)'yB(d), where 'yB is a Gaussian measure with a covariance functional B. Substituting the expression for the function fB(cp), we obtain
(Z)B = f exp
l2
f(aw(x)a(x) - m2co2(x))dx
-i f V(W(x)) dx}7B(dc2)
2.7. Continual integral in the framework of the perturbation theory. In his article [126] (see also [53]) Slavnov proposed a definition of a continual integral in the quantum field theory and in the
framework of the perturbation theory. With the aid of the Parseval equality Slavnov defined the quasi-Gaussian integrals
f
exp{i f w(x)e(x) dx}
Chapter VIII. Applications in Physics
322
x expi i f cp(x)K(x - y)cp(y) dxdy} 11 dcp(x) l
y
n
1
i 4(x1)...4(xn)
exp{
f
2
e(x)K-1(x
- y)e(y) dxdyJ
(the function f (cp) = cp(xl)...cp(xn) exp{i f w(x)e(x) dx} is a Fourier transform of the distribution n
1
IL (d77) = in
where b{ is a Dirac measure concentrated at the point , the function g(x) =
exp{-2
f
e(x)K-1(x
-
dxdyJ
is, by definition, a Fourier transform of the Feynman distribution 7(dco)
= exp{- f cp(x)K(x - y)co(y) dxdy} II dcp(x), 2 x
and relation (2.2) is rewritten as (f, ry) = (µ, g)). The expression for the generating functional (2.1) is defined in the framework of the perturbation theory. A connection constant is introduced before the potential in (2.1) and exp{-ig f V(cp(x)) dx is expanded in a series in the powers of g which is integration termwise with the use of the relation (2.2). As a result, we obtain
ZW _
E(
n=0
n9 )n
(f V(
'
n
)dx
/
x exp{ -2 f e(x)D'(x - y)e(y) dxdy where Dc(x) is a Feynman propagator (Green's causal function)
-1
/
Dc(p) =
p2
- 7n2 + io.
(2.3)
2. Transition Amplitudes
323
Note that even in a finite-dimensional case series (2.3) does not converge. Let us consider, in the framework of the perturbation theory, the integral +00
Z( ) if=o = 0000
k
1 era
f exp{ 2 02 - 29cp2n + icpe}dcp t-o
-
-00
1 (-i9)k a2nk +00 cP2 k! \ i2nk a2nk f e"°{ eXp{- 22 }
dco
-00
_
0
1 [-in+19] (2kn)!
00
-
2ir2
k=O
k!
2kn
(2.4)
(kn)!
This series diverges for n > 2. How can we interpret the formalism of distributions on a space of sources in the framework of the perturbation theory? We multiply series (2.3) by the formal expression rj de(x): X
Z(C) ll de (x) =
E 00
(
n!
)n
(Jv(_8))dx)
n
x exp{ - f e(x)D`(x - y) e(y) dxdy } 11 de (x) 2
l
(2.5)
J x
and consider
7(dk) = expS - 2 f e(x)D`(x -
dxdy } 11 de (x)
l
J
x
as a Feynman distribution with a covariance operator B = Then we can write series (2.5) in the form Z(dC)
-
'
n=0
( n9)n
(f V ( (x) )dx)
+ m2.
(2.6)
where the variational derivatives of the Feynman distribution are understood as generalized derivatives of a distribution on an infinitedimensional space.
Chapter VIII. Applications in Physics
324
Theorem 2.1 (convergence of a series from the perturbation theory of the quantum field theory in a space of distributions on an infinite-dimensional space). For any polynomial V (x) series (2.6) converges in the space of distributions M(9'(R4)).
Similar theorems hold for fermion fields, superfields, gauge fields. The corresponding series of the perturbation theory consist of generalized derivatives of Feynman distributions on an infinite-dimensional superspace.
Remark 2.1. The following example illustrates the situation with a series of the perturbation theory. Consider the series Z(l;)
_
e27rin 00
n=0
This series (consisting of "good" ordinary functions) does not converge pointwise but only converges in the framework of the theory of J (l;) being a periodic s-function. Actugeneralized functions, converges. Note that we must ally, the series Z(<) = E n=0
distinguish between two problems, namely, the problem of assigning a mathematical meaning to the sums of series of the perturbation theory in the quantum field theory and the problem of calculating the sums of series of the perturbation theory at specific points. We have solved only the first problem, i.e., we have found that the sums of series of the perturbation theory are distributions on an infinite-dimensional superspace. The question concerning the assigning of meaning to the values of generalized functions at specific points has not yet been answered even in a finite-dimensional case. The difficulties encountered in this direction are not connected with infinite dimensionality of quantum models. Indeed, let us try to calculate the value of a periodic b-function at
a point e = 0. A formal substitution of
= 0 into a trigonometric series gives a series > 1. Is it possible to obtain a finite number from this infinity? Formulas that are of the same mystical nature as those from the field theory do exist. Let us sum up a trigonometric series
2. Transition Amplitudes
325
as a geometric progression: 1/(1 - e2ai{). The function Z(C) can be continued up to a meromorphic function in a complex plane. We expand this function in a Laurent series at the point C = 0,
Z() _ - 27
1
+
1
2
+ajC+a2e2+...
and regard the value of the regular part of the Laurent series at the point C = 0 as a regularization of the sum of the series00E 1. Thus we n=0
have Zreg(0) = 2
Using the same rule to calculate the value at the point C = 0 for w the series > (which converges only in the space of generalized n=0 functions), we obtain
0
(E n=0
ne2nintll
_ =0
1
12'
and this is consistent with the answer n = - which we obtain n=0 for the main state of a boson string with the aid of the Riemann (function. By the way, even the negativity of the sum of the series consisting of positive numbers is a result that admits a physical interpretation. The main state of a boson string is a tachyon (see [25]). One of the possible points of view concerning the nature of divergences that appear in the process of the calculation of values of generalized functions at specific points (and, possibly, concerning the nature of generalized functions itself) is the following. It is possible that divergences appear because when calculating physical quantities we use only the field of real numbers (or its quadratic extension, i.e., the field of complex numbers).
2.8. Non-Archimedean hopes. The field of real numbers is used in physics for such a long time (from the time of Newton) that many researchers regard it as something given by God or something inherent in the nature of the world around us. However, real numbers are only the creation of our mind (for instance, Poincare said the
Chapter VIII. Applications in Physics
326
following: "As a result, we can say that our mind is capable of creating
symbols; due to this ability, it constructed mathematical continuity (i.e., a field of real numbers) which is only a particular system of symbols," see [51].
It should be pointed out that far from always the symbols called real numbers were regarded as something real.
"As early as Middle Ages, such combinations of symbols as f were called numeri ficti, "made-up numbers," or, in Liber abaci by Leonardo of Pisa written in 1202, they were called numeri surdi, "blanc numbers," and were not considered to be numbers at all." For the first time, in Arithmetica integra by Michel Stiefel published in Nuremberg
in 1544 they were given a conditional meaning of numbers and the corresponding name numeri irrationales. Stiefel stated: "irrationalis numerus non est verus numerus," i.e., that "irrational number is not a true number," see Florenskii [61, p. 507]. Possibly, in 400-700 years many number fields which are now regarded only as abstract mathematical constructions will be regarded as real physical objects. Just as now we use the symbols of real numbers to denote a point in space-time (actually, identifying them), in future, we shall, possibly, use the symbols of numbers from other fields to denote some physical quantities (actually, identifying them). Since Archimedean number fields (complete, normed) are exhausted by the field of real numbers (and its quadratic extension, i.e., the field of complex numbers), only non-Archimedean number fields deserve particular attention (see [21, 66]). Every process of measuring a physical quantity begins from a choice
of a unit of measurement 1 and a coefficient of increase of the measurement unit K = m (and, respectively, the coefficient of decrease k = 1/m). Actually, we can only measure numbers of the form
x=
a-n ,Mn
+...+ a-1 +ao+alm+...-i-a,m'; in
aj = 0,..., m - 1. And then our mind has two equally natural possibilities, namely, to admit that the process of decreasing the unit of measurement m times can be continued indefinitely and to admit that
2. Transition Amplitudes
327
the process of increasing the unit of measurement m times
can be
continued indefinitely. In the first case, we obtain the symbols of the form
X=...+a-n +ao+alm+...+a,.mr. Mn +...+a1 m These are real numbers in the m-adic representation. These numbers are well known in physics (although, it should be recalled that 700 years ago they were not regarded as numbers). In the second case, we obtain symbols of the form
x= a-n +ao+alm+...+armr-{-... Mn +...+a-1 m These are m-adic numbers (see, e.g., [72]). In particular, if m = p is a prime number, then these numbers form a field, a field of p-adic numbers. Since the field of p-adic numbers appeared as a result of an infinite
increase of the unit of measuremens 1 p times, we shall try to use panic numbers for describing quantities that are understood as infinities in the field of real numbers. Let us consider, for instance, a trigonometric series 00
C
ZP(S)
_
E pne21rinf
n=0
which converges in a space of generalized functions. The value of this 00 generalized function at the point = 0 is a series E pn, p = 2, 3, ... n=0
diverging in the field of real numbers. However, this series converges in the field of p-adic numbers, and the sum of the series is a rational number ZP(0) = 1/(1 - p). We can show that a one-dimensional model of the perturbation theory, series (2.4), converges in the quadratic extension of the field of p-adic numbers for any p (if p - 1 (mod 4), then the series converges even in a field of p-adic numbers).
We hope that the series from the perturbation theory for the padic quantum field theory converges not only in the space of infinite
328
Chapter VIII. Applications in Physics
p-adic-valued distributions, but also at every point of the p-adic space of sources.
See [66] for the quantum mechanics and the quantum field theory with p-adic-valued functions.
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Index Algebra Arefyeva-Volovich 294 associative 7 Banach 9 Grassmann 8 Kobayashi-Nagamashi 53 noncommutative 294 Rogers 10 De Witt 51 Amplitude of transition 315 Body of probability 248 of superspace 92
Cauchy problem 75 Cauchy-Riemann conditions 21 Chronological exponent non-Archimedean 270 Correspondence principle 197 Covariance 167 functional 228
Distribution cylindrical 240 exponential 233 Feynman 167 Gaussian 167, 269 quasi-Gaussian 167 Volkenborn 281
Duality of supermodules 128, 141 of superspaces 158 Faddeev-Popov ghosts 145 Feynman integral of boson field 170 of fermion field 318 in perturbation theory 321 of spinor field 170 of superfield 169, 319 Field boson 170, 316 fermion 170 neutral chiral 170 non-Archimedean 258 Formula Feynman-Kac 205 for integration by parts 172 Newton-Leibniz 36 Trotter 278 non-Archimedean 278 Frequency interpretation in Banach algebra 251 in quantum supertheories 315 Generalized functions analytic 63 non-Archimedean 264 noncommutative 309
346
Index Involution 136
Kolmogorov's axiomatics 245 in Banach algebras 245 Law of large numbers 232 Limit theorems 227, 254 Mean value 228 Mises theory 251 Nilpotent soul 99 subalgebra 99 Operator 86 adjoint 126 d'Alembert 86 evolution 205 heat conduction 79 Laplace 79 orthogonal 132 pseudodifferential 183 Schrodinger 86 self-adjoint 138 unitary 138 Pseudotopological superalgebra 94 superspace 96 Pseudotopology 94 Probability conditional 249 multivalued 247 Projective tensor product 301 Quantization 196 Random process 242 cylindrical 241 quasi-Gaussian 242 Wiener 243 Soul of probability 248
of superalgebra 92 of superspace 92 Spectrum of an event 247 Super algebra 7
with involution 136 Lie 199 locally convex 60
analyticity 18 conformality 20 differentiability 10 form 43 group 204 manifold 19 module 60 Banach 116 conjugate 61 covering 144 Hilbert 130 locally convex 60 topological 60 space 9 Banach 19 Hilbert 145 infinite-dimensional 222 non-Archimedean 258 noncommutative 294 pseudotopological 92 symmetry 19 Theorem central limit 228, 229 of Lyapunov 235 of Riesz 131 Topological basis 142 Transformation Fourier 74 Laplace 267
Index
Transition amplitude 315 Wiener process 242
347
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