Introduction to Random Time and Quantum Randomness

1>(\)6(q-\)d\ /

-oo

can be regarded as a linear combination of (improper) solutions of the eigenvalue problem (2.3), which suggests defining something like (Efi(\)1;)(q)=

f

tl>(x)6(q-x)dx,

J — oo

whose heuristic properties are precisely those of the r.h.s. of (2.4). It follows from (2.6), in particular, that the expectation (Q)^ of the position observable in state ip corresponds, as in equation (2.1), to the choice if = ip. A good way to remember the expression "resolution of the identity" is to check that for each given such family I?"4(A), we can write the identity

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operator as oo

/

dEA{\)

(2.7)

•OO

because /-OO

OO

/

d{1>\EA{\)1>) = /

d\\EA{X)n2.

(2.8)

J — OO

-OO

(2.7) is often called the completeness property in quantum physics textbooks. According to Axiom 2.3, the probability that the position observable of the system (say a single particle) in state tp is in the interval A is given by ||£ Q (A)y,|| 2 = / |y>(
(2.9)

JA

Notice that it would be meaningless to keep asking for the probability that Q has a sharp value since O~Q is purely continuous. As a matter of fact, the generalized eigenvectors of Q, solving (2.3) in Dirac's sense, are not even quantum states in the sense of Axiom 2.1 (although they can be approximated with any accuracy by quantum states and, for that reason, are often used by physicists as if they were regular quantum states). More generally, taking advantage of the notation (2.7), the above projection onto the subspace of eigenstates such that the eigenvalue of any observable A lies in A reduces to E£ = J^2 dEA(\). So Axiom 2.3 implies that the corresponding probability is P r o b ^ A e A) = | | £ ^ | | 2 .

(2.10)

It was in the form of (2.9) that probability originally came into the quantum picture, in a work of M. Born (1926), and it is because of (2.9) that the Hilbert space L 2 (R) is the natural home for a quantum state. It follows from this, for example, that the expectation (Q)^ of the position observable in state ip can simply be computed, without using E®(\), as /.OO

OO

/

q\lP(q)\2 dq.

i>(q)QiP(q)dq= -oo

(2.11)

J — oo

We shall come back soon to this probabilistic interpretation of state ip (or wave function, as it was called at that time).

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Since the arena of classical Hamiltonian mechanics was phase space, it is natural to expect the second fundamental observable of quantum theory to be the momentum. One defines —>W = L 2 (R)

P:VP

tP(q) H— -ih—1>(q),

(2.12)

where h is a real number, Planck's constant. Vp is made up of all ip in H which are absolutely continuous (on every compact interval) and such that Pip £ H. Now the solutions of the eigenvalue problem P are decent functions ("plane waves") proportional to eipg =
peR.

(2.13)

Whatever the proportionality constant, however, it is not in H = L 2 (R) and therefore cannot correspond to a realistic physical state (although, again, they can be approximated arbitrarily well by such quantum states). As for the position operator, we can, however, form a linear combination of solutions (2.13) which does correspond to a physical state: oo

/

c(p)
c{p)€C.

(2.14)

•oo

This means that we are representing the physical state ip in terms of the unrealistic functions (2.13) of constant momentum p = hk and amplitude C {P)- (p = hk is & relation between matter and waves, since the physical dimensions of k are the inverse of a wavelength, introduced in 1924 by L. de Broglie in his PhD thesis.) Note that (2.14) is the inverse of a Fourier transform, so c{p) = - =

/

iP(q)e-^dq

= (tpp\1>),

and that f^° \c(p)\ dp = J - ^ 1^(^)1 dq. Therefore if the probabilistic interpretation of ip{q) is taken seriously (see (2.11)), one should define the expectation of the momentum by oo

/

•oo OO

p\c(p)\2dp -1

i»00

/ -oo

pc(p)-== / ^ ( g ) e * « dqdp. yzTtn J -oo

(2.15)

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Changing the order of integration, and using again (2.14), (2.15) is equal to fOO

OO

-l

/ i,{q) • oo

J-oo

= J°°

c(p)-=ei"dPdq VZirn

$(q)(-ik^\iKq)dq />00

OO

/

(-tftVlog iftq)) |V(
^{q)Pi,{q)dq= -oo

(2.16)

J — oo

justifying a posteriori definition (2.12). The Fourier transform of ip (i-e., our above-mentioned c(p)), 1 f°° i (Ety)(p) = - _ / e-^^q)dq, V27ra 7-oo

(2.17)

often denoted V>(p) and central to the previous argument, is unitary, i.e., it maps 7i on itself, and leaves the scalar products (and therefore the norm) invariant. We shall come back soon to the key connection between unitary and self-adjoint operators (Stone Theorem). One can find formally that UP = QU i.e., P = U~lQU.

(2.18)

Using this and (2.5), we can obtain the spectral representation of P. It is easy to check that U~1E®(\)U is a resolution of the identity. So oo

/

XdEp(X),

withVP

= U-1VQ,

(2.19)

-oo

where Ep(\) = U~1E(^(X)U is the projector on those ip such that (Uip)(p) = xp(p) = 0, Vp > A. Hence the momentum, like the position observable, has a purely continuous spectrum ap = R. The complete proof of (2.18) is founded on the fact that a ip belongs to Dp if and only if (pUtp)(p) is square-integrable and, in this case, (UPip)(p) = (pUip)(p). This implies that the application of P to any state tp in Vp is equivalent to taking the Fourier transform t/j of tp, then multiplying by p and taking the inverse Fourier transform of the result. We may regard this as a new representation of the momentum observable, Pip = UPU-14>=pip,

(2.20)

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as well as of the position observable, Q4> = UQU'1^

= ih—4>. op

(2.21)

We are still describing, of course, the same elementary physical system, a one-dimensional (spinless) particle, i/> a n d 'ip a r e different mathematical "representations" of the same state in the abstract space 7i. But the result of an experiment is the expectation value of an observable, which is invariant under any such change of representations. Indeed, under ip —» Uip = ip any observable A becomes A = UAU~\

(2.22)

so that, with the appropriate restrictions on domains, (•4>\Ai>) = (U-ip\UAU-1Uip) = (Uip\UAip) = (VW)-

(2-23)

Definitions (2.2) and (2.12) in "H = L 2 (K) correspond to the position (or Schrodinger's) representation. On the basis of (2.20) and (2.21), the position and momentum operators are said to be unitarily equivalent via the Fourier transform. Let us now consider the quantum version of the two-body problem, i.e. the motion of an electron and a positively charged nucleus under the effect of potential (1.2), named after Coulomb in this context. The classical Hamiltonian of the electron with relative coordinate r = (q\ — 172), qi 6 K3, the proton being regarded as infinitely heavy, is

2m

||r||

where m is the reduced mass of the system of two particles and e a constant, the charge of the electron. Our problem is now to construct the quantum Hamiltonian observable, i.e., to quantize the classical Hamiltonian (2.24) using definitions (2.2) and (2.12) of the quantum position and momentum observables, respectively. By the Spectral Theorem we know that for any complex-valued Borel-

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measurable 3 g, if the spectral resolution of an observable is oo

/

\d^\EA{\)^p)

-oo

then (ip\g(A)tp) = f^°oog(\)d('ip\EA(\)ip), with the associated proviso on the domain. In other words, if A is the quantum observable associated with the classical observable a, then g(A) is associated with g(a). When if = ip, this allows us, for example, to compute the quantum analogue of moments of a distribution (take g(X) = An, n € N). In particular, let ip be in T>A and T>^. Then

a2(A) = - mm? is the variance of observable A in state ip, whose positive square root <J(A) = A,pA, the standard deviation of A, describes the average deviation of A from its expectation (since a2(A) = {ip\(A - (ip\A^))2ip)). Notice that iff/' is an eigenvector of A with eigenvalue A, A,/, A = 0, but, for general ip, this is not the case. The quantization procedure should preserve linearity. Using these rules, the Hamiltonian corresponding to (2.24) is the elliptic operator H = -!-P2 + V (2.25) 2m for V(r) = —e 2 /||r|| in the case of the Hydrogen atom. T>H is made up of all ip in "H = Z/2(R3) such that Vi/> is also in Ti. Since, for example, C Q ( R 3 ) , which is dense in Tt, is contained in X>#, H is densely defined. The Coulomb potential is, of course, unbounded but still small compared with the kinetic energy. This is why H can be made self-adjoint. More generally, A is called bounded below when 3a € R such that for any state tjj in T>A, {if)\Atp) > a. For example, most Hamiltonians are bounded below since this means that the system has a lowest energy level (the "ground state"). If A is Hermitian (or "symmetric") and bounded below, then it has self-adjoint extensions. But, for a given symmetric A, there may be several self-adjoint extensions or none, so the uniqueness of the extension is of special interest. In this case A is called essentially selfadjoint. It is not true that any Hermitian Hamiltonian bounded below is essentially self-adjoint. There are restrictions on the potential V. On the other hand, there are Hamiltonians unbounded below which are essentially 3

Continuous will be sufficient here.

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self-adjoint, for example (2.25) with V(q) = —q2/2. But let us come back to our Coulomb potential. Consider the case where the spectral family EH(X) of H has a discontinuity at A = E. Since EH(X) is right continuous and nondecreasing, one can define, for e > 0, PH{E) = EH{E) - EH{E - e). The range of PH (E) is not empty. Consider a non-zero tp in it. Then, EH(E)ip = V> • #(A - E), where 6 is the Heaviside distribution (9(x) = 1 for x > 0, 6(x) = 0 otherwise) and we have, symbolically, /*oo

oo

/

\dEH(\)ip= -oo

/

XS(\ - E) d\ • ip,

J — oo

i.e., Hip = Eip.

(2.26)

This is Schrodinger's time-independent equation (or Schrodinger's eigenvalue problem). Reciprocally, by definition (2.1) of the expectation of the Hamiltonian observable H in state tp, we see that H has the value E with certainty (since A^if = 0). Such a E is called an eigenvalue of H for the eigenvector ip. The collection of eigenvalues of H is its point spectrum, denoted app(H) and the closed space Hpp spanned by the eigenvectors, W

P P = {V' I PrpW = {^\EH{X)ii)

is pure point},

is often called point subspace. Such a state ip is called a bound (or stationary) state, by analogy with a bounded motion in classical mechanics (such as, for example, the elliptic orbits of the Keplerian, two-body, problem for a negative value of the conserved energy h(q,p) = e < 0). For the Hydrogen atom, the eigenvalues found by E. Schrodinger in 1926 coincide with the experimentally observed wavelengths emitted by this atom and collected into an empirical relation called the Rydberg-Balmer formula. It was to explain this remarkable regularity that N. Bohr had concocted, in 1913, his model of the Hydrogen atom, a combination of the planetary picture suggested by the classical two-body problem with non-classical ideas whereby only a discrete number of orbits were allowed. Schrodinger's discovery made Bohr's old quantum theory immediately obsolete, all the more since, as will follow from Axiom 2.4, Schrodinger's approach was not limited to stationary states.

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Since Ji = L 2 (R 3 ) is infinite dimensional, we know (from the position and momentum observables) that H may also have a continuous spectrum. The orthogonal complement of Tipp is called the continuous spectrum subspace (and the spectrum of H on this subspace is the continuous spectrum). As before, the associated eigenvectors are not in L 2 (K 3 ). Then EH(\) is continuous at A = E but not constant in any neighborhood E — e < X < E + s. When the particle (here our electron) is in such a state ip, it wanders away from its initial position, where it might never come back. Suppose that p^(dX) = d(ip\EH (X)ip) is absolutely continuous, i.e. that there is a distribution F such that dp^(X) = F(X) dX. Such a state tp is in the absolutely continuous subspace. The physical picture is that of scattering theory: the particle will never return, and therefore it makes sense to study its asymptotic properties for arbitrarily long times. Mathematically speaking, the problem is to classify systems according to their asymptotic behaviour. For example, it is natural to hope that for states in the absolutely continuous spectrum, particles will behave asymptotically like free particles (V = 0). Pi>W = \\EH(X)tp\\2, called, in general, the spectral measure of H, generates a Lebesgue-Stieltjes measure on K. According to a famous result of Lebesgue, such a measure admits a canonical decomposition into a pure point part, an absolutely continuous part and a singular continuous part, i.e. one supported by a zero Lebesgue measure set but without a pure point part. The possible existence of this last component of the spectrum is often a nuisance since the interpretation of the associated quantum states is somewhat obscure. The simplest situation is when all states of the system correspond either to bounded motions or to unlimited motion without return. Then the system is called asymptotically complete. For our Hydrogen atom there is no singular continuous spectrum (this is, actually, true of the .ZV-body problem, V7V) and the system is asymptotically complete. The interpretation is, therefore, easy and we can symbolically represent the spectral family of the Hydrogen atom as in the figure below, where En = -me4/(2h2n2), n € N (Balmer formula) and E\ = -13.6 eV is the ground state energy. This means that we need to impart +13.6 eV to the atom in order to unbind the electron from the proton ("ionization"). For an arbitrary spectral family, the picture can be much more complicated than this. In particular an eigenvalue can be embedded in the continuous spectrum, so E(X) is discontinuous there but continuously growing in the neighborhood.

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EH(\)

X = Energy Ei = - 1 3 . 6 eV

En

Axiom 2.4. Let H be the time independent Hamiltonian observable of system. Then the time development of the state tp at time to is given by Ut-to '• H —> T~C,ip <—> ip(t), where UT is a one-parameter family of unitary operators on H defined by UT = exp(—\TH). Equivalently,

ih-£(Ut-t0i>)

(2.27)

ff(tft-t„VO-

This is the axiom of quantum dynamics and (2.27) is the time-dependent Schrodinger equation. Let us stress at once that it is perfectly deterministic. More generally, Stone's Theorem states that to any continuous oneparameter family of unitary groups Ua corresponds, by the functional calculus associated with the Spectral Theorem, a unique resolution EA of the identity such that Ua

f

^dJE^A) = e"* c

Vae

(2.28)

J —(

and EA(X)Ua = UaEA(X), Va, A £ K. The domain VA of the (self-adjoint) generator A of Ua is the set of ip in TC such that (in the sense of strong convergence of operators) ih

Ua-I

A. a Notice that if the generator A is only Hermitian the unitarity of Ua is generally lost. Applying Stone's Theorem to Uaip = tp(a) in T>A, we get (2.27) when a = T and A = H.

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The family of states ip(a) is called the orbit of t/j under the action of the group Ua of automorphisms of H. Let us recall, in particular, that the self-adjointness of H (needed for the Spectral Theorem) is equivalent to the existence and uniqueness of the solution of Schrodinger's equation (2.27) for all time. So, when the selfadjointness of H is proved (a seriously nontrivial prerequisite, in concrete situations), we no longer need to worry about dynamical singularities, as in the classical case. For example, it has been known since the fifties that the quantum iV-body problem is self-adjoint (Kato-Rellich Theorem). Because of equation (2.27), this picture of the dynamics is called Schrodinger's picture. As in the case of classical Axiom 1.4, the state evolves in time and observables (1.4), such as Q, P, and H before, do not. An equivalent description is named after Heisenberg and corresponds to the quantization of the classical picture (1.13). If one forgets about the (seriously nontrivial) problem of domains, an observable A evolves in time according to the automorphism A ^ A(t) = UlAUu

(2.29)

where U} = Uf1 denotes the adjoint of Ut (cf. Axiom 2.4). As for the momentum operator (2.19), if E^(X) is the resolution of the identity of A = A(0), then U^EfrUt = Ef is that of A(t). For the same reasons as in (2.23),

m)\M(t)) = WAW)-

(2-30)

The informal quantum version of the equation of motion (1.8) of a classical observable follows easily: ih~A{t)

= ihjtA{t)

+ [A(t), H}.

(2.31)

The first term takes care of the possible explicit time dependence of A (in Schrodinger's picture), denoted by ^4s, and can be written, using (2.29), as ihUf1^-^. The second term denotes Uf1[As,H]Ut, where [•, •] is the usual commutator [B, C] = BC — CB. The similarity between Heisenberg's equation (2.31) (highlight of his "matrix mechanics") and the classical one (1.8) is at the origin of Dirac's famous correspondence principle between classical and quantum observables, {a,b}^±lA,B},

(2.32)

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which played a crucial role in the construction of quantum theory. We shall come back to this issue soon. Let us stress again that, because A and H are in general unbounded, there are delicate problems of domains associated with equations (2.29) and (2.31). Here we only consider elementary systems where such problems do not arise. The above-mentioned analogy allows us to define a quantum constant of motion A of the system under consideration by the property (cf. (2.31)) 0 = ih-A(t)

+ [A(t),H].

(2.33)

In particular, when such an operator A = As is not explicitly time-dependent, A commutes with the Hamiltonian H of a conservative system. This statement is independent of the dynamical picture, since, in this special case, A and A{t) coincide in Schrodinger's and Heisenberg's pictures. Taking the expectation of (2.31) in the (appropriate) state ip, one can compute the general time evolution ih±(A{t))i,

= ([A(t),H})^ + ih(^-(t))^.

(2.34)

In the same special case as before, (A(t))^ is a constant. We can specialize this to the simplest class of conservative systems considered in (2.25), i.e. of Hamiltonian H = ^-P2

+ V(Q),

(2.35)

for Q and P defined in (2.2) and (2.12). Using (2.31) and the fact that Q=^-[Q,H}^-P, in

rn

P=\[P,H} in

=

-VV(Q),

we find

±Q(t) = -P(t), at

jtP(t)

m

= -VV(Q(t)),

(2.36)

i.e., the quantization of Hamilton's equations of motion (1.3) for such systems. Their expectations are called Ehrenfest equations:

jt(P(t))«,

= -(VV(Q(t))^.

(2.37)

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However since, in general, (W(<5(<)))^, ^ W((Q(t)) tween classical and quantum dynamics is not trivial.

,), the relation be-

Projection postulate. If the measurement of an observable A yields a value in the interval A = ]Ai, A2], then the state of the system immediately after the measurement is an eigenvector of EA(A) = EA{\2) — EA{\{). Consider the simplest situation where A has a purely discrete, nondegenerate spectrum (i.e., a single eigenvector e^ corresponds to each eigenvalue Aj). Then the postulate says that, immediately after a measurement of A giving Aj, the state of the system is an eigenvector of the projector Pi on ej, defined by Piip = (ei\ip) ei.

(2.38)

This is von Neumann's postulate of "reduction of the quantum state" (or "collapse of the wave function ip"). We did not call it Axiom 5 in order to stress its special status. It should be regarded, instead, as an empirical prescription of the Bohr (Copenhagen or orthodox) interpretation. In this operational sense, the projection postulate does not seem to have been invalidated in 80 years of quantum experiments. This reduction of the state does not follow from the Axiom of quantum dynamics (Axiom 2.4); von Neumann stressed repeatedly the fundamental difference between the "continuous, reversible and causal change of the system in the course of time" (described by Axiom 2.4) and the "discontinuous, irreversible, noncausal and instantaneously acting measurement" involved in the projection postulate [25, pp. 349-351]. The collapse of the wave function and its interpretation is one of the central difficulties of elementary quantum mechanics. The various "solutions" to this problem (none of them universally accepted) allow, in fact, to classify the various schools of interpretation. The concept of time reversal in quantum mechanics was introduced simultaneously (in 1931) by E. P. Wigner and E. Schrodinger, from two different viewpoints. Since Schrodinger's view will be amply elaborated later on, we shall summarize here Wigner's analysis. Let us look for the quantum versions of what we found in classical mechanics, i.e. a transformation such that Q — i > Q and Q i-> — Q for systems with time independent Hamiltonians (2.25). Since those H are invariant under such a transformation, equation (2.31) applied to Q, i.e. ihQ = [Q, H], shows that the transformation cannot be unitary (otherwise Q would be invariant as well). Let ip(q) be a state; then the transformation K : ip —> ip (the com-

2. Standard quantum

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plex conjugate) does the trick since, using definitions (2.2) and (2.12), KPK~l = -P and KQK~l = Q. The transformation K is the only nonlinear one considered in this section; it is called antilinear because K(ocip + flip) = aKip + J3K

{q,t)

(2.39)

and let us define j>(q,t)=Krl>(q,-t).

(2.40)

It is easy to see that ip is a solution of the same equation (2.39), in general distinct from ip (see the initial condition). Alternatively, one may say that Ktp(q,t) = ${q,t) solves the time reversed version of (2.39), namely -ih^{q,t)

= -£-A${q,t) + V(q)$(q,t).

(2.41)

The quantum dynamics of conservative systems is invariant under time reversal because the one-parameter family UT of Axiom 2.4 form a group, as in the classical case. The time reversed evolution of the systems is always possible (as long as we do not interfere with them). In particular, Born's interpretation (2.9), true for any time t since Ut is unitary, and according to which JA "tp(q, t) ip(q, t) dq is the probability to find the particle in state ip(q, t) inside the interval A, can also be regarded as an expression of the invariance of quantum dynamics under time reversal. This has interesting conceptual consequences, rarely stressed in textbooks. For example, all students of physics have to compute the "damping of the free wave packet" (case V = 0 in (2.39)), which seems to introduce a dynamical arrow of time. It does not, however; before this damping started, the solution had contracted, in a symmetrical way, showing all the respect for the principle of invariance under time-reversal. We shall soon see other hints that, as in classical mechanics, the role of time in standard quantum dynamics hardly goes beyond that of a passive parameter.

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Probabilities in Standard Quantum Mechanics

Let us come back to a probability space, or triplet (fi,^", P), as defined in Part 1 §2, denoting now the sample space by fl. For a random variable X defined on (Q, T, P), with values on the measurable space (R, 58(R)) (where Q3(R) is the Borel tribe, i.e. the smallest a-field containing all open intervals in R), the expectation E(X) can be computed through the distribution function F(X) = P{X < A} so that E(X)=

f XdP= [ XdF(X), (3.1) Jo, Jm. where the r.h.s. is a Lebesgue-Stieltjes integral (cf. reference [3] of Part 1). If g : R —* R is Borel measurable then E(g(X))

= f g(X) dP= f 5 (A) dF(X). (3.2) Jn JR When g(X) = A™ we obtain the moments of X. In particular the first moment, denoted by m, is the mean of X and the second moment of X — m is the variance of X, denoted by a2: a2 = E{(X - E{X})2}

= E{X2}

-

{E{X})2.

The square root a of the variance is called the standard deviation of X. We shall need in particular the concept of joint probability distribution of any two random variables X\ and Xi in the same random experiment. We shall denote their joint distribution by M(Ai, A2) = P{XY < Ai, X2 < A 2 },

Ai e R.

M satisfies the following compatibility conditions (where we have introduced the notation M(Ai,oo) = lim M(Ai,A2)): M(Ai, 00) = P{Xi < AL X2 < 00} = P{X1 < Ai} = F(Ai). So Ai 1—> M{X\, 00) is called the marginal distribution function of X\, and A2 H-> M(oo,A 2 ) that of X2- In general, marginal distributions do not determine the joint distribution function since X\ and X2 are not always independent and so their joint distribution is not always a product of those of the individual variables. Now consider the situation in quantum mechanics. By Axiom 2.3, for a given state ip, we have seen (in (2.10)) that the value of an observable A is found in the real interval A with probability Prob^(A € A) = HE^^H2.

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So, for a given state ip, an observable A corresponds to a random variable and E£ is the event that the value of this random variable lies in A. Then A H-» \\E£ VII2 = P$(&) is the probability distribution of A in state ip and A H-» ||.EA(A)V>||2 = p$(A) is the distribution function of A. By von Neumann's Spectral Theorem, the quantum counterpart of (3.2) is well defined and its operational interpretation is the classical one: the expectation of g(A) is already determined by the probability distribution of A. More generally, if (2.1) holds then oo

/

g(X)d\\EA(X)^\\2

(3.3)

-OO

for any tp e H such that J^ \g(\)\2 d\\EA(X)ip\\2 < oo. As mentioned before, this is useful, for instance, for computing the moments. We have already used this functional calculus of self-adjoint operators, for example when we assumed that the quantization of the classical Hamiltonian (2.24) was (2.25). Also, after Axiom 2.4, we constructed a unitary operator Ua given the spectral resolution EA(X) of its generator A via (3.3) for g(X) — exp(—(i/h)aX). Let us note that, as a matter of fact, for the quantization of (2.24) we have assumed more than what (3.3) says, namely that a linear combination of functions of observables is an observable as well. This requires that those two observables in a given state, regarded now as random variables, have a joint probability distribution. Classically, for two random variables defined on the same sample space O, this joint distribution is always defined (cf. [5] of Part 1 for example). In quantum theory this is not obvious; for example, there is no simple relation between the spectra of the two original operators and that of their formal sum. Moreover, it is in general not even true that the sum of two self-adjoint operators is self-adjoint. It is only Hermitian on the intersection of the two domains. A more fundamental obstacle, which has nothing to do with the delicate properties of (unbounded) operators in an infinite dimensional Hilbert space will be discussed now. As a matter of fact, most of the serious conceptual difficulties of elementary quantum theory are independent of the dimension of the underlying Hilbert space H. So let us specialize to the simplest case for a while. Let our Hilbert space H be two-dimensional. By Axiom 2.2, an observable A\ acting on ~H is a 2 x 2 Hermitian matrix. So there is an orthonormal basis {ej}, j = 1,2, of eigenvectors of Ai, with (real) eigenvalues Ai, A2, distinct by hypothesis ("simplest spectrum"). A\ is diagonal with respect to this basis and the diagonal matrix elements are the eigenvalues. Any

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state tp of our system can be represented by the "superposition" 2

^ = ^<eJ#)eJ--

(3.4)

Let us denote by Pj the projection on ej = ker(Ai — Xjl) (simple spectrum!). The completeness property is I — XL=i Pj = 5Zi=i \ej) (ejl> where the handy notation of Dirac for Pj has been introduced. Let us define a new family of projectors E^1 by EAl = 0, E^1 = £ * = 1 Pj, k = 1,2. Then Pj = EAl — EA^l. So the representation of A\, familiar in linear algebra, A\ — ^2j=\XjPj, reduces to a discretized version of von Neumann's Spectral Theorem: A^^XjiE^-Efl,).

(3.5)

One could get even closer to his version by defining the continuous family {EAl(\)} so that EAl(X) = 0 for A < Aj, EAl(X) = EAl for Ax < A < A2 A and E >(\) = I for A > A2. Then dEM{\) = EM(X) - EAi(X - e) is Al clearly zero, except at A = A., where it is dE (Xj) = Pj. By (3.4), Anf, = £ 2 = 1 A,-< ej #> ei = E*=i W t f The expectation of observable A\ in any state tp is given by 2

(VIAiVHE^-lteWI2-

(3.6)

By Axiom 2.3, Aj, j = 1,2, are the only possible results of a measurement of A\. If Xj belongs only to an interval A^, j = 1,2, then the probability to find the result of measurement of A\ in Aj is |(e3|V')| • Since HV'II2 = 1J we have a normalization condition consistent with this probabilistic interpretation:

£|< ej #>| 2 = l.

(3.7)

j= l

Suppose that, by chance, state ip of the system is an eigenvector of A\: ip = e\. Then, since basis {e^} is orthonormal, we find that A\ has the value Ai with probability 1. And we are just as certain of finding A2 if rp = e 2 . Besides, this is consistent with the standard deviations A ei ^4i = A e 2 Ai = 0. But when ip is a general superposition (3.4), the best we can hope for, in a measurement

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of A\, is a probability. Now pick a second observable, say Ai, for the same system and assume that A% and A\ do not commute: [A2, A\] ^ 0. By the Spectral Theorem, A% = Y^i=i ^iPi> where Pi = |e,) (ej| and basis {ei\i=i^ has no common vectors with {e.j}j=i,2 (otherwise, A\ and A2 would commute). Now, regarding the measurement of A\, certainty is possible only when if) = ei or e2 and, for A2, only when ip — e.\ or e.2- Since the two bases have no common vectors, there is no state ip such that the result of a measurement of Ai and A2 is certain. We say that A\ and A2 are not compatible or that they cannot be "measured simultaneously". Now, clearly, the existence of the joint probability of any two random variables, in the classical sense, takes for granted the fact that, although unknown, both variables have a sharp value. Such a claim is flatly denied in quantum theory. This is why von Neumann formulated Axiom 2.3 (in its version (2.10)) for a set of observables A\,...,At with intervals A i , . . . , Ae respectively as P r o b ^ i e A x , . . . , Ae e A,) = p £ • • • EA[ V|| 2 ,

(3.8)

when all the EA\,... ,EA'{ (or equivalent^ all the EAl (X^,..., EA< (\e)) commute. Indeed he had shown before that the proper generalization of the commutativity of two matrices to the case of arbitrary, possibly unbounded, observables A\, A2 is to require that all members of the spectral family EAl(Xi) commute with all members of the spectral family EA2(\2) [25, p. 172]. On the other hand, if Ai and A2 commute in this general sense, (3.3) shows that each function g(A\) commutes with each function h{A2)This is certainly the case when A\ = A2, so two functions of the same observable always commute. The converse (proved also by von Neumann) is what actually allows extension of the functional calculus of self-adjoint operators to the family of compatible observables. It states that if such a compatible family {A,}, j = 1 , . . . ,£, is given, then there is an observable B of which Ai,A2,... ,Ag are functions. In short, a measurement of B provides a measurement of A\, A2, •.., A(. There is, on any general Hilbert space Tl, a maximal number of such compatible observables whose common eigenvectors span 7i without degeneracy. This means that if there are £ compatible observables with pure point spectra, their eigenvalues A], Af,..., A| specify completely each common eigenvector. After Dirac, we say that we, then, have a complete set of commuting observables. Only for those is the concept of joint probability distribution, defined along the lines of what was done for a single observable. In other words, A\,... ,A( can be interpreted as random variables in

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any given state ip if and only if they commute. Many observables of the system do not, however, and this is the big problem. Consider two observables A and B on H and let ip be a state in 23^2, VB2, T>AB and VBA- When T>AB n VBA is dense in H, it makes sense to assume ABip - BAtp = iCip

(3.9)

where C is an observable. Define f(a) = \\(aA + iB)ip\\2 for any real a. Then f(a) = a2 (A2).

+ ia (AB - BA)^ + (B2)

« 2 ( ^ 2t /),-a(C) , - « ( ^v , ++ (B (s22)Vj),>o Since (3.9) still holds for A - (A)^ and B - (B)^ instead of A, B, one obtains (after H. P. Robertson (1929)): A^A • A^B > \ (C)j,.

(3.10)

Consider the special case A = Q, the position operator (2.2) and B = P, the momentum operator (2.12). Then C is proportional to the identity operator, C = hi, in H = L2(R). And, using (2.8), A^Q-A,/,P> ^.

(3.11)

Another possible use of (3.10) deserves to be mentioned. We saw in §1 that energy and time are (in the extended phase space, cf. (1.9)) canonically conjugate variables, just as momentum and position are. So let us try postulating the existence of a time observable T such that ZVT-A^ff>^.

(3.12)

Now a relation like (3.9) requires that the spectra of both A and B are aA— OB = R- Indeed, a T as before would be the generator of translations in an hi the same way as P generates translations in
71=0

-

\

" /

But OH is bounded below for any realistic systems (like that of (2.25)), since only those systems have a state of lowest energy. So there is no time

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observable in quantum theory. The above argument is due to W. Pauli. The fact that time, in quantum mechanics, is nothing more than a parameter was regarded by von Neumann as the "essential weakness" of this theory [25, p. 354]. But it is true that this author had a very ambitious notion of what mathematical physicists should worry about. Nevertheless, there is another interpretation of (3.12), regarded as relating the uncertainty in the value of the energy observable to a time interval characterizing the dynamics of the (conservative) system. Let us suppose that the potential V is such that H is bounded below with an eigenvalue E (for example V € Lfoc with limi^i^oo V(q) = oo) and eigenvector ipg (cf. (2.26)). As said in §2, the energy H has the value E with certainty. Then the dynamics of such an initial state of the system is trivial: ip(q,t) = ipE(q) e - ( i / f i ) B t , by (2.27). The result of a measurement of the position, for example, is time independent since \tp(q,t)\ dq = \ipE(q)\ dq. However, if the initial state of the system is a superposition of two stationary states, say of energies E\ and E2 respectively, then, for the same reason as in the above two dimensional Hilbert space, \ip(q, t)\ oscillates in time with a period inversely proportional to the uncertainty in the energy (of order AH = \Ex - E2\), i.e. A T = -^. This line of interpretation of the time-energy uncertainty relation seems to have its origin in the work of Mandelstam and Tamm (1945). It is, however, deeply different in spirit from (3.11) since time is not an operator but a parameter. And, as remarked long before by Schrodinger [34]: "This exceptional role of time is not justified." Coming back to (3.11), let us stress that even if there are states ip such that A^Q or A^P are arbitrarily small, this Heisenberg uncertainty principle says that it is not experimentally possible to prepare a state ip violating (3.11). In other words, the experimental localization in space of a quantum particle prevents the sharp determination of its momentum. Let us return to the probabilistic interpretation of a state of superposition like (3.4). Someone familiar with statistical mechanics ensembles would intuitively guess that such a state ip describes a mixture of the states ei and e2 with weights pj — \(ej\ip)\ , j = 1,2, and Ylj=iPj = 1 t>y (3.7). Then, by construction, this mixture provides the same probability that A\ will be found to be Ai or A2. But consider another observable A2 of the same system, as before. One can easily compute that the expectation of A2 in state (3.4) is (A 2 ) v ,-p 1 (e 1 |^ 2 e 1 )+p 2 (e 2 |^2e2)+2Re{(e 1 |V)(e 2 |V)<ei|A 2 e 2 )}. (3.13)

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However, for the mixture, we obtain a weighted average of the expectations (A2)e,, namely 2

J^Pj^M^i)-

(3-14)

J=I

Let us introduce some notation here. We claim that (3.14) can be rewritten as the trace Tr(pA2), where p is the (von Neumann's) density matrix (or statistical operator) defined by p — J2j=iPj \ei) iej\- We use Dirac's notation for the projection on ej in order to avoid writing ^ - = 1 PjPj (cf. before (3.5))! Indeed, by the linearity of the trace and orthonormality of basis {ej}, 2

2

2

Tr(M2) = J2{ek I ^2P3ei){ej\A2^k) = fc=i

j=i

^Pjiej^ej). j'=i

Let us denote (3.14) by MP(A2). It is clear that (A2)^ = MP(A2) only if (ei\A2e2) = 0, i.e. if A2 is diagonal. But Ai and A2 can be diagonalized simultaneously only if they are compatible. The point is that our mixture Mp of states forgets about the phases of the complex coefficients (e7|V'} in the superposition. The extra term in (A2)^, with respect to MP(A2), is called the interference term and is responsible for some of the most puzzling interpretation problems in quantum mechanics. The consequences of Heisenberg's principle shake most of our intuitively obvious notions about classical dynamics. Let us consider only one of these. We said that automorphism (2.29) is the quantum version of the classical one described explicitly by (1.13). In the classical case, a(£) 1—> a(Ut^) can be regarded as the history of the observable a from its initial state £ = (QO:Po)- Consider the simplest (one dimensional) observable, the position a = q and the simplest dynamical case of a free particle: h(q,p) — p2/(2m). The quantum version of Hamilton's equations is (2.36) for V = 0. Since they are linear, their solution is the classical one

(28)-C.!)®for Q = Q(0), P = P(0) as in (2.2) and (2.12), respectively. Now let us

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compute [Q(t), Q(0)}. After insertion of the solution Q(t) = Q + £ t we find m),Q(0)]=--[Q,P]

so Q(t) and Q(Q) are incompatible for the same reasons that Q and P are. But, then, the meaning of the history of this position observable becomes unclear and one may be led to Bohr's positivist view that, as long as our naive notion of history does not refer to any specific measurement device, it is meaningless. Knowing (3.11), let us consider a given classical observable a(q,p), say a : M2 —* K, and the problem of quantizing a, i.e. the problem of constructing an associated self-adjoint operator A acting on TC = L 2 (K). Inspired by the quantization (2.25) of the classical Hamiltonian (2.24), one infers that A should depend linearly on a. A multiple of the identity operator corresponds to a real constant. For g : K —> R Borelian, g(A) should be associated with g(a). But the key guideline is Dirac's correspondence principle (2.32). It says that if a\,a2 are two classical observables then the following correspondence between Poisson bracket and commutator should hold: {ai,a2}^hA1,A2}.

(3.15)

in

Consider the case H = L 2 (K 3 ) with Ql, P J , i,j = 1,2,3, the coordinates of the position and momentum observables (each defined on the model of (2.2) and (2.12)). Using (1.7), the quantization of relations (1.7') becomes, on dense domains, [Pj, Pk] = [Qj,Qk] = 0,

\Qj,Pk]

= ihSjk.

(3.16)

These are Heisenberg's canonical commutation relations. The 3 components Ql or the 3 components Pj form two distinct complete sets of commuting observables. Von Neumann proved that there is essentially one unique (up to unitary equivalence) Hilbert space where [Q,P] = ih can hold for some self-adjoint operators Q and P; this is L 2 (R) with definitions (2.2) and (2.12). It is in this sense that (2.20) and (2.21) are just an equivalent representation of the same physical system, a one-dimensional particle.

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It follows clearly, now, why in general we are in trouble whenever we need to quantize an arbitrary observable a(q,p). Any reexpression of a(q,p) exploiting the classical commutativity of q and p produces a different quantized version, since [Q, P] ^ 0. More generally, one can try to make sense of the product of arbitrary observables A and B, introducing Jordan's symmetric multiplication, AoB

= ]-{AB + BA).

(3.17)

Equipped with this multiplication, the (bounded) observables form indeed a commutative algebra but we lose the classical associativity. Moreover, (3.15) cannot hold if a is a polynomial of degree > 2. This is known as van Hove-Groenewald's theorem. Also notice that if we insist, with the Copenhagen School, in interpreting \Q,P] = ih as referring to incompatible experimental measurements, then it is hard to understand anyway why any function of Q and P should make empirical sense. From a technical point of view, this circle of problems is known as the "quantization problem". The main difficulty of this problem (and the source of its great interest) is that the quantization procedure is not an algorithm. The proper ft-deformation of the laws of general classical systems (living in general phase spaces) is not known. However, in the last 30 years, the quantization problem has developed into various fields of active mathematical research, with an attractive geometrical flavour. One of them is Geometric Quantization, where the geometrical structures of the classical system to be quantized are the basic data. Another is called Deformation Quantization, because it considers the algebra of quantum observables as a deformation (in K) of the associated classical structures. One should also mention a method founded on the primacy of quantum theory and thus considering that classical mechanics should be included in it. Then the problem becomes the inverse of that considered before: for A a given quantum observable in Hilbert space, what is the associated classical observable a(q,p)7 If, in addition to the abovementioned requirements for the quantization procedure, one requires that with unitary transformations of the Heisenberg group (the group whose Lie algebra has generators satisfying (3.16)) is associated a transformation of the canonical variables in a(q,p), then the inverse problem has a unique solution. This is the basis of Weyl calculus, today embedded in the Microlocal Analysis or Pseudo-Differential Calculus. For comprehensive surveys of this beautiful area of research, see [49] and [50].

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To come back to the initial purpose of this section, two kinds of difficulties arise with the probabilistic structure of quantum mechanics: (1) Using the above definition of system observables in a given state tp as random variables, there is no way to consider all of them at once since many observables do not commute. (2) Born's probabilistic interpretation (2.9) of the wave function ip itself does not provide any mathematical clue as to the origin of randomness in the theory and therefore as to the rules of the associated stochastic calculus. Leaving aside the problem of the physical source of the randomness, one could expect that its nature would at least be specified (cf. the random walk of Part 1, §2, for example). Standard quantum mechanics does not even do that since, by item (1) above, there is no underlying sample space and, therefore, no underlying random experiment. Since ip is a pure state, known exactly, it is supposed to provide the complete knowledge of the system at a given time. The situation is, therefore, quite distinct from that of classical statistical mechanics. There, if we knew exactly the initial state, there would be no room left for probability. In short, the predictions of quantum mechanics are irreducibly probabilistic but seem to express some limitations in our interactions with the microcosm and not our lack of knowledge about it. Still, quantum theory is remarkably consistent, comprising an amazing number of mathematical structures in peaceful coexistence. Without these, twentieth century mathematics would be unrecognizable. Regarded pragmatically, as a collection of partial algorithms (von Neumann's Axioms), it is astonishingly efficient. In fact, it is rather unlikely that any alternative mathematical framework could provide as much quantitative information about the world in a simpler way. Schrodinger himself, however, was not too happy with the dominant (Copenhagen) interpretation and, especially, with its insistence on claiming that the theory, including its concept of state, applies to individual particles, although quantum probabilities are irreducible. About the von Neumann projection postulate, he wrote: This invariably entails ridiculous consequences, as, e.g., that a spherical (de Broglie) wave, which is supposed to represent "one" electron, moving in an "unknown" direction, suddenly collapses into a small wave parcel, when "that" electron is detected at a definite spot. [34]

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One should note, however, that some modern experiments involving very few quantum particles at any one time in the device seem to confirm, somehow, the validity of the theory's predictions for individual particles. But, at the very least, one must agree with Schrodinger that the projection postulate ruins any naively realistic interpretation of the theory. He concludes: If you accept the current probability view in quantum mechanics, the single-event observation becomes comparatively easy to tackle, but all the rest of physics (unfashionable at the moment) is lost to sight. [1]

4.

Feynman's Approach to Quantum Probabilities

4.1

Lagrangian mechanics

One of the most devastating consequences of the original (Copenhagen) interpretation of quantum mechanics, still propagated uncritically in classrooms, is that any notion of continuous space-time trajectories of quantum particles is self-contradictory. Of course, if one considers quantum systems without classical analogue, like the internal spin degrees of freedom of particles (which, regarded as random variables, take discrete values), there is no reason why continuous paths should help. But for systems like those considered in §2, which result from the quantization of elementary classical dynamical systems, this is harder to swallow. The origin of the above veto lies in Heisenberg's interpretation of his uncertainty relation (3.11). It is therefore interesting to come back to his original argument [2]. Heisenberg warns us against any uncritical use of the words "position" q and "velocity" for quantum particles • and explains that there are some dis• # continuities to expect and that the # track of the particle (say one dimen• sional) may very well look like the one on the right. t He adds: "It is clearly meaningless to speak about one velocity at one position (1) because one velocity can only be defined by two positions and (2), conversely, because any one point is associated with two velocities." He

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then goes on to elaborate his view that when the position of an electron is determined experimentally (i.e., when a photon is scattered by the electron) its momentum undergoes an unpredictable, discontinuous change. Heisenberg regards this as the "direct physical interpretation" of (3.11). In slightly more mathematical terms, let us note that Heisenberg's argument rules out the notion of trajectories everywhere differentiable but certainly not continuous ones. We shall see how Feynman will do the most of this missed opportunity. But first, it is necessary to return for a while to classical mechanics. The version summarized in §1, and which was the inspiration of von Neumann's axioms (§2), is Hamiltonian mechanics. There is another approach, structurally distinct; this is Lagrangian mechanics. Let us define the Lagrangian L(qi,qi,t) of a classical system with a possibly time dependent Hamiltonian h(qi,Pi,t) by the Legendre transform

L(qi,Qi,t) =maxf YlPig~

_

%i>P*)*)J

C4-1)

when the maximum is reached at a unique p, the critical point of the r.h.s. regarded as a function of p (this is the case when h is convex in the second argument). Then the following relations hold dh Qi = -Zdpi

, a n d

dL

dh

- - £ - = -£-• dqi dqi

,in.

(4-2)

For the elementary class of systems (1.4) considered here this means that L(qi,qi,t)

= ^Y^mi\qi\2

-V(q,t).

(4.3)

i

Let us come back to the one-dimensional case, for simplicity. The fundamental concept of Lagrangian mechanics is the action functional S i [7]= I ' L(q(t),q(t),t)dt,

(4.4)

where 7 denotes a curve in C 1 ^ , ^ ] ) of the form q = q(t), t\ < t < £2Between 1745 and 1760 Euler and Lagrange proved that if 7 is a critical (or extremal) point of S/,, when q(ti) = qi and 9(^2) = 12 a r e fixed, then it satisfies d fdL\

dL

*UJ-87 = 0' n

(4 5)

-

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whose left-hand side is known as the Lagrangian derivative of L. Equation (4.5) is the Euler-Lagrange equation, which started what we call, since Euler's time, the Calculus of Variations. Notice that in Lagrange's original argument, it was implicitly assumed that t»p(t)

= ^(q(t),q(t),t)

(4.6)

is in C1(\ti,t2\), but that du Bois Reymond (in modern terms, using a different class of Schwartz test functions) proved later that (4.5) follows as well without this hypothesis. For the elementary class (4.3), the second order Euler-Lagrange ordinary differential equation coincides with Newton's equation (1.2). As a fundamental principle of classical mechanics, this variational formulation of dynamics is often called Hamilton's least action principle. This is an unfortunate tradition, however, since it is not always true that the solutions of the Euler-Lagrange equation minimize (4.4). The action SL should be called Lagrangian action because there is another one, named after Hamilton and defined for continuously differentiable curves 7 : q = q(t),p = p(t),t\ < t < ti in phase space: SH[I]=

fpdq-hdt,

(4.7)

where p and h are the classical momentum and Hamiltonian variables. It is straightforward to verify that the Euler-Lagrange equations for the functional SH\J] in phase space coincide with Hamilton's equations (1.3). This is remarkable since no relation has been imposed between q and p, in contrast with (4.6). But notice that, along an extremal 7 of SL, we have dq = qdt and then, using the Legendre transform (4.1)-(4.2), /•92,*2

\ Jqi,ti

/"*2

pdq-hdt=

Ldt = SL[y], Jt\

so we can drop the index H or L for the action along extremals. Coming back to a solution 7J of the Euler-Lagrange equation (4.5) connecting {qi,t\) to (92,^2), the action S [yf] becomes a function S(qi, £1,(72, h) of these four variables ("Hamilton principal or two-point characteristic function"). It follows from what we said in §1 regarding the boundary value problem for Newton's equation that S(q\,t\,q2,t2) may not exist or be multiple valued (there may be none or many initial momenta p\ allowing a trajectory leaving ^1 at time t\ to reach q2 at time £2)- For |<2 — t\\

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small enough, however, it can be shown that Hamilton's principal function is single valued. As a matter of fact, instead of looking at a single trajectory it is often useful to consider a family (or "bundle") of solutions of Hamilton's equations. A 2-dimensional surface in extended phase space (q,p,t,h), where h = h(q, p, t) on the surface, is called Lagrangian if for all trajectories 7 in that surface starting from a fixed point (qi,t{), S[j] = /

pdq-hdt

is a locally univocal function, denoted by S*, of the final point Then we have dS*(q2,t2) = p2dq2 — h2dt2, and therefore P2 = VS*(q2,t2),

h2 = -dt2S*(q2,t2).

(q2,t2). (4.8)

Conversely, when (4.8) hold, S*(q2,t2) is univocal by Green's Theorem. This corresponds to a family of solutions of Hamilton's equations whose boundary conditions «(*i) = 9i,

p(*i)=VSJ|,1

(4.9)

define an initial Lagrangian manifold in phase space. This family is described by the action with initial condition (or its Hamiltonian counterpart) /"92,t2

S*(q2,t2)

= S*1(q1)+

Ldt

(4.10)

JqiM

where SI(
P(*2) - - VS 2 | g 2

(4.11)

corresponding to the action with final condition S2: /•92,*2

S(qi,h)

= S2(q2) +

Ldt

(4.12)

JQIM

where S2(q2) = S(q2,t2). and therefore Pl

Then we would find dS(qi,ti)

= -VSfoi.ii),

= —pidqi + h\dt\

hi = d t l S(gi,*i).

(4.13)

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Taken together, conditions (4.8) provide an equation for S*, t £ [ti,t2], dS*

—

+ hiq,vs*,t)

= o,

(4M)

S * M l ) = St(q). Equation (4.14) is called the (forward) Hamilton-Jacobi equation. Symmetrically, the backward equation is the time conjugate of (4.14)

- f + M«,-vs,«) = o, <S(q,t2)

(415)

= S2(q).

Consider the special class of conservative systems. Then, clearly, each of the PDEs (4.14) and (4.15) breaks the dynamical invariance of such systems under time reversal. But, taken together on the time interval [t\, t2], they restore this symmetry. The relation between the two HamiltonJacobi equations can be interpreted as a time reversal relation, since dS* = Ldt2 and dS = —Ldt\. Let us consider the trajectory t 1—> q(t) (unique by hypothesis, here) between q\ and q2. Since this trajectory is smooth we must have, for any q = q(t) along the trajectory, i.e. Vi € ]ti,*2[, dS P.(<7(*),*) = -gj(qi,ti,q,t)\q=s
dS = —^(q,t,q2,t2)\q=q{t)

= p(q{t),t), (4.16) where we have introduced a double notation (consistent with (4.10) and (4.12)) for the momentum regarded as function of position. The * reminds us that some past information is used in the calculation, as in definition (4.10) of the action with initial condition. Its absence indicates the symmetric situation with respect to the future, in analogy with (4.12). For the Hamilton principal function itself, the * is not needed. The simplest example is the one-dimensional free particle of mass m, whose Hamiltonian is h(q,p) = p2/(2m) and whose Lagrangian, by (4.3), is L(q,q) = (m/2) \q\ . The solution of Hamilton's equation, for t € [^1,^2], q(ti) =qi, q(t2) = 92, is 12

—

tl

t2 — t\

t2 — t\

The two-point characteristic function is S(qi,ti,q2,t2)

m (q2 - qi)2

2

t2-h

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123

And identity (4.16) expresses the smoothness of the trajectory t t-> q(t) interpolating between q\ and 92, i.e. ™ 9-91

m

t-U

12

«=«(*)

= m

*2 ~ *


The description of time reversible classical dynamics in terms of two Hamilton-Jacobi equations adjoint to each other with respect to the time parameter was familiar until the sixties (see J. L. Synge "Classical Dynamics", Encycl. of Physics Vol. 3, Part 1, Springer, 1960), but is frequently overlooked in textbooks. As is the fact that dS* = Ldt2, (4.10), i.e., the existence of Lagrangian manifolds, for a given system is, effectively, an integrability condition needed for a global solution of the Hamilton-Jacobi equation. For general Hamiltonians h (i.e. Lagrangians L), the action functional is path-dependent. If n is the dimension of the configuration space (the q-space), it can be shown that it is necessary to know n commuting first integrals in order to ensure that the action functional is path independent as before. But this defines the class of completely integrable systems used in elementary textbooks. In what follows we shall consider exclusively this class of systems, and their quantization. Besides, most of the examples mentioned by Feynman in his famous book [6] belong to this class. 4.2

Feynman's space-time reinterpretation of quantum mechanics

Standard quantum theory is very robust. There is very little room for serious alternative interpretations. In 1948 R. Feynman published his own [5], whose ambiguous success has never waned, but rather gained in strength even after its author's partial retreat to more conventional methods. Fifty years on, our understanding of a number of mathematical issues of Feynman's strategy is quite clear: see [3] and [4] for an up-to-date presentation of some descendants of Feynman's path integral method. Although many rigorous works have resulted from this method, there is still a feeling of embarrassment around it: the whole mathematical picture of Feynman's approach is clearly missing. Since these notes are devoted primarily to the conceptual content of the various theories involved, we shall come back to the original (informal) formulation of path integration, in order to show as simply as possible what Feynman was looking for. In particular, we shall see why his approach is the most systematic one along the lines of a probabilistic interpretation of quantum theory.

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The path integral approach is a rehabilitation of history, i.e., somehow of the role of the time, in quantum mechanics. Let us come back to the (time-dependent) Schrodinger equation (2.27) for the simple class of Hamiltonians (2.25). According to Axiom 2.4, its solution tp(q, t) can be written as Ut-Mti

= ( e - ^ ( t - t o ) " V o ) (
(4.18)

When the evolution operator UT has an integral kernel K, one can write the solution as tpt(q), or V>(<7-1)= J i>o(x)K(x, t0, q, t) dx.

(4.19)

For example, for the (one-dimensional) free particle of mass m, one computes easily this integral kernel

*(*,*„,**)= (^('-M)" 1/2 -ex P (if ^ 0

(4-20)

Let us consider (4.20) more carefully. Recall the Hamilton principal function of the same particle, but regarded classically, between the two space-time points (x,to) and (q,t). The unique free classical trajectory going from (x, to) to (q, t) and its associated action were computed in (4.17) (with a different notation). The result was S(x,to,q,t) = ^ t^t • So (4.20) states that K(x,t0,q,t)

/ h V1/2 t = I 2m—(t -t0) I • exp \ m J n = K(x,t-t0,q).

-S(x,t0,q,t) (4.21)

But this is a remarkable result! On the left hand side we have the key concept of quantum dynamics, which physicists call propagator. Feynman regarded the complex kernel K as the "probability amplitude" for finding the particle at q at time t if it was known to be at x at time to- On the right hand side of (4.21), except for a time-dependent normalization factor, only classical information about the system is used. It would be wrong to credit Feynman for this remarkable result. First, one can find it in the classic textbook of Dirac [7]. In point of fact, during the heroic period of the construction of quantum physics, closed by the Fifth Solvay Conference of Brussels (1927), it was quite common to manipulate classical actions and Hamilton-Jacobi equations in order to discover the

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q u a n t u m versions of physical laws (see, for example, [8] and [9]). Feynman's contribution was t o go further with t h e analogy suggested by t h e special case (4.21) and to t u r n it into t h e following functional representation of t h e integral kernel, K(x,t0,q,t)= f

e^^^-^Vw,

(4.22)

valid, as a rule, for "any" regular potential V in t h e Hamiltonian (2.25). T h e right h a n d side of (4.22) is now a sum over t h e (infinite dimensional) space of continuous p a t h s between t h e two space-time points considered: ilqx = { w

i n

C([t0,t];R)

such t h a t u(t0) = x,w(t)

= q}.

(4.23)

We have used, not accidentally, the notation UJ(T) for t h e configuration at time T in order t o suggest t h e usual XT(ui) = CV(T) of probabilists (cf. P a r t 1, after (2.1)). I n (4.22), Si[o;(-)] is the classical action functional (4.4) along such an w(-) and Vu) denotes the product r i t o < r < t dw{T)- As before, the L of SL denotes t h e Lagrangian of the underlying classical system, i.e. here (see (4.3)), L(q,q,t) = f \qf - V(q). Since VUJ is a purely symbolic object, used as a measure (or p a r t of a measure) on the p a t h space il%, one should understand in which sense (4.22) is a sum over p a t h s (or "path integral" 4 ). A way t o do it, in t e r m s of t h e Lie-Trotter product formula, was discovered by Nelson in 1964 and has been sufficiently quoted since t o allow us only t o mention t h e result (see [3], [4] and [42] for more). If the potential V is in L 2 ( R ) + L°°(R) then ^ , ( fi (t-t^~NI2 K(x,to,q,t) = lim I 2mN-too\ m \ N X

/ • • • / exp-S^(xo,xi,...,xN-i,xN;t-t0)dxi...dxN-1, JR n JR

(4.24)

JR /R

where N

't-to . N

Sf(x0,xi,...,Xiv-i,a:jv;i-to) = ^ f

m \XJ - Z j - i j

2 (t-t0)/N

2

_ .., v V[Xj)

(4.25) 4

A s noted in [10], this is a misnomer, even if one forgets the difficulty associated with the i = V - 1 m (4.22) (yet fatal as we shall see), since this functional representation is not a true path-by-path integral along a stochastic process but an informal expectation over the space H j .

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This expression is essentially that given by Feynman [6, §2.4], and he stresses that (4.22) is mostly shorthand for it. So representation (4.22) can be regarded as the limit (in L2 sense) of integrations along broken paths w between x = xo and q = x^, with linear interpolation (i.e., constant velocity) between points Xj = oj(tj), tj — j^jj2-, j = 0,.. .,N. If UJ € Q% is in C 1 ([io,^];K), then S£ is, of course, a Riemann approximation to the classical action functional (4.4) for the above classical Lagrangian. If the usual method of stationary phase could be generalized to the (infinite dimensional!) context of (4.22), and Feynman was not the kind of man to worry about this kind of issue then, when h goes to zero, the leading contribution to (4.22) should come from the critical point of the phase SL[U(-)}, i.e., by the classical least action principle of Hamilton, from the solution of (4.5) with the boundary conditions specified in 0*. Feynman regarded this aspect as so crucial to his path integral approach that he entitled his (unpublished) Princeton PhD thesis The Principle of Least Action in Quantum Mechanics (1942). Another essential aspect is that all paths w in Q | contribute to (4.22). In Feynman's words: "for every r H-> U>(T) that we could have, for every possible imaginary trajectory (in il%) we have to compute an amplitude. Then we add them all together". Each of these amplitudes is computed in terms of the action SL[W] according to (4.22). The author was quite aware of some of the difficulties of his approach, in particular of those associated with the limit s = (t — to)/N —> 0 of the time discretization involved in (4.22). For example, he stressed the fact that "in the way we have constructed the paths, the velocity is discontinuous at the various points Xj = uj(tj), that is the acceleration is infinite at these points" [6, p. 34]. But he adds that when it was necessary to evaluate such an acceleration, the "substitution

w = — (u(tj+1)

- 2uj{tj) + w(i,_i))

(4.26)

was adequate". As a matter of fact, many of the calculations done afterwards by Feynman were done, for this kind of reason, before any (informal) lim e _ 0 There is no doubt that, in this undertaking of the reinterpretation of standard quantum theory, the deepest concept introduced is that of "transition element".

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At a fixed time, say u, consider the scalar product between two states (VulVO = / $u{z)i>u{z)dz.

(4.27)

This is a complex number, and therefore cannot in general be interpreted as a probability (only its square can: see §3). Now ipu(z) can be regarded as the result of the evolution of an initial state, say ips(x) with s < u, under the integral kernel Uu-S of the evolution operator. So, using (4.19), the right hand side of (4.27) becomes / /

ips{x)K{x,u

— s,z)(pu{z)dxdz.

(4.28)

After reinsertion of (4.22) for the kernel K one obtains fff

^s(x)e^SL^{-)'u-s^u(z)VLjdxdz.

(4.29)

This the simplest "transition element", denoted ((p\l\ip)sL by Feynman (his equation (7-2) in [6]). When (pu reduces to a Dirac distribution Sq, (4.29) provides the path integral representation of the Cauchy problem for Schrodinger's equation on the interval [s,u], with initial condition tps, solved by ipu{
V-(w(*))e* SLl '" (0:t "' 1 2?w

(4.30)

where the underlying path space is now fig,t = {LJ in C ([s, t); R) such that u{t) = q) . As a matter of fact, Feynman calls (4.29) a "transition amplitude" and reserves the term "transition element" for a more general situation he has

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in mind. Denoting, as before, tj = je, j = 0 , 1 , . . . , N, e = (u — s)/N, this is the case when "any" functional F of the Xj — w(tj), j — 1,2,..., N — 1 (but not of the end points x = u>(s), z = w(u)\) is introduced into (4.29): (ip\F\^)sL ^ JJJM^)^SL[XO-'XN;U-S]

= [[[ J J Jn

F[x1,...,xN-1}^u(z)Vu)dxdz

ips(x)eis^^>u-^F[uj(-)}ipu{z)Vujdxdz, (4.31)

where (4.31) is just shorthand for the time discretized version, and the informal domain of F is now ft = {o» G C([s,u] ;K)}. The authors recommend to interpret transition elements via an analogy with the problem of describing a classical Brownian particle starting at the initial time s from point x and arriving at point z at time u. Point x for this Brownian particle is analogous to the initial wave function ips(x) in (4.29) and point z to ifu(z)- "Furthermore, the solution of the quantum mechanical problem requires integration over the variables x and z of the initial and final states, a step unnecessary in our classical problem" [6, p. 166]. When the potential V is zero, the analogy suggested here is clearly that of the process called Brownian bridge by probabilists. We shall come back to this issue soon. But even in the simplest case V — 0 there is much more in this analogy than the usual way to see the Brownian bridge as a standard Brownian motion conditioned to end at a given point in the future. Definition (4.29) plays the role of keystone for the construction of some probability measure on a path space. Notice that it involves, and this is unusual, information about two different times. Assuming that the resulting stochastic processes should be Markovian, the key difference between this and the traditional construction of Markov processes (see [19], for example) is that, besides an integral kernel K, knowledge of two functions ips, s,
|
K(x,u-

s,z)(pu(z)dxdz.

(4-32)

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If, following Bom's interpretation (2.9), we regard |
= {fps{x))

K(x,u — s,z)ifu(z)dz,

s < u, and x,y € R, (4.33) which is well defined for any (s,x) such that tps(x) 7^ 0, propagates the initial data |(t) two propagators, one from s to t and the other from t to u should be introduced in (4.31) to bridge the time gaps. If F depends on the value of the path at two different times, an extra propagator between them should be introduced, etc. . . . [6, §7.1]. The fundamental result of Feynman's functional calculus is a formula of integration by parts [6, (7.30)] for the directional derivative of "any" functional _F[w(-)] in the transition element (4.31) that contains the derivative of the action 5/,[o;(-)] (due to the derivative of the weight e%^hSL^'^T>cj in the symbolic measure used there). Since the authors present it emphatically as a "possible starting point to define the laws of quantum mechanics" we are bound to mention it. Beforehand, they observe that the looked-for relation between transition elements like (4.31) is independent of the initial and final states ip and tp, and, therefore, that any specific reference to these boundary states can be omitted. Thus (4.31) will be written simply (F)s . In consequence, the shorthand form of the integration by parts formula becomes (8F[W](SW))SL

= - l - (F6SL[U>](6U,))SL

(4.34)

where 6F[CJ](SCJ) denotes the directional derivative of the functional F at u> in the direction 5u>. For the time being, we are only going to mention two crucial applications of this formula. For this purpose, it will be more natural to consider a system whose Schrodinger equation (2.27) with Hamiltonian (2.25) is defined in L 2 (R 3 ). When F[w(-)] = 1, the left hand side of (4.34) vanishes and one obtains 5 Let us repeat, however, that this holds only for the abovementioned special pairs of boundary states.

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the Euler-Lagrange equation for SL : rn(u>)SL = -{VV(w))SL.

(4.35)

Equation (4.35) is the path integral counterpart of the Ehrenfest equations (2.37). As said before, the meaning of the acceleration on the left hand side is given by (4.26). If F[w(-)] = u)(t), the integration by parts formula yields

c

/ SL

c

\

jk

= ih6 ,

I SL

j , k = 1,2, 3,

(4.36)

a result that Feynman has no other choice than to formulate in this time discretized way since, as we shall see, the limE_>o does not make any sense here. Equation (4.36) is certainly one of the keys of the whole strategy since it is the counterpart of the canonical commutation relation (3.16) between position and momentum. Notice that, in the first term on the l.h.s. of (4.36), the momentum is computed toward the past and in the second term toward the future if t denotes the present. Since in the classical limit h = 0 those two momenta have to coincide, we must understand such a relation as a quantum deformation of (4.16). This means, in particular, that the ordering of the operators in products of observables now corresponds to some chronological ordering of the terms in the associated functional of the paths. Of course Feynman cannot systematize the rules of such calculus along the paths but he indicates through a number of concrete calculations what they should look like. For example, after some manipulations, (4.36) yields /["(« + * ) - " ( « ) 1 a \

=!»!,

£

m

\

I sL

(4.37)

where 1 denotes the 3 x 3 identity matrix. Since e occurs only to the first power in the denominator of (4.37), the authors deduce from this that "the important paths for a quantum mechanical particle are not those which have a velocity everywhere but are instead quite irregular on a very fine scale" [6, p. 176]. Relation (4.37) is that on which Feynman founded his space-time view of quantum dynamics. It is because of this relation that there is no contradiction between standard quantum mechanics and the path integral

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picture. In particular, such quantum paths are perfectly compatible with Heisenberg's argument (§4.1). The simplest of all the above-mentioned rules is that which associates some functional (or function) of ui(-) with a single quantum observable A acting on the Hilbert space. It was used implicitly in (4.36). Even this rule is not formulated explicitly in [6]. We are supposed to guess it from the examples given. A very illuminating one, for us, is the Hamiltonian H used in (4.18). Two candidates for the associated functional at time t = tj are given in [6, (7-121)], namely =

rn M* J + 1 )-u,frU 2 2 V tj+i - tj J

*

+

2i(tj+1

+ v(w(

)}

(4.38a)

- tj)

and

^=T• {w{tzl-t3)) • {u{T-ti'1])+v{uj(ti))- (4-38b) The second one displays what has been said, before (4.37), about the space-time counterpart of a product of observables, here the kinetic energy 2 ^ P 2 . But Hj is even more interesting, in our perspective: if one insists on computing naively the square of the velocity, an extra term appears. It is ^-dependent and, therefore, due to the irregularities of the quantum paths. Of course, if one computes the transition element of Hj and Hj for a pair of states {ips, Vt} such that (-),u — s] needed for (4.22), namely / (A(cj{t)),Lj(t))dt JS

= / A{u)du JS

(4.39)

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taking into account the irregularity (4.37) of the paths. Coming back to the finite sum as in (4.24)/(4.25), Feynman shows that the proper extra term has to be understood as J^jLi fA(-x^-^x^1\ (Xj - xj-1)Y Indeed, using the discretized propagator, he verifies (see also [44]) that any other choice would add unwanted terms to the corresponding Schrodinger equation: j / i ^ = _L(_j/jV + j4)V + V ^ (4.40) at 2m In particular, if A were to be evaluated at the past instant of the time interval \tj-i,tj] associated with the space increment, N

£(^fo-i), {xi-Xj-i))

(4.41)

i=i

this would add to the scalar potential of (4.40) a term ^ ( V • A)ip while its evaluation at the future instant, N

J2(A(xj), (xj-Xj-i))

(4-42)

i=i

would subtract the same term. In other words, the difference between (4.41) and (4.42) is, using (4.37),

Y-eV-A(xi)

• -

[U(V-A)dt,

but their sum is consistent with (4.40), i.e. the transition element of the extra term must be of the symmetric form

(±(A{X>}+2AiX'-l},(*,-*,->)))

•

(«3)

This result is crucial as it shows that (4.39) cannot be interpreted as a Riemann-Stieltjes integral, whose definition is independent of the choice of the intermediate Tj € [^_i,tj] where A(LJ(TJ)) is evaluated in the sum. Moreover, it provides explicit formulas for the relations between the three most natural choices of integrals, where again the distinction between past and future is essential. This brings us to a key element of the path integral strategy, which we could call Feynman's sense of history. The simplest way to illustrate this is to come back to our initial example of the one-dimensional free particle and,

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collecting together the abovementioned requirements, to formulate more carefully the classical picture suggested by Feynman. Let us consider, as before, the time interval [s, u] and pick any time t in between. We need to consider all the paths w(-) starting from x at time s and ending at z at time u. There is, of course, a unique classical (i.e., smooth) path doing this for V = 0 (cf. (4.17)), but we are allowed to use all other paths, broken at t, since they are required for the representation of the integral kernel (4.22), i.e. for the construction of the associated measure on the path space £2*.

As suggested by observation (4.21), in our simple situation it should be sufficient to take into account as "other paths" only classical ones on [s, t] (and [t, u]); this is why we have drawn only straight lines from x to position q at time t, regarded as a variable. This field of trajectories provides the past information about our free particle, with respect to time t. The future information lies in the set of straight lines leaving each q to reach the fixed point z at time u. We can use (4.14)-(4.16) to be more precise about this picture, introducing some notation which will be useful later: The past of t is made up of all solutions of the ODE d*q{T) = B*(q(r),

T)d,T,

S
t,

q(t) = q, where d* denotes the backward differential, so that d^q/dr = q(r~), the

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left hand derivative, and BJ(«(r),r) = ±-VqS(x,s,y,T)\y=q{T)

=

^ ±

m

We said "all solutions" because we look at this first order ODE as providing half the solutions of a free (Newtonian) second order boundary value problem and, from this viewpoint, the initial velocity is arbitrary. In terms of eqs. (4.9)-(4.10), the boundary condition of (4.44) corresponds to a singular initial Lagrangian manifold. For the future of t we consider, symmetrically, (dq(T)=BZ(q(T),T)dT,

t
\q(t)=q, where B'(q(T),T)

= -^qS(y,r,z,u)\y=q{T)

=

^

M

and d is now the forward differential, so that dq/dr = q(r+), the right hand derivative. The unique regular path t >—> q(t), s < t < u between x and z is characterized (see (4.16)) by its differentiability at time t, i.e. B*(q(t),t) = B*(q(t),t). It is worth noticing that none of these first order ODEs define flows on q-sp&ce (R) but only on the extended space (q,r), since their two vector fields are time-dependent. So, two time-dependent flows on (q, r) space are used to describe a time-independent flow on (q,p) space. It is the sharp separation, for each t in the time interval [s, u], between the past and the future which allows Feynman to reconcile, somehow, his broken path idea with our knowledge about classical dynamics between s and u. In this way, even when result (4.21) for the path integral involves exclusively classical information about the system, and when the construction of the associated measure on 0.% itself should not require much more, the "important paths for the quantum particle" will indeed comply with (4.37). It is essential, in this approach, that the results for simple transition elements (quadratic potentials, Gaussian case) follow from little calculation and be expressed in terms of classical paths. The example of a one-dimensional free particle going from x to z during the time interval I = [0,T] is used again [6, pp. 179,180]. It is easy to check that

(u(t))SL=x+-(z-x),

t€l,

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up to an irrelevant multiplicative factor. But the r.h.s. coincides with the associated classical trajectory q(t), t e I ((4.17) for ti = 0, ti = T). More interesting is the transition element (4.31) of the same free particle, for F[w(-)] = oj(s)u(t). The result is q(s)q(t)+i-S{Trr (w(s)w(t)) SL

t]

,

0<s
q(s)q(t)+i-HT~8),

0
whose last term on the r.h.s. is, of course, interpreted as a purely quantum mechanical deformation. On the other hand, using the additivity of the Hamilton principal function on the time interval [s,u], expressed for any t € [s,u] by SL[UJ;

u-s] = SL[w;t-s]+

SL{LJ;

u-t]

(4.47)

in the path integral representation of the integral kernel K of Schrodinger's equation, Feynman was led to interpret the group property K(x, u - s, z) =

K(x,t - s,q)K(q,u — t,z)dq

(4.48)

as meaning that "Amplitudes for events occurring in succession in time multiply" [6, p. 36]. Of course here "events" is a sloppy term for a probabilist. Coming back to the previous classical picture of kernel (4.22), this just means that one way to think about the associated path integral (or "amplitude") for the particle to go from x to z is to sum over all possible intermediate positions q, i.e. to multiply the corresponding kernels for the relevant (past and future) time intervals [s, t] and [t, u] respectively. This is why, as said before, such kernels should be introduced, for the calculation of transition elements (4.31), to bridge any time gaps between the various times t = tj involved in the (discretized version of) the functional iJ1[w(-)] under investigation. The path integral approach is much more than merely a translation of standard quantum mechanics. In many respects it goes beyond it and enables one to clarify conceptually (if not mathematically) what is left unexplained in Hilbert space. Consider the diffraction of free quantum particles of mass m through a slit, i.e. the famous "one-slit experiment" [6, pp. 47-54]. At time s = 0, a free (one-dimensional) particle starts from the origin x = 0. After a time interval T one assumes that it is known that the particle is within a distance

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±
T

0

q

space

The interesting part of this problem is not really in the solution put forward, but in Feynman's way to deal with the question: how can we know that the particle passes point q within the interval ±CJ? He says that we should "reject all trajectories which pass outside the limit ±er from <jr". In other words, we can imagine that there is a slit of width 2er between the origin and the screen where the particle arrives. So we only need to restrict the path space which should be used for such a free particle if this slit were not there. Of course, this should be done without introducing any new force in the system: the presence of a slit does not change anything to the fact that the physical potential V acting on the particle is zero. We stress that the problem cannot be solved by a single application of the quantum law of free motion (i.e. (4.19) for a free kernel K of (4.20)), since the particle is constrained by the slit. Instead, the problem is broken into two successive free motions, one from x = 0 at time 0 to q + Sq at time T, with \5q\ < a and then another from q + Sq at T to z at T + r. Using the interpretation associated with (4.48), the solution is formulated as

1>(z) = J \{-aM (Sq) K(0, T, q + Sq) K(q + Sq, r,q + z) d(Sq).

(4.49)

The integration over Sq is a sum over all the paths which qualify for the job. Since the result of (4.49) is a complicated expression in terms of Fresnel integrals, Feynman suggests replacing the indicator function l^_a^(Sq) by

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what he calls the "Gaussian slit" G(5q) = exp{-{8q)2/(2a2)). Then the integral can be evaluated and the probability P(z)dz we are looking for is proportional to \ip(z)\ dz. Of course, there is no reason why f \ip(z)\ dz should be 1, and the author uses this observation to deduce what he regards as the probability that the particle really gets through the slit. What matters is that P(z)dz is Gaussian, with mean q + VQT, where VQ is the velocity VQ = q/T of a classical free particle, and a unique /independent term in the variance, of the form h2T2/(m2a2), i.e. standard deviation hr/(ma). At r = 0, the full standard deviation reduces to a by construction. For small mass m and narrow slits, the term hr /(ma) is what makes the difference between classical and quantum free particles. In Feynman's words: "for small particles, passage through a narrow slit makes the future position uncertain". The discussion, often animated, on the physical meaning of this standard deviation hr/(ma) went on for more than 25 years after quantum theory was enthroned, even among the founders of the theory. Its first consequence is uncontroversial if not intuitive: when r is large enough, it is hard to continue referring to a (free) "particle" since its wave packet spreads indefinitely. The second consequence is even more striking. Let us call AQ(r) = hr /(ma). As we said, at r = 0, the position of the particle is known to an accuracy a = AQ. Then a short time later, denoted e, we have AQ(e) = he/(ma). So AQ • AQ(e) ~ —e. m In Pauli's vehement words, when trying to explain Einstein's objections to Max Born, this inaccuracy at the later time "destroys the possibility of using all previous positional measurements within these limits of error" (letter of W. Pauli to M. Born, March 31, 1954 [13]). This is why the history of the position observable is not defined. The above relation can be interpreted as a qualitative version of Feynman's observation (4.37). It is therefore the second time that we meet relation (3.11) of Part 1 (expressed there in discrete time). It appears that it was regarded as fundamental from the very beginning of quantum theory (when all paths were banished), then reinterpreted by Feynman as the basic kinematical property of quantum paths (for our class of Hamiltonians). Is this accidental? Let us look at the problem from a different viewpoint.

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We can regard r = 0 as the initial time of the (ordinary) free quantum evolution describing what happens after the slit. The associated Cauchy problem of Schrodinger's equation on L2(R) is dr

2m

*»<*>= (,2na w2) )

(450)

e mV Z e

" ° -'

where the imaginary part of the exponent of V'o is needed to ensure that the initial velocity is indeed vo and the real part is such that |i/>o(.z)| dz is a Gaussian of mean 0 and standard deviation Az = a. Since the initial condition of (4.50) can be written as

^^(^W'

-a (k — ko)„ikz

for fco = Po/k and po the classical momentum po = mvo, its Fourier transform xpoik) is also Gaussian, and of standard deviation Ap = h/(2a) (this remains true, in fact, Vr, a special property of free particles). So we observe that Az-Ap=^,

(4.51)

another expression of Heisenberg's uncertainty principle, in this special situation. Note that, in this form (relation between the extent of a wave and its Fourier transform), the argument is familiar since the 19th century. As we said, the interesting part of Feynman's argument is that regarding the free particle before the slit. Although it may seem unconvincing, it tries to formulate quantitatively what standard quantum mechanics carefully avoids. It is a postulate of quantum theory that passage through the narrow slit changes the state of the free particle. The slit is a measuring apparatus for the position observable Q of the free particle, whose degree of selectivity is given by its width 2a. Since the spectrum UQ is purely continuous there is, in fact, no way to set up any apparatus able to pick a single "eigenvalue" of Q but this will not matter to us. Passage through the slit is responsible for the reduction (or "collapse") of the state of the particle. Immediately after the narrow slit, indeed, the position is known quite accurately: Az is very small. But just before the slit the theory says that the standard deviation of the position was much larger: the free wave packet had all the leisure to spread after its initial localization at the origin. This is

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why von Neumann felt the need for his projection postulate, to admit such "non-causal, irreversible, instantaneous" phenomena. But is it really a phenomenon, or a theoretical dilemma in Hilbert space? If, with the realist School, we regard quantum mechanics as a theory of individual systems whose state has some kind of objective reality, how to interpret this sudden change, postulated by von Neumann? Notice that if the theory was probabilistic in nature, the idea to reactualize our knowledge on the basis of the result of new observations would be rather natural and founded on the elementary concept of conditional probabilities. Coming back to the one-slit experiment, it is worth noting that, if the final position z is read on a detector (a screen), only spots, of course, will appear there, and therefore this corresponds again to a reduction of the wave packet, one of those ridiculed before by Schrodinger. Feynman's idea of restriction of the path space for the diffraction of the free particle through a slit, hints at an explanation quite distinct from the standard one (assuming that this is one). As a matter of fact, his path integral approach is, implicitly, a kind of quantum theory of measurement. The one-slit experiment is, again, one of those "simple" situations where only classical trajectories are needed to compute the quantum result. This is why Feynman does not really need to solve any Schrodinger equation (we gave (4.50) only for greater convenience), certainly one of the strongest points of his approach. Now we could consider the same experiment with two slits at time T, say at points q\ and q2, instead of a single one. Then the probability P(z)dz of finding a particle at z on the screen would involve a sum over the two families of (classical) paths through each slit i, i = 1,2, arriving at z at time T + T, i.e. P{z)dz^\-4)l{z)

+ i)'2{z)\2 dz,

(4.52)

with ipj of the form (4.49) for q = qj, j = 1,2. Since the ipj are complex valued, this means that P(z) dz = \Pi(z) + P2(z) + 2 Re(V>i&)(z)] dz,

(4.53)

where Pj, j = 1,2, denotes the probability density for each slit j alone. Separating the modulus and the phase in each tpj, ipj{z) = P- (z) exp(i&j(z)), j = 1,2, we also have 2Re(Vi&)(*) = 2P! 1/2 P 2 1/2 (2) cos(6i - S 2 )(z).

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The last term of the r.h.s. of (4.53) describes the (in)famous "interference of probability". To try to interpret its physical meaning would force us to open the Pandora box of questions like: "How in hell does a particle going through the first slit know if the second one is open or closed?". We shall not do this. Reading Feynman's original account of the two-slit experiment [11], essential to his path integral approach, is still a thrilling experience, highly recommended to everyone. There are at least two basic difficulties with the path integral strategy. The first one is that most of the calculations made by Feynman and arguments founded thereon involve exclusively classical (or smooth) paths, i.e. the paths of the underlying classical system. This is exemplified in expression (4.21) for the free propagator, which has a generalization for any potential V smooth and bounded below (when the Hilbert space is L 2 (R n )), named after Wentzel, Kramers and Brillouin (WKB): K(x,t0,q,t)

=

(2irih)-n/2 det

d2S dxdq

-S(x,t0,q,t)-

—nj,

(4.54)

where y, is the Morse index of the trajectory (i.e., the number of focal points along the trajectory on [*o,*])- If the underlying classical boundary value problem has a finite number of distinct solutions, the propagator is a sum of terms like (4.54) for each of them. For [to,t] small enough, this is no longer the case and [i = 0. As remarked by W. Pauli [12], (4.54) cannot always be true, for otherwise quantum mechanics would be almost superfluous, since this formula is constructed from purely classical quantities. In fact, if V is a quadratic function of the position, then (4.54) is exact, but in general it is only a formula asymptotic in h. Feynman concentrated mostly on the first case, which is the counterpart of the special status granted to potentials V at most quadratic by Ehrenfest's Theorem. This is where his path integral approach is most clearly a new quantization method. But he was fully aware of the need to introduce new technical tools for the general case (see the Conclusion of [6]), where the classical information is not sufficient anymore. The second difficulty is of an incomparably more serious nature. Apart from the presence of the factor i = %/—T in the r.h.s. of (4.37), this relation is quite familiar to a probabilist, as noted before. It looks like the quadratic variation of the Brownian motion [14], cornerstone of Ito's stochastic calculus [15] for diffusion processes. We can be even more specific about the heuristic diffusion handled by Feynman. According to (4.32), its transition probability function pp(s,x,u,dz) should be given by the explicit formula

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(4.33) involving, besides the integral kernel K(x, u — s,z) of Schrodinger's equation, the value of a special solution of the same equation (of its complex conjugate, in fact) at times s and u. Assuming ip to be analytic and i/>s(:r) ^ 0, one easily obtains (in one dimension for simplicity): / (z Jsi(x)

X)PF(S,X,

U,

dz) = m

=-?-(x) (u - s) + o(u - s) i>s

(4.55)

and I

(z — x)2 PF(S,X,U,CIZ)

= —(u — s) + o(u — s),

(4.56)

JSs(x) ™ where Ss(x) is the sphere of radius 5 and center x. So equation (4.55) provides us with the initial drift (i/i/m)(Vi/' s /V' s ) of Feynman's heuristic diffusion on [s, u], while (4.56) does nothing but confirm the hunch of (4.37) about its quantum diffusion coefficient. Let us note, however, that on introducing the quantum transition kernel PF of (4.33), we have introduced as well an arrow of time which is not part of Feynman's definition of the transition amplitude (4.28) for (pu = e (-V»)(«-s)tf^ s- We could have decided to read the same amplitude as ips(x)K{x,u-

s,z)(i>u(z))

\ipu(z)\

dxdz,

/ / •

thus introducing another transition probability function PB(s,dx, u,z) = ips{x) K(x, u — s,z) (•ipu(z))~1dx

(4-57)

that we could call "backward" since its effect is to propagate the final probability |i/; u (z)| dz backwards in time to its initial expression |ips(x)| dx. In this sense, it is indeed the presence of the two wave functions tps and tyu in the fundamental transition amplitude (4.28) (restricted to the special pairs of boundary states mentioned in (4.32)) that restores the time-symmetry lost by the introduction of any quantum transition probability. Manipulations similar to (4.55) and (4.56) but using PB instead of PF give, for the associated backward (final) drift and diffusion coefficient, respectively,

JlYpL{z)

and

!»

(4.58)

m xpu m (for those z & R such that ipu(z) ^ 0). Introducing, as prescribed by Feynman, two extra integral kernels K in the above transition amplitude, one

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to bridge the gap between s and t and the other to bridge the gap between t and u, for any t G }s, u[, one easily verifies that the above functional forms of the two drifts should be preserved Vt, for the same diffusion coefficient. So we now have an accurate profile of the stochastic processes implicit in Feynman's considerations. It should be a (one dimensional here) diffusion process Xt(u>) = ui(t), t € [s,u], with diffusion coefficient ih/m, forward drift (ih/m){Vi>t/i>t){q) and backward drift (—ih/m)(Vipt/'4,t)(q), well defined for any q in R such that ipt(q) ^ 0, where tpt is a regular solution of Schrodinger's equation in L2(M) with Hamiltonian (2.25): dtb h2 ih^ = --—AiP at 2m

+ Vij.

(4.59)

By construction, of course, Bom's probabilistic interpretation (2.9) of state ipt should be preserved, P{Xt(w)

G dq} =

\MQ)\2

dq,

te[s,u).

(4.60)

since, as said before, the right hand side coincides, Vi G [s,u], with the probability density of Xt- Notice that, in this way, (2.9) would no longer be a nebulous interpretation anymore but a mathematical statement. The second "difficulty" of the path integral strategy is that the heuristic complex measure on SI = C([s, u];]R) needed for such a Xt does not exist; this was proved by R. H. Cameron forty years ago [16]. We have a finite additive complex measure on cylinder sets, i.e. on sets of the form {u : Xtl (w) G Bi,..., Xtn(uj) € Bn}, for LU G fl (denoted by W in Part 1), with s < t\ < • • • < tn < u and Bi Borelian, used in Kolmogorov's famous theorem on finite-dimensional distributions of a random process. But this complex measure is not cr-additive so it cannot be extended to the measurable space whose tribe (er-field) is generated by cylinder sets. In short, we cannot obtain a process Xt, t G [s,u], with the finite-dimensional distributions (implicitly) prescribed by Feynman. Although this could be seen as a new, and disastrous, demonstration of the "intricacy of the continuum" (Schrodinger dixit, Section 7, Part 1), it has mostly to do with the presence of the factor i = ^f—\ in the starting representation (4.22). So, at the end of the day, there is no such thing as Feynman's quantum mechanical sample paths t <—> Xt(u>). And therefore even less random times "living in a probability space where a Markov process makes its residence. .. " (end of §4, Part 1). Nevertheless, Feynman's "sense of history" is, beyond any doubt, that usually associated with a well denned stochastic process. In contrast, as

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already observed in section 3, if a free quantum particle is detected in q at time t, the problem of knowing whether it was localized around a classical intermediate position (i.e. along a straight line) between the source and q is not well posed. In any case, at this point, no serious claim can be made regarding the existence of any quantum path whatsoever, since the process does not make sense. But this does not mean that Feynman's strategy is irrelevant to our purpose, far from it! We shall now describe a probabilistic framework which is, we claim, as close as possible to his path integral approach, but still quite distinct structurally from Kac's interpretation.

5.

Schrodinger's Euclidean Q u a n t u m Mechanics

5.1

A probabilistic interpretation of Feynman's approach

Let us come back to Feynman's picture of the one dimensional free particle. Our first task is to construct an associated stochastic process (introducing a bona fide measure on a paths space) using nothing more, and this is absolutely essential, than the ingredients recommended by him. In such a simple case, this should involve exclusively the classical paths (here a straight line) drawn in the figure before (4.44) for the time interval / = [s,u]. Forewarned by the misfortune of Feynman's complex "measure", we shall replace the (free) quantum integral kernel K of (4.20) by the strictly positive Gaussian kernel, closer to the heart of probabilists:

(

ft

\ ~l/2 ex

2TT—(t-a)J

/

ui \n — xl \

t£l,x,q€R. (5-1) As observed before, the relation between kernels K and ho corresponds to an informal transformation of the original time parameter t into —it. In other words, ho(x, • — s, •) solves the (one-dimensional) free (parabolic) heat equation

rhlk

P(~2ft

= HoT1

^77*(<7,s) =

*'

t

_

3

)>

u>t>s,

(5.2)

S(q-x),

where Ho denotes the same Hamiltonian operator as in (4.50) and our somewhat peculiar notation will seem more natural in a while. The trick t —> —it, familiar to theoretical physicists as the transition to "imaginary

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time" (or to the Euclidean world, a terminology inherited from quantum field theory) is, in fact, valid for a large class of nontrivial potentials V in Hamiltonians H = Ho + V, and a large class of nontrivial initial conditions, instead of (5.2). Indeed, regarded as an element of L2(M), a solution of such a heat equation is analytic in Ret > 0, continuous for Ret > 0 and its value on the imaginary axis solves the corresponding Schrodinger equation. Of course, ho satisfies the property analogous to (4.48) and called, in this new context, the Chapman-Kolmogorov equation: ho(x,u — s, z) —

ho(x,t — s,q) • ho(q,u — t,z)dq,

s
(5.3)

The construction of Feynman's complex "measure", starting from (4.28) and illustrated in the figure above (4.44), suggests how to construct the measure for free diffusion we are looking for, say X(t), given the two boundary data at the extremities of the time interval [s,u\. In this elementary case, these are conditions X(s) = x

and

X(u) = z.

(5.4)

It is clear that what happens outside [s,u] must be irrelevant to the properties of X(t), t G [s,u]. In fact, the Euler-Lagrange equation (4.35) for the acceleration (4.26), as well as the arguments relating to the figure above (4.44), suggest that this should hold locally, for each t £ [s, u]. Such a property can be expressed precisely by introducing two tribes (or nitrations) for the time interval / = [s,u\. Let (Q.,a,P) be the probability space where each X(t), t £ / , is a real valued random variable, and Q, is the space of all continuous paths from J to R. Inside the big tribe er, let us define two sub-tribes. Here, it will be more suggestive to change slightly the notation of Part 1 (§2 and, especially, of §4) where the past filtration (respectively the future one) was denoted by Tt (respectively J-'t). So let us denote the tribe generated by {X(T),S < T < t} by Vt (for "past" at the time t) and, symmetrically, by Tt = cr{Jf(7-),i < r < u) that for the future at time tel6 6 For a physicist, a filtration (or increasing family of tr-fields, or tribe of events) like Vt represents the information about the whole history of the process, from time s to time t, i.e. the events determined by the physical (or experimental) conditions imposed on the process up to time t, and no other time. X(t) is adapted to Vt, i.e. measurable with respect to it, for any t in / . However, since here we need to impose conditions on the future of X(t) as well (for example X(u) = z), it is necessary to use another filtration, a decreasing one, representing the future of the process from t to u.

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A stochastic process X(t), s < t < u, will be called Bernstein process (or "local Markov" or "two-sided Markov") if Vs < si < t < t\ < u and any / bounded measurable, E{f(X(t)) | VS1 Vf t l }= E{f(X(t)) | X(Sl),X(h)}.

(5.5)

This property was introduced in 1932 by the probabilist S. Bernstein, who called it the "reciprocal property" [17]. The term "local Markov" is, of course, much more recent: in physics, it appeared in the programs of constructive quantum field theory and statistical mechanics of infinite lattice systems. We shall come back soon to the original motivation for the introduction of Bernstein's property, which is directly related to our subject. This author proposed, on the basis of (5.5), to construct a 3point (real valued) counterpart of Kolmogorov's Markovian transition function, say Q(s\,xi,t,A,ti,zi), with x\,z\ 6 M, A in the Borel tribe of R and s < s\ < t < t\ < u. The only difference with Kolmogorov's concept is that this new transition Q should coincide with the probability that the observed particle is in set A at time t if it was in x\ at the past time s\ and will be in z\ at the future time t\, denoted by P(X(t) € A | X(s\) = x\ and X(t\) = z\). The properties expected of Bernstein transitions were axiomatized by B. Jamison in the mid seventies [18]. By comparison with the familiar Markovian case (see, e.g., [19]), one easily guesses what those properties are, with the exception of the analogue of the Chapman-Kolmogorov equation (5.3), which is, V Borelians A, A and S < S\ < t < ti < t2 < U,

/ JA

Q(si,xi,t,A,ti,zi)Q(si,xi,ti,dzi,t2,z2) Q(si,xi,t, " /

dq, t2, z2) Q(t, q, ti,A, t2, z2). (5.6)

Notice that when the final random variable X{u) = z is fixed in Q(s, X(s), t, A, u, X(u)), the resulting function Qz{s, X(s),t, A) should have the properties of any Markovian transition function and, symmetrically, when the initial variable is fixed, X(s) = x, one should be left with a backward Markovian transition function denoted Q%{t, A,u,X(u)). As a matter of fact, (5.6) is the simplest property compatible with these two requirements. Indeed suppose that X(s) is (almost surely) constant, as in our elementary free particle case. Consider a sequence X(ti), i = 1, 2 , . . . , n in the future of

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t, i.e. s < t < t\ < ti < • • • < tn < u. Consider the conditional expectation, for any real / , bounded measurable, E{f(X(t))\X(h),X(t2),...,X(tn)}. Since X(s) is fixed, this coincides with S{/(X(t))|X(B),X(t1),...,A-(tB)}. By the Bernstein property (5.5), this reduces to E{f(X(t))

| X(s),X(tl)}

= E{f(X(t))

|

X(h)}.

So the expectation of any past behaviour of the process is not altered by any additional information about its future behaviour. This is the Markov property expressed with respect to future information instead of past information, as usual. There is no problem with that, since the Markov property is invariant under time reversal (although its formulation in terms of transition functions, forward or backward, reintroduces an arrow of time [31]). Coming back to our free process, let us rewrite (5.3) as 1 = (ho(x, u — s, z))~x

I ho(x, t — s,q) ho(q, u — t,z) dq.

(5.7)

The point of this innocuous transformation is that q(s, x, t, q, u, z) dq = (ho(x, u — s, z))~1ho(x, t — s,q) ho(q, u — t,z) dq is the density of a Bernstein transition function. Indeed, Vrr, z £ M, Q(s,x,t,A,u,z)

=

(ho(x,u — s , z ) ) _ 1 / ho(x,t — s,q)ho(q,u

— t,z)dq,

s
JA

(5.8) is a probability measure on R, for A fixed (x, z) —+ Q(s, x, t, A, u, z) is Borel measurable, and property (5.6) is satisfied. A crucial observation about (5.8) is that the second (positive) factor of its integrand solves, as a function of (q,t), dt J](q,u) =

'' 5{q-z),

s < t < u,

(5.9)

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147

i.e. the equation adjoint to (5.2) with respect to the time parameter t. So, if t represents the present time, the probability density of the (free) diffusion process we are looking for is a product of the (positive) solutions of two (free) adjoint heat equations, one with an initial boundary condition, the other with a final one. Suppose that, at that point, we decide to do backwards, with the time parameter, what we did at the beginning of this section, namely to replace t by it. Then (5.2) would reduce to the free Schrodinger equation, (5.9) to its complex conjugate and relation (5.8) to the fact that the probability (density) of finding the particle in a set B at time t should be a product of solutions of these two equations. So (5.8) can be interpreted as a probabilistic, or Euclidean, counterpart of Bom's interpretation (more precisely of a generalization of Bom's interpretation) of quantum states and our remark following (2.41) shows that the presence of (5.2) and (5.9) is required to preserve invariance under time reversal, shared indeed by Markov and Bernstein properties. However, a remark is in order here. In [6, p43], Feynman interprets the absolute square of the propagator \K(x,t — s,q)\ as the probability that the (free) particle arrives at point q when it is known to start from x at time s. He notes that the resulting probability is divergent after integration over q. Had he done with K what we do in (5.7) with ho, he would have been more consistent in his construction of a "measure" on the time interval I = [s,u], but he would have lost the positivity of the probability density (since the normalization factor, in the quantum counterpart of (5.7), is complex-valued), a physically unacceptable loss. As we shall see, our Euclidean counterpart of Feynman's approach will indeed produce many more probability measures than those needed for quantum mechanics. In any case this is highly reminiscent of what we found in §4.1 in relation with the classical two-point characteristic function of the free particle on \s,u], namely (see after (4.17)) S(x,s,Z,u) = ^{-^f.

(5.10)

We used this in §4.2 in order to understand Feynman's picture of the free particle, by introducing two classes of classical paths, one for the past at time t and the other for its future. Differentiability at any time t of the unique smooth path t — i > q(t) between x and z was characterized by the

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equality of its left-hand and right-hand derivatives (see after (4.45)) i.e., Bz(q(t),t), q(t) - x t-s

(5.11)

z ~ q(t) u—t

where B* and Bz were expressed in terms of the free two-point characteristic function. Of course, we cannot hope that (5.11) will remain true along the continuous but nowhere differentiable sample paths of the free diffusion process we are looking for. On the other hand, we know that the "Euclidean" version of Feynman's diffusion coefficient (see (4.56)-(4.58)) is h/m. So we are led to consider at once two (Ito's) stochastic differential equations (SDEs), for t G [s,u], whose drifts and diffusion coefficients are provided by Feynman's picture of the free particle: dX(t) = J — dW(t) + Bz(X(t),t)

dt,

(5.12)

X(s) = x, and d*X(t) = \ — d„W„(t) + B*{X{t),t) Vm X{u) = z.

dt,

(5.13)

In (5.12), d denotes Ito's forward stochastic differential and W(t) a onedimensional Brownian motion 7 starting from the origin and adapted to the increasing filtration Vt (the past at time t). In (5.13), * denotes the corresponding concepts with respect to the decreasing filtration Tt representing the future at time t. In the classical limit where h = 0, these two equations reduce to ODEs (4.44) and (4.45). Let us stress that (5.12) and (5.13) do not introduce any new ideas with respect to Feynman's model for the free particle. They are just its mathematical transcription. In particular, the two drifts Bz and BJ are given by the underlying classical system. The interpretation of (5.12) and (5.13) is made through integral equa7

W(t) can be regarded here as "continuum limit" of the symmetric Bernoullian random walk of Part 1.

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149

tions which are, for s < t and t < u respectively, X(t) = x+J— [ dW{r) + f Bz(X(T),T)dT V m Js Js

(5.12')

X(t) = z + J— C d*W*{r) + r Bl{X{T),r)dT. V m Jt Jt

(5.13')

The linearity of the drift and the fact that the coefficients of dW (d*W*) are of bounded variation allow us to solve explicitly these equations using the good old Newtonian calculus. Choosing the backward equation (5.13) for example, one finds , Tlr , . [rn , , , , . fm f X(t) — x\ , d»Wt(t) = J—d*X(t) - J—( —y J dt Vn \ n \ t—s

so

X(t) =x+

Z

—^(t -a)u—s

\

Vm

-Y(t),

(5.14)

u tu dd*W*{T) \ —

where Y{t) = {t-s)

Jt

T

The definition of the Tr Brownian motion W* (similar to the standard Brownian motion) is used to find the covariance ( h (*i - s)(u - t2) K(t1,t2)

=

E{Y(t1),Y(t2)}={ m _ft (*2 <m

u—s -s)(u-ti]

i

s < t\ < t2 < u,

,

S < t2 < t\ < U.

u—s

(5.15) So the solution X{t) is a continuous Gaussian process with mean (compare with (4.17)) . u—t t—s q{tj = x h zu—s u—s and covariance K(tut2) given by (5.15).

= ^{(-X"(*i) " E{X(ti)})(X(t2)

(5.16) -

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As an application of (5.15), for example, one can derive, on I = [0, T], q(s)q(t) + -S-^—t, E{X(s)X(t)}

=<

0<s
1 t(T- s) q(s)q(t) + -

{

T

"

',

0
where the q(-) are the same classical trajectories as in Feynman's result (4.46). Of course, we could as well find this by solving the forward equation, (5.12). This diffusion X(t), s
h, Vho(q,u-t,z) m ho{q,u — t,z)

=

z_zq u—t

and =

_ H Vho(xt-S,q) m n0{x,t - s,q)

=

q - x t—s

in complete ("Euclidean") analogy with Feynman's heuristic results (4.55) and (4.58) (for any time t 6 [s, u] and the special pair of positive solutions of (5.2) and (5.9) involved here). We can also take advantage of the original definition of the drifts of diffusion processes solving SDEs like (5.12) and (5.13) as limits of conditional expectations (see, e.g., [14]), namely, for £>0, E{X{t + s)-X{t)

\Vt}

=Bz(X(t),t)e

+ o{e),

s
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151

and E{X(t)

-X{t-e)\Tt}=

Bl(X{t),

t)e + o(e),

s < t < u,

to associate with the continuum limit the probabilistic counterparts of the derivatives used in (4.44) and (4.45), namely X{t + e) -

DX(t) = lim E

X(t)

t}=B*(X(t),

e—*0

t)

(5.20)

and D,X(t)

=

l

^

E

{

X

^ - ^

t

- ^ \ ^ )

B*{X{t),t).

(5.21)

These two derivatives have been used by E. Nelson [20] in a seemingly unrelated framework which will be briefly described later. The expectations of the second moments of the forward (and backward) increments yield, as e —> 0, E{(X(t

+ e) -X(t))2

| Vt} = ^e

+ o(e),

i.e. the probabilistic counterpart of Feynman's fundamental relation (4.37) and E{(X(t)

- X(t - e)f

\rt}

= -

e

+ o(e),

which can be regarded as its backward counterpart. Let us define the time reversed process X(t) = X(u + s — t), s < t < u. We were warned against excessive naivety regarding this in §6 of Part 1. If, generalizing (5.12'), one calls semimartingale a process of the form X(t) = Xs + M(t) + A(t) where M(t) is a continuous local martingale and A(t) a continuous process locally of bounded variation, 8 then its time reversal is not necessarily a semimartingale with respect to its natural filtration. This and other difficulties associated with the strong Markov property [21] will be avoided here (because of lack of random time!). Nevertheless, as observed before and as we shall see again, the time symmetry is more deeply built into the definition of Bernstein processes than into the definition of Markovian processes. I.e. 11—* A(t) is of finite length in any finite interval of time.

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Using (5.20), simple manipulations show that

E 'X(t

%{

DX(t) = lim

+

e)-X(t) £

(5.22) X-

X(t) = -D*X(u + s - t) = , ± u—t where the explicit form (5.18) has been used. Symmetrically, D*X(t) = -DX(u

+ s-t)

X

=

W~

Z

.

(5.23)

So X(t) is, not surprisingly, the Brownian bridge leaving z at time s and reaching x at time u. The invariance under time reversal of this bridge, sometimes mentioned as a curiosity in textbooks on stochastic processes, is both trivial and fundamental when X(t) is interpreted as it should, as an elementary Bernstein process. In fact, for this reason, we shall not need to consider again time reversed Bernstein diffusions. Another illuminating expression of time reversibility for X(t) appears in its variance K(t,t). By (5.15), K(t, t) = ( * - * ) ( " - * > , u—s

s

< t < u.

(5.24)

Notice that (u — t) in the numerator can be regarded as the time reversal of (t — s) on the time interval / = [s,u]. In fact, these two factors are two linearly independent solutions of the classical "Jacobi equation" of our problem, here q(t) = 0, and the denominator of (5.24) is their Wronskian, but we will not elaborate on this [39, 52]. So, by construction, variance (5.24) is invariant under time reversal. This is the specialization of the Bernstein property to the Gaussian case (central, as we know, to Feynman's strategy). As we said before, although the two drifts of the free Bernstein bridge X(t), s < t < u, are the classical (^-independent) velocities found via Feynman's classical picture, relation (5.11) cannot be true anymore along the paths of this bridge since it expresses the differentiability of the interpolating classical path at any time t. However, motivated by Feynman's observation (4.36), we can now compute an expectation which should be its Euclidean counterpart: E{X(t)

• mD*X{t) - mDX(t)

• X(t)},

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where, in contrast with Feynman, we are allowed to take the continuum limit as in (5.20) and (5.21), since we are dealing with a bona fide probability measure on path space. A short calculation using (5.18) and (5.19) and the above mean and covariance of the bridge shows that E{X{t)

• mD*X(t)

- mDX(t)

• X{t)} = h.

(5.25)

Of course, the same calculation for the R 3 -valued Bernstein bridge would replace the r.h.s. of (5.25) by M^, j,k = 1,2,3. In particular, when h = 0, SDEs (5.12) and (5.13) reduce to the pair of ODEs (4.44), (4.45), the bridge reduces to the regular trajectory 1i—» q(t), s < t < u (see (5.16)) interpolating between x and z, along which differentiability condition (5.11) holds. Notice the very close link between the preservation of Feynman's basic relation (4.36), the time-symmetric nature of the Bernstein process and the irregularity of their sample paths. In fact, D*X{t) and DX(t) are the best possible predictions of the left hand and right hand derivatives, respectively, when X(t) is known and SDEs (5.12) and (5.13) hold. Even in the quadratic Gaussian case, i.e. when the two drifts are classical velocities, quantum noise shows up inevitably in the r.h.s. of (5.25). Let us emphasize that the Wiener process by itself is not sufficient to understand Feynman's commutation relation (4.36). Its role is purely kinematic here; on the left hand side of (5.25) we need more general diffusions (such as that which solves (5.12), (5.13)). Before getting overexcited about the fact that we understand (4.36) probabilistically, we should pause for a minute and remember that, according to Feynman, the above bridge is only the starting point of his construction of the free "measure". To get a decently general counterpart of his strategy, we need to find the Euclidean analogue of his argument leading to (4.29). What plays the role of the wave functions tps and ipu required by Feynman's approach when the measures at the boundary of / = [s, u] are not concentrated at single points as in the free bridge case? On the other hand, we have shown why, when X(s) or X(u) reduce to a (almost surely) constant the resulting Bernstein process is Markovian. But what about the general case? To find the answer, we need to come back to the construction of any Bernstein process starting from the 3-point counterpart of Kolmogorov's transition function. We referred to Jamison then. Here is his result (still for one dimensional processes):

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T h e o r e m 5.1. Given a Bernstein transition function Q(si,x\,t, A, ti,Z\), with xi, z\ in K, A a Borelian, s < si < t < t\ < u, and M a probability measure on 93 x 93, for 93 the Borel tribe ofR, there is a unique probability measure Pu such that, with respect to (Q,,O,PM) the associated process X(t), t € [s,u] satisfies (1) E{f(X(t)) | VS1 V j r t j = E{f(X(t)) | X(si),X(h)}, Vs < S l < t < ti < u for any f bounded measurable. (2) PM(X(S) £ As,X(u) e Au) = M(AS x Au) for any Borelian AS,AU mQ3. (3) For any s<si
G A | X(*i),X(ti)) = Q{a1,X{a1),t,A,t1,X{h)).

(5.26)

(4) For s < si < S2 < • • • < sn < u and Ai in 93, PM{X(s)

e A„X(si) I JAsxAu

eAi,...,X{sn)

dM(x,z)

e An,X(u)

/ Q(s,x, si,dqi,u,

z)

JA1

••• JA2

s

- /

€ Au) =

n— l i Qn — li

Sni

dqn,u,z). (5.27) Notice that, in (4), the final random variable has been fixed so that, as said before, the Bernstein transition function has all the properties of a standard (forward) Markovian transition function for a given initial probability Ps(dx). We could as well leave X(s) — x fixed in each of the Bernstein functions, in which case any of those would behave as a backward Markovian transition for a given final probability Pu(dz). In this sense, the construction is completely time symmetric with respect to the probabilistic data at the boundary of the time interval / = [s, u]. On the other hand, (2) shows that one needs a joint probability measure M to start with and not only a pair Ps(dx) and Pu(dz) of probabilities. In general, as we said at the beginning of §3, the data of Ps and Pu, the marginals of M, is not sufficient to determine this joint probability measure. So, for a given pair {P 5 , Pu} a number of joint probability measures M will qualify and with each of them Jamison's Theorem will associate a Bernstein process. However, as suggested after (4.32), quantum physics according to Feynman should be Markovian, not only Bernsteinian. It is therefore gratifying to learn (from Jamison) that, given {PS,PU}, only one of the abovementioned (free) Bernstein processes will be Markovian, namely that whose

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joint measure is of the form M(AS x Au) — I J

T]*(x)h0{x,u-

s,z)r)u(z)dxdz

(5.28)

ABxAu

for rj* and rju two bounded measurable positive functions. Definition (5.28) is manifestly the probabilistic (i.e., "Euclidean") counterpart of Feynman's transition element (4.28). This is confirmed by substituting (5.28) into (5.27), for Bernstein transitions Q of the form (5.8). The finite dimensional distributions reduce to, for s < si < S2 < • • • < sn < u, PM(dqi,si,...,dqn,sn) = / / vZ(x)h0(x,si

- s,qi) ...h0(sn,u-tn,z)r)u(z)dxdq1

...dqndz, (5.29)

as proposed by Feynman. Determination of the positive pair (ri*,riu) defining the Markovian process with these finite dimensional distributions and given boundary probabilities (Ps(dx) = ps(x)dx, Pu(dz) = pu(z)dz) results from the marginals of the Markovian joint probability (5.28): r]*(x) / h0(x,u-s,

z)r]u(z)dz

=ps(x),

JR

<

(5.30)

Vuiz) / T)*(x)h0(x,u

- s, z)dx

=pu(z).

JR

The whole picture will extend quite naturally what we said about our (free) bridge. In particular, the Euclidean Born interpretation (5.8) involving the fundamental solutions of (5.2) and (5.9) is simply generalized as PM(X(t)€A)=[ri*{q,t)r)(q,t)dq,

s
A e
(5.31)

JA

for 77,, 77 positive solutions of t9V*

JT

*

-h— = Hon v*(q,s) = v*s(q)

and

+h% = Hori at

(5.32)

•q{q,u) = T)u(q)

with (T7*,77U) solving (5.30). The forward and backward drifts follow the model of (5.18) and (5.19) and coincide with the Euclidean version of Feyn-

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man's drifts (cf. (4.55) and (4.58)): B(q,t) = ~^-(q,t) m

7]

and

Bt(q,t)

= --^f(q,t).

(5.33)

m rj*

Such a Markov process X(t), s < t < u, solves a pair of Vt and Tt stochastic differential equations of the forms (5.12) and (5.13), but now for the general drifts B and B* (usually ^.-dependent, in contrast with (5.18) and (5.19)). The diffusion coefficient is, of course, unchanged. Henceforth we will assume that E {J" B2 (X (T) , T) dr | X(t) =-• q] < oo, t € [s,u] and the symmetric restriction for B*. In fact, the Brownian bridge constructed before does not comply with these conditions because of its singular boundary conditions at times s and u. But, as we shall see, these are very natural conditions (of "finite energy") in the regular case. In this way, with a single Hamiltonian (here HQ) is associated an infinite number of non-homogeneous but time-symmetric Markovian processes. Moreover, let us stress that we now have more freedom to produce probability measures than is suggested in quantum theory: any pair of positive solutions of (5.32) whose product is integrable will do. In other words, we interpret probabilistically any transition amplitude, even when it does not have a probabilistic meaning in quantum theory. Bernstein diffusions without a quantum counterpart (like the free bridge considered before) will however be most useful for understanding the relation with standard probabilistic results. Before showing that the qualitative behaviour of these diffusions is very close indeed to that anticipated by Feynman, let us discuss a generalization without which this framework would be of little relevance to theoretical physics. If one compares the above construction with Feynman's, the key word is, not surprisingly for a probabilist, positivity. Without the strict positivity of the starting Gaussian kernel ho, nothing would work. There is a large class of Borel measurable V, say from M3 —•> R, such that the integral kernel of remains e-(u-s)H/h o n £2( R 3) w i t ] l H = H0 + V, denoted by h(x,u-s,z), strictly positive (and jointly continuous in x, z and u — s > 0). This class was introduced by Kato and has proved to be quite useful in probability [19] and in mathematical physics [42, 3, 4]. Then all that we said about the Gaussian kernel ho and the Bernstein transition function Q built from it (cf. (5.8)) remains true for h(x,u — s,z). In particular, the forward transition function of the diffusion (corresponding to the Bernstein transition with fixed final random variable (see

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after (5.27))) is now of the form p(s,x,t,dq)

= (rj(s,x))~1h(x,t

— s,q) rj(q,t)dq,

s
Feynman's results revisited

Now we have the probabilistic tools to understand what Feynman was looking for in this path integral approach. Since our class of Bernstein diffusion allows to treat a large class of potentials V', and in particular all those considered by Feynman in [6], it is appropriate to start with his equation of motion (4.35). As said there, the meaning of the l.h.s. of (4.35) is given by the time discretization (4.26) of the acceleration along the paths of the heuristic diffusion. The probability measure of this diffusion is supposed to provide the regularization needed for this a priori divergent expression. Let / = f(q,t) be any smooth real valued function with compact sup-

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port on [s, u] x R. Taylor's theorem and Ito's formula show that

Df(x(t),t) = h , { ® ± ! M i ? h » )

(5.35)

and symmetrically

DJ(X(t),t) =

limE{Hm,t)-f(X(t-s),t-e) £-»0

^

£

Tt (5.36)

for B,B* defined in (5.33). So D = d/dt + g and D* = 9/9i + g, where the differential operators g and g* are called, respectively, the forward and backward infinitesimal generators of the diffusion X. Our study of the free Bernstein bridge has shown that it is natural to interpret the general drifts (5.33) as quantum deformations of velocities for the underlying classical system. Therefore we may compute, using (5.32), lfbX(t)

=

DB(X(t),t) \ot

m 7]

2m

/ \m

T) /

= VV(X(t)). Comparing with Feynman's equation of motion (4.35), we could not hope for a simpler result. The change of sign from Feynman's in the r.h.s. of (5.37) is a consequence of the Euclidean transformation t —> — it from Schrodinger to the heat equation since the l.h.s. of (4.35) is a second derivative with respect to time. Notice, however, that the (almost sure) equation (5.37) is not invariant under time reversal since, as seen in (5.22), D —> — JD» generalizes the classical relation (1.12). This is odd since the construction of X(t) is quite symmetric in this respect. But another calculation using (5.32) shows that D*D*X(t) = W(X(t)) as well. So a time-symmetric form of the law of motion is i (DDX(t) + D,D,X(t)) = W(X(t)).

(5.38)

From the technical point of view, however, the individual validity of (5.37) and its time reversal is more fundamental since each of these equations

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involves a single nitration. For systems more complicated than that considered here, some ft-deformation of the classical laws should be expected. Such a dual description is the (non trivial) probabilistic counterpart of the obvious fact that any quantum computation made with the wave function ipt could be done, without benefit, with its complex conjugate. For us, this involves changing from forward to backward Ito stochastic calculus [23]. Let us assume that X(t), DX(t), DDX(t) and their Tt counterparts are in L1. Since the sample paths are continuous, we can compute the expectations of (5.37), of its time reversed form and of (5.38). For example

jtE{X(t)}=E{DX(t)} mjtE{DX(t)}=E{VV(X(t))} is a probabilistic version of the Ehrenfest equations (2.37). Also let us observe that, although in the Gaussian quadratic case, where only classical drifts are needed (cf. the free Brownian bridge), it is always possible to interpret the construction of Bernstein processes along the historical lin3 developed since Kolmogorov (i.e. the drift and diffusion coefficients are given and one solves SDEs like (5.12)), in general it is better to look at (5.37) as an (almost sure) equation of motion, for a given V, and some boundary conditions. Of course, these equations are stronger than Feynman's (4.35), since they hold before the expectation is taken. Using them, we can make sense of Feynman's tricky calculations in discrete time using Ito calculus. Let us come to the probabilistic counterpart of Feynman's way to associate a function of w(-) with a simple quantum observable A. Consider, for example, the momentum observable P (2.12). Given the quantum expectation (P)^t of (2.16), the integrand with respect to |V"t(9)| d
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correspond, for a given pair of Euclidean states r\t,f)% at time t, to B(q,t) = ^(q), Vt

B*(q,t) = ^$(q) Vt

(5.39)

for P — —hV and p t its adjoint in L 2 (R). On the other hand, because of the positivity improving nature of the semigroups associated with (5.32), the problem of the zeroes of 77*, 77^" is much simpler to handle here. It follows from this rule that the space-time observable associated with Hamiltonian H = HQ + V = — ^ A + V in the Euclidean state r\t is h(q,t) =

^(q) Vt

(5.40)

_ ( . 2 * - ! v . i , +„)(„.,, where definition (5.33) has been used. If we remember that the time parameter T of the Schrodinger equation used by Feynman became IT in our heat equation (5.32) for rjt (with H substituted for Ho), then it is clear that (5.40) is the probabilistic counterpart of Feynman's discretized Hj in (4.38). To understand where his second candidate H? come from, let us find what it is that corresponds to the expectation {H). (i.e., for Feynman, the transition element between the special pair of states he considered there). For us, it is just the expectation E{h(X(t),t)}

= -^E{B2(X(t),t)}

- ^E{V

• B(X(t),t)}

+

E{V(X(t))}.

But it follows from (5.33) that P* and B are not independent functions. If we denote the density of the probability distribution of X(t) by p(q,t) = r)*T](q,t) (c.f. (5.31)), then P* = B — (h/m) Vlogp and we have - | P { P 2 } =- | P { P P , } + ^P{V.P}. Comparing with (5.40), this implies that h(q,t)=(-jBB* + Vyq,t),

(5.41)

i.e. the probabilistic version of Feynman's second candidate Hj in (4.38) is, indeed, another meaningful random energy variable. However, as we shall see soon, choice (5.40) is more fundamental, as far as the symmetries of the theory (i.e. its geometrical structure) are concerned.

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It is also worth noting that if the energy defined by (5.40) is a true random variable when evaluated on q = X(0) then f]t{q) is nontrivial. But if h(q,0) = E, a real number, then i]t(q) reduces to (etEr]o)(q) and, in this sense, is trivial. This is the probabilistic counterpart of our remarks, made after (3.12), that when the energy of the initial quantum state is known with certainty, the dynamics of this state is trivial. In the general case, because H is self-adjoint, the same energy of state 7)1 can be equally expressed by h*(q,t) = HT]*/^. Instead of calculating it as before, it is simpler to observe that h„ should be the time reversal of h and, because D — i > — £>» under such reversal, we immediately get from (5.40) h*(q,t) = {-^Bl

+ ^V • B* + V)(q,t).

(5.42)

Some symmetry between time and energy was stressed in §3 of Part 1 for Brownian motion. It was of a kinematic nature, in the sense of Feynman's formula (4.37). Another expression of symmetry, now of a dynamical nature and valid for any Bernstein diffusion, results from noting that (5.40) reduces also to h(q,t) = h^(q,t)

(5.40')

and (5.42) to

h*(q,t) = -h^f(q,t),

(5.42')

since r\ and if solve (5.32) for the Hamiltonian H. The relation between time and energy is here analogous to the one between position and momentum expressed in (5.33). This will be needed later. Now let us come back to Feynman's one-slit experiment, in order to see if the existence of well defined measures on path spaces helps, or not, to understand what happens. As we said in section 4, the interesting part of the problem is the way the author deals with the free particle before the slit. In probabilistic terms, we are given the free kernel ho of (5.1) to start with, and the information that the particle leaves the origin at time t = 0. If this was all, the diffusion would be a standard Brownian motion (solving (5.12) for the drift B = 0 and (5.13) with B*(q,t) = q/t). But we are also told that the particle, whose constant velocity is vo = q/T, "passes the point q within the interval ±
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deviation of a "Gaussian slit" whose distribution at this time T is, therefore, the normal N{q,cr2). For a probabilist, this problem makes perfect sense: one should construct a standard Brownian motion but conditioned to be normally distributed with N(q, a2) at time T. This is a restriction of the sample paths of the original process, as suggested by Feynman. The determination of the infinitesimal parameters of the new process, given those of the original process, is familiar since, at least, Doob's works in the fifties. Since this calculation is given as a problem in Karlin and Taylor [40, ex. 25, p. 388], we feel free to give the result and concentrate on its interpretation. The diffusion coefficient (h/m) is unaltered; this is physically obvious since the kinematic characterization (4.37) of the paths is independent of the presence of the slit. But the forward drift cannot be zero anymore because of the extra information we are given about the future variance of X(t). One finds DX

® = T+1wT-l)X®

+

T + tWT-l)>

°^^T-

(5.43) On the other hand, since the past boundary condition of the Brownian motion on the time interval / = [0, T] has not changed, the backward drift is unaltered: D.X(t)

= X^-,

t € /.

(5.44)

It is easy to verify that X(t), t £ / , is a Gaussian process with distribution

called henceforth the one-slit process. Of course, by construction, Af(m(T), CT2(T)) reduces to what we want. Notice that we did not cheat about the dynamics; it is still true that, after this conditioning, as required by (5.37) and its time reversal, DDX(t)

= D*D*X(t) = 0,

tel.

(5.46)

Now consider the formal limit a = 0 of what we did. This means that we are conditioning with respect to a set of sample paths of probability zero. In spite of the technical complications associated with such conditioning by events of zero probability, the resulting diffusion is well defined. Since,

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163

then,

DX(t) =

q

-j4r

and

D.X(t) = ^ - ,

(5.47)

this limiting diffusion coincides with the free bridge (5.12)-(5.13) between x = 0 a n d z = q, t G [0,T]. As said after (5.24), the variance of this free bridge displays the time symmetry of Bernstein diffusions. But this property is lost in the one-slit process, whose variance in (5.45) reveals a time-arrow. So the introduction of the slit, implying the collapse of the wave packet of the free particle, which provides a measurement of its position observable, is indeed the source of a dynamical irreversibility. Our conditioning has introduced such an asymmetry at the boundary of the time interval [0, T] that the resulting free dynamics is necessarily irreversible. This is why Schrodinger's equation (in real time) cannot describe it. Of course, in our probabilistic analogy we have taken full advantage of the fact that both without and with the slit, we are dealing with well defined random experiments, defined by their probability spaces. Now, it follows from the basic principles of probability theory that the conditioning of the free Bernstein bridge leading to the one-slit process corresponds indeed to a new random experiment built from the original one. New probabilities are assigned to some events (some continuous paths w : / —> M) of the original experiment. The one-slit process X(t), t G [0, T], can also be regarded, if needed, as a regularization of the free bridge, where the singular boundary condition of (5.9) at time T, TJT — Sz, is replaced by a positive Gaussian boundary condition proportional to »7T(0 =

exp

2\

To* J*

+

<j^

After the slit, we can consider the probabilistic counterpart (5.32) of the Cauchy problem (4.50) for the Schrodinger equation. Then the dynamics of the associated Bernstein diffusion, with initial Gaussian distribution Af{q,
K2^2 One should remember that the real initial velocity VQ of (4.50) becomes purely imaginary since the time parameter itself becomes imaginary in the

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transition from (4.50) to (5.32). In this sense our variance at time T is consistent with Feynman's result (see after (4.49)). But note that T clearly cannot be arbitrary anymore. In fact,

M < ^f,

(5-49)

so our Bernstein diffusion is only locally defined in time. As a matter of fact, this aspect is an unavoidable effect of the definition of this class of processes, already stressed by S. Bernstein himself [17]. In general, they cannot be extended arbitrarily beyond their initial time interval of definition. But, paradoxically, this is what makes them so close, in their properties, to what we associate with quantum dynamics. For example, the change of sign in the variance (5.48) with respect to the quantum mechanical one, is also responsible for its manifestly time-symmetric form (a + T) • (a — — T). \ zmcr / \ 2ma ) for T G [—2mcr2/fi, 2mcr2/ti\. So we know that this process is built indeed from a quantum, unitary, evolution, in contrast with that describing what happens before the slit. Now let us come to the probabilistic counterpart of the quantum interference of probability (4.53). For us, a positive solution of (5.32), say 7?^, is a Euclidean state at time t. We know that it is associated with a Bernstein diffusion on some time interval / = [s, w], with probability density p(q,t)dq = i]*rit(q)dq, t € I. So the counterpart of the separation of the modulus and phase for a wave function becomes %*(?) = P1/2(q,t)exp(-&(q,t)),

(5.50)

where &(q,t) is of the form &(q, t) = \ \og(r]t/r]*)(q). As we said, the relation between rj* and rjt on which is built our probabilistic analogy with quantum mechanics is a relation of time reversal. So p is even and 6 odd under time reversal and representation (5.50) makes sense. Once again, we observe that, somehow, we mimic the properties of complex valued functions with real valued ones equipped with a special operation of time reversal. If ??t* (q) is the Euclidean state of a free particle, then rj^ {q — £) + r\X (q + €) is another one, by the linearity of the heat equation (5.32), for any fixed constant £. It is an elementary exercise to verify that the probability density of the Bernstein diffusion associated with the superposition satisfies

5. Schrodinger's Euclidean quantum mechanics p{q,t)=p(q-t,t)+p{q 1/2

165

+ i,t) 1/2

+ 2p (q - t,t)p {q

+ t,t) cosh[6(g -t,t)-G(q

+ t,t)],

(5.51)

in complete analogy with (4.53) since our phase © is now real-valued. Although it is not a complete surprise, in a construction whose data are two probabilities at the boundary of a given time interval, (5.51) does certainly suggest fresh thoughts about the interpretation of standard quantum theory, in particular on Bohr's concept of the "wholeness" of a quantum experiment (cf. [8]). This may be the right time to pause and elaborate slightly on the meaning of what we have done. What are actually the status and purpose of this "Euclidean Quantum Mechanics" summarized here? In this section, we have mentioned various scientists who have contributed to the first steps of this construction: S. Bernstein, A. Beurling, B. Jamison. We have not yet mentioned the one at the origin of what we call Euclidean Quantum Mechanics. It is E. Schrodinger, in 1931-32 [34] and the history of his idea is quite interesting. Schrodinger himself credited A. S. Eddington for the following suggestive observation regarding quantum mechanics: The whole interpretation is very obscure, but it seems to depend on whether you are considering the probability after you know what has happened or the probability for the purpose of prediction. The tfit'ipt is obtained by introducing two symmetrical systems of ip waves traveling in opposite directions in time; one of these must presumably correspond to probable inference from what is known (or is stated) to have been the condition at a later time. (Gifford Lectures, Cambridge, 1928). In Schrodinger's original perspective, relation (5.31) provides the solution to his problem of finding the most probable free evolution of a probability density on a time interval I = [s,u], given the boundary probability distributions ps(dx) and pu(dz). He also formulated nonlinear system (5.30) as part of the solution, supplying the boundary conditions of the two relevant adjoint heat equations (5.32). When Schrodinger wrote [34], quantum mechanics as we know it today was built. J. von Neumann had already published his book laying the axiomatic foundations of quantum theory in Hilbert space, and providing a plentiful source of problems in functional analysis for several generations

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of mathematical physicists. But Schrodinger was never convinced by the dominant interpretation of the theory and what we just mentioned was one of the many attempts he made to show that one should be more critical about it. Still, Schrodinger was also very sharp in his criticism of the various "solutions" suggested. In particular, he never accepted the statistical interpretation of quantum mechanics advocated by Einstein, according to which the wave function ip does not describe an individual quantum system but an ensemble of systems. He never supported the research program known as "hidden variables" launched by this point of view. However, he disagreed quite early with the Copenhagen point of view claiming that irrefutable facts force us to renounce any continuous space-time approach to quantum theory. In this sense, he was not so far from Feynman, and it is all the more remarkable that some consequences of his idea of 1931-32 are instrumental to our mathematical reinterpretation of the path integral approach. The point of what we call Euclidean Quantum Mechanics is to construct the closest possible probabilistic analogy with standard quantum theory. This strategy is grounded in Schrodinger's conviction (implicit in his abovementioned idea) that any direct probabilistic interpretation of the theory would be frustrating and sterile. Seventy years after Schrodinger, it seems that we have accumulated enough evidence to claim that there is indeed a lot to be learned from such a program, both from the mathematical and the physical points of view. On the mathematical side, it is striking that standard quantum theory only uses the most elementary notions of probability theory. Of course, we have seen that there are excellent reasons for such a situation: quantum mechanics is simply not a regular probabilistic theory! And everything suggests it will never be. The integral kernel K(x, u — s,z) of the Schrodinger equation used by Feynman as the building block of his path integral approach will remain complex whatever happens and, consequently, the associated analogy with the theory of Brownian motion will always be unreliable. It is embarrassing, a half century after Feynman's work [5], to see papers published by physicists using exactly the same symbolic methods, as if the whole point of this approach was to do dreadful time discretized computations and not to benefit from the elegance and simplicity of well defined probability measures on path spaces. Nevertheless, the path integral approach has so far been the most systematic approach to relate quantum physics and stochastic analysis. All

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167

attempts to go beyond it, in real time (i.e., along the lines of some hidden variable theories) have had to sacrifice, willingly or not, some basic aspects of standard quantum theory. Moreover, through the years there have been unexpected convergences between the various difficulties met by these, often independent, approaches. Comparison with what we did with the Feynman-Kac formula [41] is interesting. As is well known, this formula is a path integral representation of the solution of the heat equation (5.32) (the equation for 77*, generally), regarded as the probabilistic version of Feynman's relation (4.30), and any initial condition in Cb(M) n L 2 (R), for example. The underlying stochastic process is a Brownian motion ending in q at time t. What remains of Feynman's original action functional SL appears in / V(W(r))dT Js for V in the Kato class. This multiplicative functional of the Brownian process is also named after Feynman-Kac [42, 19]. The classical kinetic energy term in the action SL is informally absorbed into the construction of the Wiener measure; this is a singular expression along the nowhere differentiable paths of the Brownian motion but compensated by Feynman's symbolic product T>u>. The Feynman-Kac formula has been very useful in mathematical physics [3, 4, 42], especially for the study of the spectral properties of the Hamiltonian H. It reappears in our framework when we need to regard the law of Bernstein diffusions as absolutely continuous w.r.t. that of Brownian diffusions (cf. §6.2 and [35]). But even Feynman's interpretation (4.36) of the commutation relations requires, as we said, going beyond Brownian motion, whose role is only kinematic and not dynamical in the path integral strategy. It should be clear, however, that there is no contradiction between what we did and the Feynman-Kac viewpoint. For example, the traditional identification of free quantum mechanical motion with the Wiener measure starting at x, which originated in the Feynman-Kac viewpoint, does not go wrong, of course. It is just insufficient for a full understanding of the free dynamics according to Feynman. At this point, the framework created in the mid-sixties by E. Nelson and known as "stochastic mechanics" deserves to be mentioned [20], since our Euclidean approach shares some kinematic features with it. For the same class of Hamiltonians H = HQ + V as before, a well defined real-valued diffusion process £(£) is built, compatible, like Feynman's, with the Born exp

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interpretation (2.9): \ipt(q)\ dq = p(q,t)dq, t € / . Its diffusion coefficient is not, however, (4.37) (as we said, such a diffusion does not make mathematical sense) but coincides with our Euclidean one h/m. On the other hand, its two drifts are neither of the Feynman form given in (4.55) and (4.58), nor of our Euclidean form (5.33), but of the somewhat intermediate type

6 ±(gt t) = A R a ( ^ ( , ) ) ± A I m ( ^ ( g ) ) .

(5.52)

The dynamics of these diffusions is profoundly different from that discussed before. Their time-symmetric law of motion does not involve the same acceleration as (5.38) (so the kinematics is also distinct) but uses the physical force —VV(£(i)), as is natural here since we are in real time (this was the whole point of Nelson's strategy). However, in spite of what is suggested by this apparently simple law of motion, it has been proved that the Feynman-Kac multiplicative functional underlying Nelson's process does not involve only the given physical potential V but also an additional time-dependent, non-local, term of the form h2 A o 1 / 2 Q(q,t) = --T7T (9,*)-

(5-53)

Q has been called the "quantum potential" and is emblematic of the most famous (non-local) hidden variable theory, that created by D. Bohm in the early fifties [8]. In this framework, the hidden variable (whose inaccessible knowledge would complete the knowledge encapsulated in the wave function about the state of the system) was the initial position of the particle and the potential Q the only origin of its non classical dynamical features.9 No stochastic processes were needed. The influence of the quantum potential is essential to the dynamics of Nelson diffusions. Any conditioning of such a diffusion, like that needed for the one-slit process, changes the dynamics of this process. For example, neither standard Brownian motion nor the Brownian bridge describe anymore some free quantum particle. They are both associated with (distinct) linear and time-dependent forces. So simple relations like (5.17), for instance, are not available in stochastic mechanics. Moreover, and in striking contrast with the path integral strategy, it is never true, in this real time 9

Bohm's potential allows action-at-a-distance when the physical system consists of two particles in a general state, and therefore contradicts relativistic causality. But so does standard quantum mechanics.

5. Schrodinger's

Euclidean quantum

mechanics

169

framework, that some diffusions can be entirely determined from classical paths. So, even for at most quadratic potentials and Gaussian probabilities, it is always necessary to know the solution of Schrodinger's equation in order to be able to use the diffusion.10 Still, what stochastic mechanics has shown is interesting: it is impossible to construct a real valued diffusion process associated with a Hamiltonian as before, and satisfying the Born interpretation, without reintroducing by the back door the Bohm potential, known to be responsible for the most puzzling (non-local) features of quantum theory. 11 Therefore, in spite of what many thought, including myself at some stage, the unexpected convergence of Nelson's and Bohm's theories shows that stochastic mechanics cannot be regarded as a rigorous approach to Feynman's ideas, considerably less dependent on the Schrodinger equation but much closer to all aspects of quantum dynamics. Consequently, stochastic mechanics has also very little to do with our Euclidean framework, as the next section will again illustrate. On the physical side, Euclidean Quantum Mechanics (EQM) seems to be a new expression of a line of thought which has reappeared periodically since the early days and is still remarkably consistent with modern reflections on the foundations of quantum physics. In the early days, V. A. Fock, for example, expressed the view that, regarding the quantum notion of measurement, one should think about an "initial experiment", which includes a certain preparation of the quantum system (for instance of a monochromatic beam of electrons), the installation of slits, etc. . . . "The initial experiment is always addressed to the future". But we are also told that there is a "final experiment" in which the potential possibilities materialize and which may be conducted in different ways 10 T h e existence of Nelson's diffusions under a finite energy condition has been proved by E. Carlen in 1984 [53]. His ingenious proof is analytical and the quantum potential does not show up in it. A probabilistic construction, under the same regularity condition, is due to P. Cattiaux and C. Leonard [51]. There the role of Bohm's potential is quite explicit. u I n 1981 I was a graduate student in Geneva and asked J. Bell (already famous for his inequalities showing that any hidden variable theory with the same predictions as quantum mechanics must be non-local) why he did not care about probabilistic approaches to this theory. "Why should I", was his answer, "since most of the difficulties are in the quantum potential, which is available without the slightest knowledge of the theory of stochastic process". J. Bell was an ardent supporter of Bohm's perspective (cf. [48] for a recent exposition). Some of his principles inspire, nowadays, real experiments made by "quantum engineers", for instance in quantum cryptography (cf. [58] for a lively account).

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(we can choose to measure different observables). "The final experiment is always addressed to the past". This latter one allows us to verify the predictions of the initial experiment. For such a given verifying experiment the potential possibilities are expressed in terms of probability distributions of some observables. So a great number of repetitions of the same experiment (with the same preparation and the same external conditions) is needed. This whole collection makes up the complete quantum experiment and the point of the theory is "to describe the initial state of a system in such a way as to make it possible to obtain probability distributions for any type of final experiment from this state" [9]. What is interesting here, in the perspective of EQM, is the complete symmetry between the initial and final experiments, reducing our natural ("classical"!) tendency to associate any physical experiment with a Cauchy problem. And we know that, in EQM, such a symmetric perspective is precisely at the origin of the invariance under time reversal of the theory, when no measurement is made, and of "spectacular" consequences like the Euclidean superposition principle (5.51). More recently, the question of time symmetry and asymmetry in quantum dynamics has been discussed often, especially in the context of quantum cosmology. In 1964, for example, Aharonov, Bergmann and Lebowitz published a version of quantum mechanics relevant to cosmology where initial and final conditions (of von Neumann's density matrix) were used to produce a theory invariant under time reversal [30]. Different cosmological scenarios are specified, in such a framework, by different choices of initial and final conditions. Closely related ideas are still discussed today [26], together with the associated weakening of the naive notion of causality. Even closer to us, in a discussion on Mathematical Physics in the 20th and 21st centuries, R. Penrose calls for a new quantum theory which could "make real sense of the process of measurement" and stresses that, in quantum gravity, the proper theory should be time-asymmetrical (because of the existence of space-time singularities). And this author links this "timeasymmetric requirement of the missing union of quantum physics with general relativity to the need to make the reduction of the wave function into an objective phenomenon" [27]. The present bringing together of physics and mathematics seems, curiously, to stimulate critical reflections on the foundations of quantum theory [45]. In these respects, EQM could help, at least as a toy model, to clarifysome of these issues since it suggests new perspectives on the role of time

6. Beyond Feynman's

approach

171

symmetry in quantum mechanics.

6.

Beyond Feynman's Approach

6.1

More quantum symmetries

Up to now a theoretical physicist reading these lines may still suspect us of translating into an unfamiliar language what he knows quite well already. The purpose of this section is to reassure those readers in this respect and to illustrate what can really be expected from EQM as a research program. Let us come back to the functionals of the quantum paths (4.38) mentioned by R. Feynman as candidates for the space-time observable of energy and their well defined probabilistic counterparts (5.40)-(5.41), two functions of our Bernstein diffusion X(t). As we noted, after Feynman, both random variables make sense since their expectations coincide. As a matter of fact, when Feynman mentions (4.38) [6, §7.7], he complains that his handling of discretized symbolic random variables "does not exhibit the important relationship between the Hamiltonian and time". What he is alluding to here, expressed in the classical Lagrangian terms of his path integral approach, is to a special case of Ncether's theorem, the second most important result of Lagrangian mechanics after the least action principle of Hamilton. For a given system, this structural theorem relates the symmetries of the action functional SL (or equivalently, of the Lagrangian L) to the existence of first integrals (or conservation laws) of this system. Let us formulate Ncether's theorem (for a one dimensional configuration space) in a form which is not the most familiar in physics but will be quite appropriate in our random (and quantum) perspective. A smooth vector field v\ defined by

^ = nt)Wt+xiq,t)-

+

(--q-)-

is called a divergence symmetry of the Lagrangian L = L(q,q,t) a smooth ("divergence" ) ^ : R x R - » R such that

(6.D if there is

In Lie group theory, v\ is called the first prolongation for the (infinitesimal)

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randomness

generator « = X(«,t)^+T(t)^

(6.3)

of the following one-parameter group of transformations of the extended configuration space (i.e. the space-time) L/Q:RxM->RxR (q, t)^{Qa

= q + aX(q, t) + o(a), ra = t + aT(t) + o(a)).

(6.4)

The meaning of the terminology for the vector field v\ is that it prolongs the effect of our change of space-time variables (6.4) to the q dependence of L. Notice that the time is regarded as an extra coordinate here. This is needed in order to formulate the property Feynman was referring to above. So we have Theorem 6.1 (Noether's Theorem). When the (regular) Lagrangian L admits a divergence symmetry such that (6.2) holds, then, along every extremal of the action SL, the following conservation law is valid: jt(pX

- hT -cj>) = 0,

(6.5)

where p and h denote, as before, the momentum and the Hamiltonian of our system. The starting vector fields v used to construct v\ are infinitesimal generators of the symmetry groups of the Euler-Lagrange equation (4.5) for the action SL- This means that each of these symmetry groups, say Ua, transforms any given solution of (4.5), say q(t), into another (a-dependent) solution Ua{q{t)) of the same equation. This and (6.2) implies (6.5). The version generally given in textbooks is that where <j> and T vanish so the variation v\ (L) of the Lagrangian L (as physicists say) reduces to /r.

„dL

dX dL

n

and pX is constant. Consider again our class of one-dimensional elementary Lagrangians of the form (4.3): L(q,q,t)

= j\q\2-V(q,t).

(6.6)

6. Beyond Feynman's

173

approach

When the scalar potential V is time-independent, the system is conservative, i.e. invariant under time translation. Then L admits, of course, v\ = d/dt, i.e. the case T = 1, X = (j> = 0 m. (6.1), so Ncether's theorem says that the Hamiltonian h(q,p) = p2/(2m) + V(q) is a first integral of the system. This is the "important relation between the Hamiltonian and time" alluded to by Feynman. So a natural question arises: What is the probabilistic counterpart of the first integral of a classical dynamical system? Is it the case that one of (or both?) our candidates (5.40) and (5.41) for the energy represents such a random first integral? It is at this point that the concept of martingale with respect to a given tribe (Part 1, §3) is going to reappear. Let us come back to the standard definition of a quantum first integral (2.33), i.e., a perhaps time-dependent, observable A(t) such that (on a dense domain of H. = L 2 (R)) ih^A(t) + [A(t),H]=0.

(6.7)

Since our probabilistic analogy (5.32) corresponds to t —> —it, it is not very surprising that the counterpart of (6.7) becomes h^N(t) + [N(t),H]=0

(6.8)

for N(t) = N one of the normal (not necessarily self-adjoint!) operators associated with quantum observables in our framework. One can show that (6.8) is, indeed, analytically well defined [28]. Now let us denote by n the Euclidean space-time observable associated with any N such that (6.8) holds. A simple calculation with (5.35) shows that, as long as X(t) is defined,

O (x<,)

"

'" = sxko(l4 ff )<"'>™'"

(69)

where the r.h.s. of the first line is a very useful and general representation of operator D [29] in terms of the associated heat equation. Now, by definition (5.35) of D, this means that, under the condition E{\n(X(t),t)\} < oo, Vi G 7, we have E{n(X(t),t)\Vs}=n(X(s),s),

Vs < tin I.

(6.10)

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So n is a martingale with respect to tribe Vt (the "past"), as defined in §3 of Part 1, and denoted {nt,Vt} there. In particular, if N coincides with the Hamiltonian H = Ho + V of a conservative system, then it satisfies trivially condition (6.8) and therefore the space-time observable associated with it by h = Hr]/r] (see (5.40)) is a TVmartingale. So the fact that h(X(t),t) is a 'Pt-martingale, when X(-) solves equation of motion (5.37) is the counterpart of the fact that the Hamiltonian h(q(t),p(t)) is a classical first integral when q(-) solves the Euler-Lagrange equation for (6.6), i.e. mq(t) = -VV(q{t)). Since the second candidate h mentioned in (5.41) (i.e., the well defined version of Feynman's Hj in (4.38)) does not satisfy this martingale property, we can conclude that Feynman's candidate Hj was more appropriate to his needs. Let us stress that such a conclusion was not accessible to the father of path integrals because, as we noted, the "continuum limit" (u — s)/N —* 0 does not make any mathematical sense in his framework. This is another example where the existence of a well defined measure on path space helps to understand elementary physical principles (here the conservation of energy!). In relation with the tribe Tt (the "future"), the energy space-time observable takes the form (5.42). Now operator D* of (5.36) admits a representation symmetric to that used in (6.9), involving instead the adjoint heat equation for r\*. Therefore it can be shown, in the same way, that h*(X(t),t) is a .Ft-martingale, another expression of the time-symmetry of the formalism. Of course, the principle of conservation of the energy is not exactly breaking news, neither classically nor quantum mechanically! But let us reconsider in a Feynmanian perspective what we have found; since the classical conservation of the energy is a very special case of Ncether's theorem along smooth paths, what about a probabilistic counterpart of (6.5) along the paths of Bernstein diffusions, for H = HQ + V given as before? The first observation is that our diffusions are entirely built in terms of (positive) solutions of heat equations (5.32) (with Ho replaced by H). So the study of the symmetry group of our random equation of motion (5.37), for example, amounts to the study of the symmetry group of the partial differential equation hdr)/dt = Hrj. But this has been well known since the time of S. Lie: A (local) one-parameter group of transformations of this

6. Beyond Feynman's

approach

175

heat equation is denned by Va : (77, q, t)^{r]a

= ri+ -(q, t) + o(a),

Qa = q + aX(q,t)

+ o(a),

(6-n)

Ta = t + aT(t) + o{a)), where the starting solution 77 itself is transformed, and the coefficients <j>, X and T are assumed to be analytic in q and t. We have chosen a notation stressing, as we shall see, the analogy with the classical Noether's theorem. For the same reason, let us call N the infinitesimal generator of (6.11), i.e., informally,

N=x +T

(612)

h li-\*-

As is clear, we treat space and time variables on an equal footing here. This suggests that we think about our heat equation as Hi] — 0, for

H (or, better, hH) is a (Euclidean) quantization of the classical Hamiltonian observable h acting on the extended phase space (see (1.9)). We can complete this analogy by allowing, from now on, potential V to be (smoothly) time-dependent. The symmetry group of the heat equation is defined by the property that if Hr\ = 0 then HNr] = 0. This is possible if and only if the coefficients X, T and 4> of the generator N satisfy the following (linear) "determining equations" of the symmetry group: dX _ d<j> ~dt ~ dq'

d_l dt

+

±^l 2m 8q2 dT dt

=

dTv dt _2dX dq

+ xd_V+TdV

dq

dt'

(614)

V

'

Although this way to determine all the (X, T, <j>) tangent to Va and transform any given solution 77 of our heat equation into another is not well known in theoretical physics, the method was designed by Lie in the 19th century. For a given potential V, the collection of such infinitesimal generators AT forms the ("symmetry") Lie algebra of our system.

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randomness

What is the probabilistic counterpart of the classical divergence symmetry condition (6.2)? Here we face two basic difficulties. The first is that we do not have, up to now, a Lagrangian L to start our construction. This is quite regrettable since we claim to follow Feynman's strategy. The other problem is that, if the first two terms of generator (6.12) can be interpreted as the analogues of the classical generator v of (6.3), the counterpart of the extra term in its first prolongation (cf. (6.1)) is unclear. But the Bernstein processes X(t) that we have associated with positive solutions of the heat equation HTJ = 0 will give the key to the solution of both difficulties. As we said at the beginning of section 4.1, for the elementary class of completely integrable classical systems that we want to quantize "a la Feynman", the relation between the momentum and energy observables on the one hand and the Hamiltonian (or Lagrangian) action functional on the other hand is very simple. Equation (4.13), for example, says that the differential form —L(q,q,t)dt is exact, namely of the form dSx(gr, £), where SL is the action regarded as a function of the trajectories leaving point q at time t and with final condition, say Su(z), at a later time u. But we know the random variables representing the momentum and energy as functions of process X(t). They are given by (5.33) and (5.40'), respectively: mB(X(t),t)

= hVlogr)(q,t),

h(X(t),t)

= hdt logr](X(t),t).

(6.15)

Consequently, we should define the action function (up to an additive normalization constant) as SL(q,t)

=-Mogri(q,t),

(6.16)

so that, in analogy with (4.13), and denoting by p(q,t) the momentum space-time observable, p(q,t) = mB(q,t)

= -VSL(q,t),

h(q,t) = -dtSL(q,t).

(6.17)

In order to find the generalization of <1SL = —Ldt along the paths of Bernstein diffusions, let us recall that Ito (forward and backward) calculus associated with the stochastic differential equations (5.12) and (5.13) does not follow the rule of classical calculus. In the explicit case ((5.12'), (5.13')), this does not show but when the coefficient of dW (or d*W*) is not locally of bounded variation, there are extra terms with respect to Newton's calculus. This explains the /i-dependent terms in (5.35) or (5.36), for

6. Beyond Feynman's approach

177

example, interpreted along Feynman's line as a deformation of the classical derivatives. As an illustration, let X(t) be a Brownian motion with X(0) = 0 and diffusion coefficient h/m. Then, since the drift, B, is 0, (5.35) shows that DX2(t)

=- . (6.18) m In terms of Ito's (forward) stochastic differential, this corresponds to the fact that dX2(t) = 2X(t) dX(t) + —dt, and then E{dX2{t)

\ Vt} = (h/m)dt

E{2X{t)dX{t)

(6.19)

since

| Vt} = 2X{t)E{dX{t)

\ Vt} = 0.

After integration, this last relation stresses the fundamental property of Ito's integral: as a function of the upper time u, this integral is a Vumartingale. More generally, for any real valued / bounded with bounded derivative as in (5.35) and any one-dimensional diffusion X(t) of (finite energy) drift B and diffusion coefficient h/m, the (forward) stochastic differential is obtained from the Taylor expansion f(X(t + e),t + e) — f(X(t),t) around (X(t),t) and up to second order derivatives. Then, all terms of the form (dt)2, dW(t)dt are neglected, and (dW(t))2 is replaced by (h/m)dt (see footnote 9 of Part 1 (page 22)). This last substitution is, as already noted after (5.20) and (5.21), the probabilistic counterpart of Feynman's fundamental relation (4.37). So we obtain Ito's famous formula

df(X(t),t) =(^(X{t),t)

+ B • Vf(X(t),t)

+ Vf(X(t),t)J-dW(t). V m

+

^-Af(X(t),t))dt (6.20)

Comparison with (5.35) shows that D, as defined there, retains the bounded variation term of the r.h.s. of (6.20), but annuls its TVmartingale part. The property dSx = —Ldt, however, is of a differential geometric nature and is preserved by Stratonovich instead of Ito calculus. (On the other hand, the above martingale property is lost by the associated stochastic integral.) The relations between those two versions of stochastic calculus are well known [20, 23]. For any r G I = [s, u], the time interval of definition

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of a general Bernstein diffusion X(T), and for p, h regular enough, equations (6.16) and (6.17) imply that -dSL(X(T),

T) = P(X(T),

T) O dX(r) + h(X(r), T) dr,

(6.21)

where o denotes the (Fisk) Stratonovich integral [23]. So, for s < t < u, and assuming integrability,

-EUU = EifU

dSL(X(r),T) P(X(T),

T) O

Vt\ dX{r) + h(X(r),

T)

dr

Vt\

or SL(X(t),t)

+ EtMUL(X(T),DX(T),T)dr\,

= Et{SL(X(u),u)}

(6.22)

where we have used the shorthand Et{-} for E{- • • | Pt} and the passage from Hamiltonian action (6.21) generalizing (4.7) to Lagrangian action (6.22) uses, besides (6.17), the differential relation between Stratonovich and Ito integrals [23] (denoted simply p • dX(t)), EtlTpodX{T)\=EtirP-dX(T) = Et{(mB2

+ +

^dpdx\

^VB)dTy

as well as the above-mentioned rules of stochastic calculus. The last relation is the probabilistic version of what Feynman found for the transition element of the vector potential since, when p = p(q, r) as in (4.41), (4.42), (4.43), [ podX(r)=

f

(p-dX(T)

+ —V-pdT).

(6.23)

If, instead of the usual forward Ito integral, one uses the backward one, denoted p * dX, one obtains instead f podX(r)=

J

(p*dX(r)-^-V

pdrj.

(6. 24)

6. Beyond Feynman's approach

179

The origin of these two relations lies in the definitions of the forward ltd integral with respect to the past filtration as the limit in probability /

p.dX(r)=

Jt

Li.p.

^(Xfo.OKXfoJ-Xfa-x)),

max|i-j-T,--i|-»0 J • , l<j
and of the backward integral with respect to the future filtration N

I

p*dX(r)

=

t

Li. p.

^(XfoOHXfaO-Xfo-i)).

max|Tj— Tj-i|—*0 -_-i J 1<J
Their sum defines the symmetric (Fisk) Stratonovich integral [23]

I

i

PodX(T)

Li. p. max|Tj—TJ_I|-*0

\<j
f ^ ^ . ! ) ) J

+ P ( X ( T , - ) ) ) (X(TJ)

-

X{r^)).

•_,

So, Feynman's symmetric "integral" in (4.43) corresponds to the Stratonovich integral, and (4.41), (4.42) to the forward and backward Ito integrals, respectively. Relations (6.23) (and (6.24)) between Stratonovich and forward (backward) Ito integrals are the well defined counterparts of what Feynman found. In particular, after introducing the energy (5.40), the action (6.22) reduces indeed to SL(X(t),t)

= Et{SL(X(u),u)}

+ EtiJU{^(DX(T))2

+ y(X(r)))drj

(6.25) For the elementary class of systems considered here, whose classical Lagrangian is given by (6.6), we could not hope, again, for a simpler result: the Lagrangian is the classical one, up to the usual Euclidean change of sign of the potential V. What plays the role of the time derivative along smooth trajectories is, now, DX(T), along the continuous paths of unbounded variation of our diffusion processes. So we know the Lagrangian and can come back to our problem of what is the probabilistic counterpart of the classical divergence symmetry (6.2). Before doing so, however, let us remark on two important consequences of (6.25). Denoting by Eqj the expectation used there when X(t) = q, it

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randomness

follows from our definition (6.16) that U

V(q,t)=exp[~Eq,t^

(^(DX(T))2

+ V(X(T)))dT + Su(X(u))}

,

(6.26) where Su(z) = SL(Z,U). This is the probabilistic counterpart of Feynman's path integral representation of a solution of the complex conjugate Schrodinger equation. Equation (6.26) also shows why the finite energy condition mentioned after (5.33) is natural in the construction of Bernstein diffusions. With the help of the time reversed diffusion X(T) = X(U+S—T), it is also easy to find •q*{q,t) = e x p

l^ {/(?( D * X ^ 2 + yWT)))dT + ^(*(«))}] ,

(6.26') where Eq,t is shorthand for E{-•• | X(t) = q} when the conditioning is in the future and S*(x) = —hlogT]*(x,s). So (6.26') is the counterpart of Feynman's symbolic representation (4.30). Notice that there is only one (classical) Lagrangian involved in (6.26) and (6.26'), as it should be, but this is evaluated on the forward or backward time derivatives, respectively. In particular, because the probability measures exist, there are non singular path integral representations of our two regular and positive solutions of the adjoint heat equations. The second noteworthy consequence follows from (6.21). Since the stochastic ("Poincare-Cartan") one-form defined by its r.h.s. is exact, then

TT-SI^I' + S;* 5 '^ 0 -

(6 27

- >

This can be regarded as a uniformly parabolic non-linear PDE for the scalar field SL — SL{q, t), known as the Hamilton-Jacobi-Bellman equation. Given its origin here, one may wonder if it does not provide the solution of a stochastic version of Hamilton's least action principle. This is indeed the case, and in this sense our diffusions with (forward) drift B = —m~1VSL are critical points of the action (6.25) in complete analogy with the smooth case [35].12 We shall come back to this issue in §6.2. Clearly (6.27) is the (probabilistic) quantum deformation of the classical Hamilton-Jacobi equation (4.15). 12 In general, interesting solutions of (6.27) are not regular. The method of viscosity (M. G. Crandall, P.-L. Lions [32]) has proved, in the last 15 years, to be the proper notion of a weak solution for the Hamilton-Jacobi-Bellman equation.

6. Beyond Feynman's

approach

181

In addition, (6.27) is the starting point of a stochastic symplectic geometry, which is a fa-deformation of Cartan's classical theory. But this is another story (to be continued [54]). The starting change of variable (6.16) from the linear heat equation to the non-linear Hamilton-Jacobi-Bellman equation (6.27) seemed to come out of the blue. It did not, however, since it was introduced in 1926 by Schrodinger when he discovered the partial differential equation named after him (Collected Papers on Wave Mechanics (Chelsea: 1978)). Our definition (6.16) is just the probabilistic counterpart of this transformation. As we said after (6.26'), the probabilistic analogy of the complex conjugate solution is also useful. So let us come back to the counterpart of symmetry condition (6.2). With our preparation, is should not come as a surprise that this condition becomes, for L = L(q, q, t) the (Euclidean) Lagrangian of representation (6.26) (where V may also depend smoothly on the time parameter), T^-+ X~ + (DX - BDT)^-+ LDT = D4> a.s. (6.28) ot oq oB This should hold along Bernstein diffusions. Because we denoted such processes by X(T) and by X(q, t) as well the coefficient of the space transformation in (6.12), we have now used B for the second variable of the Lagrangian instead of DX(T) as in (6.25), in order to limit the risk of confusion. When the paths r —» X(T) are smooth (i.e. in the singular limit h = 0 of the SDE), condition (6.28) reduces exactly to the classical (6.2). Then the probabilistic counterpart of (6.5) is the Theorem 6.2 (Stochastic Ncether's Theorem [33, 43]). When the Lagrangian integrand of our action (6.25) satisfies condition (6.28) for any X,T,4> solving the determining equations (6.14) of the symmetry group of Hrj = 0 then, along Bernstein diffusions Z(t), s < t < u, of momentum and energy defined in (6.17), D{pX + hT-4>)(Z(t),t)=0

a.s.

(6.29)

Here we denoted, exceptionally, the diffusions by Z(t) to avoid the above-mentioned confusion. The proof, which is a long exercise in Ito calculus, can be found in [33]. In particular, if the system is conservative (i.e. if the potential V is not time-dependent), our Euclidean Lagrangian clearly satisfies (6.28) for T = 1 and X = = 0, a trivial solution of (6.14). Then Ncether confirms

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the fact that the energy space-time variable ft is a P t -martingale of our conservative system. Let us consider the whole collection of Bernstein diffusions associated with the free case V = 0. The symmetry Lie algebra of the (one-dimensional, still) free heat equation is of dimension 6 ( 13 ) and was calculated by S. Lie around 1880. One verifies directly that X(q, t) = t, T = 0 and (p(q,t) = q are solutions of (6.14), so that, by Ncether's Theorem m(Z(t),t) = p(Z{t), t) t - Z(t) is a TVmartingale. Another solution of (6.14) is given by X(q,t) T{t) = 2t2, 4> = q2 - ht, so that n2(Z(t),t)

= p(Z(t),t)

It Z(t) + h{Z{t), t) 2t2 - (Z2(t) - ht)

(6.30) = 2tq,

(6.31)

is also a 7-t-martingale. Three additional nontrivial martingales belong to the basis of this symmetry Lie algebra. On the other hand, the functional forms of p = mB and h in terms of any regular positive solution rj of the heat equation are given by (6.17). Let us consider a Brownian motion starting from x at time 0. It can be regarded as a (very) special Bernstein diffusion associated with the free case V = 0, whose relevant solutions of the adjoint heat equations (5.32) are rj(q,t) = l,

ri*(q,t) = ho(x,t,q).

(6.32)

In particular, (5.31) then reduces to the usual probability distribution of a (one-dimensional) Brownian motion. According to (5.33), this means that B(q,t)=0,

B*(q,t) = ?-^.

(6.33)

Notice that this process is defined Vi € / = M + , in contrast with most Bernstein diffusions. On the other hand, there is nothing quantum about this process. Its variance is (h/m)t and reflects its dynamical irreversibility (this example was, in fact, already mentioned in [34]). It is clear from (6.32) that the Brownian motion corresponds to a transition amplitude and not to a quantum unitary evolution. In fact, we now can be more specific about our previous remark that the Brownian has no quantum dynamical meaning. 13 Besides a trivial infinite dimensional subalgebra reflecting the linearity of the equation (but implying the highly nontrivial interference of probability (5.51)!).

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According to (5.37) and (6.33) this process is a solution of the free equation DDX(t)

= 0

with

X(0) = x

and

DX(0) = 0.

In the formal classical limit h = 0 (where D reduces to an ordinary time derivative) the solution of the Cauchy problem for the resulting ODE is the trivial one, X(t) = x, Vi > 0. So the Brownian motion is just the quantum deformation of this trivial solution. All the same, it is fundamental since any Bernstein diffusion is built from it (via an absolutely continuous change of probability measure: Cf. §6.2). Let us come back to the two martingales of the Brownian used in Part 1 (§3) to solve both questions in the Prologue. As hinted on page 23, those martingales are actually the first two of a family generated by the Brownian X(t). More precisely, for each A e i (cf. §6.2, [14]), the exponential

px{t) =

H

expl\X(t)--\2t

is a martingale (relative to the standard filtration Vt). By successive differentiations with respect to the parameter A, we generate polynomials in X and t which are martingales. In particular, 14 dpx(t) dX

A=0

d2px(t) = X\t)-t, d\2 A=o

etc..

The polynomials Hn(X(t),t)=

^Px{t)

n = 1,2,3,. A=o are closely related to the Hermite polynomials, an old favourite of students of quantum physics. But, there, they come to light for particles subject to harmonic forces whereas our Brownian motion is free. Our problem is, therefore, to understand the relation, if any, between these Brownian martingales and those predicted by our stochastic Ncether's Theorem. 14

A quantum physicist would identify p\ as the Wick (ordered or normal) exponential of the Brownian motion and the polynomials Hn(X(t), t), n = 1, 2, 3 , . . . as the renormalization of its ordinary power Xn(t). The Wick ordering, introduced in 1950 by G. C. Wick in terms of creation and annihilation operators in Hilbert space and for the study of Dirac's correspondence principle for polynomial observables in q and p, provides the simplest renormalization procedure of quantum physics.

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Using (6.12), we can reformulate the remark leading to martingale (6.30), for instance, as the fact that dq is one of the infinitesimal generators of the free Lie algebra. Equivalently, after reintroduction of Planck's constant,

V\(q,t)=(e*Nri)(q,t),

A

provides a one-parameter family of solutions of Ht] = 0, for V = 0, when r] is a given (positive) solution. More explicitly, each 1 V\ (q,t) = e ^-^)r1(q-Xt,t),

X

is a positive solution of the same free heat equation. But, then, there is a (free) Bernstein Zx(t) associated with this solution, in the same way as there is one, called Z(t), associated with the given rj. The relation between these two diffusions is a (deterministic) translation: Zx{t) = Z{t) + \t,

AGK.

(6.34)

Indeed, we have —h\nr]x(q,t) = —hlnr](q,t) — hlnha where h\ = r)\/r]. In other words, the relation between the respective actions (defined in (6.16)) reduces to S£{q,t) =

SL(q,t)-hhxhx.

Then, from (6.17), follows the relation between the drifts and energies of Zx{t) and Z{t): Bx(q,t)

= B(q,t) +

V\oghx(q,t),

£\(q, t) = s(q, t) + dt log h\(q, t). All this is valid, of course, for any solution Z(t) of the free equation (any given positive solution rj). In particular it should hold for the trivial one represented by the Brownian Z(t) = X(t). Since, then, T] = 1, we have ftA(?,*)=c*^-*A't)

6. Beyond Feynman's

approach

185

and therefore the drifts and energy of Zx(t) are the two constants Bx(q,t)

= X,

ex(q,t) =

-^\2.

The exponential martingale h\ of the Brownian coincides with p\ above and provides us with all its polynomial martingales. We have, therefore, reinterpreted them in terms of one of Ncether's symmetries allowed by our free dynamics, namely the translation (6.34). See [54] for the complete explanation. More generally, Ncether's Theorem refers, in fact, only to the space-time observable associated with the generator N, not its exponential. Some of the higher order martingales coming from an exponential martingale may also reappear in connection with another Ncether symmetry. For instance, since the momentum and energy random variables of the Brownian vanish, ri2(X(t),t) as given in (6.31) is proportional to H2(X(t),t) but associated with a symmetry distinct from that behind (6.30). So it seems indeed that our stochastic theorem of Ncether does have a nontrivial probabilistic content. It relates, however, to a dynamical concept of symmetry of diffusion processes which is not at all familiar in probability theory. But what about quantum dynamics? As we just said, Brownian motion has nothing really reminiscent of quantum dynamics. Should we accept that, for example, the rich probabilistic information contained in the Brownian exponential martingale has nothing to do with the motion of a free quantum particle? Then the suspicious theoretical physicist opening this section would certainly be right... In order to think about the quantum meaning of what we found we have to do backward with the time parameter what we did to go from Feynman's path integral approach to our probabilistic framework. In many respects, of course, this attempt is doomed to disaster. Consider, for example, the forward transition function of a Bernstein diffusion given in (5.34). This is the Euclidean counterpart of Feynman's heuristic expression (4.33). There is no Markovian diffusion on the measurable space (R, 93) with a transition function like (4.33) because such a function has to be non-negative, by definition. The situation does not improve if we replace R with the complex field C. The resulting diffusions are, in fact, Feynman's symbolic ones, described in §4. Appealing as they are, they refuse to live in the Hilbert space with all the properties required by Feynman for the purpose of his path integral reinterpretation of standard quantum theory.

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But a lot of what we did still makes sense. For instance, the quantum counterpart of (5.35) is defined (when ipt(q) ¥" 0) by Dt = — — • V - -—A (6.35) at m ipt 2m for ipt solving the Schrodinger equation (2.27) with Hamiltonian H — HQ + V as before. Dt is called the quantum derivative along ipt- One shows that it is a densely defined differential operator in L2(R, 1^(9)1 dq) [24]. As noted before (5.39), the idea of a space-time observable associated with a quantum observable A in a state ipt is implicitly (and in discrete time) already in Feynman. Since A may depend on time, we can define

a t) =

^

^w-

(6 36)

-

for all q e R such that ipt(q) ^ 0. Notice that the problem of the zeroes of wave functions is considerably more serious now than in the probabilistic framework since we lose the positivity preserving (and even the reality preserving) property of the heat kernel. However this problem, which amounts to the investigation of the wave front of K(x,t — s,q) = keTnel{exp(—(i/h)H(t — s))}(x, q) for fixed initial configuration x, is under control for a large class of potentials V which are perturbations of integrable potentials [24, and references therein]. Then one can assume that the null set Nf = {x e R | ipt(x) = 0} has zero Lebesgue measure and prove that, for any tp in T>H (where ipt = Utip) ±(t).

(6.37)

It is natural to call the space-time observable ad,t a quantum martingale when Dta$t = 0, since, in this case, the associated (family of) observable(s) A(t) is a standard quantum constant of motion. The complex conjugate of (6.35), i.e., the quantum derivative along ipt, = d ih Vi/>t „ ih . Dt = - + - ^ - V + — A , at m ipt 2m

, „. 6.38

now plays the role of the counterpart of D*, denned in (5.36). Consider the space of (complex) functions / = (f^)teR in the domain of Dt which, in addition, are bounded and continuous. They form a commutative algebra. It is straightforward to verify that Dt (or Dt) introduces a quantum deformation of the Leibniz rule in this algebra, which is the counterpart of the deformation introduced by Ito's formula (see (6.19), for

6. Beyond Feynman's

approach

187

example). Under appropriate restrictions on the domains, we have, for instance, Dt(f • g) = (Dtf)g + f(Dtg) - ^ V / • Vg.

(6.39)

Then, one uses Feynman's two integral kernels PF and pg ((4.33) and (4.57)) to define operators of quantum conditional expectations in a state, given a configuration in the past or future respectively. Except for the positivity, these operators have, in respect of the "absolute" expectation associated with Born's interpretation (4.60), all the properties of usual conditional expectations. Yet, this lack of positivity is sufficient to spoil, as we know, the very existence of Feynman's diffusions. The calculus of quantum space-time observables (6.36) with (6.39) still makes sense. Notice that our method can be regarded as a contribution along the line of the quantization problem (cf. Weyl calculus, etc. in §3). The difference is that, by (6.36), a complex valued function a in the extended configuration space (and not in phase space) is associated with an observable A, and the key deformation of the classical commutative structure is given by (6.39), i.e. by the deformation corresponding to Feynman's diffusion (see (4.58)) if this was well defined. This brings together the geometrical methods of pseudo-differential calculus and Ito's stochastic calculus. As a matter of fact we should emphasize that the quantization problem now takes on quite a new shape, since the mathematical rules of the hdeformation are precisely defined by Ito's calculus. In connection with our definition (6.36), let us note that if ipA is an eigenvector of A with eigenvalue o, then a^A (q, t) = a. So our definition generalizes this idea to an arbitrary state V such that the standard deviation A,/,(v4) is non-zero. A famous theorem of Kochen and Specker is devoted to the properties of any such "value functions" V^{A). If, inspired by von Neumann's spectral theorem (cf. (3.3)), for any complex-valued Borel measurable g, one assumes the property V^,{g{A)) = g(Vlp(A)), then their theorem shows that, in any Hilbert space of dimension greater than 2, no value function can exist. This result is, together with the Bell inequalities, a strong argument against hidden variables theories (see Lectures on Quantum Theory, C. J. Isham, Imperial College Press, 1997, for an interesting discussion). Let us note that (6.36) does not satisfy the above property and that, if it did, our stochastic Ncether Theorem, for example, would not hold. Now let us see why that would be a pity.

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The symmetry group of the Schrodinger equation mentioned before is very closely related to that of the associated heat equation. This allows one to find the quantum counterpart of the infinitesimal generator of this group of transformations. It is not symmetric for the same reason that (6.12) is not, but can be made so using Jordan's multiplication o of (3.17); this amounts to redefining the "phase"
+ i{4>Q + ^VXQyQ(t),t),

(6.40)

where XQ, TQ and (f>Q solve the counterpart of the determining equations (6.14) for the Schrodinger equation, namely

i

dQ

dXQ _ .d(j)Q dt dq 2 h d Q dTQ V -

dTQ dt

8V

+ XQ^+TQ-^T,

dV (6.14Q)

dXQ dq

and Q(t), P(t) and H(t) denote, respectively, the quantum observables of position, momentum and energy in Heisenberg's sense (2.29). When the starting Hamiltonian H = HQ + V is a polynomial of degree < 2 in Q and P (V may be time-dependent), the calculations can be done explicitly; all the generators N(t) of the symmetry group are essentially self-adjoint and defined on a common dense invariant domain in L2(M). Notice that this is the situation where Dirac's correspondence (2.32) holds true. For general H, this is no longer the case, but it is still true that (6.40) defines quantum constants of motion [24]. Now we can come back to our elementary application of the stochastic Ncether theorem: the first two martingales {X(t)} and {X2(t) — ht} of the Brownian motion X(t) which are essential to the solution of both questions in the Prologue of Part 1. Do they have a quantum mechanical meaning? In §3 we already mentioned the fact that the solution of the free Heisenberg equation (here for m = 1) is given by

($)-C.:)G)-

6. Beyond Feynman's

approach

189

for Q and P defined in (2.2) and (2.12), respectively. It follows, in particular, that Vn e N, Nn(t) = (Q(t)-tP(t))n

(6.42)

is a trivial quantum constant of motion, the nth power of the initial position observable. In particular, JVi(t) = -tP(t)

+ Q(t)

(6.43)

is a Ncetherian observable associated with the solutions XQ(q,t)

= -t,

of the determining equations ten as

TQ(t)=0, (6.14Q).

N2(t) = -2tQ{t)P(t)

Q(q,t) =-iq

(6.44)

In the same way, iV*2(t) can be rewrit-

+ t2P2(t) + (Q2(t) + iht),

i.e. as the Ncetherian observable associated with the solutions XQ(q,t)

=-2tq,

TQ(t) = -2t2,

<j>Q{q,t) = -iq2 + H.

(6.45)

Quantum constants of all powers of the initial position follow, as before, from exponentiation of the infinitesimal generator. The quantum analogue of the probability measure of the Brownian motion associated with (6.32) is the unnormalized solution ipt = 1 oi the free Schrodinger equation. This is only an analogy, since then V>t = 1 as well and certainly not the free integral kernel k${x, t, q) corresponding to (6.32). In any case, on tpt = 1 the space-time observables (6.36) of momentum and energy vanish and the constant quantum observables we are left with are proportional to the (multiplication) Heisenberg operators for the free quantum particle: {Q(t)}

and

{Q2(t)+iM}.

(6.46)

These are the quantum counterparts of the first two martingales of the Brownian motion. In contrast, neither in the Feynman-Kac approach nor in that of Nelson do the quantum constant observables have a probabilistic counterpart. 15 Is is striking to see that Euclidean quantum mechanics, where it is indispensable to handle time inhomogeneous situations, guided us to look 15

In particular, the remark of [20, p. 77] suggesting the contrary in stochastic mechanics is incorrect.

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for explicitly time-dependent quantum observables, generally disregarded in standard quantum theory. As a matter of fact, even in the one-dimensional free case, this method allows to find more symmetries than those provided by standard quantum theory. Of course, all the familiar (time-independent) constant observables are also part of the collection predicted by (6.40) [24]. 6.2

Introduction to functional calculus

Let us come back to Feynman's integration by parts formula (4.34), which is the origin of most of the qualitative results summarized in §4.2. We already know the (Euclidean) meaning to be given to the "expectation" (-) s . And to a functional of the continuous paths (or trajectories) T H-» LJ{T) = X(T,W) of the process. Two typical such functional are given by action (6.25) f

L(X(T),DX(T),T)<1T

(6.47)

or the expression of part of this action (cf. (6.21)) in Hamiltonian terms, whose integrand is the Euclidean counterpart of the classical Poincare oneform[49]: JUp(X(T),T)odX(T).

(6.48)

Although (6.47) can be regarded as a familiar Riemann-Stieltjes integral, this is not the case with this second integral, if X{T) is a solution of a stochastic differential equation like (5.12), as already observed after (6.23). If the Brownian motion (or Wiener process) W(-) in (5.12) had differentiable (or bounded variation) trajectories such an equation could be treated by the theory of ordinary differential equations. But such "quantum paths", in Feynman's sense, have infinite variation in every non-empty interval. This means, in particular, that integral (6.48) cannot be defined path-bypath in the usual (Riemann-Stieltjes) way. This difficulty is solved by Ito's stochastic differential and integral calculus alluded to in §6.1. Ito's theory is based on the Wiener space, introduced by N. Wiener, in the early 1920's, for his mathematical construction of the Brownian motion [55]. Informally, the Wiener space is a typical domain of a path integral (see §4.2), namely Qo = { w € C(R + ; R) such that u>(0) = 0 }

(6.49)

6. Beyond Feynman's

approach

191

equipped with a probability measure Pw induced by a Brownian motion. In other words, Pw is denned by the two following properties: (1) The distribution of w(i) under Pw is given by Pw(u(t)

€ dq) =

ho{0,t,q)dq

where ho is the Gaussian kernel (5.1), i.e. the Euclidean version of Feynman's free kernel (4.20). (2) Using the notation of (4.25), for any 0 = to < ti < ... < t^, the increments w{tj) — u(tj-i) are independent under Pw{du>). If
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respect to the scalar product ( fio such that o> >—> u> +
^-{f>>^)-U°>>f

dr } .

(6.50)

Now the Cameron-Martin space does not support the Wiener measure, in fact, PwCH) = 0: this measure "sees" exclusively quantum paths. Still, with a

5F[U](V) = D„FM = lim n ' + qri-J'M.

(651)

This is manifestly an almost sure analogue of the Gateaux derivative. The linearity of this limit in

M by VF[w](p) = D„F[u].

(6.52)

Now let us restrict the existence of the processes to a compact time interval, say [0, T] instead of the whole real line. Let us pick an orthonormal 17 T h e probability measure P' of a random process Z[ is absolutely continuous with respect to the probability P of another one Z., taking values in the same measurable space, if there is a nonnegative density h = i^- such that for any measurable / : E '[f(z')] = E[f{Z) h(Z.)] where E' and E denote respectively the expectations for P' and P.

6. Beyond Feynman's

approach

193

basis {ej} of the Cameron-Martin subspace on [0, T], and define

||VF||2M = £|D e t FM| 2 and the Sobolev space Wftn,,, J V ; R ) = {F 6 L 2 ( f t 0 , i V ; K ) ; £ p J | V F | | 2 < o o } .

(6.53)

For any regular, adapted <j> in L2(Cl0,Pw, H) and functional F in W 2 (fio, Pw;K), the following integration by parts formula holds: EPw [VF[u](4>)] =

EPw

FMjf g[u,]<Mr)

(6.54)

where, of course, w(t) = W(t,u>) are Wiener paths and the associated integral appears in the r.h.s.. The origin of this formula is simple: its l.h.s. means E Pw

lim

F[w + e<j>] - F[u]

e-»0

= lim -EPw lim

-EPU

[F[LJ + ecf>] - F[u]]

F\LJ] exp< e /

I 7o

e—0 e

•E'PW

FM-(exp

—duM

dr

H"57

dw{r)

risiw

(\dn,

Ffwl

W £= 0

which is the r.h.s. of (6.54). The key point is the substitution of the Cameron-Martin-Girsanov expression (6.50) for dP^/dPwConsider, for instance, our translation (6.34) of free Bernstein diffusions in the simplest case where Z(t) is the standard Brownian motion W(t). It is clearly a (deterministic) Cameron-Martin translation with tp(t) = (f>(t) = Xt for any t € [0,T]. Then j ^ reduces to exp{XW{t) - {X2/2)t} i.e. the positive martingale h\(W(t),t) defined there, and the integration by parts formula to EPw [VF[W}(Xt)} = EPw

[F[W}XW(T)].

For a general Bernstein diffusion X(-), the integration by parts formula is slightly more complicated than the above one for the Wiener. But it

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follows easily from (5.34) and the Feynman-Kac multiplicative functional that the measure P\ is absolutely continuous with respect to Pw, with dPx dPw

v(X(T),T) < 1 'T e x p | - | f V(X(T))dA. ij(X(0),0)

(6.55)

Calling p this density, and Eppw the associated expectation for X, one proves that, under regularity conditions on p, (j> and F (cf. [35]) the relevant generalization of the integration by parts formula is EpPw [ V F M M ] = EpPw

F[u] Q f

^ [ w ] Mr)

- V log p[w](ft

(6.56) This is the rigorous version of Feynman's formula (4.34), although not in its explicit form. Instead of giving this one, let us illustrate a typical use of the directional derivative for the action functional (6.25) controlling our class of elementary systems: F[X] = Etj

L(X(r),DX(T))dT

+

EtSL(X(T),T),

where L{q, q) = &?+ V(q). Let us assume that the domain Dp of F is made up of diffusions, with measures absolutely continuous with respect to Pw, and solving stochastic differential equations like (5.12) for regular but arbitrary drifts. We say that X in DF is an extremal for F if E[VF[X]()] = 0 V<£ : Q0 -> H regular enough to preserve the absolute continuity under the shift X >—» X+. Then we compute, for t = 0 and a common fixed initial condition X(0) = x,

E0,x [VF[X](0] = £0,x £ (H<£ + -W5XD*) dT +

Eo,x[VSL(X(T),T)(T)].

Now take f(X(r), r) = g(X(r), r) -i/?(r) where ip is differentiable. Then, by (5.35), D(g-tp) = Dg-ip-\-g-^ and we can integrate by parts the second term under the integral sign (since Dcf> = £p(f> by hypothesis). Our final result is 0 = E0tX[VF[X}(
^-D^jdr Eo,x

f dL + VS L \~dDZ

)(X(T),T)-4>(T) (6.57)

6. Beyond Feynman's

approach

195

Since <j) is arbitrary, this means that an extremal of the action, for the above elementary L, almost surely solves 'DDX(T)=W(X{T)),

< X(0) = x

DX(T) =

0
(6.58)

-VSL{X(T),T)

namely (5.37) with its proper boundary conditions. All the qualitative results of Feynman mentioned in §4.2 (and many others not considered here) can be given a rigorous meaning using the stochastic calculus of variations sketched here. For instance, it yields a compact and elegant version of the stochastic Ncether's theorem. It is likely that a number of theoretical aspects of the path integral approach will appear more clearly in this context than through complicated manipulations of time discretized expressions. This remark illustrates well what can be realistically expected from Euclidean quantum mechanics. This approach is certainly not another reductionist attempt to claim that quantum theory is nothing more than a particularly obscure chapter of the theory of stochastic processes. All the possible mistakes along this line of thought seem to have been made already, from the claim of existence of probability measures which never existed to those founded on probabilistic interpretations of a few quantum equations (notably Schrodinger's) and to which the totality of quantum physics is arbitrarily reduced afterwards. As a research program, Euclidean quantum mechanics is both modest and ambitious. It is modest because it provides only an analogy with standard quantum theory. It is founded on the same conviction, implicit in [34], that any direct probabilistic approach would be frustrating or sterile. Almost seventy years of history have, indeed, confirmed Schrodinger's intuition. But the program is also ambitious since it seems to give the closest classical analogy with quantum theory, close enough to suggest new physical results. In particular, all the heuristic ideas of Feynman are preserved and often sharpened in this Euclidean context, as we have seen. It is also clear that the basic idea (see (5.31), (5.32)) of this probabilistic construction is independent of the form of the starting Hamiltonian H, although we limited ourselves here to the class considered by Feynman. We claim that it is always possible to replace the quantum problem by another one involving the construction of probability measures on path space, on

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the model summarized here. Of course, the underlying Bernstein processes are then considerably more general than our simple diffusions, and depend on the representation adopted. For instance, in the momentum representation (cf. §2) the relevant Bernstein processes are a class of time reversible diffusions with jumps, driven by Levy processes, as can be guessed from the form of the integro- (pseudo-) differential Hamiltonian H in this case, and expression (6.9) for D in terms of H (cf. [57]). A credible probabilistic counterpart of quantum theory should start from the relevant time-symmetric processes (or measures) associated with each representation and construct the probabilistic counterpart of the unitary equivalence between them in Hilbert space. New relations between the associated processes will show up, which are inaccessible to the traditional probabilistic analysis. It seems that we now have some evidence that this strategy may also be more than a sterile exercise for quantum physics. However, one may feel frustrated that Euclidean quantum mechanics does not help to answer the question: What is the true origin of probabilities in quantum theory? This frustration may be stronger for those who tend to regard theoretical physics as an ontological science. We could adopt the ironical viewpoint of Feynman claiming that "The analyses of such problems are, of course, in the nature of philosophical questions. They are not necessary for the further development of physics" [6, p. 23]. In any case, we expect that our probabilistic analogy will also be of some use for the foundations of quantum theory. Indeed, when all the terms of this analogy will have been completely understood, in the various representations available in this theory, we will be able to transfer concepts from stochastic analysis to quantum physics and back, far beyond a literal translation since the mathematical structures involved in each framework are quite distinct.

7.

Time for a Dialogue

In sections 5 and 6 we considered exclusively processes defined on R. So we did not need any of those beautiful optional stopping times described in Part 1. But it will be very interesting to introduce the proper concept of random time in this framework. Let us explain why, since this illustrates another aspect of what can be expected from Euclidean Quantum Mechanics, as well as the kind of difficulties we meet with when implementing this program. Let us come back, first, to quantum physics. As we said after (3.12),

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the lack of a time observable in quantum theory has been regarded, from the early days, as a fundamental weakness of the theory (cf. [34] for example). Here "observable" refers, of course, to von Neumann's Axiom 2.2. This is a rather unfortunate state of affairs, especially when relativistic considerations are needed. It is fair to say that the problem of the construction of a proper time observable is still open, since the creation of quantum theory. But this does not mean that theoretical physicists are not thinking about the issue. For a recent review see [36]. Generally, such attempts concentrate on the definition of a time of arrival of a quantum particle at a given location, and usually the looked for operator fails to exist, except, perhaps, for unbounded (unphysical) Hamiltonians. Reading any serious review on this problem one is convinced that if it has resisted so long it is because it is not of a purely technical nature. The problem is conceptual, not well posed mathematically, and its roots reach our fundamental notions of space and time. It is therefore quite unlikely that the solution of the puzzle will be found using exclusively the familiar tools of standard quantum mechanics, when physicists are still arguing about the very meaning of probability in this theory. Now, what is the situation in Euclidean Quantum Mechanics? There, at least we have well defined probability measures on path space to start with. They should guide us in our quest for time. Let us consider, for example, the Euclidean version of the one dimensional quantum plane wave il>(q,t) + e*(vq~1£t),

«eR\{0}.

(7.1)

rp represents the monochromatic plane wave of a mass one free particle traveling with constant phase velocity v and energy E = v2/2 to the right or the left according to the sign of v. Although ip solves the free timedependent Schrodinger equation, it is not a realistic physical state since it does not live in L2(W), but a superposition in v of such solutions indeed represents a general free state. Unrealistic as it is, (7.1) has always been used as a precious tool for physical intuition. The Euclidean counterpart of this plane wave is the pair T,(9)t) = e * ( » « - ^ ) , r,*(q,t)=e~^(vq~^t)

(7.2)

of positive solutions of the two adjoint heat equations (5.32) associated with

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this free quantum evolution. Notice that, for the purpose of comparison with (7.1), v should be thought of as a purely imaginary parameter, as the time parameter itself. By (5.31), the associated Bernstein diffusion X(t) has an unnormalizable probability density, since rj*r](q, t) dq = dq,\/t, but there is no problem with this. According to (5.33) and (5.40), the momentum and energy random variables of this process are, in fact, constant: B = B*=v,

v2 h = h, = -—+Eo,

(7.3)

as expected of a probabilistic reinterpretation of the quantum plane wave. Notice that we have added to — \ a positive constant EQ in (7.3), irrelevant to the dynamics, but large enough to ensure that the value of h coincides with a possible value (E > 0) of the energy of a free particle. This will provide us with the probabilistic reinterpretation of the quantum plane wave. Now let us consider, in the spirit of §1, Part 1, the following elementary problem. For z > q fixed, what is the earliest time Tz, if any, at which the process reaches z? The problem is only slightly more complicated than that of §1. It follows easily from Ito's formula that for any A € K, and when X(0) = 0,

M(t) = expl\X(t)-

(\v+ yftW}

(7.4)

is a 'Pt-martingale. Let us call E/h the constant Xv + ^-h. If we choose A > 0 large enough, this constant is positive. By Doob's optional stopping Theorem, discussed in §3 Part 1, one shows that E{e~%T>} = e-Xz .

(7.5)

The requirement A > 0 and our definition of ^ imply that A = i (y/v2 +2E-vY

(7.6)

When the velocity v is > 0, P{TZ < oo} = 1 and we can find the expectation of Tz by differentiating (7.5) with respect to the parameter E and subsequently setting E = 0. We obtain E{TZ} =

Z

-.

(7.7)

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However, if v < 0 there is a positive probability to obtain an infinite Tz. As a matter of fact, P{Tz
= e"£ zM .

(7.8)

When our Brownian motion with positive drift starts from x at time 0 instead of from 0, the expectation of Tz becomes the very classical looking EX{TZ} =

Z

-^, z>x. (7.9) v When v < 0 but z > x, our classical intuition about the free particle is also justified. Let us recall that, for simple Brownian motion, it was found (Example 2, §1, Part 1) that EX{TZ} = oo, z > x. This is a further indication that Brownian motion is totally devoid of dynamical interpretation in quantum theory. Its role is, as we said, purely kinematic and its relation with dynamics shows itself only in the drift of the process. 18 Finally, when our Brownian motion with positive drift starts from q at time t, it is easy to see that the generalization of (7.9) becomes Eq,t{Tz-t} = ^

.

(7.10)

wq{t)=Eq,t{Tz}-t.

(7.11)

Let us call wq(t) the left hand side, i.e.

Introducing the generator of our diffusion with constant drift (see (5.35)), we observe that wq (t) solves the elliptic boundary value problem (Dwq(t) = -1

qe)z,oo[,

\wz(t)=0. This problem is clearly the continuum limit and generalization of that solved by the random walk of Example 2 above, except for the boundary conditions given there for a compact interval. Notice that the form of equation (7.12) is still consistent with our interpretation of D as a time derivative, since its solution then has to represent a (random) time. Another interesting remark following from (7.11) is that EQtt{Tz} is a TVmartingale of the process X(t) = q. This is quite natural. The solution 18

This is also why, in the wording of the above example, a physicist would have difficulties surmising such a strange property of Brownian motion.

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of the classical counterpart of our problem, i.e. to find the (deterministic) time a classical free particle needs to reach the level z > q starting from q(t) = q with a positive velocity, is given by a "quadrature", known since Newton,

T , - * = r{T*)=zjL Jq(t)

=

V%e

i^m,

(7.13)

y/2e

where the value e of the conserved energy has been introduced. The momentum P(T),T > t, is constant, p(r) = \/2e, since the particle is free. Now let us interpret

r = + £

* ' ( 7i[)

(7 14)

-

as a classical, time dependent, observable n(q,p,t). Then it is straightforward to verify, using definition (1.8), that n = Tz is a classical constant of motion of the free system and the fact that, for our Euclidean plane wave, Eqtt[Tz] is a martingale can be regarded as its quantum deformation. Notice that the operator counterpart of this remark has indeed been used in [36]. Of course, because the probability measure of our underlying diffusion is trivial, it is not surprising, in Feynman's perspective, that result (7.10) is essentially the classical one (7.13). In fact, it is the very special nature of the pair of solutions (7.2), for which momentum and energy (7.3) are not random variables but real numbers, which turns this example into an almost classical one. Another way to see this is to note that the solution of the HamiltonJacobi-Bellman equation (6.27) relevant here (for V = 0, L = \ \q\ ) is 2

SL = —vq+ \ t + const., which coincides with the solution of the classical limit h = 0 of this equation. It is also for this reason that the underlying diffusion is defined for any t, a rather unusual feature for Bernstein processes. As noted in (5.48), for example, when a free diffusion results from a superposition of plane waves, it is generally defined locally in time. Moreover, its momentum and energy are genuine random variables and the above calculations become nontrivial. In particular, since such processes are, in general, inhomogeneous in time, most of the elementary examples of random times mentioned in textbooks of probability theory are physically irrelevant. Of course, there is a trick, in the theory of stochastic processes, to reduce inhomogeneous diffusions to homogeneous ones. This is the probabilistic

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counterpart of the introduction of an extended configuration space in classical mechanics, in which the time parameter is regarded as an extra spatial coordinate along which motion (to the right) is deterministic and with unit velocity [31]. But a number of precious analytic tools disappear along the way when the resulting degenerate process is used. The proper "analytic dual structure" needed for quantum dynamics does not seem to have been discovered, as yet, by probabilists. For Markov chains, this may have to do with the duality between the last exit from state i and the first entrance into state j [10, p. 93]. On the other hand, the Bernstein property (5.5) should provide extra structures that help to calculate some random times. For example, D. Slepian (quoted in [18]) explicitly worked out the first passage time (a stopping time) for the stationary Gaussian Bernstein (but not Markovian!) process defined by X(t) = W(t + 1) — W(t), where W{t) is the Brownian motion. So, often the tools needed for the development of our probabilistic counterpart of quantum theory are not directly available. One needs, first, to "symmetrize" the traditional constructions, too rigidly associated with the increasing filtration Vt (the past). Only afterwards does the structure fully exploiting the time-symmetry of the Markov property and Bernstein measures appear clearly. It is easy to do this for our Euclidean plane wave. The time reversed version of boundary value problem (7.12) is, for x < q, / ^ : w

=

1

(7.i5)

KW=o and its solution can be written as £«•'{*-Tx} =

v = «£(*),

q1

(7-16)

where the notation E ' reminds us that the conditioning is now in the future. Moreover, here Eq,t{Tx) is a ^-martingale of the diffusion. The dual structure we are looking for involves the solutions of the two boundary value problems (7.12) and (7.15). In this sense, this research program requires more than the straightforward application of standard probabilistic ideas. This is why it calls for a serious dialogue between theoretical (or mathematical) physicists and specialists in probability theory. Both communities

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should clearly benefit from it. Some specialists in probability theory should find it interesting to look for the above dual structure, often alluded to in their literature [21, for instance] in the light of Feynman's approach to quantum dynamics. If, as suggested by Euclidean quantum mechanics, this structure has a quantum analogue, such result could also be very important for the conceptual (and mathematical) foundations of quantum mechanics and their extensions in relativity theory. A closely related, more physical, formulation of the problem is as follows. Given the energy space-time observable (5.40) as well as its time reversal (5.42), how should we define general random times such that the counterpart of relation (5.25) holds? This would correspond, for time and energy, to Feynman's interpretation (4.36) of Heisenberg's commutation relations (3.16), which is missing in standard quantum theory. It is likely that such a random time observable would not be built on the standard model of (6.36). Given this time observable, we could reconsider the original problem in quantum mechanics and check if what happened for our stochastic Ncether theorem does not re-occur. Sure enough, the probabilistic content will be lost in such a "translation" but one could be led to introduce a quantum time, probably not an observable in von Neumman's sense, which could serve the physical purpose. What is indeed interesting about the above treatment of the plane wave is that it provides us with an elementary example of a Euclidean time observable, so natural that it is hard to believe that it means nothing to quantum theory. Again, the existence of well defined probability measures allows us to go beyond Feynman's path integral approach. Many ideas expressed in Part 1 should survive as well in our new context. As soon as the strong Markov property is available for the physically relevant random times, "incredibly rich and varied developments" should follow (cf. p. 34 of Part 1). A general concept of time observable would definitively convince quantum physicists that it is a mistake to ignore altogether probability theory and stochastic analysis. In the last 20 years, these fields have provided a huge amount of new tools (see for instance [15] and [46]). Handled properly, they are not only quite natural for quantum physics but should help to discover new conceptual results. Along the way, what has been done in [19] for the time-independent Schrodinger equation (2.26) should give precious indications for the general case. Indeed, although it makes no difference, in a stationary state, to interpret probabilistically (2.26) or the heat equation, the situation is com-

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pletely different for (2.27). Then, as we have seen, the only way to preserve very close relations with all aspects of quantum dynamics is to switch from (2.27) to a pair of associated adjoint heat equations, as in (5.32). It is in this context that pairs of random times relevant to a kind of time-dependent quantum potential theory, founded on the Hamiltonian H of (6.13) instead of the original one, should appear naturally. There is a thin line between sense and nonsense regarding what probability theory can bring to quantum physics and reciprocally. This explains, in part, the sulphurous reputation of this area of research, where an amazing number of fatal mistakes have already been made. It seems clear, seventy years after the birth of quantum mechanics, that the best we can hope for is an analogy. But the consequences of the proper analogy, still not completely explored, should have more good surprises in store. It is intriguing to see that, from the beginning, very few sound ideas have been available in this area. And that some of them, regarded for a while as independent, are not after all, and bring us back to other forms of the same old difficulties. This is not necessarily a sign of poverty. Given the number of constraints to be respected by a serious probabilistic counterpart of quantum theory, the existence of many independent and consistent perspectives would be a surprise. The one advocated here, for instance, is as close as possible to Feynman's path integral approach, although its origin is much older. In fact, most of the basic ideas used in these notes are, indeed, quite old. It is both remarkable and symptomatic that some of them are still not completely explored or understood, such a long time after their original publication. 19 The exciting issues of the foundations of quantum theory can be very sterile when they are not grounded in research programs providing new perspectives and new tools. Euclidean quantum mechanics allows us to reconsider some of the puzzles of the interpretation of standard quantum theory from a fresh viewpoint, but always with the proviso that we are referring only to a classical analogy. Actually, it is precisely as an analogy that 19 Schr6dinger's remarkable idea ([34]) which is the root of Euclidean quantum mechanics has been especially cursed. Although it was known for a very long time by specialists in the history of quantum mechanics (see [47]), its relevance to mathematical physics has been studied only since 1984-85 (see references in [28]). And it is still nowadays afflicted with bizarre misinterpretations. For example, a probabilist retired from the University of Zurich used it in recent years to rediscover many times Nelson's diffusions and to conclude, among other puzzling things, that Schrodinger's equation is a kind of Boltzmann equation! It is certainly not this type of surrealistic "dialogue" between communities that we are asking for.

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it is most useful in this respect, freeing us from the traditional prejudices against some of these puzzles. For example, this approach allows us to understand what is specifically quantum in the difficulties. At the very least, this analogy suggests that standard quantum mechanics is a mathematical framework organizing, in an optimal way, the results of micro-experiments which have already been done, and certainly not in progress. The adjective "optimal" here refers to the variational component of Schrodinger's original argument, which has been only alluded to here in §6.2 [35, 37]. Although our Euclidean counterpart is, a priori, the solution of some problem of classical statistical mechanics, the "simple" alteration of the "usual" boundary conditions conjures up qualitative properties which are regarded, erroneously, as the exclusive privileges of quantum physics. We share the view, expressed long ago [34] by Schrodinger, that this is not accidental, all the more so since the idea of time symmetry or asymmetry of quantum evolutions according to the choice of boundary conditions has reappeared in various other contexts throughout the years and has even been presented recently as a necessary condition of future progress [27]. On the other hand, it is clear that we have limited ourselves to the most elementary physical systems in these notes. Many of the puzzling aspects of quantum theory, regarding composite systems (quantum "entanglement", non-locality, . . . ) , for example, have hardly been mentioned. Other consistent perspectives on the role of probability in quantum mechanics (apart from that denying its existence) are possible. But not so many. An intriguing one uses the difficulties of Section 3 as motivation to renounce the classical axiomatic of probability theory (due to Kolmogorov) in order to construct a non-commutative version, regarded as the quantum one. For its relations with some of the ideas summarized here, see [38]. Although this approach has proved to be interesting for quantum open systems, it should be clearly said that as far as the elementary theory of systems in pure states (considered here) is concerned, it does not add anything new to the traditional viewpoint on quantum randomness. Whatever may be the final interpretation of Euclidean quantum mechanics in the future (if any), this program should be judged on the basis of the new physical results it allows to discover. This brings us to a question which goes far beyond what we said here. Given the present level of mathematical sophistication in the standard formulation of quantum theory in Hilbert space, is it more likely to guess qualitatively new theoretical results in this framework or in others, closely related but using very different conceptual and mathematical tools?

References

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For the time being, we will leave the reader, and especially the young and ambitious researcher, on this question.

References (The (*) indicate references already given in Part 1.) 1. E. Schrodinger, "Are there quantum jumps", The British Journal for the Philosophy of Science, Part II (November 1952). 2. W. Heisenberg, "The physical content of quantum kinematics and mechanics". Translation of Zeitschrift fiir Physik 4 3 (1927), 172-198 (Reprinted in [8]-) 3. S. Albeverio, "Wiener and Feynman path integrals and their applications", in Proceedings of Symposium in Applied Mathematics 52 (Providence, Rhode I.: AMS), p. 163. 4. G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman's Operational Calculus (Oxford Mathem., 2000). 5. R. Feynman, Review of Modern Physics 20 (1948), 367. 6. R. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (New York: McGraw-Hill, 1965). 7. P. A. M. Dirac, The Principles of Quantum Mechanics, Fourth Edition (Oxford) (First Edition: 1930). 8. J. A. Wheeler and W. H. Zurek (editors), Quantum Theory and Measurement (Princeton University Press, 1983). 9. V. A. Fock, Fundamentals of Quantum Mechanics, translated by E. Yankovski (1976) of the original Russian version (1931). 10. (*) Kai Lai Chung, Green, Brown, and probability & Brownian Motion on the line, 2nd ed. (World Scientific Publishing, 2001). 1st ed.: 1995. 11. R. P. Feynman, "The concept of probability in quantum mechanics", in Second Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, California: University of California Press, 1951), p. 533. 12. W. Pauli, Lectures on Physics Volume 6, Selected Topics in Field Quantization, Chapter 7, edited by C. P. Enz (Cambridge, Massachusetts: MIT Press, 1973). 13. Born-Einstein Letters (London: MacMillan Press, 1971). 14. K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, Second Edition, Probability and its Applications Series (Birkhauser, 1990). 15. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes (Amsterdam: North Holland, 1981). 16. R. H. Cameron, "A family of integrals serving to connect the Wiener and Feynman integrals", Journal of Mathematical Physics 39 (1960), 126. 17. S. Bernstein, "Sur les liaisons entre les grandeurs aleatoires", Verh. der Intern. Mathematikerkongr. Band 1 (Zurich, 1932). 18. B. Jamison, "Reciprocal processes", Z. Wahrsch. Gebiete 30 (1974), 65.

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19. Kai Lai Chung and Z. Zhao, From Brownian Motion to Schrodinger's Equation (Springer- Verlag, 1995). 20. E. Nelson, Quantum fluctuations, Princeton Series in Physics (Princeton, N. J.: Princeton Univ. Press, 1985). 21. (*) Kai Lai Chung and J. B. Walsh, "To reverse a Markov process", Acta Mathematica 123 (1969), 225. 22. A. Beurling, Annate of Mathematics (2) 72 (1) (1960), 189. 23. K. Ito, "Extension of Stochastic Integrals", K. Ito Edit., Kinokuniya, Tokyo, in Proc. of Intern. Symp. SDE (Kyoto: 1976), p. 95. 24. S. Albeverio, J Rezende and J. C. Zambrini, "Probability and quantum symmetries II. The Theorem of Ncether in quantum mechanics", to appear. 25. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton, N. J.: Princeton University Press, 1955). 26. M. Gell-Mann and J. B. Hartle, "Time symmetry and asymmetry in quantum mechanics and quantum cosmology", in Physical Origins of Time Asymmetry, edited by J. J. Halliwell, J. Perez Mercader and W. H. Zurek (Cambridge University Press, 1996). 27. R. Penrose, "Mathematical physics of the 20th and 21st centuries", in Mathematics, Frontiers and Perspectives 2000, edited by V. Arnold et al., (Providence: AMS, 2000). 28. S. Albeverio, K. Yasue, J. C. Zambrini, "Euclidean quantum mechanics: analytical approach", in Ann. Inst. Henri Poincare 50 (3) (1989), 259. 29. T. Kolsrud and J. C. Zambrini, "The general mathematical framework of Euclidean quantum mechanics: an outline", in Stochastic Analysis and Applications, edited by A. B. Cruzeiro and J. C. Zambrini, Progress in Probability Series Vol. 26 (1991), p. 123. 30. Aharonov Y., Bergmann P. G., Lebowitz J. L., Phys. Rev. 134 (B) (1964), 1410. 31. A. D. Wentzell, A course in the theory of Stochastic processes (McGraw-Hill, 1981). 32. W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions (Springer, 1993). 33. M. Thieullen and J. C. Zambrini, "Probability and quantum symmetries I. The Theorem of Ncether in Schrodinger's Euclidean Quantum Mechanics" Ann. Inst. H. Poincare (Phys. Th.) 67 (3) (1997), 297. 34. E. Schrodinger, "Sur la theorie relativiste de l'electron et Interpretation de la mecanique quantique", Ann. Inst. H. Poincare 2 (1932), 269. 35. A. B. Cruzeiro and J. C. Zambrini, "Malliavin Calculus and Euclidean Quantum Mechanics. I. Functional Calculus", Journal of Functional Analysis 96 (1) (1991), 62. 36. J. Oppenheim, B. Reznik and W. G. Unruh, "Time as an observable" in Quantum Future, Xth Max Born Symposium, Led. Notes in Physics 517 (Springer, 1999), p. 204. 37. A. B. Cruzeiro, W. Liming and J. C. Zambrini, "Bernstein Processes associated with a Markov process", in Stochastic Analysis and Mathematical Physics, edited by R. Rebolledo (Birkhauser, 2000).

References

207

38. R. Rebolledo, "Complete positivity and open quantum systems", in Stochastic Analysis and Mathematical Physics, edited by A. B. Cruzeiro and J. C. Zambrini, Birkhauser Progress in Probability Series, Vol. 50 (Boston: Birkhauser, 2001). 39. T. Kolsrud and J. C. Zambrini, J. Math. Phys. 3 3 (4) (1992), 1301. 40. S. Karlin and H. M. Taylor, A second course in Stochastic Processes, (New York: Academic Press, 1981). 41. M. Kac, in Proceedings of 2nd Berkeley Symposium on Probability and Statistics, edited by J. Neyman, (Berkeley: Univ. of California Press, 1951). 42. B. Simon, Functional Integration and Quantum Physics, (Academic Press, 1979). 43. M. Thieullen and J. C. Zambrini, "Symmetries in the stochastic calculus of variations", Probability Theory and Related Fields 107 (1997), 401. 44. L. S. Schulman, Techniques and Applications of Path Integration, (New York: John Wiley & Sons, 1981). 45. A. Connes, A. Lichnerowicz, M. P. Schiitzenberger, Triangie de Pensee, Edited by O. Jacob (2000). 46. P. Malliavin, Stochastic Analysis, Grund. der Math. Wiss. 313 (Springer, 1997). 47. M. Jammer, The Philosophy of Quantum Mechanics, (New York: McGrawHill, 1974). 48. P. Holland, The Quantum Theory of Motion, (Cambridge Univ. Press, 1993). 49. A. A. Kirillov, Geometric Quantization in Dynamical Systems IV, edited by V. I. Arnol'd and S. P. Novikov, Encyclopedia of Mathematical Sciences 4 (New York: Springer-Verlag, 1990). 50. A. Cannas da Silva, A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes Series (AMS, 1999). 51. P. Cattiaux, C. Leonard, "Large deviations and Nelson processes" Forum Math. 7 (1995), 95. 52. A. B. Cruzeiro, J.C. Zambrini, "Ornstein-Uhlenbeck processes as Bernstein processes" in Barcelona Seminar on Stochastic Analysis, D. Nualart and M. Sanz Sole Ed., ProgTess in Prob. Series 32 (1991) 40. 53. E. Carlen, Comm. Math. Phys. 94 (1984) 293. 54. P. Lescot, J. C. Zambrini, C. R. Acad. Sci. Paris, Ser I 335 (2002) 263. 55. N. Wiener, "Differential Space" J. Math. Phys. 2 (1923) 131. 56. N. Wiener, A. Siegel, Phys. Rev. 91 (1953) 1551. 57. N. Privault, J. C. Zambrini, Markovian Bridges and Reversible Diffusion Processes with Jumps, to appear. 58. N. Gisin, "Sundays in a Quantum Engineer's life", in Proceedings of Conference in Commemoration of John Bell, Vienna, 10-14 Nov. 2000. Published in Quantum [Unjspeakable, pp 199-208 (Ed. R. A. Bertlmann and A. Zeilinger, Springer 2002). Also at h t t p : / / a r x i v . o r g / a b s / q u a n t - p h / 0 1 0 4 1 4 0 . Nota bene: Some references given in t h e text are not repeated here.

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Index

almost surely, 6, 37, 38, 47, 63 amplitude (probability), 124

quantum, 105, 186, 188-190 convergence in any Lp, 38, 41 in probability, 38, 39 with probability one, 6, 37, 38, 47, 63 cylinder sets, 29, 142

Bernoullian random walk, 4, 6, 15, 21, 22, 35, 37, 48, 52, 57, 148 Bernstein process (diffusions), 145, 150-154, 157, 159, 161, 163, 164, 167, 171, 174, 176, 178, 180-182, 185, 198, 201 Borel tribe, 13, 108, 145, 154, 191 Born interpretation, 96, 107, 117, 129, 142, 168, 187 Euclidean, 155, 157 Brownian bridge, 128, 150, 152, 156, 159, 168 Brownian motion, 6, 16, 22, 63, 64, 66, 128, 140, 148-150, 161, 162, 166-168, 177, 182, 185, 188-190

determinism, 82, 90, 103, 201 diffusion coefficient, 141, 142, 148, 150, 156, 159, 162, 168, 177 Dirac's correspondence principle, 104, 115, 188 Doob's optional stopping Theorem, 20, 198 Ehrenfest equations, 105, 130, 159 energy-time commutation relations, 202 expectation conditional, 19, 30, 34, 146, 150, 187 of an observable, 94, 99, 100, 108, 110

Canonical commutation relations classical, 88 quantum, 115, 202 Chapman-Kolmogorov equation, 30, 73, 144, 145 collapse, 106, 117, 138, 163 conditional expectation, 19, 30, 34, 146, 150, 187 configuration space, 69, 86, 123, 171, 172 extended, 187, 201 constant of motion classical, 88, 90

Feynman's (complex valued) diffusions, 187 Feynman's commutation relation, 153 Feynman-Kac, 167, 168, 189 filtration (past and future), 11, 144, 148, 151, 159, 179, 201 first entrance (or hitting) time, 3, 8, 209

210

16, 48, 65, 201 decomposition formula, 52 first integral classical, 88, 90 quantum, 105, 186, 188-190 flow, 89-91, 134 Hamilton-Jacobi equation, 122-124, 180 Hamilton-Jacobi-Bellman equation, 180, 181, 200 Heisenberg equation, 104, 188 picture (of quantum dynamics), 104, 105 uncertainty principle, 113, 118, 138 history Feynman's sense of, 132, 142 of an observable, 92, 114, 115, 137 homogeneous temporally, 29, 31, 46, 50, 66, 200 incompatible observables, 115, 116 integration by parts formula, 129, 190, 192-194 interference, 114, 140, 164 Ito calculus, 192 Ito calculus, 140, 159, 176, 177, 181, 187 formula, 158, 177, 186 forward stochastic differential, 148, 177 integral (forward and backward), 177-179 stochastic differential equations, 148, 192 joint probability distribution, 29, 62, 67, 108, 109, 111, 154, 155 last exit decomposition formula, 54 time, 48, 53, 201 law of large numbers

Index strong, 13, 37, 39, 47, 49 weak, 26, 38, 39 limit in any Lp, 38, 41 limit in probability, 38, 39, 179 marginal distributions, 108, 154, 155 Markov process, 13, 16, 30, 31, 36, 45-47, 49, 52, 57, 59, 63, 64, 80, 128, 129, 142, 150, 155-157, 201 homogeneous, 29, 32, 46, 51, 64 inhomogeneous, 31, 156 reverse, 50, 57 property, 30, 31, 34, 46, 49, 50, 53, 68, 70, 72, 146, 147, 151 symmetry (temporal), 51, 201 strong property, 31, 34, 50, 202 martingale, 13, 18-27, 30, 69, 151, 173, 174, 177, 182, 183, 186, 188, 189, 199-201 exponential, 185 Newton's time, 92 non-recurrent (or transient) state, 56 Ncether's Theorem, 172, 181, 188, 202 observable, 79, 87-89, 91-94, 96, 98-100, 104, 106, 108, 109, 111, 116, 117, 130, 131, 170, 173, 202 as random variable, 117 classical, 104 energy, 113 expectation of, 94, 100, 108, 110 history, 114, 137 Ncetherian, 188, 189 space-time, 159, 160, 171-174, 176, 186, 187, 189, 202 standard deviation, 100, 108, 110 time, 79, 112, 197, 202 variance, 100 one-slit experiment, 135, 139, 161 process, 162, 163, 168

Index optional time, 13, 16, 19-22, 27, 29, 31, 32, 36, 41, 46-49, 80, 196 path right continuous, 45 sample, 44, 61-63, 65-67, 69, 74, 75, 142, 148, 153, 162 path integral, 80-82, 123-125, 127, 130, 132, 134, 135, 139, 140, 142, 143, 157, 166-168, 171, 174, 180, 185, 203 phase space, 30, 36, 86-88, 97, 120, 121, 187 extended, 90, 112, 121, 134, 175 Poisson bracket, 88, 89, 115 Poisson process, 44, 45, 75 position-momentum commutation relations, 115, 116, 130 probabilistic interpretation of the state ip, 96, 97, 117, 142 probability amplitude, 124 projection postulate, 106, 117, 139 propagator (or probability amplitude), 124, 129, 132, 140, 147 quantization, 93, 100, 104, 105, 109, 115, 116, 118, 123, 140, 175, 187 random walk, 3, 5-8, 11, 13, 33, 34, 49, 57, 58, 85 Bernoullian, 4, 6, 15, 21, 22, 35, 37, 48, 52, 57, 148 recurrent state, 35, 37, 56 reduction (of wave packet), 106, 117, 138, 139, 163, 170 reversible (time), 91, 157 right continuous paths, 45 sample function (path), 44, 61-63, 65-67, 69, 74, 75, 142, 148, 153, 162 sample space, 12, 61, 108, 109, 117 Schrodinger picture (of quantum dynamics), 67, 104, 105 time-dependent equation, 103, 107, 124

211 time-independent equation, 101, 202 sojourn time, 65, 67, 68, 73 St. Petersburg game, 25-27, 33 state bound, 101 pure, 86, 87, 89, 93, 117 stochastic analysis, 191, 192, 202 stochastic calculus of variations, 192, 195 stochastic differential equations, 148, 156, 176, 191, 192 forward, 148, 177 Stratonovich (or symmetric) integral, 178, 179 strong Markov property, 31, 34, 50, 202 symmetry group, 172, 174, 175, 181, 188 time observable, 79, 112, 197, 202 time reversal, 50, 91, 106, 107, 122, 123, 146, 151, 152, 158, 161, 162, 164, 170, 202 invariance under, 107, 122, 127, 147, 152, 170 transition element, 126-129, 131, 132, 134, 135, 155, 160, 178 transition matrix of a chain, 51, 53, 63, 64, 66 tribe, 12, 144 Borel, 13, 108, 145, 154 future, 28, 29, 174 past, 28, 174 Wiener integral, 192 Wiener process (see also Brownian motion), 190 Wiener space, 190, 191

Introduction To Random Time and Quantum Randomness This book is made up of two essays on the role of time in probability and quantum physics. In the first one, K L Chung explains why, in his view, probability theory starts where random time appears. This idea is illustrated in various probability schemes and the deep impact of those random times on the theory of the stochastic process is shown. In the second essay J-C Zambrini shows why quantum physics is not a regular probabilistic theory, but also why stochastic analysis provides new tools for analyzing further the meaning of Feynman's path integral approach and a number of foundational issues of quantum physics far beyond what is generally considered. The role of the time parameter, in this theory, is critically re-examined and a fresh way to approach the long-standing problem of the quantum time observable is suggested.

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ISBN 981-238-388-3

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