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= < P i ( x n ) * P.(xk) P i ( x n )> > O, (*i,<('j) > O. A little elementary geometry shows that (3.27) and (3.28) yield the following
Proposition 3.1; If the constant 5 is smaller than some constant 6 , m (of purely geometric origin) then the m vectors {i|i .}•_•• are linearly independent and dim It > m. This shows that, for 6 < S , the state < - > is a m mixture (convex combination) of at least m pure phases. The Peierls argument supplies a systematic way of estimating
167
Since
m I P (x) = I, for all x, *=1 * m
(I
.
V
z n
)
z^x,y
= I < TT C
z e zvn A
(3.30)
C(z) (z)) (x,y)
Given A, . , we define a family r, . of connected x (x,y) (x,y) surfaces Y in D (=all faces of unit cubes in C ) with the properties that "- (x,y)
' "
(2) Y decomposes A,
0
(x,y)
. into precisely two
disjoint subsets B (Y) ^> D
and B
(Y) ? •
M3-31) YJ
An element of r, . is called a contour. (x,y) If Y is a contour in r,
. w e define N ( Y ) to be the
(x,y) set of all nearest neighbor cubes in £
with one common
face in Y • Recalling that in (3.30) i?^j, we conclude that, given a configuration C, there exists a contour Y (C) £ r (x,y) with the properties that
168
(1) for all (D Dz
c
Bx
, D z ,)eN(Y(C)) with (Y(C))
C(z) = i,
C(z') / i > (3.32)
(2) There exists a connected set B C B such that D n
z
C B
C'
x
and
c B , and C(z) = ±, c
3B
C
D
Y(C)
(Y (C) )
for all
-
(See also Fig. 3 below)
We may now derive an upper bound for the r.h.s. of (3.30)
( P i ( x ) P (y)> = I ( c
Y € r
(x,y)
{C:
I Y e r
e (r x , y ) ,
L
*(C)
I C ( Z
TT' z£ ZvnA ( x y ) r
">^ c(z')jt±
, <
=
Y}
(z)
:(z)
pc(z) («);
TT
zezvnA
^
(x,y)
/ TT \ (nz.Dz,)6N(Y)
p
i
( z ) p
c(z')
( 2 , )
>
(3.33)
16
For the proof of (3.33) we apply the definition of y(C), use the fact that {P^ (z) :z e 1V n A
, i=1 ,
is a family of commuting,
,m} positive, selfadjoint
operators with 0 < P (z) < I, and
finally recall that
A B <; A 1 B' if A,B, A' and B' are commuting, positive bounded operators with 0 < A < A' and 0 < B < B'. Let|y | denote the total surface of y, Theorem 3.2: (see
(i.e. the number of faces in
y).
[GJS2])
Assume that, for all A, . such as defined above and (x,y) all contours Y e r , ,, and with (x,y) C(z) = i ± c ( z ' ) , f o r a l l
/
X
D z # ) e N( Y )
p±(Z) p C ( 2 ' ) ( 2 , ) >
n (Dz,
(• ^
K Y i e" I I
(3.34)
Dz,)eN(Y)
for some constant K > K (v,m,e,6), where K(v,m,e,6) is a fixed constant only depending on the dimension v of space, the number m of projections and two positive numbers e and 6 (see (3.23) and (3.24)). Then
(P^X)
P.(y)^
For K > K(v,m,e,6
< 6 e 2 , for all j?£i.
(3.35)
) the state < - > is a mixture of
170
at least m pure phases; (see Proposition 3.1).
Remarks: For the purposes of these lectures it suffices to prove Theorem 3.1 for the case v = m = 2 considered in [GJS2]. We remark however that the analysis of the general case is perhaps a little more than an academic exercise: It is important in the analysis of multidimensional systems which are expected to have tripleor m-tuple points (i.e., for certain choices of the parameters of the system,' it is expected to have at least three, or m, pure phases). We also note that method 3.1
(infrared domination)
generally only proves existence of phase transitions without
proving more than obvious information (derived
from the structure of the internal symmetries) about the manifold of pure phases. This is not so in method 3.2 (the Peierls argument), as one learns from Proposition 3.1 and Theorem 3.2. Sometimes it is necessary to combine (among others) both methods to get the desired information on the structure of the pure phases of a system. Proof of Theorem 3.2 (for \>=m=2) ; For the purpose of good intuition we write P
=
P + , P 2 = P_ ("spin up-down"). 2 We now pick two arbitrary points x and y in % . It is then to be proved that (3.34) implies (3.35). The family C
is now a family of unit squares, the set 1Q
171
of their faces a family of bonds. Let R , , be the (x,y) smallest rectangle of unit squares containing the squares • x and • y', and let 3R (x,y) be the length of its boundary, (i.e. the number of bonds in 3R, , ) . (x,y)' Let now A vx,y; . . be an arbitrary rectangle of unit squares with the property that all contours Y G r , , , (x,y)' where r.vx ,y;. has been defined in (3.31), which have distance 0 from 3A, . (i.e. touch 3A, .) have at least length
3R (x,y)
There are two adjacent faces f.,,f9 of R. (of i ^ ix ,y) total length -^ |3R,„ „,|) such that each contour y£V , , c ^ ix,y) (x,y) contains at least one bond b.(y) (a "bond of entrance into R, ,") contained in the interior of R, . that ix,y) (x,y) touches f u f but none of the other two faces of 3 R , l 1 (x,y) Figure 3 below shows two sites x and y, the corresponding rectangles R, . (the shaded region) and A, 3 (x,y) (x,y) the faces f11 ,f and three contours in r, . with some ^ ix,y) bonds of entrance f«
2 A
JL. •i
1
l
--
-, r -
—j r
""
--
:;
W£?&Y*3 P r
Rr.
*>y)
(x,y)
ti
3
W— -~MW? s z £ ^ T^=: i'
i — . —i
ii
—J| 1
'i i1
*ir-
, 7 ^ 11
: [ — n_ i!
2
f?
3
Fig.3
172
There are TT |3R, ^
> | different bonds of entrance
ix tY)
for the contours in r, ,. This follows from the de(x,y) finition (3.31) of r. . and the definition of the class ix ,y) of bonds of entrance. For the class of contours in r, > \X r Y /
of length n there are no more than min {n, ^ |3R (x y ) I>
(3.36)
different bonds of entrance, a consequence of (3.31), the definitions of R, , and of the bonds of entrance. (x,y) Using now the definitions of 1, . and r , . we con^ (x,y) (x,y) elude from (3.36) that, for fixed n, there are at most n different bonds of entrance into R, ., for the class (x,y) of all contours in r, . of length n. (x,y) ^ If we now choose the bond of entrance as the first bond of a contour in r, . and apply a standard (x,y) ^ x argument we conclude that, for fixed n, there are at most n 3
n-1
(3.37)
contours y€ T.vx ,y; . with |y| = n. Finally we note that the length of a closed contour Y € r,
. is even. For closed contours (3.37)
may obviously be replaced by n 3
.
(3.38) (3.37')
173
Lemma 3.3 Let K be the constant introduced in Theorem 3.2, (3.34) , and assume that K > log 3. Then (p (x) P (y))
I ( TT N( "^ (xfy) °z'nz'> r
P+(z) P (z')\ " '
oo
V i < 2 2 n n=2
,2n-2 3
-K-2n e
,0 _., (3.39)
°
Proof: The first inequality in Lemma 3.3 is inequality (3.33). Now I Yer
(
TT
P.(Z)
P_(Z'))
(xfy) HD z ,a 2 ,)eN( Y )
= I n
I
TT
P + ( Z ) p_(z-)}
^ r (x,y) ; lY|=nl \ ( D z , D Z ,) £ N( Y )
{
Y n + ^ r (x,y) : |closed } ^ ( D z , d z , )eN (Y)
= I n
/
{
£ || =
/
TT
P (z)P_(z')}
+ E
/ (A
(x,y)>'
where E(A, . ) represents the sum over all terms (x,y)' ^ labelled by contours Y£T, , that are not closed. But, (x,y) by assumption (3.34) and (3.37), E(/\. . )*0, ix, y) 2 v / as A, fL (x,y)^/ "
174
The proof now follows immediately from (3.37), (3.38) and the assumed inequality (3.34) . Q.E.D. Remark: For v>2 and m arbitrary (3.39) is replaced by (PI(X)P
(y))
<
I Cn(m-1)2n e"K"2n, n=v
(3.39')
where C is the number of closed contours in r, xr -, and-, C „ ^-e const.n surface 2n, n
. of J
It is obvious that, as K/"°°, the r.h.s. of (3.39) and (3.39') tend to 0. This completes the proof of Theorem 3.2. _. „ p, (J .
CJ
.L)
.
Summary of Section 3; We have proven that the existence of a phase transition in a system described by (0i,<->),
in the sense
that <-> is a mixture of at least two pure phases (dim
SL > 2) follows from oo
—
3.1 inequalities (3.16) and (3.19), o£ 3.2 inequalities (3.23) and (3.34), and that information on the number of pure phases can be obtained from combining 3.1 and 3.2 with the structure of the internal symmetries.
175
In Sections 4-8 we prove these inequalities for various model systems (Models 1-6), thus establishing the results announced in Section 2.
Section 4: INFRARED (GAUSSIAN) DOMINATION AND THERMODYNAMIC LIMIT In this section we prove inequalities (3.16) and (3.16') in the form of Theorem B, (2.23), for the correlation functions of the classical, ferromagnetic systems introduced in Section 2, (2 .8)- (2 .11) , and for a class of Euclidean field theory models. These estimates, in particular Theorem B are basic for (1) taking the thermodynamic limit A / 2 , (2) applying method 3.1 to proving the occurence of phase transitions, see [FSS]; (3) the proof of the basic inequality (3.34), Theorem 3.2, in method 3.2 (the Peierls argument). The inequality we are going to prove asserts that the correlation functions of such systems are dominated by Gaussian correlations with covariance (= two point _2 function) in momentum space ~0(k ) . It is related to and motivated by the grad *- bounds of canonical quantum field theory, [GJ2, He].
176
4 .0 The 9 1 a- (grad (ft-) bounds: Theorem B Let H ({a}) (Hamilton function) and dp ({a}) (Gibbs state) be as defined in (2 .8) , (2.10) . We pick some direction % , hence forth called Tdirection (the direction of transfer). Without loss of generality this may be the 1-direction. The hyper plane in Z v perpendicular to T and passing through the origin is denoted T x . Let the space cutoff A be a rectangle of the form A.= [-L,L] x A. (4.1) W l th
Kx = [-L2,L2] x....x [-Lv,Lv]c
J T
The point a = (Jlj,...., L .+1 ,...., X. ) is identified with a = (l 1 , . . . . , - Z , . , . . . . , 2. ) ,
(4.2)
(corresponding to periodic boundary conditions). Let <->, = <->.' A
denote expectation in dp,. c
A
A
Theorem 4.1: (H Theorem B, [FSS]) For v=1,2,...., N=1,2,.... and all single spin
dis-
tributions dX in the class K, defined in (2.20) r
u
/
\
">%•
T
^ „ hl „ ( a ) - 3 aa) ^ , AJ (e a.'*
1
<
e
2JTfa
E
|h„(a) 2
'..1 . ' H
,
(4.3)
17
independently of dX ,h, and A. The proof of Theorem 3.1 is in three steps. The basic tool is 4.1 The Transfer-Matrix formalism, [T] Without loss of generality we may first assume that dX(a) = g (a) d N o,
(4.4)
for some non-negative function g of integral 1 and compact support X. The general case follows later by a limiting argument. We set a=(i,l), i e T . I e T * , and define x.'i = o - l{a, ( i^,TI1 )) 'T i, e,A l clearly x. £ $
-=
x
W© eA
'1
a,. T , ; °
(4.5)
, with p = N•|A x |; we write
«1 = X X l A l l , G(x ) = 1
(4.6)
T T 9 ( a ( 1 x)) I£Aj
U , J J
x exp[-f
I
eh
I
a(i
'I}
(3* a
)2] ,
(4.7)
and define F(x) = G ( x ) 2 . In these new notations
Acta Physica Austriaca, Suppl. XV
(4.8)
178
Z A duN{a}) = JT A A . ' 1=-L
G(x.) e " J / 2 1
(x
i X i+1 }
dP x. 1
(4.9)
Let T denote the operator on L («x) = L (fi^d? x) defined by the kernel K(x,x')= F(x) e" (J/ ^ (x " X ' )2 F(x')
(4.10)
T is called the transfer matrix. For the purposes of taking the thermodynamic limit and proving the existence of phase transitions we must consider perturbed transfer matrices. Let F 1 ( x ) , F_(x) be two functions in L (fti) and a some vector in $ p . We define T->-(F.,,F_) to be the operator on L
2
(ill)
with kernel K+
( F
^>
(x,x') - e
^
Fl(x-) e C j / ^ x - x , ) y • ( - - ' ) x F 2 (x')
->-. a .2
= Fl(x) and T(F X ) = T> ( F ^ F ^ .
- 1 e ^ / ^— 'V 'T j '
F2(x),
(4.11) (4.12)
Lemma 4.2: (1) T is a positive, selfadjoint, trace-class, 2 positivity improving operator on L (ft..); T+
(2)
{F1,F2)
is trace-class.
| | T | I is an eigenvalue of T of multiplicity 1, and the corresponding eigenvector <j> can be chosen to be non-negative with
I I
= 1 .
Proof; (1) W e first show that T->- (F 1 ,F 2 ) is trace-class, for all a e $
P
, F
W
F
2
in hz{Q1)
. Since
T = T-v (F,F) = T(F) ,
this also yields that T is trace-class. For this purpose w e must show that
T r ( | T j ( F l f F 2 ) |) a.
is finite, where
|Tj(F!,F 2 )| = / r ^ ( F l f F 2 ) * Tj(Fx,F2) is the absolute value of T-»- (F 1 ,F 2 ) . 3.
Let {h } _
be some complete, orthonormal system
for L 2 ( Q x ) . Then on
Tr(|Tj(F lf F 2 )|) = a
I (h_,lT£(F lr F 2 )|h ) n a n n=o
(4.1
180
We now estimate each term in the series on the r.h.s. of (4.13): Let \j> and cj> be arbitrary vectors in Z,2 (nl)
. Then
(i|),T?(F1,F2)
P
P
2
x d x' (F^) (x) e
= |(§2-,P/2
d
Pk
(Fl<) ,(£,e
J
2J
J
e *2
< | (2]L)P/2 dHt| (Fn(.) (k) [2
x |(§1)P
(F2*)<£) /
2J ,V2 it 2
/2
(F24>) (x1)
JdPk|(F2*)(£)|2 e
V2
2J
T
(4.14)
(by the Schwarz inequality) = (<|,,T(Fl)i|>)1/2
(t, T(F2)
(4.15)
Polar decomposition tells us that Tj(F!,F2) = 0 T + ( F l f F 2 ) | , a.
(4.16)
a
for some unitary operator U on L
2
{til).
Combining (4.15) with (4.16) we obtain l U j T j (P 1# F 2 ) |*) | = |(U
Cl
<(<(1,TU(F1)4>)1/2 (^,T(F2)ii;)'l/2,
(4.17)
where T (F^ = U* T(F L ) U
(4.18)
181
Thus (h n r |Tj(F 1 # F 2 )|h n ) < (Uh n , T(F X ) U h n ) 1 / 2 x (h n , T(F 2 ) h n ) 1 / 2 , and, by the Schwarz inequality for sequences and (4.13), Tr(|T^(F 1 ,F 2 )|) < [ I (Uh , T(Fi) Uh )] 1/2 n=o oo
x [ I (hn,T(F2) h n ) ] 1 / 2 n=o
(4.19)
Since U is unitary and {h } is a complete orthonoon=o normal system, so is {Uh }"_ . Thus, the r.h.s. of (4.19) is equal to {TrCTtFj))} 1/2 (Tr(T(F 2 ))} 1 / 2
(4.20)
Next, we remark, that T(F..) is positive: From the definition (4.12) it is obvious that T(F 1 ) is symmetric, and positivity follows from the fact that 2 e is a function of positive type. If A is a traceclass operator on L (fij) with a kernel C(x,x') that is bounded and jointly continuous in x and x' then Tr A = ldP x
C(x,x)
(4.21)
From (4.21) and a limiting argument we now conclude that, for i=1,2,
182
Tr (T(F..)) = ]dPx |F ± (x)| 2 = I IF-L I I f
(4.22)
Thus we obtain from (4.19), (4.20) and (4.22) that T r l l ^ l F ^ F j ) |) < | |Fi| | 2 • | |F 2 | | 2 < ». Since T = T(F), T is positive and trace-class. This completes the proof of (1). Part (2) follows directly from the (generalized) PerronFroboenius theorem. Q.E.D. Let H T ,....,H T
be arbitrary functions on fli with
the property that H. G is integrable, for all j , where G is defined in (4.7), and set F. s (H.G) 1 ^ 2 Let ot=(i,I) label the points of A, and set a, = {h(i,I)} T _.
=
©
h(i,I),
(4.23)
where h(i,I) €. $ N , for all (i,I)e A. Finally <->. denotes expectation with respect to du ; see (4.9) .
Lemma 4.3: ("Feynman-Kac formula") T
-Z
((TTLH., .«»
h(a)3 : a
) \
183
J
= e
j
J
(4.24)
Tr (T 2 L + 1 )
Proof; If we integrate (4.9) and use (4.10) and (4.21) we obtain Z A = Tr (T 2 L + 1 )
(4.25)
Writing out the numerator on the r.h.s. of (4.24) explicitly in terms of integrals, using the fact that L JT T-»- (F.,F. 1 ) is trace-class, by Lemma 4.2, (1) and j=-L a j ^ D+l applying (4.21), next writing out the l.h.s. of (4.24) explicitly, using (4.9), finally applying (4.25) we immediately see that (4.24) holds. Q.E.D. Remark: If we combine Lemmata 4.2, (2) and 4.3 we see that the cutoff can always be removed in one direction ; (L -»•«•) . We also note that ||T|| - lim (Tr
(T 2L+1 )} 1/2L + 1
(4.26)
L-+- 00
These observations are sometimes useful in the analysis of two dimensional systems, as is well known.
184
4.2 A trace inequality The following is one of the most basic estimates of Section 4. Lemma 4.4: (see [FSSj) L |Tr ( TT
Tj
(F ,F
)) | 1
<
TT {Tr(T(F.) 1 i=-L
2L+1
)}
2L+1
(4.27)
Proof: By HSlder's inequality for traces; see e.g. [ D S ] L |Tr ( JT
Tj
(F
L < "JT (Tr (|T+ a j=-L j
F,
)) |
(F ,F ^ 3+1
1 2L+1 ) |2L+I)}
Lemma 4.4 therefore follows from the inequality Tr(|T+ ( F r F 2 ) | 2 L + 1 ) < (Tr ( T ^ ) 2 ^ 1 ) } 1 ' 2
{Tr(T(F2)|2L+1}1/2
Let A = | T + (F!,F 2 )I, B = Ty ( F L ) ; see (4.18), and C = T (F2) Using the definition of the trace, see (4.13), inequality (4.17) and the Schwarz inequality for sequences
we get Tr (A 2 L + 1 ) < {Tr ( A L B A L ) } 1 / 2 {Tr ( A L C A L ) } 1 / 2 Using again Holder's inequality we obtain {Tr (A L BA L )} 1 / 2 < { T r ( B 2 L + 1 ) } 2 ( 2 L + 1 ) { T r ( A
2 L + 1
)^
and similarly for {Tr(A L CA L } 1 / 2 . Hence 2L 0 < T r ( A 2 L + 1 ) < { T r ( A 2 L + 1 ) } 2 L + 1 {Tr(B 2 L + 1 ) } 2 ( 2 L + 1 : ,„2L+1.,2(2L+1) x Im {Tr(C ) } 2L 2L+1
2L+1
If we divide this inequality by { T r ( A )} and st raise it then to the (2L+1) power we arrive at Tr(A 2 L + 1 ) < {Tr(B 2 L + 1 )} 1 /2
{Tr(C2L+1}}1/2
= {Tr(Tu{F1)2L+1)}1/2{Tr(T(F2)2L+1)}1/2 = {TrCTCF^2^1)}1/2 {Tr(T(F2)2L+1)}1/2, and we have used the unitary invariance of the trace. Q.E.D.
4.3 "Nelson's symmetry" We now assume that the functions H T ,....,H T introduced above, see Lemma 4.3, are of the form
186
H.(x.) = TT l l x£ h
f.. T . (a ,. T. (i,D (i,I)
x exp[ I I h (i,I) 3*of. z . ] IeA1 1=2 for some non-negative functions f,. T> on in.
(4.28)
compatible
\ i 11)
with the requirement that H.G be integrable, for all =
1
—
J-1/....fJj.
We want to estimate
x
a£A
=
/( TT x
i =-T
i=-L
a£A X,=1
H ) exp[- I x
~^ A
a€A
h (a) 3 1 a ] ) a
'
(4.29)
'
By Lemma 4.3 this is equal to L
if J A I V ° H 2
Tr(
.TT
T
(F F
a\
j'
3=~L 3 Tr (T 2 L + 1 )
J
j+ i>>
(4.30)
which is non-negative. From Lemma 4.4 we know that (4.30) is bounded above by
jj
J A I.V-H 2
Lj.
Tr(T(F.) 2 L + 1 ) ^ 2 L + 1
(4.31)
i=-L Tr(T2L+1) Since all directions in Z v (parallel to a face of A)
187
can be chosen as directions of transfer, we may choose next the 2-direction as our new direction of transfer. A minute of reflection which involves rewriting Lemma 4.3 and (4.31) in terms of transfer matrices in the 2-direction shows that
Tr(T2L+1) is precisely of the form of the l.h.s. of (4.27) (with transfer matrices in the 2-direction) and can therefore be estimated by using Lemma 4.4; (see also [FSS]; in Euclidean field theory this observation is called "Nelson's symmetry", since Nelson first applied it there. See [Si]/ [Gu]) . Repeating these observations another (v-2) times we arrive at Lemma 4.5;
(.from [FSS] U [FS] )
If we define a.
=
A
log Z,, and, for f a non-negative
lAl
A
function on ft™, ft ,
V » " 777 109 tz
(4.32)
188
( I T f o (a o ) exp[- I a£ A aeA
1_
I
< e2J
aeA
I h£(a)a\]) J «.= 1
v
I l\(a)| 2
*=1
I [«A(fa)-«A] eaeA
(4.33)
This beautiful inequality is a more general form of Theorem 4.1, so that we have now completed the proof of the latter. Following [ F S ] , we will later refer to (4.33) as "chess board estimate". Of course this estimate extends to complex functions f (a ) if we take on the a a l.h.s. of (4.33) the absolute value and, on the r.h.s. of (4.33), replace f
by If I.
4.4 The termodynamic limit Next we want to apply the chessboard estimate (4.33) to prove existence of the thermodynamic limit. For this purpose we first derive bounds on the "pressures" a
A
and a (f) x. . . .xu |~-L ,L 1 we write L 1,L.l For A = r-L 1 1J v v o.(f) A
= a TL ..,L (f) T lf v
Lemma 4.6: (1) Let f be any non-negative function. Then a.(f) is monotone decreasing in A, in particular a. (f)
< a,
,
(f)
189
where a
1
(f) = log ]f(a) e
dx (a)
l #•••-/ i
(2) a.(f)
,~, - ,~v _ (f) (2 v -1 ) o,
U > 2,v a0
A
Z/..../Z
-(f),
I;•••*/ I
for L. > 2, all i=1, i -
Proof: We set H(x. ) = 1
TT
f (a,. Tx ) G(x ) , (x,1>
I£A
1
in (4.7), and F(x) = /H(x). where G is as ir We then note that a,(f) = —
1 „ 2L +1) 2L +1 log (Tr(T(F) )}
a consequence of (4.25) and (4.3 2) Using3 the definition of the trace (with {h } all the n n=o orthogonal eigenvectors of the positive operator T(F)) and the fact that x p > 1 , we see that
p
is concave on {x > o } , for all
{Tr (T(F)P) } p
(4.34)
is monotone decreasing in p. Since the logarithm is monotone, and |A | is independent of L 1 , a (f) is monotone decreasing in 1 v L 1 . Part (1) now follows from Nelson's symmetry.
190
Proof of (2): By Holder's inequality for the trace 2L1 + 1 2L-| 0 < Tr(T(F) 2 ) < {Tr(T(F)2L1
1
} 2 L l + 1 {Tr(T(F)2L1+1)}2Ll+1
By taking logarithms, dividing by |Ai| and using (4.34) we get
T
L..,....,L
(f) > 2 <x0 T T (f) -a, T T (f) 2,L„,..-.,L (jL-z-.-./L v (4.35)
From Lemma 4 . 6 , (1) we h a v e a
i
T
' 2
T
Repeating «T Jj w
( f )
-
a
i
1
now ( 4 . 3 5 )
T
. . . . t1-1
( f }
T
•
I, i , . . . . ,ii
' • • • • ' v
we g e t
(f) > 4 a , _ —
• ^ »
z
/
T J
-
,
- j ' ' " " * '
> 8 a, , ,
i
(f)-3a
T, -
(f) > i * rlj2
v
' ' ' ' ' '
\i
(f) - 4 a , , ,
T V
f
»
(f) r
f
y
- 3 a
> 8 a, , , £t
I £t j & I m m •
(f)
(f) - 7 a , f ±J
and we h a v e u s e d
\ f \ f \ f m t m f
(4.34)
again,
L V
(f)
> 2Va-
(f)-(2 V -1) ai
.(f)
This completes the proof of (2). Q.E.D. The following inequality is sometimes more convenient (see Section 6 ) : Suppose that, for all L.. , . . . . ,L aT
(f) < 5(f)
T
1
V
Then aT
T
(f) > 2Va„
Estimating a
(f) - (2V-1)o(f)
(4.36)
_ (f) for our lattice systems is a
straightforward task which we leave to the reader. An immediate consequence of Lemma 4.6 is Corollary 4.7; If
f(0) e h '°dx(o) > 0
lim a.(f) = in f a.(f) = a A A7>ZV A A
(f) exists.
We are now prepared to extend Theorem 4.1 and Lemma 4.5 to arbitrary single spin distributions in the class: ,~ . , K, = {dX: £j e > o such that e eI'a 'I e h'tr dx(a) < <» }
192
and then to prove the for the
existence of a thermodynamic limit
correlation functions.
We recall that, so far, Theorem 4.1 and Lemma 4.5 have only been proven for single spin distributions dX (a) = g (a) d a , where g is some non-negative, measurable function of integral 1 and compact support. The extension of our estimates to K
is a straight-
forward consequence of the chessboard estimate (4.33), Lemma 4.6 and Corollary 4.7: Let f a (a) = e g ( a ) " a « for some function g on A with values in % . Let B denote the support of g; (B c A ) . Then the chessborad estimate (4.33) yields I J
(exp [ I g(a)-aj} a£A
< e
0 A (expg(a) -a)-«AJ
a6B
(4.37)
By Lemma 4.6 the r.h.s. of (4.37) is bounded by ex
P [I
(ai
.(expgUl-oJ^V
so t h a t we o b t a i n , u s i n g Lemma 4 . 6 ,
(exp[ I a€ A
g(«) -a ] }
J
<
0 +(2
(1)
v
-1)ai
,
193
e g(a)-a e h.a
^^-j
a€B
x
[ [eh'° dX(a)] ^ - 1 ) | B | . e - 2 > | a 2
2
(4>38)
The following two observations are important: 1. The r.h.s. of (4.38) is independent of A; 2. for |g(a) | < e, a € A, the r.h.s. of (4.38) is uniformly bounded on the class of all single spin distributions dX for which elcl h* a ~ , . , e ' 'e dx(a) < const. Next, let dX (a) be some fixed single spin distribution in K, . h Then there exists some e >o such that CO
( e \a \ h * a ~ , . I e »' ' e dX^ (a) < «° Using now (4.38) it is seen by a simple limiting argument that there exists a sequence {g } _
of non-negative,
measurable functions of compact support (and integral 1) such that, for N dX n (a) = g n (a) d a, and for all functions g(a) with |g(a)[<e
, for all
a e A, the expectations of exp[rg(a) *o ] in the Gibbs
Acta Physica Austriaca, Suppl. XV
13
194
measures corresponding to (A,{dX }) converge to the one of exp [Eg (a)'a ] in the Gibbs measure corresponding to a
(A/dX^). Moreover the pressures introduced in (4.32) corresponding to (A,{dX }) converge to the ones corresponding to (A,dX ), Therefore Theorem 4.1 and Lemma 4.5 immediately extend to any expectation <-> corresponding to a single spin distribution dX^ in K, . We may now pass to the thermodynamic limit: From the above considerations, (4.37) and Corollary 4.7 we conclude that, for g some function of compact support B bounded in modulus by e and single spin distribution dX ,
lim sup
En
/exp[ Z
g(a) 'a ] \
[a (exp g ( a ) ' a ) -
a 1
< e
(4.39)
From definition (4.32), Holder's inequality and Corollary 4.7 we conclude that a
(exp g-a) is convex in g.
Therefore the r.h.s. of (4.39) is finite and continuous on the class of all functions g on 1 of rapid decrease bounded in modulus by e / 2 . If we combine this with (4.38) and apply a standard compactness argument (Cantor's diagonal procedure) we obtain
195
Theorem 4.8: There exists a sequence {A } _
of rectangles in-
creasing to 5?v such that, for all dX £ K , dp ({a}) = lim dp ({a}) exists, n-v«> n in the sense that all the moments of the measures ^ n = 0 (i-e- the correlation functions) converge to n h the moments of some probability measure, denoted dp , on E . {dp
Theorem 4.1 and the chessboard estimate, Lemma 4.5, hold,
mutatis mutandis ,
<-> '
determined by dp . We note that if <->
for the expectation <->
=
is a pure phase state (if not
decompose it into pure phase states, [Fo]) then / §g(a)aa\ J (e / where a^ = a
1/2J E g(a) [(-A)"1g](a) e
-
(4.40) /a \
, and A is
the finite difference
Laplacean; (4.40) follows directly from Theorem 4.1; see [FSS]. It justifies the term: "Gaussian domination".
Theorem 4.9: (see Theorem C; [FSS])
<( I g £ ( - ) 3 1 a a ) 2 ) J < 1 I a,I
a,I
\giM\2
196
Proof: This form of Theorem 4.9 is proven in [FSSj. If we replace — by -% ^ then Theorem 4.9 follows directly from Theorem 4.1:
1/2j|Rez|2 ^ J g ^ U ) |2
. Z E„ g„(a)3* a , a (e a '* * ) J| <
Z Efc gfc(«)3 a Q and the fact that / e a ' " 1 is an entire function of Z, by the Cauchy estimate Q.E.D. Theorem B, Section 2, is an immediate consequence of Theorem 4.9 and (3.16), (3.17). The final issue of this section is to discuss the dependence of all our estimates on the lattice spacing 6 If the underlying cubic lattice of the system has lattice constant 5 we make the following substitutions: We redefine 3 , 3
V = "4 [°a "a*]? a I
a., I
•+ I
Sv,
see
(2.7), (2.16)
at all places
^ K.
(4.41)
a, i
As 6 ^ 0 ,
I
<5V
- * £ d
V
a
We leave it to the reader to check that if we take into account (4.41) all our estimates, Theorem 4.1 and 4.9,
197
and Lemmata 4.5, (4.6) are uniform in 6. These estimates form the technical core of the remainder of Part I and make the following sections reasonably short.
4.5 The decomposition into pure phases (This subsection may be skipped) The final issue of Section 4 is to establish the connection between the general framework introduced in Section 3 and the models considered in Sections 2 and 4, v Given a finite set B e ! , let I bra of Borel sets on TT $ . . . aeB
denote the o-alge-
An interesting consequence of the chessboard estimate Lemma 4.5 (in a slightly more general form), and Lemma 4.6 is the following Theorem: For the measures constructed in Theorem 4.9 dy / L , the restriction of du to £_ ,is absolutely conB B tinuous with respect to TT • dX(a ) (4.42) aeB An analogue of this theorem for the P (<(>)„ models in the continuum limit 6=0, [E,Si], has been obtained in [FS]. The case considered here follows from the same arguments if one takes into account Lemmata 4.5 and 4.6. This theorem permits us to define (X(B) = L~( X %**,., a€B ^a>
TT cU(a )) a a£B
(4.43)
198
The algebra Ot is then defined as in Section 3. Given a measure dp of the type constructed in Theorem 4.9, we may then define 01 (~B) and Ot as in Section 3, (3.1). (In this case the G.N.S. Hilbert space 5L is = L
(X ac Z V to be the smallest o-algebra
h
iH i \ »^/dp )) .
We define ^
such that all functions in 01
are I -measurable. Then OO
dy
is the restriction of dy
CO
to E . Let X be some con-
venient model for the support of dp . We now recall a general result concerning the decomposition jsition of a state <-> cof the type constructed in Theorem 4.9 into pure phases,
Theorem 4.10: [Fo] For all A£(X J U =
x d " - ( X > (A)
X
and, for dy -almost all x £ X , <->
is a pure phase state. A
The measures corresponding to different pure phase states are mutually singular. If dim
X
= m there exist m different pure phase
states, (a consequence of the definition of Ot
and (3.2))
Remark: For the interested reader we remark that, for dy OO
almost all x £ X , the states <->
and <->
satisfy the
199
same "Dobrushin-Lanford-Ruelle equations"; [R,GRSj. Theorem 4.10 has been extended to the continuum limit (6=0; ffv -v flV) , i.e. the case of (Nelson-Symanzik positive) Euclidean field theory. More precisely, let du be some probability measure defined on the a-algebra generated by the Borel cylinder sets of $
'
, ( )H )
real which satisfies Osterwalder-Schrader positivity
[OS] in
the form of [F5] and the moments of which are Euclidean Green's functions (Wightman functions restricted to the Euclidean points) of some relativistic quantum field theory. Such a measure is called a quantum measure, [F5]. Let Z
be the smallest a-algebra such that all Tt
functions in du T = du|
(see (3.11) — (3.12)) are Z -measurable,
. Let X be a convenient model for the support
of du .
Theorem 4 . 1Q' :
[F5]
For all A££ u(A) =
du T (x) U V (A) , , A
1
X
where, for du-almost all xeX, u
is a Euclidean in-
variant quantum measure ergodic under the action of the translation group. Different "pure phase" quantum measures are mutually singular. Theorems 4.10 and 4.10' will be used freely in the subsequent sections without explicit reference. The statement "there are n <_ °° pure phases" is to be understood in the sense of the integral decompositions of
94 200
Theorems 4.10 and 4.10', with the precision that dim n, (dim d. dl. =n) , i.e. the support of dy^ (du ) contains n points.
Section 5 : PHASE TRANSITIONS AND SPONTANEOUS SYMMETRY BREAKING FOR THE CLASSICAL N-VECTOR MODELS (MODELS 1-3,6) In this section we combine method 3.1
(Infrared
Domination, see inequalities (3.16) and (3.19)) with Theorems 4.1 and 4.9 to derive the existence of phase transitions for some class of the states <->
construc-
ted in Theorem 4.8, in particular for Models 1-3 with v > 3 and Model 6 introduced in Section 2. Theorem 5.1
(= Theorem D, [FSS])
Let v > 3 and N = 1,2,3,....,. Assume that d\ is an 0(N) invariant single spin distribution in the class K defined in (2.20), and d\(a) f
6 (a)dNa.
Let the Hamilton function be given by HA =
v J I J a -a i ; see (2.10). aeA 1=1 +
Then there exists some finite J
> 0 such that, for
201
J > J
the state <->
(constructed in Theorem 4.8) is not
a pure phase state; (see 4.5). The long range order a
,
(Section 3 ) , is strictly positive. N-1 There are at least S many pure phases, and for a set of pure phases of non-vanishing ^-measure (a x
o /\
X
^ 0, (i.e. <-> is "spontaneously magnetized"). X
Proof: We prove this theorem for the special case where there exists some <5>0 such that X({a: |a|<6})
0
(5.1)
The general case is proven in [FSS] by the same methods but involves some more detailed estimates on the pressure a (J) = lim
1 i'i log Z,, as a function of J
We apply the strategy of Section 3.1 For the observable A we choose a . o Then (5.1) gives
<^>
(5.2)
= <
Let dai (k) be the Fourier transform of
/ a a > . From \ o a' Theorem 4.9 we obtain, (see also Theorem C, Section 2 and (3.17)) doi(k) = [y60(k) + F C (k)]d V k, where y = I (o ) I + a , and 'No'1 a0 C 0 < F (k) < ^T [2v - 2 { I cos k * } ] - 1 Jl=1
> I
(5.3)
202
Integration of (5.3) and use of (5.2) gives ,1-1 V 6 2 < y +. I 7f J R d.v,.^ k [ 2 v - 2„ r{ ^E ___ c o s k, . 1 }] J B £=1 1 = Y + const. —
(5.4)
Here B is the first Brillouin zone and the constant on the r.h.s. of (5.4) is finite, provided v >_ 3; (for v=2 the integral diverges logarithmically at k=0). Next we note that Theorem 4.9, since
^
CT
(a ) = 0 . This follows fr om J = 0, for all finite A. 0)
Thus, using (5.3) and (5.4) 2 > 6
a o
1 - const. —
—
(5.5)
j
0
This is obviously positive for large enough J. We may now apply Theorem 4.101 In a pure phase there is n£ long range order. Thus / \ J \ o o /
/ \ o
a
\ J,T o'
o du
i.e.
( ){ (a -a ) J - | (a > J | 2 } <» AX \o o/ x ' \ o/ x du (x) I ( o )
for J > J"
| = a
> 0, Q.E.D
20
Theorem 5.2: F Let \) = N = 4 and let H„, dX, X and o be as xn A Section 2, Model 6; see (2.9), (2.19). Let <->a be the corresponding state constructed in Theorem 4.8. e.g. Then, for J = 1 and each fixed X>0, there exists some finite o (X) such that, for a>a (X), <-> c c ' a pure phase state.
is not
Proof For the observalbe A (method 3.1) we may e.g. choose
Let da) (k) be the Fourier transform of / A A \ \ o a/
a
.
Then Theorem 4.9 (= Theorem C) adapted to this 3 , , i \,S.»o shows that v v I g (a) 3 a + I g (a) 3 A , a a l 4=1 l
model:
dw(k) = |>6o(k) + F C (k)]d 4 k, where y = |<^A°) a | 2 + « A o ' 0 < F C (k) < 1 [3- f ~ J L 4=1
cos
and
>
i,-1 ^]
J
As in the proof of Theorem 5.1 we show that
By i n t e g r a t i n g
(5.6) we t h e r e f o r e
find
(5.6)
98 204
< ( A o ) 2 ) ° = a A°
+ C
(5
'
- 7)
where the constant C is independent of X and a; and J = 1. Next we note that
by Nelson's symmetry. Let a (J,X,a) be the pressure of the system:
-H*f({A}) a (J,A,a) = lim A / Z"
|A| ,-*-
x
T~T 1
' aeA
log I e > ->
-X (A -A a a e
. 2
(5.9) -+
-*•
+aA -A ua
. d A
01,4,
a
From Jensen's inequality and the chess board estimate (Lemma 4.5 and Theorem 4.8) we get -2a (A -A )
° °
-2aA -A
i /e
a
° °\
a (1 ,X,-a)-a (1 ,X,a) (5.10) From (5.9) and (2.9) we get (1) a
(J,A,-1) < a (0,0,-1)
(2) a
(J,X,1)
> Const. X
, as X ^ 0,
uniformly in J £ [0,1] . ,1 ^r,1) A n ,+ 1 „ (3) a„ (1,A,a) = ara(^,
205
Combining
a
(1,X,-o)
(1)
-
-
a
CO
(3)
we g e t
(for
a>1)
(1,X,a) 0°
< aoo(0,0,-1) -a„
(1, -^,1)
< - Const, a 2 , as a / <*>
(5.11)
Taking the logarithm of (5.10) and using (5.11) we conclude : (A-A)>0(a),asa;"°° x o o' Theorem 5.2 now follows from (5.7), (5.8) and
(5.12) (5.12). Q.E.D.
Remark: Of course, (5.12) agrees with the prediction of the naive Goldstone picture, [Co]. A physically more interesting result of the type of Theorem 5.2 is discussed
in [F3] . Theorem 5.3: ( [FSS]) Let v>3, N=1 and dX some Borel probability, measure on % with the properties that 1 . dX has no symmetries 2. dX£ /O K, . h
h
3. There exist positive numbers e and 6 such that X ( (—,-6]) > e, X ([6,«) ) > e.
206
Let HB ({a}) =
v 7 I { I a aeA 1=1
a . + h a }; a
(see (2.10)) .
Then there exists some finite J„ such that, for all J > J ,
(a
\
'
is discontinuous in h at some h=h
(J), for at least one h
.,
(J), and there is a phase
., (J)) which is not accompanied c crit. by the spontaneous breaking of any symmetry. transition at (J, h
Proof: We prove Theorem 5.3 for the special case, where X([-6,5]) =-- 0
(5.13)
For the general case see [FSS] . Cleary estimates (5.3) and (5.4) apply to this case, thus using (5.13) and (5.4),
<52 1
(ao)h'J
=
I ( a o) h ' J ! 2
+
%
+ COnst
-
J
o As a consequence of the FKG inequalities the state <-> ' may always be constructed in such a way that = 0; see [FS,FSS] .
a o Thus I<
a 0
) h ' J | - / «2-const. 1
(5.14)
Given some positive e < 6, there exists therefore a finite Jr,(e') such that
207
\/a 1
) h,J \ o'
I > e', for all J > J_(e) - C
(5.15)
Using now hypotheses 2. and 3. we see by a very simple argument that lim
h/>-
(a X
\
°
h
' J > o, lim
h^ —
(a X
\h'J < 0
(5.16)
o/
h J Combination of (5.15) and (5.16) proves that \/a )\ ' has at least one discontinuity in h. Thus , for some h = h . , (J) , there exist at least two differentr pure crxt. phase states.
By hypothesis 1. this phase transition is not accompanied by spontaneous symmetry breaking. Q.E.D. Remark: Theorem 5.3 also holds in v=2 dimensions. The proof is then based on the Peierls argument. See Section 8. Remarks: The results proven in Theorems 5.1 and 5.3 are both very satisfactory and insufficient: They are very satisfactory, because they show that in v > 3 dimensions spontaneous breaking of continuous internal symmetries does occur and that phase transitions are not always accompanied by spontaneous symmetry breaking, (i.e. the concept of phase transitions is more fundamental than and independent of the one of symmetry breaking. This can also be shown in 2 dimensions; see Section 8 ) .
102 208
They are insufficient, because no more than trivial information, derived from the structure of the internal symmetries, about the manifold of pure phases has been achieved. By choosing suitable single spin distributions dX essentially any manifold of pure phases compatible with the internal symmetries can be achieved. This weakness of method 3.1 can often be cured by combining it with the Peierls argument, (method 3.2); see Section 8. The relevance of results of the form of Theorem 5.2 must be discussed elsewhere.
Section 6: THE (£•!') | - QUANTUM FIELD MODEL
In this section we prove the existence of a phase transition and the spontaneous breaking of the internal 0(N)-symmetry for the (<)>•<(>)? - quantum field model (4> = ($ *,. . . . ,
103 209
existence and Wightman axioms for the {§'$)':
- models.
These results are very deep and difficult to prove, so that all proofs will be omitted. First we recall the definition of these models, in the case of a positive lattice constant <5. The single spin distribution is given by _L ( ) dX(
(6.1) 2
2
L U ) = \(
(6.2)
3 Let A be a rectangle in Z. of the form on [-1^,1^] x [-L2,L2] x [-L 3 ,L 3 ], where L. is an arbitrary, fixed positive number (e.g. an integer) independent of n, for i = 1,2,3. We redefine |A| : |A| = 8 L 1 L 2 L 3
(6.3)
The finite volume "Gibbs measure" is then given by 3 ,q n/t\ n /, ?. r \~1 2 xcA duA((()) = Z A (A,a,h,6 n ) e
n £=1
y
x (6.4)
"xiA
6
Acta Physica Austriaca, Supp). XV
n [LI(
104 210
-C 2 U,a,6 n )
[Ll L 2 + L 1 L 3
+
L 2 L3] (b . 4)
where 3 is defined as in (2.16) and Z(X,a,h,6 ) is chosen such that dp is a probability measure.
Theorem 6.1 F o r e a c h t i x e d , b o u n d e d r e c t a n g l e A and a p r o p e r c h o i c e of
{M2U,6
specified
in
du A
n
) }°° , {C.(A,a,<5 }°° , i = n=o 1 n n=o
1,2,
[ G J 1 , P a , Se S i ]
(*) = l i m du A
(*)
n-t-o°
exists, in the sense that all the moments and the characteristic functionals of the measures {du,} A n=o converge, as n->-°°. Remarks: A convenient model for the measure space on which dy)'a is defined is 1 ' , ( ^ 3 ) x N equipped with the A y reaJ. 0-algebra generated by all Borel cylinder sets; points in this space are denoted <(>(.), and the "observables" (see Section 3) may be built up from the functions
, iJ(?)
{e
~% r
0.
,(fi)3.xN
: f €. ^ r e a l (V/l )
-% , ,
, supp f c. A}
Theorem 6.1 is due to Park; see [Pa] and refs. given there. In his proof Park uses the fundamental techniques and results of [GJ1] and of [Fe].
211
We let <-> ' a denote the expectation determined by dy
,a
. We then define the finite volume pressure (vacuum
energy density) . = - ? - log
aAU,a)
where
: <> j • c(S :
": (x)
(x«)
= w-lim
a : 4 -6 : ( x . ) , X , o ) A
( e
=
I,
d x:f-i|:(x),
(6.5)
and
{ 4 ( x ) • <> f (X+E ) -
I e I -+o
};
(6.6)
4 IT I e I
The precise meaning of (6.6) is explained in [FSS] Theorem 6.2: (1) For N = 1,2,3 and arbitrary, fixed X > o, a > o and all h ^ o . \.a.Z,i, ,. dy (4) = lim
e
-|A|a (X.a) a:*-*:(xA) Xp l t , e dy ' (4)
A^fft3
A
exists, in the sense specified in Theorem 6.1. There exists a sequence {h } converging to 0 such that ^ n n=o ^ ^ for all unit vectors e e %~ , X,a,e,->. ,. , X,a,h e ,-*-. dy (
'»3 "
a AA (X,a) exists,
106 212
(for all h, and, for h = o, is independent of e ) . (2) The moments of the measures constructed in part (1) are the Euclidean Green's functions of some unique relativistic quantum field theory satisfying all Wightman-Osterwalder-Schrader axioms. Remarks: In the form stated here part (1) has been proven in [F6]. The difficult portions of the proof are however due to [MS] and [FeO 1,2]. The basic results of these refs. are combined in [F6] with the Lee-Yang theorems of [SG, DN] (or, for N=1,2, with correlation inequalities of [GRS, DN]) to derive (1). For N=1 independent proofs have been given in [Pa,Fe 0 2 ] . In [MS] and [Fe 01] it is also shown that, for a=o, {M2(X,6 ) } _ can be chosen such that d\x ' ' is independent of e and continuous in X in a weak sense, A. O
6
even at X=o+; the weak limit of {dp ' ' }, as U o , is the Gaussian measure du field.
describing a free, massless (6.7)
For h^o the physical mass of the theory constructed in Theorem 6.2, (1) is strictly positive; see [F6] and
[Fe 02] .
6.2 Phase transitions and Goldstone bosons Let <-> ' ' , , X,a ,e to dp
denote the expectation with respect
Theorem 6.3: (JI Theorem E , [FSs] ) For N=1,2,3 and each X>o, there exists some finite a (X) such t h a t , for a>a ( X ) , ( <j> (o) )
= M e,
with M * f 0; the internal 0(N) symmetry of the Lagran,CT
gean is spontaneously broken by <-> exist N-1 Goldstone bosons.
' , and there
Proof: From
[F6 ] a n d
(:?-|:(o))"'°
[ F S S ] we know
that
= lim { < J (o) 4 ( x ) ) | x | -»-o
A
'°
5— } 4 IT | x |
(6.8)
exists and is f i n i t e , for all finite a. Next
<*(o).*(x>>
X
>°'*
= (M*)2 + — ! — (2TT)
3
f d \ e i k x F C (k) >
a consequence of Theorem 4.9 (see also (3.17)) and Theorem 6.2, ( 2 ) ; (in particular, the cluster property , a',)e .. of.c <-> X ' Theorem 4.9 (in the limit 6=o, see (4.41)) gives
214
O < ( k X ) 2 F C (k) < N, for all i=1,2,3. From this one can conclude that 0 < F C (k) < ^ k
, (see [FSS] )
(6.9)
(This can also be concluded from (6.8) and the KallenLehmann representation) Thus , \\ (o)X),a ,e t•
,,,*, 2 ), +1 [ ,3,dr_,kC [F ,, ,(k) N --i- J = (M 1 T 3 k (2w) '
X
< (M*) 2
(6.10) / : <j> • <j> : (o) \
It now suffices to show that
positive, for a sufficiently large. Then M since <-> ' ' - * • - * •
->
' '
is
> o and,
is invariant under the substitution
-+
e y\ <j> -»• -e/\
(t(o))
X
' a ' e = M* e.
Let Q be the characteristic function of some unit cube. By translation invariance -y
^ :<j) -<j>: ( o ) y
-*•
=
^:
Jensen's inequality and the chessboard estimate (Lemma 4.5, Theorem 4.8) give -a
(:|"4:(D) ) X ' a ' e
-a:M:(Q) <
(e
A,a,e ;
a (X,o)-a (A .a) CO
<
e
*
OO
'
215
Hence (:•••:(•>>
X#a
->' e > aJl.oJ-aJHrO) ,
(6.11)
and it suffices now to prove positivity of aoo(A,a) - a
a:|.|:(XB) A ,o <^e )
>
(6>12)
for all bounded cubes B cz y/[ . By scaling lengths, X and a it is easily seen that it now suffices to show that, for fixed a>o and some bounded cube B,
a:$•£:(x R ). X,o (e
\
> 1
(6.13)
for sufficiently small X. By the chess board estimate -a :<|> • : (XTJI
^/O
[B| a (X,-a)
< e
(6.14)
and the r.h.s. is uniformly bounded in X € [o,X ] , for each finite X ; by
[GJ1J,
From [MS,Fe 01] we know that < [_:<j> - : (x B ) J
)
converges to
110 216
([:£-$: (x B )J m } 0 , as X 'H o, for all m < «. Here<->
is the Euclidean (Gaussian) expectation of a
free massless field; see (6.7). Finally
M I m=o
2m ^
„ < [:?•?: (xB> ] 2 m >
(6.15)
Q
diverges to +°° , as M->-°°, for large enough | B | , as o n e varifies by a n e x p l i c i t calculation. Combining these facts w e conclude that / a :<(. -<|) : (x B ) V A ,o
(e
>
which diverges to +°°, as X \ o , by (6.14) and (6.15). This proves (6.13); (more details can be found in [FSS]) The existence of N-1 Goldstone bosons, (massless one particle states coupled to the physical vacuum by the field e A $ ) , follows essentially from the Goldstone theorem in the form of
[ESw]• Q.E.D.
Remarks: (1) For N>3 the occurence of phase transitions can still be proven (by almost identical arguments), but Lorentz covariance of the theory is unknown. (2) For N=2,3 and o>o (.\),
the physical mass of the
217
theory tends to 0, as h ^ 0, linearly in h; see [F6 , LP] . (3) It is shown in [DN] by means of correlation inequalities that
<J(e4) (o) (e-f) (x))
X
' ° ' e = 0(( ((e/\+) (o) • /"*" t\
i \ \ X ,a (eA*) (x)^
as
,e.2.
) ) ,
I x I -*-» .
Section 7: PHASE TRANSITIONS IN TWO DIMENSIONAL QUANTUM FIELD MODELS In this section we consider the P(<|>) -, [E,Si], X cos(e<j>)2-f
[F1 , FSe] , and the pseudoscalar Yukawa.,
quantum field models, [ Y ? ] . We prove that, for certain values of the bare couplings, these models have phase transitions which are sometimes, but not necessarily, accompanied by the breaking of an internal symmetry transforming <j> into -<|>. We consider the P(<|))2 models in some detail, whereas for X cos (e<|>) 2
(Model 5) and the pseudo scalar
Yukawa model we just state the results; see [F3] for detailed proofs. Our methods are inspired by [GJS2] and [FSSJ .
218
We assume that the reader is somewhat familiar with the P(ij))
Euclidean field theory; see [E,Si]. We
just recall the m o s t important definitions: Let A = [-L ,L ]J x L[-L ,L J1 c 1' 1 2'"2
)2
where L 1 , L _ are positive integers. The free field Gaussian measure o n & ' , (VA. ) r ea -L with m e a n 0 , bare m a s s = 1 and periodic boundary c o n ditions at 3 A is denoted d u , , and <->. is the expectation determined by dy . T h e colons :-: denote Wick ordering of polynomials in the (pseudo-)scalar Euclidean field <(> w i t h respect to bare mass = 1,(i.e. w i t h respect to <-> s < - > „ 2 ; [ s i ] ) . If Q is some polynomial and B c ill some bounded rectangle then :Q(
B :Q(*):(x)d
2
x
(7.1)
The Euclidean action of the P (((>)- theory is given by U A (P) E L : P U ) : (x) d 2 x,
(7.2)
where P is some positive polynomial. We define dji*(*) H { (e
-U (P) _1 -U (P) A } °} e A du°(*),
(7.3)
(the cutoff interacting m e a s u r e ; see [E,Si]), p
r^ « A Ch) =
'
i / "UA(P)+h*(A)\ o log <^e J A,
(7.4)
219
and, for fixed, positive R,
.*(...) , - i - l o g
(.
»
2
>;
(7.5)
A
(the finite volume vacuum energy densities; [GRS]) It is known that, for a. as in (7.4) , (7.5) , a^ =
1
^mm
a
A
exists, see [GRS] ,- a (h) is a convex function of h and hence it is continuously differentiable in h except possibly at countably many values of h. We now state without proof the basic result concerning existence of the infinite volume limit. Theorem 7.1: P (1) If a (h) is continuously differentiable in h at some point h then r o , P+h n x ,,. ,. dy ° (cj>) = lxm
Ml
2
, P+h n x ,,. dp ° (
exists, in the sense of Section 6, (and the limiting measure is independent of "classical boundary conditions"; see [FS] ) . (2) &VP+ (<{,) = lim dv. P+hX h* o
(•)
exists, and P P P d u ^ = dv iff a (h) is continuously differentiable at h = o.
114 220
(3) The moments of the infinite volume measures constructed in (1) and (2) are the Euclidean Green's functions of a unique, relativistic quantum field theory satisfying all the axioms of Wightman and Osterwalder-Schrader. Remarks: In this form Theorem 7.1 is due to [ F S ] . However the results and methods of this ref. are based in an essential way on earlier results; see [E,SiJ and refs. given there. Theorem 7.1 not only asserts existence of the infinite volume limit, but it also shows that, in a very strong sense (see (1) and [ F S ] ) , phase transitions do P not occur if a (h) is continuously differentiable in h at h=o (see (2) , (3)) . We now want to prove that, for certain choices of P A
p
-i
A
p
+ da^(h) (<=>
is discontinuous at h=o
(7.6)
dh <=> there is a phase transition). Our proof is based on the Peierls argument in the form of [GJS2]; see Section 3, method 3.2. Some knowledge of Section 3.2 is henceforth assumed.
221
Theorem 7.2: For an arbitrary, even, positive polynomial R and a fixed a e (1 ,2 TT 2 ) there exists a positive X
such that,
for P(x) = XR(x)- -| x 2 , and 0 < X < X , there is a phase transition, in the sense of (7.6) , and p the states (expectations) < - > + determined by d)i+ break the <)>->—$ symmetry of P. Remarks 1. Using scaling and re-Wick ordering it is seen that this theorem extends to polynomials P(x) = XR(x) , X>X .,_. , (a=o) ; see LfGJS2] . J critical, 2. Theorem 7.2 is due to [GJS2]. A simplified proof and a generalization to non-even P's, as well as extensions to A cos (e<j>)„ and pseudo-scalar Yukawa. are presented in [F3] . Proof: Cover %
with a grid of mesh 1. Let •
be some
unit square of this grid. According to method 3.2 we must define m
mutually orthogonal projections on
T = L2(J real ( ^ 2 ) ' ^ t * w h i c h a r e "supported" on D , i.e. Zn - measurable functions on / ' . (% ) , and the Lj real sum of which is the identity:
222
We let x + ( D ) be the characteristic functions of the measurable sets U e supp dy 0 : * ( D > 7
0 } c
^'real
(
^2)"
(7
'7)
In the notation of 3.2 we have m = 2, P = x , ( Q ) / P 2 = X _ ( D ) ; obviously X + ( D ) + x_(D) = 1. 2 Suppose now that, for all x e 2? , <X+(Oo>
X_(Dx))
±
< 1-
6,
(7.8)
for some 6>o. Since
<_
> + is a pure phase state, by Theorem 7.1, (3),
<x + (D 0 ) x _ ( D x ) > ±
- <x+(Do)}±
<x_(D0))±.
as ]x| ->- oo.
But < x _ ( n o ) > We set
(7.9)
±
=1-
<x+(Do))±
( x + ( D o ) } ± = M*
(7.10)
By construction of the state <->,, see Theorem 7.1, (2), and the assumed evenness of P M* > 1/2
(7.11)
Combining (7.8)-(7.11) we conclude M* - (M*)2
-t~4
+
—
+ ~ Z
(7.12)
22
Furthermore M_ = 1-M
< -j - /«"
(7.13)
which proves the phase transition, and, since, for a state <-> invariant under 4>-»— <J», /x
(•)/
= 1/2, this
also shows that the states <->+ break the <(>-»— $
symmetry
of the dynamics. (The chain of arguments (7.8) - (7.13) is taken from [GJS2J). We are now left with proving (7.8): By Theorem 3.2, inequalities (3.34), (3.35), (7.8) follows from the inequality / TT X . ( O ) X (D')\ + < e " K l Y | \ ( D , D')e N( Y ) / _ for all contours y g r ,
(7.14)
, (see Section 3) and some
(x,y) sufficiently large constant K. (By Lemma 3.3, K^log 3.5 yields (7.8)). Let J be some positive number to be chosen later. We let x + ( D ) be the characteristic functions of > J {*:•(•) < _j>, 2 and x + ( D ) the ones of {*:* (D ) £ [0,±J]}, Then
x±
= X+ + xj
(7.15)
We insert (7.15) into the l.h.s. of (7.14) and expand. This yields
118 224
/
TT
x.(D) X _(D") ) /
< ( • , D')€N(Y)
I
(7.16)
(
e (D )=1,2
(D) ,,_,. elU; ,-. x:E (D) (D)xI (D')(D') \+
TT
MD,D')£N(Y)
/ "
E(D')=1,2
Let N.(y) be some maximal subset of N(y) with the property that if (•.,,•.]) and ( D 2 , D ^ )
are two dif-
ferent elements of N.. (y) then • . f D'. , for i=1,2. Obviously Y
1 . |N1 (y) I > j |N(Y) I = '-j-' ,
(7.17)
where |N.(y)| is the number of pairs (•,•') in N (y) . We now label the pairs (•,•') in N 1 (y) by some index i e {1 , 2 ,...., |N1 (y) |} = I . We then define Xl(i)
= xl(D) xl(D'), x,(i) = x+(D) z
(7.18)
x3(i) = x?(D')i x4(D = x+(d) xf(O') S i n c e 0 <_ x + ( d ) 1 1/ f o r a l l • g e t from ( 7 . 1 6 )
/
TT
\ ( D , n')eN(y)
-
I {P(i)=l,...,4}
x+(0)x_(D')\ +
/
and e = 1 , 2 , we now
+ ±
( TT x o m ( D \ \ i eI
p U )
+
/ *
(7.19)
225
Next we note that, for i = (• , D ') , < F l ( i) , e - 2 J e * ( Q ) - * ( a , )
O <
X l (i)
0 <
Xo (i) < F n (i) = e
oJ
2- d- jj M D ) " )
^ 0 < x3(i) 1
0 < X ,Ci) 1
F
(i) 3
F
4(i)
=
F
? (i>
F-^ti)
a n i m m e d i a t e c o n s e q u e n c e of Combining
(7.19)
/
II
\(D,
D")€N(Y)
±
(7.20)
(1- JJ *(D')2)
I {P(i)=i,...,4}
and
(7.18); see a l s o
x J+ D ) x (D X
( TT F \iei
[GJS2
( 7 . 2 0 ) we g e t
'\>
( i )
ii)\
'
/
p l 1
(7.21)
+
"
The idea is now to estimate the r.h.s. of (7.21) by means of "Gaussian domination" and the chess board estimate (Lemma 4.5, Theorem 4.8, (4.41)). Let h.(x) be the function on W • u D
with support in
the graph of which is given by Fig. 4: jraph hj Fig. 4
Acta Physica Austriaca, Suppl. XV
226
£(i) is the direction perpendicular to the common face of •
and D1. Then
(•)-*(a'> = -|hi(x) 3 i ( i ) *(x) d 2 x. (7.22) where 3 l ( l )
$(x) = ( d t(i) 3x
•)
(x) .
2 2 Obviously | |h. | |_ = -_- , and (7.23) supp h. D supp h. , = <j>, for i ^ i 1 By (7.20) and (7.22) lll)
F l ( i) = e " 2 J
B*
*lhL)
(7.24)
Next we define a.U,a) = A - 1 - log I A. I
where a
(7.25) /
-XU A (R)
f [:* 2 :(A)- D E CA
:*(D)2:]\0
\
) A
+
«
is the thermodynamic pressure per unit volume
of the free field of bare mass 1 in a periodic box of length 2L.. at inverse temperature 2L_. From Corollary 4.7 we infer existence of a (A,a) and from Lemma 4.6.(1) «„(X'CT) 1 « A U , a ) , A Q = [-1 ,1]X[-1 #1]
We now define weights
(7.26)
121 227
W1 = 2J - 1
W
2 =
W
3 = 1
,
W
aJ2 2
4 =
+a
+ f (•(D)2)
(X ,a) -a
(X,a)
Lemma 7.3:
TT F P(i) :D\
\i6I
-.zr e
leI
p(i) Y
(7.27)
Proof: We note that oJ' F 2 (i)
2
(*(D)2)
§:<|,(D)2: (7.28)
similarly for F-. (i) , (replace •
by • ' ) ,
and F 4 (i) = F 2 (i) F 3 (i) If we inspect (7.24) and (7.28) and use the fact that iei labels a subset of N( Y ) consisting of disjoint pairs, (i.e.pairs with mutually disjoint supports), see (7.17), we immediately conclude that the chess board estimate can be applied to the l.h.s. of (7.27); (see also [GJS2, FS, F3]). But the chess board estimate and the definition of the weights W.. , . . . , W. yield (7 .27) . This completes the proof. Finally we propose to prove that, for each finite J, and some fixed aeCl ,2TT2)
122 228
W. -* », as A^l O,
(7.29)
for i = 2,3,4. This follows from Lemma 7.4: Let a be some fixed number in the open internal (1,2ir2). Then, for each finite X >0, (1) a (A,, a) is bounded uniformly in Xefo,X 1 , and L
oo
O
(2) a ( A , a ) / » , a s X \ 0 . Proof; By ( 7 . 2 6 ) a (X , a )
a,
oo
e
a n d ( 7 . 2 5 ) (and c o n d i t i o n i n g ,
[GRS])
(A. , a )
n
< e
v
o
o
. -2XU. (P) . „ . A < (<e o >°Q)1 (
/Q / 8
^ e a [ : * 2 : ( D ) - : 0 » ( D ) 2 : ] ^ o}1/2e«A0
(7>3(
50)
I t i s w e l l known t h a t t h e f i r s t f a c t o r on t h e r . h . s . of (7.30) i s bounded unformly i n Xefo,A. 1 . The second L oJ factor can be calculated explicitly:
/ea[:^2:(D)-:*(D)2:]\ D o
(2a) n = e x P { I A£LL_ I [kz+lj-n} n=2 o#kern
(7.31)
229
where r n
is the momentum space lattice dual to • . If 2 2 and k^O then, obviously, k > 4 IT , so that
ke r
{(2a) n
I o^ker
[k2+ l ] ~ n } 1 / n - 2a
<
^
4TT2 + 1
as n-+°°, which proves that the series in the exponent of the r.h.s. of (7.31) converges. This proves (1). Part (2) follows from exactly the same arguments used in Section 6, (6 .13)- (6 .15) . This completes the proof of Lemma 7.4. If we now combine (7.21), (7.27) and (7.29) we see that there exists some X . ^ > 0 such that, for crit. all xe(0,X ) , inequality (7.14) is true, for some K ^ log 3.5. This implies (7.8) and completes the proof of the theorem. Q.E.D. Remarks (1) The phase transition for the pseudoscalar Y theory (which has a symmetry taking $ to -<(>) follows by the same proof. One just must verify that, in a periodic box, grad $- bounds and the chess board estimate apply. This is straightforward, but see [F3]. The arguments proving that a. (X ,a) -a (X ,a) /" <*>, as X >* 0 (or X/0) , for fixed a>a are as in the case of the P(<j>) models r (Lemma 7.4 & Section 6 ) . (2) For the X cos (e<|>) --theory (i.e. Model 5 of Section 2) the proof of existence of a phase transition is based on the same strategy as the one of Theorem 7.2, i.e. the Peierls argument, grad §-bounds and the chessboard esti-
i.e. the analogue of Lemma 4.5, Theorem 4.8.
2 30
m a t e . These estimates follow directly from Section 4 and the convergence of the lattice approximation proven for 0 < e 2 < — in T F I ] . T h e estimates o n the statistical IT
L
-1
weight of contours (inequ. (7.14)) are then obtained as in (7.20), (7.21) and (7.27), but in (7.20) we must set J = j — (this is a convenient but not the only possible choice), and the functions F (i) , F (i) (see (7.20)) are replaced by - X /J e X:cos e ,).(a (,; ) . e */ Q-,-1
where X = x[ (cose<j»(D)/) °]
= X exp[f- ( D , (-A+1)
1
D)],
and Wick ordering is always done with respect to a bare mass = 1. The required estimates on the vacuum energy density a (X,e,m ) and on a (X ,e ,m ) < a. (X ,e ,m ) •» o A o o X[:COSE*:(A
)- E
:cose*(D):]
0
o , .
1 , / L ° ncAo > +a. (m ) = 1 lo<3 <e ° / A ,m A o' where m \ is the bare mass in the free Lagrangean o o and o the Gaussian measure with expectation <->«o / are however o'' o c " more delicate; but see [F3], also [F1] We obtain Theorem 7.5: For the X cos (e<(>) --theory, there exists some X
. >0
231
such that, for all X>X ., , all e 2 <e 2 . . (X)<— and ' crit. crit. IT 2 for m o-
(X)>o) there is a phase transition, and the ty •* -<(> symmetry of the Lagrangean is spontaneously broken. (3) Let R(x), X and a be defined as in Theorem 7.2. Let Q be some arbitrary, real, not necessarily even polynimial with deg Q .<, deg R. We define P(x) = X(R(x) + eQ(x))- j x 2 + hx, (7.32) 2
Where a £ ( 1 , 2 i r ) , 0 < X < X c , with X have
such as in Theorem 7.2, and h € [1,1]. Then we
Theorem 7.6; There exists some e . , >0 such that, for all e £ crit. ' (-e . , ,e . . ) there are at least two disjoint P (<(>),.states (see Theorem 7.1) , i.e. a phase transition, for at least one h = h .. (X,a,e)Gf-1,ll. L crit. 'J Remark; This is a field theoretic version of Theorem 5.3 and shows that phase transitions may occur without being accompanied by symmetry breaking. The proof of Theorem 7.6 follows by first noting that all estimates established in the proof of Theorem 7.2, in particular inequality (7.8) , hold uniformly in
232
E and h, for -e
< e < e , -h < h < h and some suffio— o o— — o ciently small, positive e , h . From Theorem 7.1, (2) -* o o (cluster property for <->+!) and inequality (7.8) we now conclude that
I < x + ( D ) - x_ = lim
(D)>"' h | 2
(( x + (0 0 )- x_
(D0))(x+(Dx) -x_
(Dx))) ±'h
> min (y2+(1-y)2) - 2 (1 - 6) y€[o,i] = 26,
(7.33) <->
where
+'
are the clustering states corresponding to
the polynomial P defined in (7.32) which have been constructed in Theorem 7.1, (2). Next, Theorem 5.2 of ref. [FSj tells us that, for all but countably many h ^ o, lim
( X + ( D ) - x-
(D))^'h=
( x + ( D ) - x_ (•))
°' h (7.34)
and the r.h.s. of (7.34) is strictly positive, for h>o, and strictly negative, for h<0, as a consequence of Theorem 7.2, inequality (7.12) and the FKG-inequalities that give monotonicity of / x . ( D ) _ x_ (•)/ °'
in h.
Combining this fact with (7.33) and (7.34) we conclude that, for sufficiently small |e| f 0, the function (x±(d)-
x_ ( D ) / + '
has a discontinuity in h at at
233
least one h
• . G [*-h ,h 1 , whence c n t . L o oJ
<">: ,h * <">!'h . for h = h c r i f c >
Q.E.D.
Conjecture: For all positive polynomials R and fixed ae(1,2ir2) there exists some X
= \
(R,a) > 0 with the property that
for O < X < X , there is an h_ such that, for P(x) = XR(x)- | x 2 + h c x, P +
,
P
r
Our present estimates on a
and a
are not sharp enough
to provide a proof of this conjecture. Remark.: The results of Section 7 are the basic input for our construction of soliton sectors in Part 2.
Section 8; QUANTUM CRYSTALS AND MORE ABOUT THE PEIERLS ARGUMENT 8.1 Quantum Crystals (a straightforward application of [FSS]
)
234
Here we briefly sketch an extension of our results of Sections 4 and 5 to systems which may be interpreted as simple models of
anharmonic (quantum) crystals.
We consider a cubic lattice 2. = 2 . x....x 2. , where o o -i o — 1 v 1. = {x:x = 6-m; m £ Z } ,
(8.1)
o
i.e. 6. is the lattice constant in the direction i. i
We set „i
i
1
%
1
a
+
with a^, 3 1 as in (2.6), (2.7). v We abbrviate JJ 6 . by TT_6; i=1 1
(8.2)
Z,. o
A denotes a rectangle in
The Hamilton function associated with A is of the form H, ({a}) = 4r
I a£A
TT6
{ J J. (31a ) +ha } i=1
(8.3)
with periodic boundary conditions at 3A. Let dX be some single spin distribution in the class K introduced in (2.20), and let <->. denote the expectation associated with H. and dX. Then, for all Ac?^, A
—
0
the following slightly more general version of Theorem 4.1 holds:
h
129 235
I u(S 3g. (a) •3 1 a '. - l a\ (e 0 " 1 )
1
-j A
< e
7 - 1 i ,. 2 ir6 — |g. (a) i ' "- J i ' ^ (8.4)
/ •
2 a
The proof is almost as in Section 4 (Steps 4.1-4.3), details are left to the reader. An interesting example for this somewhat more general situation is an anharmonic quantum crystal: Here v=4, the number N of components of o is arbitrary; (a is now interpreted as an N tuple of oscillator coordinates). We let J 1 , J ? , J be arbitrary positive numbers; (for simplicity we may suppose that J =J = J = J > 0 ) , and e.g.
6
1
=
6
2
=
6
3
=
K
Furthermore
^4
=
M' ^4
=
2L+1' ^
=
1'2'3''"*"'
(8.5)
for some positive constants M and g. The rectangle A has the form A = A1 x [o,g] .3 with A1 a rectangle c. 2' The single spin distribution is given by
(8.6)
236
d\(a)
= e
- & *
Via) dNa
(8.7)
where V is some r e a l , measurable function on W f e - B V(a)
d
Na
<
^
with (8.8)
The equilibrium measure corresponding to H ° - see (8.3) - &A = - ^ j " see (8.5) - and dX is denoted d\i. T Aj ,L Then, for
|AI|<0°/
one can prove: ± I
lxm L-n»
J
dy, 1
*
0
g(o)o„
({a}) e
i x | Al fB dtg(t,I)a (t)
=
J0 £ A V1 l ({a}) [{a}) e I"' >°
dy^
x
*
(8.9)
Aj ,»
for all bounded, periodic functions g(t,I) on [o,gJ * (A H Z ) which are continuous in t. Here dp
^
is a perturbed Wiener measure with N J A J dimensional state space and periodic boundary conditions at t=0,@. The proof of convergence in (8.9) follows by combining the Trotter product formula and the Golden-Thompson inequality (using (8.8)) with the Feynman-Kac formula. a
The measure dy
is the equilibrium path space
measure at inverse temperature 3 of a quantum crystal with quantum mechanical Hamiltonian Xj\,
l
(
2M & o
+
V
(o T )+ X
^
j(oy-aT,)2}
{I':|I-I'|=1} (8.10)
131 237
where A. a
is the Laplacean in the variables a,.. J
I
The moments of dy
are the imaginary time ("tempera-
ture-ordered") Green's functions of the system. The point we want to make is that inequality (8.4) combined with a lower bound on the second moment of
dy,, Z J ,°°
({a}) a -a o
> const. > 0,
(8.11)
o —
uniformly in B, (which must be derived from the properties of V; see e.g. Section 5, Theorem 5.2) yields existence of phase transitions for sufficiently large B, for many (physically interesting) choices of V; (note that the truncated second moment is bounded above by OCB~ 1 ) by (8.4)) . We leave it to the reader to make a list of potentials V for which (8.11) is valid and to exploit consequences of the Lee-Yang theorems of [SG,DN], for the case where V is a quartic polynomial.
8.2 Some more consequences of the Peierls argument (a) Theorem 5.3 remains true in v=2 dimensions: This is an extension of a recent result of Sylvester and van Bejeren [SvB] to single spin distributions that are not necessarily invariant under a-+-a.
132 238
The (somewhat complicated) proof follows from the Peierls argument and is very similar in spirit to the proofs of Theorems 7.2 and 7.6. Rather than proving discontinuity of ' some h = hcrit. .. (as in Theorem tinuity x j of <x,(o)A+ A _ (°) >
at
5.3) one proves discon^
at some h = h c r i..t . (see
Theorem 7.6). Details are contained in some unpublished work of the author. (b) Anisotropic N-vector models; [Ma,KuJ Let dX (a) ^ S (a) d a be some 0(N)-invariant Borel probability measure on f/l • We define dX_(a) = dx (2-) , J /J and v > 2.
(8.12)
Let a finite volume Gibbs measure be given by dvl'J({a})
=
Z A ( E ,J)" 1 exp[-l
I aeA |a'-a|=1
W_ (a..) x ]J e e a dXj(a a ), a£A
{(a
_0
)2
+
e(a1
1 )2}-[
(8.13)
12 E J where W (a) = 2e(a ) - za • a, and let dy ' be some infinite volume limit of (dy^' }.
239
Theorem: For arbitrarily small, fixed e>0 (or for e=e J 1 log J, with e > -j(N-1)) there exists some J < ~ such F
7
that, for all J > J , there is a phase transition: dy ' is a superposition of at least two pure phase (ergodic) measures some of which are spontaneously magnetized. Remark: This theorem extends work of Malishew [Ma] and Kunz [Ku]. The proof is again based on the Peierls argument and is contained in (unpublished) work of E. Lieb and the author.
(c) Triple points Let v = 2,3,..., N = 1, and dX(o)e=g'2w1+1 J
(Wj 6(a-J)+6(a) + W j 6(o+J)}da,
(8.14)
where W j is e.g. a monotone increasing function with the property that W_ = 1, for J = J
.
> J , (for a
certain J^ < °°) . Let H A be the usual Hamilton function and du
some infinite volume Gibbs measure corresponding
to H A and dX
(as in
(8.14)).
Then
( d ) for sufficiently small J>0 du
is ergodic and
unique (in the sense that it is independent of boundary conditions used in the construction of the thermodynamic limit). There is no spontaneous magnetization. The proof
240
is by means of a "high temperature expansion". (c2) for J=J . dy is the superposition of at least three different pure phase (ergodic) measures, at least two of which have spontaneous magnetization. Proof: This follows by combining the results of Section 3.2 (Proposition 3.1 and Theorem 3.2, with m=3 orthogonal projections at each site: onto a=±J, a=o),
the
lattice version of the first inequality in (7.20) (see also (7.24)) and Gaussian domination (Theorem 4.1). Q.E.D. (c3) Suppose e.g. that W T / , « , as J / » . Then we conjecture that, for J>>J . there are precisely two different pure phase equilibrium states with nonvanishing, opposite spontaneous magnetization. (Presumably this follows from low temperature methods. It is known that under these conditions there are at least two different pure phases.) Remark: Examples of the type of (c) with an arbitrary degree of sophistication may be invented; (also for the case where N>1; e.g. dX (0) a convex combination of {6(|a[-j JO J)d 0} Jo — i , 0 < j.i <....<j n , 0<J). The point is that the Peierls argument in the form of method 3.2 and Theorem 4.1 combined with (7.20), (7.21), (7.24) supply a systematic approach to their analysis.
24
The interest in such examples depends on our ability of inventing and analyzing models that may describe certain aspects of real physical systems. One interesting open problem of the type discussed here is to prove or disprove the existence of triple points in P((j>)„ quantum field models, (with deg P > 6) .
Part 2: Section 9: PHASE TRANSITIONS AND THE SPONTANEOUS OCCURRENCE OF CHARGED SUPER-SELECTION (SOLITON-) SECTORS In this section we explain a basic connection between phase transitions and the existence of non-trivial super-selection rules ("topological charges") in two dimensional quantum field models. We show that (under suitable hypotheses specified in [F7]) the following holds. Theorem 9.1: [ F 7 ] In two space-time dimensions, any quantum field theory with an n-fold degenerate physical vacuum has n(n-1) charged super-selection (soliton) sectors (disjoint from the vacuum sectors). These sectors are labelled by the elements of a finite group, (the so called soliton-group).
Acta Physica Austriaca, Suppl. XV
242
We shall not present a general proof of this theorem at this place. This has been done in [F7,§6]; see also [Ro]. We rather illustrate the general theory of solitonsectors in two dimensions by studiing one specific model, (the anisotropic (*•<(>) -model) . But we remark 2 that Theorem 9.1 applies to all the models analyzed in Section 7 and proves the existence of non-trivial topological charges and soliton states for those models; [F7J. Particularly clear interpretations of the physical mechanism responsible for the existence of soliton states can be given in the context of the sine-Gordon theory, with or without mass term; see [Co 2,3, F1, F Se, F7j; Perhaps the most interesting model is the pseudo scalar Yukawa model. Mathematically, the P (<j>) 2 -models (with non-even P) are most fascinating from the point of view of solitons: For these models the existence of solitonstates is still unknown. Here we propose to study the anisotropic (<)>•<£)
-
model, with $ = (
" §
!
•?
:
'
x >
°'
whence the Euclidean action
tf£(1/2ir2),
(9.1)
24
U
A =
J X : (ht)2-
(x)- § :<|>f:(x)]d2x
(9.2)
Existence of this model in the infinite volume limit has been proven in [F4,6]. With
U i (D ) ) ' • • • Theorem 7.2 immediately extends
to this model and proves the existence of a phase transition and spontaneous <(> 1 ->• -
<->_
(9.3)
satisfying all the axioms of Osterwalder and Schrader
[OS] , and <*i) + = " (
(9.4)
The Osterwalder-Schrader reconstruction gives two pure Wightman states u and m_ with the property that
= wj n ) (f r ...,f n )
(9.5)
-»• "*" 1 2 Here <j> (f) = <(>1 (f ) +
are the
(pure phase) n-point Wightman distributions(obtained as a boundary value of the analytic continuation in the time arguments of the n point Euclidean Green's functions, i.e. the n
moments of <-> ) .
244
In the following we propose to construct two states a) and u- interpolating between u
and OJ_ (the soliton-,
anti-soliton state, resp.). Our construction is best explained in the framework of the theory of algebras of local observables [HKJ adapted to the two dimensional models; see [GJ 2,3] Let $ (x) =
(x,o)
(9.6)
As the ($•<(>) - model is a canonical quantum field theory, TT1 ,TT are the momenta canonically conjugate to $..,*„, respectively. Let O be some finite union of compact intervals on space. We define a real, local Sobolev space of distributions supported in O by <^-1/2 (0)
=
{f:f
real-valued
' supp f c O, | |f | l_ 1 /2
where | [f[ | . = (f , (——-1/2 d^2
?0- Z-U2
+ 1)
'
f) L
2
<
and
t°) * »-i/2<°>
A local von Neumann algebra OC (O) is defined as the weak closure of the operators { e ia(fV^(g)) : ^ € 7
0J
on the Fock space "3- of the free field fy(A=a=0) ,with
245
bare mass 1. A C -algebra 01 is now obtained by taking the closure of
\J
01 (0) in the operator norm.
Oct The following results, due to Glimm and Jaffe [GJ2,3], are known for the (<(>•<)>)
- theory:
F1) The restrictions of the states
w
to the algebras
01 (0) are ultraweakly continuous, for all bounded 0. This is the so called locally Fock property established in [GJ2] . As a consequence, the vacua w ,to_ are well defined states on the algebra F2) The Hilbert spaces %+ obtained from the Wightman distributions (9.5) by means of Wightman reconstruction coincide with the Hilbert spaces X " * " obtained from {(a+,0t ) by means of the Gel'fandNaimark-Segal construction; see [GJ2,3]. (Note that, a priori, %+ could be a proper subspace of X + " " *) F3) The representations of 01 on
% , <£_ are irreducible
(a consequence of the fact that OJ and OJ are pure phase vacuum states). They are disjoint from each other. F4) Let 0 be the union of diamonds (intersection of a forward and a backward light cone) with base 0, and let 0,.
. be the space-time region obtained
from 0 by a Poincare transformation (A,a).
246
Since one knows the quantum fields are selfadjoint, one can define local algebras Oi (0) in the obvious manner. Glimm and Jaffe prove [GJ2] that 01(0) = 06(0)
(9.8)
The Poincare automorphisms of 01 , formally defined by (T(A
a ) J)
(x,t) = J(A~ 1 (x-a 1 ,t-a Q ))
(9.9)
have the property that T
(A,a)(<*<5>> - ^ ( 5 ( A f a ) ) .
They are implemented the representation U obtained by Wightman McB] and refs. given
(9-10)
on X + by the unitaries of of the Poincare group on
Definition: Given some bounded diamond 0 we let ~0 denote its causal complement. In two space-time dimensions - 0 = 0 T u 0_, where 0 T . . is a wedge region opening to the left (the right). Construction of soliton states: We are now prepared to explain the construction of the states w
and w-, (a C -algebra construction).
We want these states to have the following two basic
properties; (the reader is advised to consult Section 1 for heuristic motivation): (S1) There is some bounded diamond 0 such that us (A) = u (A) , for all A £ 0i{0' S
) , for all 0' c 0 T ;
~"
= oo, ( A ) ,
S
Li
for
all
A£0t(O"),
+
for
allO"cO
n XX
(For
oo- satisThe idea of how to construct states to s fying (S1) and (S2) is inspired by algebraic field theory in the form of [ D H R ] : We propose to construct automorphisms a and a- of the algebra OL with the s s property that OJ (A) = in, o o
s
+
s
(A) E to, (a (A)) + s (9.11)
to-(A) = a) o a- (A) = to o a (A) , s + s — s
for all
A€0t
142 248
In two space-time dimensions existence of morphisms a , a- with the property that u
auto-
and oi- satis
fy (S1), (S2) is a very general fact; see [F7,§5-6]. We attempt to define the automorphisms a , aexplicitly and then outline the verification of (S1), (S2). Our explicit definition is motivated by the heuristic considerations presented in the Introduction (Section 1 ) ; (see Fig. 1,2). Definition 9.2: Consider the (<|>-
(=3,14...)
and,for some finite L, > a (x) =
IT,
(9.12)
for x < -L, a (x) = 0, for x > L;
^-«(x)£L2(|). dx (9.12) is called the "soliton-condition". The automorphism a
is given by the following Bogoliubov
transformation
2
^
C9.13)
Let 9
denote the
to
-IT .
We set
a-
=
automorphism taking
a o ? S
(9.14) J TT
(Obviously ai = GJ O Q ) ±
-
•> IT
Next we explain why the transformations defined in (9.13), (9.14) are well defined automorphisms of 01 • Let 0 be a finite union of compact intervals, and let T (x) be a C -function of compact support such that f
(x) = 1 , for all x € 0 .
We set a L(a 0 ) =
= aip
and define the operator
dx a Q (x) [*1(x)if2(x)-irl(x)*2(x)J
(9.15)
Lemma 9.3: (1) The operator M<* n ) is self adjoint on Tl , X _, and e±iL(a0)£
^ ( s u p p ^^j
(2) For all A in some subalgebra weakly dense in ^LUQ) e
A
e -iL(a 0 ) =
Q
01(0)
(A)/
s with a
given by (9.13) .
The proof of Lemma 9.3 is given in [F7,§3]. It tells us that the formal Definition 9.2 really determines
144 250
a well defined
automorphism of 01 .
If we now define the states to , u- by (9.11) we easily see that condition (S1) is satisfied- Furthermore it is straight forward to show that the representations of 01 on the spaces % , H_, X a n d * ~ a r e disjoint irreducible representations. Parity is the conjugation mapping "][ onto Tt- and vice versa, (i.e. parity is spontaneously broken on the soliton sectors A. , X-) . For proofs see [F7,§3] . A heuristic argument for all that is as follows: Define the conserved "topological"charge dx (-23x
• ) (x,t) (9.16)
= |dx (2dx
*..) (x) '
(associated with the conserved current (j°(x) ,j1(x)) = ((!_ + 1 ) (x), ( - ^ ^ H x ) ) ) oX
Then Q \
= {3}, 2$
Q«jj
if), for all i> £
= - 2 $ c <|i, f o r a l l ij. £
Finally, parity takes Q
£
(9.17)
/ -
to -Q.
Important remarks: (1) The spaces on dl , .s
_, di- and the representations of 01 are independent of the specific choice
2
of the function a: All functions a satisfying the soliton-condition
(9.12) (for some finite L) yield
identical soliton-sectors. (2) As a special case of a general theorem proven in [F7,§6] we note that P (the identity automorphism) o , a and a- form an abelian group (the soliton J 7T
S
S
g r o u p of
^2 a
s
= o
-}
(J-J)
): {9 18)
s=°i-ye'
o P = a-, aJ IT s ' s
-
op
= a
3TT
, a s ' s
o a-
S
= p
JTT
Next we propose to discuss and partly prove, for the the special case of the anisotropic (^•<\>) - theory defined in (9.1)-(9.2) the following main result Theorem 9.4; (see [F7j) Let if be s or s, and suppose that u +
(*-,)
=
-">_ ^-j)
=
$
> c
°'
Then (1) There exists a continuous unitary U„ of the Poincare group on automorphism group { T ,
H,
representation
implementing the
. } of <%; i.e. condition
( A , a) (S2) holds. (2) The spectrum of the energy-momentum
operator
(H„ , Pj,) , the infinitesimal generator of spacetime translations, is (a) contained in the forward light cone (b) purely continuous; (in particular
din does not
252
contain any vacuum states). (3) There exist
(Q-) charged soliton fields s(x),(s(x))
with non-vanishing matrix elements between
£
and
Xs
( df and It-).These fields can be constructed + s to be "almost local" fields and such that they are "almost local" relative to even functions of the fundamental field
(4) If
X
[F7,§4j).
contains a one-particle-state (the quantum
soliton) then JC- contains a one-particle-state (the anti-soliton) with equal quantum numbers except opposite Q-charge. For sufficently small \€{o,\
),
the mass of the soliton is positive. If N (N-) denotes the number of solitons (antis s solitons) in a scattering state (constructed according to the Haag-Ruelle theory) then (a) N - N- is even, for all scattering states in s s (b) N
- N- is odd, for all scattering states in % „
Remark: The proof of Theorem 9.4 is given in
[F7,F7'J
. It
is technically rather difficult. In the next section we sketch what we feel are the main ideas of the proof.
253
Section 10: SOME REMARKS ON THE PROOF OF THEOREM 9.4; CONCLUSIONS The main physical as well as technical idea behind the proof of Theorem 9.4 is the following: Construct the soliton state u as a weak limit of states res presenting a pair of a soliton and an anti-soliton, i.e. of vector states in TC or r3t_ with Q-charge 0, by sending the anti-soliton off "to behind the moon", (mathematically to spatial °°) . Derive properties of u , in particular Theorem 9.4, by analyzing in detail these approximating vector states in It . •#-
it
Formulated in terms of
automorphisms of the
algebra 0L this program is expressed as follows: Represent as (see (9.13) and Lemma 9.3) as a limit of inner automorphisms of OL • (these are then unitarily implementable on lt+) . Next we outline how this idea is realized technically. Let a, be a function satisfying the soliton-condition (9.12), and let 0 be the smallest compact interval containing supp (— a ) , 0 the diamond with base 0. dx Let 0-£ = {y:y-x£ 0} and 0-»- denote the translates of 0, 0, respectively, by the vector x. We define a->- = .
-
+
-
*
.
-
>
x
a (y-x) and pick, x such that 0 and 0-*- are disjoint. Obviously, the support of a-a->- is compact, (denote it by A-*) , so that
254
iL(a-a-t-) X
z = e
V
£ a(Aj) ;
a ,X
(10.1)
x
clearly V
Let Q vector V
%., ±
+ is a well defined unitary operator on a ,x
denote the physical vacuum in
JL+. The
+fi+ can be thought of as representing a pair
of a soliton and an anti-soliton, and
a) (•) s
= w * - lim
(V
-»• fl , • V + n.) a ,x + a ,x +
(10.2)
X-»•-"»
Moreover
a
(A) = n-lim
V
s
•> A V •+, a ,x a ,x
(10.3)
• >
x+-»
for all A£0t Equations
.
(10.2) and
(10.3) make precise how the
announced program is to be understood. Let
£ = (A,a)
denote the elements of the Poincare group
We define a "propagator"
(a so called 1-cocycle; see
e.g. [RoJ )
r
+i e ) = fr (v
a ,x
£
-o v*
a ,x
do.4)
z
a ,x
(= u*U ) v j u . ( u v * j , ±
a,x
as operators on
±
a ,x
j[ ) . Here T
automorphism defined in
(9.9),
forward calculation shows that
denotes the Poincare (9.10). A straight-
25
r
+( 5 • C ') = x c , (r a ,x
^
^
£
+(U ) T
a ,x
+ ( 5 ')
(10.5)
a ,x
This is called the cocycle identity.
10.1 The role of local 1-cocycles in proving Theorem 9.4 In the following we propose to analyze the 1-cocycles r ->-(£;) more closely and we show how they are used in the proof of Theorem 9.4. Let A->- be the diamond _ _ x with base A-+, and let A-», (0 ) be the space-time x x, t, 5 region obtained by applying the Poincare transformation E to A+, (0, resp.) Since A-*- is compact, so is A->- ,and X
X
X / c,
r +(5)£fl(A+un+ ) , a ,x s x x, 5 i.e. r
(10.6)
-»-( 5 ) is a strictly local observable. By (F2)
and the locally Fock property (F1) we may therefore analyze r
+( f;) as an operator acting on the Fock
Ct / X
space ^ of the free field. This permits us to appeal to the following theorem due to Cannon and Jaffe [cj] (see also Mc Bryan C M C B J ) : Given some compact neighborhood N of the identityl of the Poincare group p * there exist ( 5 ) on j
UN ' x r
such that, as an operator equation on.| "
a,x(?) = Un,A^(5)
for all 5 £ N.
unitary operators
V
a,xUN,A.
( 5 )
V
!,x '
(10
"7>
256
The advantage of this representation of the cocycle r -•(£;) is that the operators U and detailed estimates
. are known explicitly,
(in particular Trotter product
formulas) are available; see [CJ, McB] , and § 3 , Lemma 3 of
[F7] for an application.
These estimates combined with (1) duality for the free field, first proven by Araki [A],
(see Lemma 1, §3 of [F7] ) ,
(2) the energy - momentum - and boost densities of the anisotropic
(<|> •<)>)_-theory are invariant under j>
(i.e. under the substitution ^ - ^ j a r e used in [F7] , (see Lemma 3, §3 of [F7] and [F7 1 ]) to prove
Theorem 10.1:
[F7, F7']
Given 0 and N, t h e r e are compact diamonds B(0,N)
2
(J 5 £ £ N 5
a n d B(0->-,N) x
= B(0,N)+ X
2
?
£
U N
0+ x ' ?
such that if |x| is so large that B(0,N) and B(0->-,N) are space-like separated
r
a,x(?) =
for all
r a
U ) T _ ^ < x
then
€
) ,
(10.8)
C C N , and
(1) for all 5 e N, r ( ? ) £ 0£(B(O,N)) is a local cocycle that is strongly continuous in E, , (2) r -aj
( e ) £ 01 (B(0+,N) ) is a local cocycle that is •»
x
strongly continuous in E, .
25
Remark. (1) Theorem 10.1 is the technical core in the proof of Theorem
9.4.
(2) In § 3 , Lemmas 3 and 4 of [F7j Theorem 10.1 has been proven for the special case where the group elements £
are space-time translations. The proof of the
general result is very similar to the one of this special case is however technically a little more complicated. It is too long to be reproduced here. (3) We emphasize again that the existence of locally correct generators for the Poincare automorphisms, i.e. the results of
[CJ, M c B ] , and duality
(for the
free field only) are basic ingedients of the proof; technically the Trotter product formula and finite propagation speed, i.e.
(9.10),play a basic role.
(4) The main point of Theorem 10.1
(in the realization
of the announced program) is that, by approximating the
automorphism a
by inner
automorphisms
implemented by the local operators V i.e. by approximating u
-+, (as x->— °° ;
by vector states in X . ) ,
we achieved to construct strictly local r
1-cocycles
( £ ) , a direct construction of which appears to
be difficult. For the expert it is now clear that Theorem 10.1 immediately yields a proof of Theorem 9.4. The last step is the following.
Corollary 10.2:
[F7,F7']
(1) The unitary operators {U + ( £)r
(? ) : ? £ p | } form
a continuous, unitary representation U
Acta Physica Austriaca, Suppl. XV
of the
258
Poincare group na , on
£+.
(2) For all A d 01 OS(TC
(A)) = U ± U
)ra(C )as(A) (U ± (5)r a (C))*,
^ (10.9) as an operator equation on
a
((I,a)) : a £ $ 2 }
is contained in the forward light cone. The proof of Corollary 10.2 is exactly as in §3, Lemmas 4 and 7 of [F7J. The proof of (3) relies in an essential manner on (10.2), (10.3) and the fact that w are vacuum states satisfying spatial cluster decomposition properties . (This stresses once more the usefulness of our approach: (10.2), (10.3)). Given Corollary 10.2 the proof of Theorem 9.4 is now simple: Let T be an intertwining operator mapping 9f, 1-1 and isometrically onto Tfc . Then a continuous unitary representation U of the Poincare group n on di is defined by the equation U*(€ )Ts s
* = T
=
U* (C ) * s
(10.10)
r* (c ) u*(c ) * .
T
s
a
This yields Theorem 9.4, (1). For the proof of Theorem 9.4, (2) see Lemma 7 and "Proof of Theorem 3" in §3 of ref. [F7], and for the
259
one of (3) & (4) see §4 of [F7]. We feel that it has been shown here what the main ideas of the proof of Theorem 9.4 are and in which manner local 1-cocycles enter the analysis. We should emphasize that the present techniques may not apply to the P (<)>)_theory if P is not even. The reason is that, for this case, it is not known, yet, whether different pure phase vacua are related to each other by
automorphisms of 01
(of the sort of j> ) that commute with the energy-momentum - and boost densities. For a general analysis we refer the reader to §6 of [F7]; (see also [RoJ).
10.2 Conclusions (1) Presently we are checking whether directly
that the spectrum of the
operator
(H. ,P ) on
M s , (for sufficiently
3?
one can
prove
energy-momentum
has a positive mass
small \ € ( 0 , \
gap
) ) . Heuristic
c arguments suggest that M is bounded below by some analogue of what one knows as surface tension x in the statistical mechanics of classical spin systems; (see e.g. [ G M ] ) . This surface tension can be esti-
mated directly by means of the Peierls argument; (see Section 7 ) . The author has shown that it diverges to +» (i.e.x* +») as X*Or (with X the coupling constant of the (J-|")2 - term, and a>1 fixed). (2) There seems to be an interesting the existence
connection
of soliton states and the
between
soectrum
154 260
of bound states in such two dimensional quantum field theories. (Example: For fixed CT>1, ^sA„ and 0< [ y | << 1 the X C$ • J)2-acf>2 + ji^j model in two dimensions should have a rich spectrum of bound states
(of "would-be" solitons and
-anti-solitons)
which decay into soliton-anti-soliton pairs, as ij->-0) . This has been briefly discussed in remark 5) §6 of ref.
[F7] (and, independently, by Glimm, Jaffe
and Spencer in a forthcoming p a p e r ) . (3) The next urgent task in the theory of quantum solitons might be the study of quantum vortices in scalar QED in three space-time dimensions;
(abelian
Higgs m o d e l ) . On a formal level this looks rather promising. In the case of non-abelian Yang-Mills theories in four space-time dimensions
(perhaps the
only candidates for soliton-behaviour in four d i mensions) the situation does not seem to be well understood, y e t , even on a purely formal level, although the existence of solutions of the classical field equations like the t'Hooft monopole is very intriguing. (4) For refs. to many original papers on quantum see
solitons
[Co4, F 7 j . Since the number of such papers is
rapidly diverging a proper list of refs. could not be included here.
155 261
References: [A]
Araki,H.: J. Math. Phys.4_, 1343, (1963), J.
Math. Phys.5_, 1
, (1964),
see also K. Osterwalder, Commun. Math. Phys. 29_, (1973) [AHR]
Araki, H. K. Hepp and D. Ruelle, Helv. Phys. Acta J35_, 164, (1962).
[CJ]
Cannon, J.T. and A. Jaffe, Commun. Math. Phys., V7_, (1970).
[Co1]
Coleman, S., "Secret Symmetry" 1973 Erice Lectures (Int. School of Subnuclear Physics, "Ettore Majorana").
[Co2]
Coleman, S. Phys. Rev. D11 , 2088, (1975).
[Co3]
Coleman, S., R. Jackiv and L. Susskind, Annals of Physics 9_3, 267, (1975).
[Co4]
Coleman,S., "Classical Lumps and Their Quantum Descendants" 197 5 Erice Lectures (Int. School of Subnuclear Physics "Ettore Majorana").
[CQFT]
Symanzik, K. , J. Math. Phys. 1_, 510, (1966) Symanzik, K., "A Modified Model of Euclidean Quantum Field Theory", N.Y.U. Preprint, 1964. Nelson, E., "Quantum Fields and Markoff Fields' in: Proceedings of the Summer Inst, on Partial Diff. Equ., D. Spencer (ed), Berkeley 1971, A. Math. S o c , Providence 1973. Nelson, E., J. Funct. Anal. J_2_, 97, (1973). Guerra, F., Phys. Rev. Letters 28, 1213,(1972).
Glimm, J. and A. Jaffe, refs. [GJ1,3] Glimm, J., A. Jaffe and T. Spencer, refs. [GJS1,2] Guerra, F., L. Rosen and B. Simon, ref. [GRS] Osterwalder, K. and R. Schrader, ref. [OSj. See also refs. [E] [Si] [Y ] . [DHR]
Doplicher, S., R. Haag and J. Roberts , Commun. math. Phys. _2_3, 199, (1971) and _3_5 49, (1974).
[DS]
Dunford, N. and J. Schwartz, "Linear Operators Interscience, New York, 1963.
[DN]
Dunlop, F. and C. Newman, Commun. math. Phys. 44, 223, (1975) .
[EEF]
Eckmann, J.-P., H. Epstein and J. Frohlich, "Asymptotic Perturbation Expansion for the S-Matrix and ...." to appear in Ann. Inst. H. Poincare, 1976.
[E]
"Constructive Quantum Field Theory", G.Velo and A.S. Wightman (eds.), Springer Lecture Notes in Physics, vol. 25., (1973) .
[ESw]
Ezawa, H. and J. Swieca, Commun. math. Phys. 5,
330, (1967) .
[Fe]
Feldman, J., Commun. math. Phys. 3_7, 93, (1974).
[Fed]
Feldman, J., and K. Osterwalder, "The Wightman Axioms and the Mass Gap for Weakly Coupled (^L Quantum Field Theories", to appear in Annals of Physics.
263
[Fe02]
Feldman, J. and K. Osterwalder, "The Construction of X C<}>^) o Quantum Field Models" Proceedings of the Int. Colloqu. on Math. Methods of QFT, Marseille 1975.
[FO]
Follmer, H., "Phase Transition and Martin Boundary", in: Seminaire de Probabilites IX, Universite de Strasbourg, p. 305, Springer Lecture Notes in Math..vol. 465, (1975).
[F1]
Frohlich, J., "Quantum Sine-Gordon Equation and Quantum Solitons in Two Space-Time Dimensions", in: "Renormalization Theory" Int. School of Math. Physics" Ettore Majorana". 1975, Reidel, Dordrecht - Boston, 1976.
[F2]
Frohlich, J., "Classical and Quantum Statistical Mechanics in One and Two Dimensions: ", to appear in Commun. math. Phys. 1976.
[F3]
Frohlich, J., "Phase Transitions in Two Dimensional Quantum Field Models", ZiF, University of Bielefeld, Preprint 1976; (extended version in preparation).
[F4]
Frohlich,J., "Poetic Phenomena in Two Dimensional Quantum Field Theory: ", Proceedings of the Int. Colloqu. on Math.Methods of QFT, Marseille 1975.
[F5]
Frohlich, J., "The Pure Phases, the Irreducible Quantum Field ", to appear in Annals of Physics, 1976.
[F6]
Frohlich, J., "Existence and Analyticity in the Bare Parameters of the [x (<j> -<j>) 2-a<j> ^-ut,] Quantum Field Models" to appear.
158
[F7]
Frohlich, J., "New Super-Selection Sectors ("Soliton-States") in Two Dimensional Bose Quantum Field Models", to appear in Commun. math. Phys., 1976, and paper in preparation (referred to as [F7'J).
[FSeJ
Frohlich, J. and E. Seiler, "The Massive Thirring-Schwinger Model (QED_): Convergence of Perturbation Theory and Particle Structure", to appear in Helv. Phys. Acta.
[FS]
Frohlich, J. and B. Simon, "Pure States for General P (<)>) _ Theories: Construction, Regularity and Variational Equality", submitted to Ann. Math.
[FSS]
Frohlich, J., B.Simon and T. Spencer, "Infrared Bounds, Phase Transitions and Continuous Symmetry Breaking", Princeton University, Preprint 1976; results announced in Phys. Rev. Letters 3_6, 804, (1976).
[GM]
Gallavotti, G. and A. Martin-Lof, Commun. math. Phys. 2S.i
87
' (1972) .
[GJ1]
Glimm, J. and A. Jaffe, Fortschritte d. Physik 2A_, 327, (1973) .
[GJ2]
Glimm, J. and A. Jaffe, Commun. math. Phys. ££ 1, (1971)
[GJ3]
Glimm, J. and A. Jaffe, (<j>^ II,III, I V ) , Ann. Math. 9_1_, 362, (1970), Acta Math. 125, 203, (1970), J. Math. Phys. J_3, 1568, (1972).
[GJS1]
Glimm, J., A. Jaffe and T. Spencer, "The Particle Structure of The Weakly Coupled P(<|>)2 Model and Other Applications of High
265
Temperature Expansions, Part II: The Cluster Expansion", contribution in ref. [E] and refs. given there. Glimm, J., A. Jaffe and T. Spencer, Commun. math. Phys. 4_5, 203, (1975), and paper in preparation. Guerra, F., "Exponential Bounds in Lattice Field Theory", Proceedings of the Int. Colloqu. on Math. Methods of QFT, Marseille 1975. Guerra, F. , L. Rosen and B. Simon, Ann. Math. 101 , 111, (1975). Haag, R. and D. Kastler, J. Math. Phys.5_, 848, (1964) . Herbst, I. "Remarks on Canonical Quantum Field Theory", Princeton University, Preprint 1975. "Statistical Mechanics and Quantum Field Theory", Les Houches 1970 C. De Witt and R. Stora (eds.), Gordon and Breach, New York, 1971. Jost, R., "The General Theory of Quantized Fields", A. Math. Soc., Providence, R.I., 1965. Kac, M., "On Applying Mathematics: Reflections and Examples", Quarterly of Appl. Math. 2P_/ 17 » (1972) • Kunz, H., private communication• Lamb Jr. G., Rev. Modern Physics £3, 99, (1971) • Lebowitz, J,, and 0. Penrose, Phys. Rev. Letters 35, 549, (1975) •
160
[MaSeJ
Magnen, J. and R. Seneor, "The Infinite Volume Limit of the (ty1*)^ Model", to appear in Ann. Inst. H. Poincare.
[Ma]
Malishev, V.A. Commun. math. Phys. £0, 75, (1975) .
[McB]
McBryan, 0., Nuovo Cimento, 18A, 654, (1973).
[M]
Mermin, N.D., J. Math. Phys. 8_, 1061, (1967).
[N]
Nelson, E., "Probability Theory and Euclidean Field Theory", contribution in ref. [EJ.
[OSJ
Osterwalder, K. and R. Schrader, Commun. math. Phys. 2 1 '
[OSe]
83
' (1973) and £2, 281, (1975).
Osteralder, K. and R. Seneor, "The S-Matrix is Non-Trivial in ($**)....",
to appear in
Helv. Phys. Acta. [Pa]
Park, Y.M., "Convergence of Lattice Approximations and Infinite Volume Limit in the (A.<J>^-atji 2 -u<J>) 3 Field Theory", Schladming Lectures, 1976, and J. Math. Phys. 16,1065, (1975), and to appear in J. Math. Phys.
[Pe]
Peierls, R. , 477, (1936).
[PT]
general refs. on phase transitions: See refs. [H],
Poc. Cambridge Phil. Soc. , 32,
[R], [FSS], [GM] and refs. given there.
Results that are somewhat complementary to what is discussed in these lectures may be found in L. Onsager, Phys. Rev. 6_5, 117, (1944), (Ising model), H. Araki and E.J. Woods, J. Math. Phys. _4, 637, (1963), (free Bose gas) T.H. Berlin and M. Kac Phys. Rev. 8£, 821, (1952),
267
E. Lieb and C.J. Thompson, J. Math. Phys. 10, 1403, (1969), and refs given there; (spherical model). N.N. Bogolinbov, Soviet Physics JETP, 7_, 41, (1958), W. Thirring and A. Wehrl, Commun. math. Phys. £, 301, (1966); (BCS model). K. Hepp and E. Lieb, Ann. Phys. 7_6, 3 60, (1972), and contribution to ref. [ E ] ; (Laser models). R. Griffiths, Phys. Rev. 136A, 437, (1967), R. Dobrushin, Funct. Anal. Appl. 2_, 44, (1968), (Peierls argument) R. Israel, Princeton University Series in Physics, Monograph, to appear; (general results on phase transitions), and Commun. math. Phys. 43_, 59, (1975) . F. Dyson, Commun. math. Phys. Jl_2_, 91, (1969). Roberts, J., "Local Cohomology and Superselection Structure", Centre de Physique Theorique, C.N.R.S. - Marseille, Preprint 1976. For results concerning the solitons of the free massless, scalar fields in two dimensions, see R.F. Streater and I.F. Wilde Nuclear Physics B24, 561, (1970); and P. Bonnard and R.F. Streater, ZiF, University of Bielefeld, Preprint 1975. Ruelle, P. "Statistical Mechanics - Rigorous Results", Benjamin, New York, 1969. Seller, E. and B. Simon, "Nelson's Symmetry and All That in the (Yukawa), and (
268
Field Theories", to appear in Annals of Phys. [Si]
Simon B., "The P (
[SG]
Simon, B. and R. Griffiths, Commun. math. Phys. 32/ (1973) .
[St E]
St. Exupery, A. "Le Petit Prince".
[St W]
Streater, R.F. and A.S. Wightman, "PCT, Spin and Statistics and All That", Benjamin, New York, 1964.
[SvB]
Sylvester G. and H. van Beijeren, "Phase Transitions for Continuous Spin Ising Ferromagnets", Yeshiva University, Preprint 1975.
[T]
See ref s. [H] [RJ ; also K. Huang, "Statistical Mechanics", Wiley and Sons, New York-London 1963.
[Y2]
Refs. on the Euclidean description of the (Yukawa)- quantum field model are: E. Seller, Commun. math. Phys. £2, 163, (1975) . E. Seller and B. Simon, J. Math. Phys. to appear ("finite mass renormalizations") and ref. [SeSi] . E. Seller and B. Simon, Commun. math. Phys. .45, 99, (1975) . O. McBryan, Commun. math. Phys. £4_, 237, (1975) . 0. McBryan, Commun. math. Phys. £5, 279, (1975) .
2
O. McBryan, Contribution to the Proceedings of the Int. Colloqu. on Math. Methods of QFT, Marseille, 1975. J. Magnen and R. Seneor, "The Wightman Axioms for the Weakly Coupled Yukawa Model in Two Dimensions", ZiF, University of Bielefeld, Preprint, 1975. A. Cooper and L. Rosen, "The Weakly Coupled (Yukawa)
Field Theory: Cluster Expansion
and Wightman Axioms", Princeton University, Princeton, 1976.
THE PURE PHASES (HARMONIC FUNCTIONS) OF GENERALIZED PROCESSES1 OR: MATHEMATICAL PHYSICS OF PHASE TRANSITIONS AND SYMMETRY BREAKING BY J. FROHLICH2
Contents I. Introduction. (Description of the problem, symmetries and broken symmetries). II. Lattice Systems and Generalized Processes. (The mathematical structure, interactions, finite systems, thermodynamic functions, equilibrium states, uniqueness theorems). III. The General Notion of Phase Transitions. (Definition of phase transitions, strategy for proving the existence of a phase transition, connection with symmetry breaking). IV. Reflection Positivity. (The cone of "reflection positive interactions", a generalization of the Holder inequality for traces and the "chessboard estimates", examples of reflection positive interactions, infrared bounds). V. Application to Classical Lattice Systems: Phase Transitions for Gibbs Random Fields. (Application of results of §§III and IV to the proof of existence of phase transitions for a large class of systems, conclusions). I. Introduction. This paper is written for mathematicians and mathematical physicists with some knowledge of stochastic processes and of the basic notions of statistical mechanics, but I have tried to explain what I believe are all major concepts, notions and definitions required for the understanding of the main results, i.e. I have tried to write these notes for the nonexpert at the risk of boring the expert and, perhaps, being a little imprecise here and there. (The expert may find some new results in §§IV and V.) All major recent or new results I am describing in this paper were obtained in collaboration with B. Simon, T. Spencer, E. H. Lieb and R. Israel. The reader is advised to consult references [l]-[4] for statements of the original results and complete proofs. Reviews of some of the material contained in these references and applications to relativistic quantum field theory may be found in [5], [6]. The reader may consult [7], [8], [1] for the original results on phase transitions in This is an expanded version of an invited address presented at the 83rd Annual Meeting of the American Mathematical Society under the title, Mathematical physics of phase transitions and symmetry breaking: New rigorous results in St. Louis, Missouri on January 28, 1977; received by the editors May 16, 1977. AMS (MOS) subject classifications (1970). Primary 60G20, 81A18. 'Work supported in part by the National Science Foundation under Grant MPS 75-11864. 2 A. P. Sloan Foundation Fellow. © American Mathematical Society 1978 165
Reprinted from "The Pure Phases (Harmonic Functions) of Generalized Processes" or "Mathematical Physics of Phase Transitions and Symmetry Breaking", by Jiirg Frohlich, Bulletin of the American Mathematical Society, Volume 84, Number 1-3 (1978), pp. 165 193, by permission of the American Mathematical Society.
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relativistic quantum field theory and their proofs. The general point of view adopted in this paper is developed in [9]-[ll] and references given there (see also [12]-[14]). Some of the weighting of different concepts and a few results have grown out of a course I have taught at Princeton University in the fall semester 1976. ACKNOWLEDGEMENTS. I am very much indebted to E. H. Lieb, B. Simon and T. Spencer for all they have taught me and for the joy of collaboration and to these colleagues and R. Israel for permission to present results that are in part not yet published. 1.1. Description of the problem. In these notes I try to outline a new mathematically rigorous theory of phase transitions and symmetry breaking which is rather general. It applies to Gibbs random fields and noncommutative generalizations of these, namely some class of quantum lattice systems and Fermion (Grassmann) lattice systems; the general concepts and methods involved may however equally well be applied to other physical theories, in particular relativistic quantum field theory. Many of the results I am going to describe were actually first obtained in the context of relativistic quantum field theory or at least motivated by it. This illustrates once again that mathematics can sometimes profit a lot from theoretical physics. The few proofs contained in these notes also show that, to use some words of Mark Kac, "in the right hands, Schwarz's inequality and integration by parts are still among the most powerful tools of analysis".3 Mathematically speaking, we shall be concerned, in this talk, with certain aspects of the theory of stochastic processes and their noncommutative versions; aspects that are somewhat related to probabilistic potential theory. In particular I want to discuss an analogy between phase transitions and the existence of nonconstant harmonic functions of a generalized process. The simplest example of a generalized process is a (multi-time) Markov process (or a multi-dimensional Markov chain), but the concept of generalized processes such as has emerged from the work of the past few years [9], [10], [15], [16], [12], [13], [11], [17] is more general and includes "Gibbs lattice fields" which are some sort of noncommutative random fields. The general problem I shall discuss may be posed as follows: Suppose we are given the local characteristics of a generalized process, in the commutative case e.g. a system of conditional probabilities or some equilibrium equations of the Dobrushin-Lanford-Ruelle (DLR) type [9HH], m the noncommutative case e.g. a "Gibbs condition" [13] or a "Gibbs variational equality" (all cases), can we prove general theorems giving a complete description of all harmonic functions of the generalized process (i.e. all Gibbs lattice fields with given local characteristics) or, in a physicist's language, the pure phases of the process"} Our results are two-fold: 1. Uniqueness theorems: Dobrushin's theorem [15], [11], [18] and its noncommutative versions [19], [20]. 2. A general method for proving the existence of "nonconstant" harmonic 3
M. Kac, On applying mathematics: Reflections and examples, Quart. Appl. Math. 30 (1972).
PURE PHASES OF GENERALIZED PROCESSES
167
functions, or, in other words, of several distinct pure phases with identical local characteristics. For aesthetical and educational reasons I shall emphasize the discussion of symmetries and symmetry breaking, by which I mean that the local characteristics of a generalized process (and in particular its "Gibbs potential"* resp. its Hamiltonian) may be invariant under a symmetry group which does not leave invariant some or all its nonconstant harmonic functions, i.e. which permutes the pure phases of the process among themselves. I briefly want to motivate this emphasis on symmetries and symmetry breaking. 1.2. The role of symmetry in mathematics and physics. For a beautiful discussion of the significance and the history of symmetry as a concept in mathematics, the natural sciences and the arts I refer the reader to Hermann Weyl's book entitled Symmetry [21]. (A new, somewhat more modern treatise on this subject would be desirable.) There are two aspects of "symmetry" of direct relevance to this paper: 1. A purely geometric aspect. 2. A dynamical aspect. The symmetries of geometric objects lead to the mathematical concept of symmetry and are one of several major roots for the development of group theory. (One might recall, here, Felix Klein's program of characterizing geometries by their invariance groups.) Geometric symmetries played and still play an important role in chemistry, crystallography, biology and other natural sciences. Historically they also played a role in dynamics, especially celestial mechanics. The Greeks believed that the motions of the planets and the moon would necessarily have to be circular or a superposition of circular motions, as the circle is a geometric object of maximal symmetry. The Platonic solids, namely the tetrahedron, the cube, the octahedron, the pentagon dodecahedron and the icosahedron found their way into celestial mechanics: Kepler tried to reduce the distances in the planetary system to the shapes of these solids which he alternatingly inscribed and circumscribed to spheres. The six spheres correspond to the six planets Saturn, Jupiter, Mars, Earth, Venus and Mercurius, known at that time, separated in this order by cube, tetrahedron, dodecahedron, octahedron and icosahedron. Kepler's attempt to understand laws of nature in terms of static, geometric symmetries is typical for natural philosophy in pre-Galilean and preNewtonian times. One of the most significant steps in the history of human thinking may well have occurred when the static, geometric concept of symmetry and its rather successless applications to dynamics were abandoned in favour of a dynamical concept of symmetry. With Newton physicists started to conceive the idea that it is the laws of physics describing the motion of particles, e.g. the planets, which are invariant under certain symmetry groups rather than the orbits of the particles themselves. This dynamical concept of "symmetry" is at the basis of some
168
J. FROHLICH
of the major revolutions in the physics of this century and is the one of main importance for the following. 1.3. Spontaneously broken symmetries. The idea that the symmetry group leaving invariant the laws that describe a physical system may be broken in the space of states of the system is in some sense the main theme of these notes. This idea will take a mathematically precise shape in the following discussion. The statement that some symmetry of the laws describing a physical system is broken means that the states of the system fall into equivalence classes invariant under the time evolution and under all possible measurements one can do at such systems which are however not invariant under a symmetry operation. Rather, symmetry operations permute these equivalence classes among themselves. A very striking example of a broken symmetry is found in biology: Living organisms contain the dextro-rotatory form of glucose and the laevo-rotatory form of fructose. There seems to be no a priori reason why it should not be the opposite or why living organisms of both kinds should not coexist (though perhaps coexistence might necessarily result in the extinction of one kind). We all know that this striking asymmetry in the chemical constitution of living organisms has been preserved over centuries and has so far not been destroyed by any changes of the environmental conditions. Thus it really presents an example of a broken symmetry. A trivial example of a broken symmetry in every day physics is a dumbbell shaped balloon with two distinct, asymmetric equilibrium shapes.
The physical laws describing the balloon do not distinguish between left and right. Without going into any detail I want to recall the fundamental role played by dynamical symmetries (spatial or internal) and their breaking in elementary particles physics. The idea that symmetries of physical laws may be broken spontaneously (or dynamically) in the state space of a system is a fundamental ingredient in all recent theories of elementary particle physics [22]. To conclude this introduction let me mention some examples of symmetry breaking in solid state physics that have a certain bearing on the subject of my talk: The first example is a ferromagnet, i.e. a system of bulk matter (e.g. iron) that has the property that when an external magnetic field in a fixed direction is turned off and the temperature is low enough it remains magnetized in the direction of the turned off field. Quantum mechanically, this phenomenon is not yet well understood. Another theoretically closely related phenomenon is Bose-Einstein condensation. In a quantum gas of particles satisfying Bose-Einstein statistics the ground state of the gas may have, at low temperatures, a macroscopic occupation. This is accompanied by the spontaneous breaking of a gauge
PURE PHASES OF GENERALIZED PROCESSES
169
group of the first kind isomorphic to SO (2) which leaves the physical laws describing the gas invariant. We shall also meet examples where a discrete symmetry is spontaneously broken. One of the most striking and fundamental phenomena is however, no doubt, the existence of crystals in nature, that is to say of states of matter which break the translational invariance of all physical laws. In the past two years mathematically rigorous theoretical understanding of the phenomenon of phase transitions and symmetry breaking in the framework of admittedly somewhat too simple models has made great progress. What I intend to do is to describe some of the mathematical and analytical aspects of this progress. I hope this introduction has convinced the reader that the problems I am going to discuss are important and that it has indicated what kind of mathematics is involved (generalized processes, Gibbs random fields, probabilistic potential theory). II. Lattice systems and generalized processes. II. 1. Description of the mathematical structure. Let £ denote some p-dimensional lattice. For simplicity I shall in general assume in these notes that £ = Z"; the simple cubic lattice. Many of the results I am going to indicate in the following depend however only on a certain reflection invariance property of the lattice £, i.e. a geometric symmetry property of £. (Some of the results, e.g. the uniqueness theorems, do not depend on any special properties of the lattice, at all.) Since there are only finitely many crystallographic groups in v dimensions (17 for v = 2, 230 for v = 3), it is a matter of consulting a table of these groups in order to give a complete list of all lattices having the required reflection invariance. At each site / G £ we are given an algebra 31, of operators. We must distinguish two cases (if we included Fermions it would be three): (C) classical case
% * c(a,-), where S2, is a copy of some fixed, compact Hausdorff space J20. In these notes Q0 c R^4, N = 1, 2, 3 , . . . , but in interesting cases (lattice gauge theories) Q0 may be a nonabelian compact group. We equip 21, with the sup norm and complex conjugation as an involution*. (QM) quantum mechanical case here 3C, is an isomorphic copy of some fixed, finite dimensional Hilbert space %j. The norm and the * operation on 2t, are defined in the usual way. For X G 9f(t) (the algebra of bounded subsets of the lattice £), we define STjr = ® 21,. H X c A" we consider %x to be the subalgebra of %x. defined by 4
Here RN is the one-point compactification of Rw.
170
J. FROHLICH
( ® «,-) ® ( ®
0
where 1, is the identity element in 91,. Technically speaking, {21*: X & ^ ( E ^ i s a family of C* algebras (in case (QM) von Neumann algebras), and they are called "local algebras". If a is a translation in the lattice £ and * is a finite subset of £ then X + a denotes the translate of X by a. The natural identification of 21* with 21*+,, is denoted ra. The * algebra 21 = U x ^ x i s called the algebra of all local observables; 21 is normed in the obvious way with the norm denoted ]| • ||. The group ( v a e £ } acts as a * automorphism group on 21. The completion of 21 in the norm || • || is denoted 2t and is called the algebra of all quasi-local observables. This algebra is a C* algebra. In the classical case, 21 is isomorphic to C(fl), with
a- x a,. zee
(Stone-Weierstrass theorem). A state p on 21 is a positive linear functional on 21 normalized such that P(l) = 1. with 1 = ®,-6eli ( t n e identity in 2t). The space of states on 21 is denoted 21*. The structure of 21* is analyzed in [14]. In the classical case, where 21 = C(Q), 21* is simply the class of all regular Borel probability measures on U. For all A e 21*, define
(C)
tr04)=f
II 4i«MK)>
a
x iex where fl* = X ,<=*&„ to* = (w,: i £ Jf}, and dfi is some probability measure on fi0; (QM)
tr(A) =
(\/d)Ti%x(A),
where %x = ® , e * %t, d is the dimension of DC*, i.e. d = (dim %y)w, with \X\ the number of sites in X, and T % is the usual trace on B (%x) = In the definition of tr, A' is an arbitrary finite subset of £. Hence tr extends by continuity to a state on 21 (note that tr is linear, tr(,4M) > 0, for all A 6 21, tr(l) = 1). II.2. Interactions ("Gibbs potentials"). An interaction $ is a function on the class %(£) of all finite subsets of £ with values in 21 such that, for X G #,(£), ($1)
$: X |-> $(X) G 21*.
($2)
(*3)
$(X)* = $ ( * ) ,
(translation invariance).
for all X.
The interactions <& form a real Banach space <© with norm ||$||_ = 2 * a o l l * W I I (0 is the origin in £).
PURE PHASES OF GENERALIZED PROCESSES
171
An interaction $ is of finite range if $(X) = 0 when diam X > r, for some finite r > 0. The iteractions of finite range are dense in <$. Symmetries. Let G be some compact, topological group acting as a group of continuous * automorphisms on 2I0. Clearly the action of G as a * automorphism group of 9t0 has a natural extension to a representation [rg: g G G } of G by continuous * automorphisms of the algebra 3f. We say that the interaction $ is G-invariant (or: G is a symmetry of 3>) if r g ($(X)) = $(X),
for all * e # / £ ) .
II.3. F/w'/e systems. We are now prepared to define the systems considered in the following. They represent a class of dynamical systems characterized by -a C* algebra of observables, —a one parameter * automorphism group of "time-translations" on this algebra, -the "states of interest" on the algebra of observables (in these notes we concentrate on the analysis of equilibrium states, defined below, see also [9], [13], [14]). First we study finite (dynamical) systems: For simplicity we assume that £ = Z ^ / 2 = Z" + ( l / 2 , . . . , l / 2 ) . Let A be some finite rectangle in Z"l/2. We identify opposite faces of A, i.e. we wrap A on a torus (recovering in this way a group of translations). We then regard 21A as the algebra of observables of a finite dynamical system in the region A. Given an interaction $ we construct the time-translation automorphisms of the system by means of a Hamiltonian XcA
Property ($3) guarantees that, for all finite regions A, / / * is selfadjoint. One may therefore define the time-evolution of an observable A G SIA by A b->af(A) =
ei,H*Ae-"H*.
In case (C) a,A is trivial, but in case (QM) it is in general not. The equilibrium state ("state of interest" in these notes) of a finite, dynamical system specified by the region A and the interaction $ is then defined by
pg;*(A) = tvie-^yhrie-^U),
(A e 3TA).
It is unique and it is invariant under all * automorphism groups of 9tA commuting with H® and leaving tr invariant. This means that "finite systems do not have phase transitions or symmetry breaking". Here some examples of interactions that are interesting for physics: (CI) fi0 = { - 1 , 1}; S0 is the function on fi0 defined by 5 0 (±1) = ± 1; 5, = T,(5 0 ); S, is called the (Jsing) spin at site /. $({/}) = - hSif h real; $({',./'}) = - J,-JS,SJ, with J,_j > 0 and /,_, > 0, for |« - j \ = 1; $(X) = 0, for |jr| > 3;
M{±1})~5-
J. FROHLICH
172
This is the so-called ferromagnetic Ising model. We note that, for h = 0, //* has a discrete symmetry H
A(SA)
=
#A(-5A)>
i.e. the dynamics is invariant under flipping all spins in A. This symmetry is shared by d\i and tr. (C2) Here S20 = SN~', the unit sphere in R*; SQ is the function assigning to a unit vector in S20 its /th component, and S 0 = (S0\ . . . , SQ), S, = T,(S 0 ). $({/}) = - h S„ h e R"; *({/J}) = - / , _ , S , - S,, with e.g. /,_, > 0, and Jt_j > 0, for |/ -j\ = 1; $ ( * ) = 0, for |A"| > 3; rfM(So) = fi(|So| -
l ^ X
This is the classical, ferromagnetic TV-vector model (when N = 3 it is frequently called the classical Heisenberg model). For h = 0, its symmetry group is obviously 0 ( A ) . REMARK. If in the definition of the interaction <& in models (CI) and (C2) we set Jj_j = 0, for |/ — j \ > 2, we obtain examples of multidimensional Markov chains. (The equilibrium expectations of these models have the local Markov property, [16], [17], [23].) (QM1)
%> = C 2 S + 1 ;
S = 1/2, 1, 3 / 2 , . . . ;
{SQ: i — x,y, z) a 2S + 1 dimensional, irreducible representation of the Lie algebra of SU(2); S 0 = (S$, S&, 50z); S(- = T,(S 0 ); $ as in (C2). This is the spin-S Heisenberg ferromagnet. For h = 0 it has 0(3) as its symmetry group. This is a difficult model which is not completely understood yet. (QM2) The same as (QM1), but in the definition of $ we require./,- • < 0, for |/ — j \ = 1, Jj_j = 0, otherwise. This is the spin-S Heisenberg antiferromagnet. Again the symmetry group is O (3) (for h = 0). For results see [2]. The main part of these notes is devoted to the discussion of new results concerning the equilibrium statistical mechanics of a dynamical system specified by the observable algebra 3lA and the dynamics //*. We are mainly interested in the systems obtained by taking the thermodynamic limit A—» Z>\/2- This limit must be taken before one can start to discuss phase transitions and symmetry breaking. (Finite systems never exhibit symmetry breaking!) I shall now introduce some basic objects of thermodynamics and statistical mechanics, in particular the so called thermodynamic functions. They are needed to define the "states of interest" for the infinite systems (A = Z\/2)11.4. Thermodynamic functions. We define the canonical partition function for a system in the region A with interaction <J> by Z A (/?,0) = t r ( e - ^ ) , and the free energy / A (/S, $) per unit volume by #A(J8,*)=
"(1/|A|) In Z A (/?,$).
Here /? is the inverse temperature. Let p be a translation invariant state of the infinite system, i.e.
PURE PHASES OF GENERALIZED PROCESSES P{T,{A))
=
173
p(A),
for all / e Z", all A e 21. We set pA = p/2I A
(the restriction of p to 21A).
We define the internal energy per site by
«A(P. *) = (l/|A|)p(//*), and the specific entropy by 5A(p)=-(l/|A|)tr(pAlnpA). (In case (QM) we consider arbitrary translation-invariant states on 21, in case (C) only those states p with the property that pA is absolutely continuous with respect to 11,,=A dn{u,). The class of all these states is denoted E'.) The following results (see [9] and references given there) summarize some rigorous thermodynamics for lattice systems with interactions 4> €E $ . THEOREM 1. For all real ft $ £ S , p £ E', the following limits exist {and are "independent of boundary conditions'" and of the sequence {A} —* Z"; A —> X" "in the sense of van Hove").
/ ( £ < ! > ) = Iiir. / A (/?,4>), A—*U
u(p,
|AT|-' p(HX)),
sA(p).
A—>Z'
(2) The function s{p) is affine and upper semicontinuous. (3) Gibbs {variational) ienqualtiy; s{p)< pu{p, *) - / ? / ( & $ ) . THEOREM 2. For a// rea/ /? awrf $ E <S f/iere existe a/ /ear? one translation invariant state ponty. such that s{p) = pu(p,<S>) - 0 / ( 0 , * ) {Gibbs variational equality [9]). Any cluster point of the sequence of states in E' ( Z A ( A 4»)-'tr A (e-*"J - ) ® M - ) } { A K Z , 2 satisfies the Gibbs variational equality, i.e. any limiting state of the equilibrium states of the finite systems, as A ^ Z ' / 2 satisfies this equality. REMARK 3. Existence of at least one such limiting state follows from a standard compactness argument. Moreover, under suitable assumptions on the interaction $, one can prove that a.(A) = n-lim a*(A) exists, for all A €E 2t; A^-Zi/2
see [9]. A different construction of the time-translations for infinite systems is discussed in [24].
174
J. FROHLICH
We are now prepared to define the "states of interest" for the infinite systems. II.5. Equilibrium states, uniqueness theorems. Any state p G E1 satisfying the Gibbs variational equality for an interaction 4> G $ and inverse temperature B is called an equilibrium state of the infinite system (specified by <&) at inverse temperature B. These are the "states of interest" in these notes. For people familiar with thermodynamics this definition looks most reasonable. I should however emphasize that the justification of this definition and its consequences is the subject of deep and difficult work that is still partly in "statu nascendi" [9], [10], [13], [14]. In particular it is possible to prove the equivalence of this definition with a characterization of equilibrium states in terms of local characteristics (systems of conditional probabilities in case (C)). This establishes a connection to the theory of generalized processes. Moreover there is a deep connection of the theory of equilibrium states and Tomita-Takesaki theory [25]. Since s(p) and w(p, $) are affine in p, we immediately conclude that the set A^* of all equilibrium states with given 0 and B is convex. As a matter of fact one has THEOREM 4. A^'* is a Choquet simplex, i.e. each equilibrium state p G A^'* is the resultant of (a unique probability measure supported on) extremal elements in Aft* {see e.g. [9], [31]; a simple proof in the classical case (C) is outlined in [19]). REMARKS. The extremal elements of A^* are called the pure phases of the infinite system with interaction 4>, at inverse temperature fi. If A'5'* happens to be the family of all stationary states of a multidimensional Markov chain (see the Remark in §11.3, Example (C2))-or some more general stochastic process-then the probability measures supported on the extreme points of A"-* are in 1-1 correspondence with the harmonic functions of the Markov chain. The next result asserts that, for small B, A'3-* contains typically only one state.
THEOREM 5. Let 3> be of finite range. Then, for sufficiently small fi ("high temperature"), A^* contains precisely one state p^*. If A and B are arbitrary operators in some local algebra then \p^(ArJ(B))-p^(A)p^(B)\ decays exponentially in \j\ ("exponential clustering"). REMARKS. 1. A more general result has been groven in the classical case (C) by Dobrushin [15]: For any interaction $ G ® (||$||_ < °°) A'3'* contains precisely one state, for B. small enough (but generally no exponential clustering); see also [11], [18]. 2. In the quantum case (QM) Theorem 5 is due to the author [19]. In one dimension (v = 1) Araki has proven a more general result of this type, for all B [20]. Preliminary results of this genre were obtained in [26], [27]. 3. Let G be a connected Lie group acting as a nontrivial, local, continuous * automorphism group on 21. Let $ be an interaction of finite range that is G-invariant. For v = 1 and 2 Dobrushin and Shlosman [28] have proven that
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175
in the classical case (C) all states in A'3'* are G-invariant. Hence there is no spontaneous symmetry breaking. We shall show that if $ has very long range then this conclusion is in general false. 4. The proofs of Theorem 5 and the results mentioned in Remarks 1-3 involve a great deal of concrete, hard analysis, in particular expansion methods, fixed point theorems, trace-inequalities, etc. HI. The general notion of phase transition. We consider an infinite lattice system characterized by an interaction 4> and the family A'3-* of all its translation-invariant equilibrium states at inverse temperature /?. DEFINITION OF PHASE TRANSITIONS. We say that a system with interaction <1> has a phase transition if the number of extremal equilibrium states in A^'* is not constant as a function of /?. From Theorem 5 we already know that, under suitable assumptions on $ and for \fi\ small enough, A^'* contains precisely one state. In this situation we speak of a phase transition if, for sufficiently large | /? |, Aft* contains more than one extremal state. Thus we will have proven the existence of a phase transition if we can find some /? and a state p^'* £ AA* which is not an extremal state. We must therefore formulate a criterion which permits us to decide whether some state p^'* is extremal or not. Since in the following p'5'* is some fixed element of AA*, we do not need the labels ft and 4> and write <<4> for p^'%4), A e 31. We now state this criterion, then indicate why it is correct and finally discuss the connections between phase transitions and symmetry breaking. Let A e 21 be some quasi-local observable and At = Tj(A) the translate of A by the vector /. We define
c = c(A):=
\im
-J-
*-*Z\/l
2
|A|
2
2
,eA
yeA
We define a truncated expectation A *A by (A*A)r =(A*A) - c. The state < — ) is not an extremal invariant equilibrium state (extreme point of A'3'*) if and only if, for some A e 91, (A*A)T<
(PT)
(A* A)-
\
Clearly this inequality is equivalent to c > \
(MF)
lim
A<-*Z1 /2
A theoretical physicist interprets this inequality as showing the presence of macroscopic fluctuations (or long range order) in the state < - ) , see e.g. [2]. If < — ) were extremal invariant then, clearly,
I. FROHLICH
176
lim
(AA%AAAy=0,
for all/I G 31.
Therefore the state < —) is not extremal invariant if and only if there exists some quasi-local observable A satisfying inequality (PT) or, equivalently, inequality (MF). We now develop some geometric notions that may be used to make this criterion plausible (and, as a matter of fact, to prove it; see [9], [10], [1], [2], [5]). Given any translation-invariant state < - > on the algebra 31, there exists a Hilbert space %, a cyclic vector Q G %, a representation -IT of 31 on % and a continuous, unitary representation {Ua: a 6 Z ' } of the translation group such that (A)
{Ara{B))
=
(2,TT(A)Q),
= {Q,ir{A)UaTr{B)$),
UaQ = 2,
for all a G Z".
This is simply the Gel'fand-Naimark-Segal construction; see e.g. [31]. Let %' be the closed subspace of % consisting of all translation-invariant vectors in %, i.e. for * G %', Ua* = ¥ . Clearly dim %' > 1, as Q G %'. THEOREM 6. < —) = p ^ * is extremal (moreprecisely: extremal invariant) if and only if dim %' = 1, i.e. %' = {&}.
The proof of this result involves using the definition of equilibrium states in terms of local characteristics (the DLR-equations in the classical case, the KMS- or Araki's Gibbs condition in the quantum case). Thus, in order to prove that < — > is not extremal, it suffices to show that dim %' > 2. Let P' denote the orthogonal projection onto the orthogonal complement ol%'. For A and B in 31 we define a "truncated expectation" of A and B by {BAY
=(7r(B)*a, P'ir(A)Q).
(For B = A* this definition coincides with the one given previously. This follows from the spectral theorem.) If dim %' = 1 then P1 is simply the orthogonal projection onto {£2}^ which we denote by 1 — Pu. Hence <^>r=<^(B>-<^><JB>. Since fi G %', P' < 1 - Pa, so that
{A*A)T^(A*A)-\{Ay\\ Thus, in order to show that < - > is not extremal, it suffices to show that <^>r<<^>-|<^>|2, for some suitable A G 31, i.e. inequality (PT)! Let At = T,(/4). The Fourier transform of
+ (ir(A0)Q, (1 -
P')*(A$l).
PURE PHASES OF GENERALIZED PROCESSES
177
The second term is independent of /, since (1 — P') projects onto the space of translation-invariant vectors. Therefore doi{k) = c8{k)dpk + doiT(k), for some c > 0; if c > |<^>|2 then (PT) and (MF) hold. Our strategy for the proof of the existence of a phase transition can now be formulated as follows: (0) Choose a suitable local observable A. (1) Derive an upper bound for (A*A}r: (A*A}T
= f duT(k)
< c,.
(II) Derive a lower bound on (A*A} — \(A)\2: (A*A) - \(A)\2 > c2. If c, < c2 then (A*A)-\(A)]2>(A*A)T, therefore dim %' > 2, i.e. < —> = p ^ * is not extremal (that means that < - > has macroscopic fluctuations in the sense of inequality (MF)). Next we want to show why a phase transition may be accompanied by the spontaneous breaking of a symmetry of the system. Let G be some compact topological group acting as a nontrivial, local * automorphism group {rg: g E G) on the algebra 3( of all quasi-local observables, and rg(%x) = 31*, for all X G 9f{£). Suppose now that the interaction 4> of the system is (7-invariant, i.e. Tg(
= <*>(*), for all X G «?/£).
This is a precise expression for "C is a symmetry of the system" (dynamical concept of symmetry). Next, assume that the equilibrium state < —) is (7-invariant, i.e.
rg(A) dg * 0. •'G
Suppose now that A satisfies Estimates (I) and (II) with c{ < c2, i.e. (A*A)
>(A*A)T.
(Estimate (II) is simplified because (A} = 0.) In this case < —) is not extremal, so that by Theorem 4, (A)=f
dp(xKA)x
(for all A G 31),
where S(A'5*) denotes the set of extremal states in A"'*, and dp is a proba-
178
J. FROHLICH
bility measure on S(A^*) with at least two different extremal states in its support. For Jp-almost all x> < — ) x is extremal, i.e.
(A*Ayx=(A*A)x-\(Ayx\\ and therefore dp(X)\(,A)x\2 = f
f
dp{x)l(A*Ayx-(A*Ayx]
= (A*A) - (A*A)T>
0,
i.e. (A}x ¥= 0, for a set T of x's of positive dp-measure. From
{^(-)V«eG,x£r} are equilibrium states of the system (they satisfy the Gibbs variational equality!). Thus the symmetry group G of the interaction $ (the "dynamics of the system") is broken by the states {
*(*)
for some probability measure dju. on G. In this case information on the structure of A'*'* is obviously rather complete. As proven by Slawny [29] and Lebowitz [30] this special situation is met in the ferromagnetic Ising models (of the form of Example (CI), §11.3) at all but possibly countably many temperatures. It follows from our discussion that the concept of phase transition is in principle more basic than the one of symmetry breaking (see [1], [6], [28] for a precise discussion of this point). In many important physical theories phase transitions and symmetry breaking come however in pairs. The remaining part of these notes is devoted to an outline of a general theory for the derivation of Estimate (I) and the application of our strategy (Estimates (I) and (II)) to the proof of existence of phase transitions in specific models. The starting point for our proofs of Estimate (I) is the following chain of simple observations: To get an upper bound on (A*A~)T = fB duT(k) it suffices to prove a pointwise upper bound on du T(k). Let / (k) be some continuous function on the first Brillouin zone B with J (k = 0) = 0. Then / (kf du{k) = / (A:)2 du
(1) 2
T
(k),
because J(k) S(k) = 0, as a measure on B. Taking the Fourier transform of equation (1) we conclude that
178 PURE PHASES OF GENERALIZED PROCESSES
(2)
<(/ * A«%(/ * A),) = ((J*A•)„(/
*
179
A)f.
Let C(h) = SQAC) = 2 T,(C)/r(/), where C is a local observable and h(i) is some function on Z'i/2. Then we obtain from (2) ( ( / *A*)(h)(J
* A){hj) = ( ( / *A*){h)(J
*
A){h))\
Hence if we can find some / such that its Fourier transform J is nonpositive, —J(k)~' is */"A:-integrable, and for all summable functions h ({J * A*)(h)(J * A)(h)) < -const 2
J
W)
i^jKj)
•J
then J (kf dtc T(k) < - const J (k) d'k, duT(k)
or
< - c o n s t / (k)~x d'k
which is the required upper bound. These are the estimates derived-under suitable assumptions on the interaction 4>-in the subsequent sections. IV. Reflection positivity. In this section we consider a certain cone of interactions which satisfy a positivity property called reflection positivity [4]. This property can only be formulated for lattices which have a certain reflection in variance (alluded to in §11.1). For simplicity we only consider simple, cubic lattices; but see [4] for more general results in the classical case (Q. In the language of a mathematical physicist reflection positivity expresses the existence of a selfadjoint transfer matrix. IV. 1. Reflection positivity infinite systems. Unless otherwise stated all the following results are proven in [4]. We let A c Z\/2 = Z" + ( 1 / 2 , . . . , 1/2) be the rectangle X [ - / , + 1 / 2 , / , - 1/2], i=
/,. = 1 , 2 , 3 , . . . ,
I
and we identify lt + 1/2 with - /, + 1/2, i.e. we wrap A on a torus. We then define A4 = X [-/,. + 1/2, ( - 1/2] x [ ± l / 2 , ±lt+
1/2].
The following figure represents a cross section of A (viewed as a torus): 5/2
0'
3/2
(,-1/2
/
-//+1/2 s
/
/ %
/
y
/
J K112
1-1/2 -3/2
180
J. FROHLICH
For / = ( / ' „ . . . , /„) G Z^ /2 we define OH = (/„ . . . , - / } , . . . ; , / „ ) . Thus if X is some subset of Z\/2, 9jX will denote the reflection of X at the plane /} = 0. We now define & to be the * automorphism of 21A which when restricted to 21,. is the identification map: 21,-* 21^, and @J(WX)= ® , e * &(%)• Obviously e>siA4 = 2t M . In the following we generally suppress the superscript^'. Any statement that does not contain explicit reference to a distinct j is true for ally = 1 , . . . , v. LEMMA 7. For all A G 2tA+, tr(AQ(A*)) > 0. PROOF.
t r ( / J 6 ^ * ) ) = tr A+ (/()tr A (©(/(*)) = tr A ^(^)tr A+ (^*) = tvA+(A)T^jA)
> 0. Q.E.D.
Note that because of translation invariance of A and tr the choice of an origin on the torus A is arbitrary, as indicated in the figure! DEFINITION. An interaction $ satisfies reflection positivity iff, for all finite X, 0(
* ( * ) = - 2 BiQ(Bi)
(or = - / BX@(BX) rfx),
JfnA-#0
where all the operators Bt (resp. Bx) are selfadjoint elements of 2lA+. ($4) quantum mechanical case (QM): For all X G ^ ( Z ^ ) , $(X) is real with respect to some complex conjugation of 21 (i.e.
2
* ( * ) = - 2 5,0(5,)
(or = - / BXB{BX) dx\
A"nA + ¥=0 JfnA_^0
where all operators B: (resp. Bx) are rea/ elements of 21A and B* = ± 5, (B*=±fi x );see[2],[4]. REMARKS. 1. A similar (somewhat more complicated) condition defines reflection positivity in Fermion lattice systems; see [19], [4]. 2. From now on we shall only consider the classical case (C). All results extend however to the quantum mechanical case (QM) if the interaction $ satisfies (<M) and if we only consider real observables A G 21, replacing Q(A*) systematically by ®(A). These conditions seem to exclude the treatment of the quantum mechanical ferromagnet (model (QM1)), permit however the analysis of the antiferromagnet (model (QM2)) and the so-called
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181
quantum mechanical x-y model within the gerneral theory described in this and the next section; see [2]. LEMMA
8. / / $ satisfies reflection positivity ($4) then
(H)
Ht = C + 0 ( C ) - 2
5/6(5.-)
with C = C* G 9fA+ and B, G 9t A+ ,/or a// /. PROOF.
= 2 *(*)+ 2 *(*) + 2 *eA +
xeA_
*(*)•
A-nA + y=0 AT)A_,<=0
We set C = Z * e A +
@(c) = 2
©(*(*)) = 2 <s>(ox)
= 2 *(*)• -YeA_ This and reflection positivity ($4) yield equation (H). The remaining part of Lemma 8 follows from the selfadjointness of <&(X), for all X, and from Q(X)e%x. Q.E.D. THEOREM
9. If $ satisfies ($4) then (A@(A*))A=
ZA(/3, 4))-' ir(e-^
A@{A*)) > 0,
/or a// A G 3tA+, arbitrary A 0. Before we prove Theorem 9 we pause for a REMARK. In the classical case (C) we can derive Theorem 9 from a more general notion of reflection positivity: ($4') An interaction <J> is reflection positive (in the generalized sense) if ©($(*)) = <&(0X) and, for all finite rectangles A, 2 tr(A
for all A e 8tA+ with tr(A) = 0. This version of reflection positivity is weaker than condition ($4). One can prove that ($4') is equivalent to the inequality <^@(/l*)>A > 0, for all A G 3lA+, arbitrary A and a// fl > 0. Finally there is a third (even more general) notion of reflection positivity that is applicable and useful in the classical case: Without loss of generality we may assume that the interaction $ is normalized such that
182
J. FROHLICH
(T)
tr($(X)) = 0, for all X e
9/Z\/2).
(If (T) does not hold we may introduce a physically equivalent interaction $ defined by $(X) =
2
tr(^$(Z)9(/l*)) > 0,
foralM £ U A finite 9tA+. For a large class of classical interactions 4> one can use correlation inequalities to prove that-under the hypothesis that $ satisfy (4")-Theorem 9 holds in the thermodynamic limit A = Z\/2; (in this case Theorem 9 may fail for finite A). Our general theory of phase transitions applies under these circumstances (§§IV.3 and IV.4; see [4] for proofs). PROOF OF THEOREM 9. We only consider the classical case which is somewhat simpler than the quantum case, since all operators commute. Because ZA( fi, $) > 0 we must only show that tr(e~PH- A
t^e-Wi
2
-**n
tr
•0,
oo,
all/
Vm~J. /
Each term in the sum on the r.h.s. is positive, by Lemma 7. Q.E.D. COROLLARY 10 [7], [34], [3] ("CHESSBOARD ESTIMATE"). Assume that $ satisfies reflection positivity ($4). Let At £ 31, be self adjoint, for all i e A. Then
I/
\ I
/
I * I'EA
'A|
I ' E A VyeA
\,/|A|
( n ^ ) U n ( n ^w) • 'A
The proof of Corollary 10 is based on the following, inequality (one of the fundamental inequalities of Lp- and *5p -theory). THEOREM [3] (GENERALIZED HOLDER INEQUALITY). Let V be some complex vector space (e.g. an algebra) and * a conjugation on V. Let ube a multilinear functional on Vx21 with the following three properties
(A)
( 0 ( 5 , , . . . , Bv) - u(B2l, Bx,...,
(cyclic invariance)
B^x
)
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183
(B)for arbitrary B[, ... , Bj in V, i = 1, 2, 3, . . . , Mu =
u{(B!)*,...,{Bi)*,Bi,...,B{)
defines a positive semidejinite matrix; in particular (C)
u(Ar, ...,A*x,Bv...,Bt)
= u(Br,
...,Bt,Ai,...,A,).
Then B*a ) , / v ,
|«o(*„ . . . , Bv)\ < IT
and 1/2/
\B\\2l~co(B,B*,...,B,B*) is a seminorm on V. We give the proof for the case 1 = 2:
By properties (B) and (C) of co we have the Schwarz inequality (S)
«(/»„ B2, B3, B4) <
B4)l/2.
Using (S) and the cyclic invariance of w, i.e. (A), we obtain w(A, B,C,D)<
u(A, B, B*,A*)i/2
«(/)*, C"\ C, Z)) , / 2
= a(A*, A, B, B*)l/2 co(A D*, C*, C ) , / 2 < co(A*,AfA*,A)l/4
w{B, B*, B, B*)i/4
X co(D, D*, D, D*)l/4 co(C*, C, C*. C ) , / 4 - w(/l, /I*, A,A*)l/4 u(B, B*, B, B*),/4 X
D*)l/4
which completes the proof for 1 = 2. The proof for general / is similar, but involves an additional maximization argument; see [3]. APPLICATION 1. For a = 1 , . . . , 2£ and /, they'th component of / 6 Z* /2 , define Ba=
II ij*°-lj-l/2
^(B,,..
At,
with Ai £ 3f„ and
+a
., B2lj) = ( II 4-) , »iSA
' A
where < - >A is the equilibrium state of a finite system with an interaction 4> satisfying reflection positivity. Then, by Theorem 9 and the translation invariance of <-> A , w, has properties (A)-(Q- If we now apply the generalized Holder inequality to «, and let.;' vary from 1 to v we immediately obtain Corollary 10. APPLICATION 2 {Holder inequality for traces). Let C„ . . . , C„ be arbitrary matrices (or functions) and tr the usual trace (integral), as above. Let
184
J. FROHLICH
m,, . . . , mn be rational numbers > 1 with
£ /«,-= i. *=i
Choose a positive, even integer 2/ such that mf '2/ is a positive integer, for all /'. We define (for arbitrary matrices (functions) Bx . .. B2/) « ( f i „ . . . , 5 2 / ) = trf II B, which has obviously properties (A)-(C). Let C, = \J\C\ be the polar decomposition of C, and define Di = Ci|C,.|m'/z',
Z>, = IC,!^ 2 ';
then C, = DlD(2,/m')-\
for all / = 1, . . . , n.
Hence tr(C1...C„) = t r | n
AA(2'/m)-')
12// < n [ 4 4 A*,... )l/2'T "
n trdQr)'/"' = n iic, m, and we have used that
«(A, A*, • • •) =«(4 A*, • • •) = tr(|c,r). The inequality just proven and a simple continuity argument yield the general Holder inequality for traces (resp. arbitrary central states). Other applications of the generalized Holder inequality include proofs of most of the important inequalities for traces and KMS states. IV.2. Examples of interactions satisfying reflection positivity. Again, we restrict ourselves to the classical case. THEOREM
11. Let $ be the interaction defined by
*({«}) = hrn(A),
<&({«, m}) = - 2
J^mT„(Ba)Tm(Ba),
a
where n, m are lattice sites, and A, Ba selfadjoint operators in 2t0. Furthermore
for\X\>3.
Assume, in addition, that for arbitrary {cn} C C,
PURE PHASES O F G E N E R A L I Z E D PROCESSES
(u\ V
Jna-&mCncm > 0, for allj and all a.
2
'
185
n,->0 mj>0
Then $ satisfies reflection positivity ($4). REMARK. In the classical case this theorem was first proven by the author [32]; see also [33]. The general case appears in [4]. PROOF. We only consider the case v = 1 (the general case is hardly more difficult). Then condition (R) is
2
Jll)ctCj>Q.
!>0
By a version of Bochner's theorem and the fact that Jja) —»0, as |y'| —» oo, this implies that for |«| > 1 J(na)= r + l Al"l-' dpM(X),
(R')
for some positive measure dp^ on [— 1, 1]. We now consider a fixed a and set Aa)=Jn,
dp^ = dp and
r„(Ba) = S„.
Then, using (R'),
2 /. +
W = 2 W ^ , + 2 J2,-o+J)S,sej
,//-l/2
-/
2
\//-'/
2 A"-/ 5j
•'-l
\«-l/2
+r y
-I
2
2
/\m=l/2
A'-,/2-^J
s \n-I/2
\
Xm-,/2^mUP(A) /
2
A'-'/2-%mU(A)
/ \ m=l/2
/
which, in view of ®(Sm) = S ^ , is precisely of the form ($4). REMARK. Conditions analogous to (R) and (R') can also be derived for many body interactions (^(A') ¥= 0, for some X with \X\ > 3); see [4]. An explicit example of a y,- • in v dimensions satisfying condition (R) is j
=
' ""•'
[const |/ -y|-<"~ 2 + "), \ const',
/^=/, / =j,
where i) is an arbitrary, positive number. For J to define an admissible interaction (i.e. $ G <$) we must require TJ > 2; see [4]. For a field theorist the verification of condition (R) for this choice of 7,_y is a rather easy exercise (catch word: conformal invariant two point functions). IV.3. Extension to the thermodynamic limit. Let < —> denote some equilibrium state which is an arbitrary cluster point of the family { < - ) A ® tr A r (-)} A c Z . of equilibrium states of finite systems with an interaction <S? satisfying (<^1H$3) and 2x3 (1 /2....../2)II^WII < « .
J. FROHLICH
186
A standard compactness argument shows that such a family of states has always at least one cluster point. Any cluster point is automatically translation invariant. We let 9l ± denote the normclosure of U ACZT/2 ^ A ± THEOREM
12. Assume, in addition, that <E> satisfies reflection positivity ($4).
Then (1)
(AQ(A*))
> 0, for all A G 3t+-
(2) For any finite subset B C Z^ /2 and operators A, = A* G 3t„ / G B, I
(n
\
A)
< n
'
i£B
Mes
\1/|A|
I
- n
I™ A z
(n
- 'i/2
\,eA ,
iim
(n
T,.,.^,-)) 'A v i/|A|
TJ^A,))
.
REMARKS. Part (1) is an immediate consequence of Theorem 9. Part (2) follows from Corollary 10 and a well-known theorem which asserts that the free energy per site f(B, $) (see Theorem 1, (1)) of infinite lattice systems with interactions $ of the type considered here is independent of "boundary conditions." See [3] for a detailed proof. IV.4. Infrared bounds. In this section we prove a basic estimate on the Fourier transform dwT(k) of the truncated expectation (A^A/)7 considered in §111. This estimate is a precise version of Estimate (I) in the strategy for proving the existence of phase transitions described there. Let $ be an interaction satisfying reflection positivity (4>4) and
*({«,./•})=-./,•_, S.-S,, where S, = T,(B), B = (/?„ . . . , BN) is an AMuple of selfadjoint operators in 3l0, and J satisfies condition (R). THEOREM 13 (INFRARED BOUND). Let J denote the Fourier transform of J and dum(k) the one of <S0 • S,) (r) . Then, under the assumptions on $ stated above and in the classical case (C),
duT{k)
<
,. N . d% 2B(J (0) - J (k))
where B~l is the temperature. In the quantum mechanical case (QM) the analogous estimate is somewhat more complicated, but see [2], [4]. PROOF. We prove Theorem 13 only for a special class of classical models (the general case is treated in [4]). They are defined as follows: fi0 = R^, S 0 = (S 0 \ . . . , SQ), with S0(x) = x' (the ith component of x), for all x G R"; S, = T,.(S0), for;
G
Z\/2.
The a priori measure dp used in the definition of the expectation tr (tr^-(-) =fQxUiEx ^M(S,)) is assumed to be quasi-invariant under the translations of R^ (this is no loss of generality, since the general case will follow from the one considered here by a limiting argument). The interaction $ is given by $({«}) = - h - S „ , h £ R " (independent of ri); $({n, m}) = / n _ m S n -S„ I ,$(X) = 0 , f o r | X | > 3 .
PURE PHASES OF GENERALIZED PROCESSES
187
Without loss of generality we may normalize Jn_m such that
(N)
2 - / n - m = i(o) = o.
(This can be achieved by a suitable choice of J0 and hence amounts to a trivial redefinition of dp..) For each g G R^, we define =
g"
s
+«>4i(s + 8 )
=
hg
^ ( s + g)
Consider now the equilibrium expectation /e-)8Sn.„/„_„(2S„g„-g„-gJ>.
We note that
- 2 [^- m (2S n • gm - g„ • gm) - $({«, m})] = 2 -/n-m(s„ - g„) • (sm - gm). This suggests a change of variables S^, = S„ — g„. Then ehS»^(S„)-^(S>hS"^(S;). Let < — >A denote, as usual, the equilibrium expectation of the system with interaction <£ in the region A C Z"i/2, at inverse temperature p. Then the substitution S„ -^ S'„ 4- g„ yields /e-PZ„.^n~m(2S„gm-g„-gm)\ A
( II F g: (S,)\ , f o r a l l A Q Z ^ . * zeA 'A We now apply the chessboard estimate (Corollary 10, Theorem 12) to the r.h.s. of this identity. This yields
( n Fg,(s,))
1/|A|
FJS,))
»zeA ' A / e A > yeA ' Aj for all bounded A. Now we undo the substitution S„ -»S^, + g„ on the r.h.s. of this inequality and obtain
(n
F^SJ))
= <,-^,^^(2s, g ,- g , g ,) >A
Using the normalization (N) ofJ„_m we find 2 ^ - m ( 2 S „ - g , - g , - g , ) = 0, whence the r.h.s. of the equation above equals 1. This still holds in the limit A = Z"l/2 (use Theorem 12!). If we put everything together we conclude (e-fi-z„.„J„-m(2s„-gm-g„-gm)y
<
]_
188
J. FROHLICH
Next, scale g, to eg, and expand the l.h.s. of this inequality in powers of e. This yields 1 - 2e/3 2 /„_m<S„> • gm
+ \^n2
2
/B_m/B._M.<(sfl.gm)(s,-gM.)>
+ e2P 2 ^ ^ m g n - g m + 0 ( £ 3 ) < 1. Now note that S„/„_ m <S n >- gm = 0. This follows from (N) and the fact that <S„>- gm is independent of n, by translation invariance of < — >. If we divide the inequality above by e2 and let e tend to 0 we obtain
2
^- m ^- m <(s n -gj(s„.- gm ,)>
(IR')
Next choose g„ such that its Fourier transform is peaked near some momentum k £ B. Then (Fourier transformation of) (IR') gives
-N(2BylJ(k).
J (kfdu(k) < Since J (k = 0) = 0, by (N), we obtain
(IR) dco(k) = cS(k)d"k + dcoT(k) < c8(k)
-
N 2pj{k)
d'k.
Q.E.D.
V. Applications to classical lattice systems: phase transitions for Gibbs random fields. In this final section we consider classical lattice systems (the so-called classical ferromagnets), i.e. models (CI) and (C2) defined in §11.3. More results on these systems and a detailed study of a class of quantum lattice systems may be found in [2], [4]; for applications to Fermion lattice systems we refer the reader to [4], [19]. Some of our results {Ising models with long range interactions in one dimension) have earlier been obtained by Dyson [35]. In models (CI) and (C2), £20 = SN~\ the unit sphere in R", N = 1, 2, 3 , . . . . The measure d\x. on fi0 used in the definition of the trace tr is given by rf/i(S) =
fi(|S|-
\)dNS.
We assume that the interaction 4> satisfies reflection positivity (4>4); see §IV.l. Moreover $ ( { « } ) = - h • S„,
h a fixed vector in R",
*({«,«})= -/ n - m s„-s m , where J satisfies the reflection positivity condition (R) (resp. (R')) of §IV.2. For 1^1 > 4, $(X) is an O(N) invariant polynomial in Sx = {St}iex (compatible with (<M)). Obviously <S, • S,> = 1, for any state < —>.
PURE PHASES O F G E N E R A L I Z E D PROCESSES
189
If h = 0 these models have O (N) as their symmetry group. Let < —) denote some infinite volume (thermodynamic) limit of the equilibrium states {< —>A = P A ' * ( - ) } , where {A} is a sequence of rectangles increasing to Z ^ 2 ( se e Theorem 2). Under these assumptions we may apply the infrared bound of §4. Thus (IR)
du(k) < (c8(k) + N^lp(j
(0) - J
(k))yl)d'k.
This inequality gives THEOREM 14 [1], [4]. If
d'k J(0)-J
(k)
is finite then c is positive for fi > NI(v, J)/2. For h = 0 c > 0 implies the existence of macroscopic fluctuations in the state < —> and hence of a phase transition and O (N)-symmetry breaking in pure phases. PROOF.
If we integrate inequality (IR) we obtain <S,. • S,> = 1 = JT dco(k)
I{v, J).
This proves the first part of the theorem. If h = 0 then the state < — > is O (Af)-invariant, so that <S,-> = 0
and c = <S, • S,> - <S, • S,> r > 0
implies the existence of a phase transition. The remaining assertions therefore follow from the results of §111. Q.E.D. Finally we derive conditions for the finiteness of I(v, J). For simplicity we only consider the case where /, > 0, for all / ^ 0 (but see [4] for more general results). Let Sj be the unit vector with components SM,j — 1 , . . . , v. 15 [1], [4]. Suppose that JnjSj > 0, for some nJ - 1, 2, 3 , . . . ; e.g.
PROPOSITION
j = \,...,v;
JSj = /
> 0,
for all]
=
\,...,P
("ferromagnetic nearest neighbor coupling"; see [1], [4]). Then I(y, J) is finite, for all v > 3 and hence there is a phase transition. PROOF.
/ (0) - J (k) = 2
//(I - cos(A: • /))
> 2 /^(l-cosC*'-**)) > \
min (j„,Sj (nJf)\k\2
> const \k\2
190
J. FROHLICH
Thus (J(IS) ~ J(k))~' is 3. Q.E.D. Next we briefly discuss phase transitions in v = 1 and 2 dimensions. From §IV.2 we know that fc|i|-,
y
1*0,
satisfies reflection positivity, for all a > 0 (v = 1 or 2). We assume that J, > 0, for all / =£ 0, and (1) /, > c|/|~°, as|/|-»oo, for some positive constant c. Existence of the thermodynamic limit requires (2) a > v. The angle between a vector / £ Z" and a vector k G B is denoted * (/, &). 16 [4]. I(v, J) is finite ifa< 2v. For v < a < 2v and h = 0 there exists a phase transition and O(N)symmetry breaking, for (i sufficiently large {low temperatures). THEOREM
PROOF. By Theorem 14 it suffices to show that I(v, J) < oo, for a < 2v. We first estimate /(0) - J(k): Clearly J(0) - J(k) > 0, for k =£ 0, under our assumptions on /,. It therefore suffices to estimate the behaviour near k = 0. We assume that \k\ < 1. Then
J (0) - J (k) = 2
4 0 - cosO • /)) >
ieZ'
2
4 (1 - cos(A: • i))
sc (U) < »/4
> ct\k\2
2
m2,
for e small enough,
~"dx,
fore|/c|-' » 1 ,
3C (i,k) < ir/4
(3)
|/|<e|*|-'
>
c2|Afj
i
2 2+a = c 3 |/c| |/c|-- ,
for a < v + 2,
= c3\k\"~'. Hence (7(0) - /(A:)) -1 isrf"A:-integrableif (4) REMARKS.
« < 2v.
Q.E.D.
1. Let S„ be an ^-dimensional, classical spin. Let $ be defined
by H{n,m})=-J„_mSn-Sm, where / satisfies the reflection positivity condition (R) (resp. (R')) of §IV.2; $(A-) = 0, for \X\ *= 2. THEOREM
14'. Under these assumptions and for N>2the
condition
PURE PHASES OF GENERALIZED PROCESSES
I(",J)=f
~
191
J
/ (0) - y (k) is necessary and sufficient for the existence of spontaneous magnetization. B
Theorem 14' follows directly from Theorem 14 and the Mermin-Wagner theorem [36]. 2. For N = 1 and $(X) = - JxIliEX S„ Jx > 0, Slawny [29] has obtained sharp upper bounds on the number of equilibrium states in A^*, for small /?. Using recent results of Lebowitz [30] one can extend these bounds to all but possibly countably many values of the inverse temperature /?. As an example we mention that for such models for which $({«, m}) = — J„-mS„ • S„, with J„ > 0, for |«| = 1, there exist at most two translation invariant equilibrium states at all but possibly countably many values of /? [30]. In the two dimensional Ising model with nearest neighbor ferromagnetic interaction one has: (1) In an external magnetic field (i.e. <£({«}) = - h- S„, with h =£ 0) there exists precisely one translation invariant equilibrium state. The same is true for h = 0 and all fi < fic, where fi~' > 0 is the critical temperature. (2) For h = 0 and fi > (ic there exist precisely two translation invariant equilibrium states with opposite spontaneous magnetization. The proofs of these results are surprisingly simple. We urge the reader to consult [29], [30]. It has recently been shown in [37] that the plane rotator model in zero external field (N = 2, J-,_j > 0, h = 0) has a unique, extremal, translation invariant equilibrium state whenever there is no spontaneous magnetization. (3) For application and adaptations of the general theory of §§IV and V to quantum lattice systems, including non-relativistic fermions, see [2], [4]. We conclude with some open problems: 1. Is there a generalization of the theory described in §§IV and V which is applicable to the quantum mechanical Heisenberg ferromagnet? (The present theory only covers the anti-ferromagnet and the x-y model; see [2].) 2. How can one analyze phase transitions in systems on irregular lattices and with impurities {other than Ising type models with discrete spins?) 3. How close is the connection between reflection positivity (Theorem 9) and the validity of infrared bounds (Theorem 13, §IV.4)? It is known that there exist classical lattice systems which violate reflection positivity (and translation invariance) for which infrared bounds are true (E. H. Lieb, private communication). The only known such examples are however systems without phase transitions. 4. Is there a generalization of the Slawny-Lebowitz theory [29], [30] concerning the number of translation invariant equilibrium states to general classical or quantum lattice systems? 5. Consider a lattice system at the critical temperature (clustering, but not exponential). Is there a connection between reflection positivity and scaling behaviour at large distances? It is easy to see that if there is scaling, the scaling limit of the correlation functions of a system satisfying reflection positivity are the Euclidean Green's functions of a relativistic quantum field theory satisfying the Osterwalder-Schrader axioms. (For results concerning
192
J. FROHLICH
phase transitions and the critical point in relativistic quantum field theory see [7], [8], [1], [4].) REFERENCES 1 J. Frohlich, B. Simon and T. Spencer, Comm. Math. Phys. SO (1976), 79. 2. F. Dyson, E. H. Lieb and B. Simon, Phase transitions in quantum spin systems with isotropic and non-isotropic interactions (to appear). See also Phys. Rev. Lett. 37 (1976), 120. 3. J. Frohlich and E. H. Lieb, Existence of phase transitions for anisotropic Heisenberg models, (to appear). See also Phys. Rev. Lett. 38 (1977), 440. (Note: The proofs of the results on quantum mechanical ferromagnets announced in references 2 and 3 contain a gap, as they were based on an incorrect lemma of 2.) 4. J. Frohlich, R. Israel, E. H. Lieb and B. Simon, papers concerning phase transitions in lattice systems with long range interactions (to appear). 5. J. Frohlich, Phase transitions, Goldstone bosons and topological superselection rules, Acta Phys. Austriaca 15 (1976), 133. 6. J. Frohlich and T. Spencer, Phase transitions in statistical mechanics and quantum field theory, Proc. 1976 Cargese Summer School (to appear). 7. J. Glimm, A. Jaffe and T. Spencer, Comm. Math. Phys. 45 (1975), 203. 8. , Ann. Physics 101 (1976), 610; 101 (1976), 631. 9. D. Ruelle, Statistical mechanics, Math. Phys. Monograph Ser., Benjamin, London and Amsterdam, 1969. 10. , in Mechanique Statistique et Theorie Quantique des Champs, Les Houches 1970, C. DeWitt and R. Stora, editors, Gordon and Breach, New York, 1971. 11. O. E. Lanford III, in Statistical Mechanics and Mathematical Problems, Battelle Recontres 1971, A Lenard, editor, Lecture Notes in Phys., vol. 20, Springer-Verlag, Berlin and New York, 1973. 12. H. Araki and P. D. F. Ion, Comm. Math. Phys. 35 (1974), 1. 13. H. Araki, Comm. Math. Phys. 38 (1974), 1. 14. R. Israel, Tangents to the pressure as invariant equilibrium states in statistical mechanics of lattice systems, Thesis, Princeton Univ., 1975; Princeton Ser. in Physics, Princeton Univ. Press, Princeton, N.J. (to appear). See also Comm. Math. Phys. 43 (1975), 59, and D. Ruelle, On manifolds of phase coexistence, I.H.E.S. Preprint, 1975. 15. R. L. Dobrushin, Functional Anal. Appl. 2 (1968), 302; see also Theor. Probability Appl. 13 (1968), 197. 16. E. Nelson, J. Functional Analysis 12 (1973), 97; 12, (1973), 211. 17. F. Spitzer, Amer. Math. Monthly 78 (1971), 142. 18. R. Israel, Comm. Math. Phys. 50 (1976), 245. D. RueUe, Comm. Math. Phys. 9 (1968), 267. 19. J. Frohlich, Lectures on equilibrium statistical mechanics, Princeton Univ., 1976/77 and paper in preparation. 20. H. Araki, Comm. Math. Phys. 44 (1975), 1 (also 14 (1968), 120). 21. H. Weyl, Symmetry, Princeton Univ. Press, Princeton, N.J., 1952. 22. S. Coleman, Secret symmetry: An introduction to spontaneous symmetry breakdown and gauge fields, Proc. of the 1973 Internat. Summer School of Physics, Ettore Majorana, A. Zicchichi, editors. 23. J. Frohlich, Advances in Math. 23 (1977), 119. 24. D. Ruelle, J. Math. Phys. 12 (1971), 901; Helv. Phys. Acta 45 (1972), 215. J. Frohlich, Helv. Phys. Acta 48 (1975), 3"55and paper in preparation. 25. M. Takesaki, in Statistical Mechanics and Mathematical Problems, see reference 11, Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes in Math.,vol. 128, Springer-Verlag, Berlin and New York, 1970. See also: R. Haag, N. Hugenholtz and M. Winnink, Coram. Math. Phys. 5 (1967), 215. 26. W. Greenberg, Comm. Math. Phys. 11 (1969), 314. 2 7 . 0 . E. Lanford III, Cargese Lectures, Gordon and Breach, New York, 1969. 28. R. L. Dobrushin and S. B. Shlosman, Comm. Math. Phys. 42 (1975), 31. 29. J. Slawny, Comm. Math. Phys. 35 (1974), 297; see also Comm. Math. Phys. 34 (1973), 271
PURE PHASES O F G E N E R A L I Z E D PROCESSES
193
and W. Holsztynski and J. Slawny, Phase transitions in ferromagnetic spin systems at low temperatures, preprint, 1976, also Lett. Nuovo Cimento 13 (1975), 534. 30. J. L. Lebowitz and A. Martin-Lof, Comm. Math. Phys. 25 (1972), 276. J. L. Lebowitz, Coexistence ofphases in Ising ferromagnets, I.H.E.S. Preprint, 1976. 31. O. E. Lanford HI, in Mechanique Statistique et Theorie Quantique des Champs, see reference 10. 32. J. Frohlich, unpublished notes, 1976. 33. G. Hegerfeldt and C. Nappi, ZiF-Univ. of Bielefeld, preprint, 1976, Comm. Math. Phys. (to appear). 34. E. Seiler and B. Simon, Ann. Physics 97 (1976), 470. (For earlier estimates of this type see reference 23. A general version of these estimates also appears in J. Frohlich and B. Simon, Ann. of Math. 105 (1977), 493, and in reference 3. The methods of reference 3 are inspired by the paper of E. Seiler and B. Simon.) 35. F. Dyson, Comm. Math. Phys. 12 (1969), 91; 21 (1971), 269. See also H. Kunz and C.-E. Pfister, ZiF-Univ. of Bielefeld, preprint, 1976, for a recent application of Dyson's methods. 36. N. D. Mermin, J. Math. Phys. 8 (1967), 1061. See also M. Kac, Quart. Appl. Math. 30 (1972), 17; O. McBryan and T. Spencer, Comm. Math. Phys. 53 (1977), 299. 37. J. Bricmont, J. R. Fontaine and L. J. Landau, On the uniqueness of the equilibrium state in plane rotators, Univ. Louvain, preprint UCL-IPT-77/03. DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY 08540
193 SPONTANEOUSLY BROKEN AND DYNAMICALLY ENHANCED GLOBAL AND LOCAL SYMMETRIES
Jiirg Frohlich Institut des Hautes Etudes Scientifiques 35, route de Chartres F-91440 Bures-sur-Yvette
Abstract. I re-examine the notions of spontaneously broken, global and local symmetries and discuss them in terms of some examples in quantum field theory or statistical mechanics. I then briefly recall some basic ideas and facts about the renormalization group. They are used to introduce and discuss the concept of dynamically enhanced (or "generated") asymptotic symmetries.
Contents. 1. Remarks on the historical development of symmetry concepts; basic definitions. 2. Spontaneous breaking of global symmetries. 3. "Spontaneous breaking" of local symmetries. 4. Renormalization group ideas. 5. Symmetry enhancement : Generation of asymptotic, global and local symmetries.
Remark. Most of the ideas and results described in these notes have been worked out in collaboration with T. Spencer to whom I am deeply indebted for his clear insights and his generosity. I have also benefitted from collaborations with D. Brydges, G. Morchio, C. Pfister, E. Seiler, B. Simon and F. Strocchi to whom I extend my gratitude. These notes are intended for light reading. The interested reader is advised to consult the following references for further study : Sect. 1
[1,2,3,4] .
Sect. 2
[5,6,7]
Sect. 3
[2,10,11]
Sect. 4
[13]
Sect. 5
[14,15,16]
and the reviews [2,3,4,8,9] . and in particular [12] .
(see also the contribution of H.P. Diirr).
Reprinted from Unified Theories of Elementary Particles. Critical Assessment and Prospects, P. Breitenlohner and H.P. Diirr (eds.), Proceedings of the Heisenberg Symposium in Miinchen, July 16 - 21, 1981, Lecture Notes in Physics 160, Springer-Verlag Berlin, Heidelberg, 1982. © Springer-Verlag 1982
194 118
1. Remarks on the historical development of symmetry concepts; basic definitions.
1.1. We distinguish two aspects of symmetry a) a geometrical, static aspect, and b) a dynamical aspect. Moreover, dynamical symmetries can either be bl) global symmetries, or b2) local "symmetries". a) The physics
of the antique and of the middle ages (Ptolemy, Kepler...),
before Galilei and Newton, only knew the static, geometrical aspect of the concept of symmetries. (The orbits of the planets were believed to have high symmetry. An attempt was made by Kepler to explain the interplanetary distances in terms of the platonic solids. Matter was conceived as being built of highly symmetric "elementary bodies", etc.). The geometrical aspect of symmetry is of course still extremely important in molecular physics, crystallography, condensed matter physics, biophysics. Historically, it has been an important root in the development of group theory. In a more algebraic outfit, it still appears in every branch of physics in discussions of the symmetries of invariant states (vacuum - or equilibrium states) of physical systems. b) The idea that it is not the orbits of physical systems which necessarily exhibit high symmetries, but that it is the dynamical laws of physical systems which, may admit invariance or symmetry groups could of course appear only after dynamics was introduced into physics, i.e. after the discovery of Newtonian mechanics. I now briefly characterize static and global dynamical symmetries of physical systems abstractly : We consider a physical system , S . The family of all possible states of
S
is called
action of a group x
£ X ,
the image of
a) A state group
X , its elements
G , i.e. with each
x
x
x,y,... . We assume that
x £ X
under the transformation
g 6 G
X
carries the
we associate a state
g .
of a physical system has a static symmetry, described by a sub-
H c C , if x, = x , for all h
(H
and each
is the stability group of
stability groups).
h £ H .
x . All states on the same
G-orbit have conjugate
An orbit of a dynamical, physical system, S , is a mapping from
the real line, the time axis, into the space of states, X , of
S , i.e.
195 119
i s t i—»• x(t) e x. The family of all possible orbits is denoted bl) with each
S
0 .
has a global (dynamical) invariance - or symmetry group
x( - ) € 0
H <= G
if
the trajectory
{x(t)h : t em} is again an orbit of
S , i.e. an element of
0 , for all
h £ H . This means that
x ( t ) h = x h (t) , (where x = x(0) , x, = image of tion x, ) . h
x
under
h ,
Xj^t)
=
orbit with initial condi-
b2) Next, we give a somewhat misleading, abstract characterization of local (dynamical) symmetries : Suppose constituents,
S, ,... ,S I n
An orbit, x(t) , of x.(t)
S
consists of (spatially separated, but coupled)
, n = 2,3,..., S
i.e.
S ~ S, x S„ x ... x s 1 L n
then consists of an n-tuple
is an orbit of the constituent system
is a "local symmetry group" of trary orbit,
x(t) , of
S
S. ,
if, for arbitrary
(x (t),...,x (t)) , where
i = l,...,n . A subgroup h.,.. . ,h
in
H
H <= G
and an arbi-
S , x(t) - (x l( t) ~ 1
xn(t)h) , n
h = (h, h ) , is again a possible orbit of ~ 1 n h be time-independent.
S . Usually, it is not assumed that
"Local symmetries" have little in common with global symmetries, since for physical systems admitting a local symmetry group (an invariance group of gauge transformations of the second kind) one cannot devise any experiment which would distinguish between the two orbits
x(t)
and
x
( t ) h • Thus, x(t)
really correspond to the same, physical orbit of nal
and x ^ ) ^
S , expressed in different "inter-
coordinates". The idea of "local symmetries" has led to the development of gauge theories,
(H. Weyl, 0. Klein, W. Pauli, Yang and Mills), in physics, and in mathematics it gave rise to the theory of fibre - and principal bundles. The present trend in particle physics is to eliminate global symmetries from the fundamental dynamical laws, but to find dynamical laws admitting a (usually nonabelian) "local symmetry group", although the spatial symmetries (Poincare covariance) are still global symmetries, unless gravity is included. Such dynamical laws
196 120
are expressed in the form of gauge theories. That trend poses the interesting, theoretical problem of deriving the global, internal symmetries of phenomenological models or macroscopic descriptions as dynamically generated, asymptotic symmetries; (see Sect. 5 ) .
1.2. We now turn to the discussion of symmetry breaking. We start with the breaking of global symmetries : We distinguish between i)
Explicit breaking,
ii)
Spontaneous breaking,
iii) Dynamical breaking. We speak of an explicitly broken symmetry if the dynamics of a physical system contains a "small" term which is not invariant under a symmetry group leaving invariant the other terms of the dynamics. This concept is of importance when one tries to understand small masses (like the pion mass). It often arises in quantum field theories which, in the classical limit (tree approximation), have a symmetry that does not survive on the quantum level, because of anomalies. (Examples may be chiral symmetries (PCAC), dilation or conformal invariance). A different example for i) is the eightfold way. This topic is not discussed in my notes, although it is very important. We say that a global symmetry group, H , of a physical system, S , is broken spontaneously (or dynamically) if
H
is impossible to transform an orbit,
is an invariance group of the dynamics, but it x(t) , of
S
into the orbit
*(')». » f ° r some
h € H , by a sequence of local operations (local » local in space) without passing through states (or "configurations") of the system which have infinite energy, (or infinite action). By considering the example of an infinite, three-dimensional ferromagnet one convinces oneself that the definition sketched above is appropriate. Spontaneously broken, global symmetries can in general be characterized by a local order parameter, (in the example of the ferromagnet by the spontaneous magnetization - expectation value of the spin observable associated with a point or some microscopically small region). It is not easy to distinguish between the concepts of spontaneous and dynamical symmetry breaking on an abstract level. (One might say that dynamical symmetry breaking does not manifest itself on the tree level and is non-perturbative, while spontaneous breaking appears already on the tree level and may be discussed perturbativeiy>. It is much more difficult to give an abstract characterization of the "sponta-
197 121
neous or dynamical breaking of local symmetries".
This notion cannot mean that the
physics of a system admitting a local symmetry group, i.e. a general covariance under changes of the internal coordinates, depends on the choice of local, internal coordinates. Broken, local symmetries cannot be characterized by (gauge-invariant) local order parameters. The concept of a broken, local symmetry is intrinsically dynamical and must, in the author's opinion, be discussed in the context of a specific formulation of specific theories. One might vaguely describe it as follows : A system with a local, internal symmetry group
H
(assumed to be a compact Lie group) has always dynamical
degrees of freedom carried by gauge fields which are indexed by generators of the Lie algebra of
H . One might say that
down to a subgroup H/H^
L c H
H
is "broken spontaneously (or dynamically)"
if the degrees of freedom associated with the coset space
are "frozen out at small energies", i.e. are invisible at large distances.
(But see Sect. 3 ) .
A discussion of the concept of symmetry enhancement is postponed to Sects. 4 and 5.
198 122
2. Spontaneous breaking of global symmetries. General references are [2,3,4,5,6]. General, group theoretical discussions of the possible symmetry breaking patterns in physical systems can be found e.g. in [17]. The breaking of spatial symmetries is a general phenomenon in the physics of matter at positive density (and temperature), in the thermodynamic limit : The boosts are always broken, rotation invariance is often broken (directional long range order; e.g. in liquid crystals), translation invariance is broken in systems with a crystalline equilibrium state (translational long range order). The mathematical understanding of systems with broken rotation - or translation invariance is however rudimentaryThere is a prominent example of the breaking of Lorentz invariance in a system at zero temperature and density : The boost symmetries are broken on the charged sectors of quantum electrodynamics [18], although the term "broken" is used here in a slightly different (weaker) sense. An important feature of the breaking of continuous, internal symmetries and translation invariance is that it is always accompanied by the appearence of zero mass excitations (the Goldstone bosons, the phonons, respectively) which manifest themselves in the slow decay of correlations of suitably chosen observables; [6,20], (The breaking of rotation invariance also implies the existence of correlations with slow decay but not the existence of Goldstone excitations). Starting from our definition in Sect. 1, one can easily convince oneself - at least heuristically - that, in one and two dimensions (space dimensions, in equilibrium statistical mechanics; space-time dimensions in quantum field theory), continuous internal symmetries or translation invariance cannot be broken spontaneously or dynamically [6,20,21,22]
(I recall the droplet argument, made rigorous in [22]),
except in systems with interactions of extremely long range [23,24]. It may not be generally appreciated that we do not know any example of a system with translation invariant dynamics for which we can prove mathematically that the breaking of translation invariance occurs, (except in the rather unphysical, one-dimensional jellium model which exhibits a Wigner lattice or in idealized models of systems at positive density, but at zero temperature). Moreover, there are no known continuum models of liquid crystals for which we can rigorously establish directional long range order. These are serious gaps in the mathematical foundations of condensed matter physics. Up until 1976 there were no known examples of models with a continuous, internal symmetry group for which spontaneous symmetry breaking, and hence the existence of Goldstone excitations, was established rigorously (except for the somewhat trivial spherical model of Berlin and Kac). That situation changed with the appearence of
199 123 [7], where a class of three - or higher dimensional lattice models of statistical mechanics, including the classical Heisenberg model, and the well known
X|ip| -
quantum field model in three space-time dimensions were discussed which exhibit spontaneous breaking of internal
0(N)
symmetries at suitably chosen values of the
thermodynamic parameters, the coupling constant
X and the bare mass, respectively.
The method in [7] involves a rigorous version of spin wave theory. It was extended in [25] to quantum-mechanical lattice models, including the quantum
XX
model and
the Heisenberg anti-ferromagnet. The method was generalized considerably in [24]. However, the Heisenberg ferromagnet, for example, has so far resisted all attempts at a mathematically rigorous understanding. All these developments have been reviewed pedagogically e.g. in [3,8,9,26] , and I do not repeat that here. A new method in the theory of phase transitions and continuous symmetry breaking has been introduced recently by Spencer and myself [27]. Unfortunately, it can only be applied to systems with an abelian symmetry group, so far, (although it is conceivable that one might recover the results on The method
0(N)
lattice
a-models of [7]).
has the advantage of extending to abelian lattice gauge theories, permit-
ting us to give fairly simple, new proofs of the results in [28,29]. It relies on a representation of abelian lattice spin systems or - field theories as gases of defects of co-dimension 2 carrying integer flux numbers. In three dimensions those defects are closely related to the Abrikosov vortices in a super conductor. They interact through long range forces. The broken symmetry appears when the defects have low effective activity , z , and form a dilute gas. The symmetry is restored when the defects condense. A heuristic discussion of the transition is based on an energy-entropy argument : The entropy , S, of a defect line (or-network) of length
L
is bounded by
S < cL , where
c
is a geometric constant, the energy is bounded by E > c'|tp|"L .
where
z
[
c' > 0 . The effec-
is therefore bounded above by z < e-Cec'-OL _ g .
and is exponentially small in
L
for large
1/fcT
(
B(> c/c') . Thus, for small temperatures
T , the defects have small sizes and form a dilute gas. Hence the medium is ordered, and the symmetry is spontaneously broken. This argument is clearly reminiscent of the Peierls argument, e.g. [26], where one considers a gas of Bloch walls. It might have interesting applications to the theory of melting of solids. For a rigorous version see [27].
200 124 3. "Spontaneous breaking" of local, internal symmetries. This topic is huge and less well understood than the breaking of global symmetries. It would require a series of lectures of its own. I therefore concentrate my attention on the discussion of a few specific aspects which are probably not typical for the whole circle of problems. Systems admitting a local, internal symmetry group (i.e. general covariance under changes of internal "coordinates") are always described in the form of gauge (Yang-Mills) theories. Presently, the most widely used, non-perturbative ultraviolet regularization of gauge theories consists of putting these theories on a lattice. This preserves gauge-invariance, translation invariance and positivity of the metric in the physical Hilbert space, [30,31,32,33]. We now consider an example : The gauge group gauge field is denoted by g ™ {g } , where xy d ~ xy neighbors in 7L , and g is an element of G A
Z g = P(e xy
is chosen to be
SU(2) . The
formally given by
,.(5)d5v "
*
We also introduce a Higgs field
G
is an arbitrary pair of nearest
) , for all
xy.
,
$ : x £ E -»-£ £ E , with isospin 1 . Let
x
De tne
spin 1/2 character of
SU(2) , and
U
the spin 1
representation. The Euclidean functional measure which determines the vacuum state is given by SVH -1 x du(g,*) = Z n e X (xy)
UVB
x V "yV ^ (1)
Bx(g3vJ - ne n dg P
(xy)
^
n ax(* ) , x
dX(?) e=8- exp[-!|?| 4 + -|H£| 2 ]d 3 * , g
•= 3p
II 0 xyc3p
8 . ( p a unit square = plaquette), **
B,C
and X
are positive
constants. The r.s. of (1) is defined as the thermodynamic limit of the measures associated with finite sublattices, with arbitrary boundary conditions (b.c.) imposed at the boundary of each sublattice. Let that
h
h : x -•• h
be an arbitrary function from 7L
= 1 , except for finitely many sites
into
G with the property
x . We make the change of variables
201 125
*xy * V x y h y X '=
l
* \ (2)
+x
*
D(h
x>*x
This change of variables leaves
S
< * \
dy(g,4>)
invariant, i.e. dy(g,<|>) - dy(g ^
matter what b.c. have been used to construct
du
. Let
<•> = J(-)dii(g,.(i) . It clearly follows that gauge-dependent observables, like
g
or $ , have zero
expectation, i.e.
'W
• <*x> = ° •
(3)
independent of the b.c. used in the construction of <•> . This is an immediate consequence of a definition of
<•> by means of the Dobrushin-Lanford-Ruelle equa-
tions, for example. It really expresses the triviality that if one does not fix a gauge by hand, the gauge is not fixed, and therefore gauge-dependent variables are averaged out when one computes their expectation (an observation made e.g. in [10,11]) . Let us now fix a gauge and see whether we obtain a useful notion of "breaking of local symmetries". Gauge fixing is achieved by multiplying
du by a function
F(g,i|>) with the property / F(g h ,+ h )n dh x - i . x
(4)
Let <•> be the expectation determined by F(g,i(>)du(g,
.
(5)
However, it is now possible, a priori, that
4X>F t o , if suitable b.c. are imposed. One possible, partial gauge fixing is to turn all Higgs variables, $ , paralx 3 lei to the 3-axis, e , of E . (Choose F to be proportional to
* ^ V ^ x ^ • Then 4X>F " 'lfJV-3 ' ^ x ^ F > ° •
<6>
202 126
(no matter whether F(<j>)du(g,ij>) invariant
u
2
is positive or negative). For this choice of
F ,
has a residual, local invariance group : If in (2) every (U(h )e
» e , Vx) , hence belonging to a
U(l)
h
subgroup of
leaves
e
G , then
F(*h)dp(gh,*h) = FCtOdpCg,*) .
(7)
Thus, in a sense the coupling of the gauge field to the Higgs field has broken the 2 gauge group down to a residual U(l), but since this happens no matter whether y is positive or negative, it does not provide a terribly useful notion of "spontaneous, local symmetry breaking". (This becomes particularly evident in a theory with a large gauge group action of
G , = G
several inequivalent of
SU(3) , and a Higgs field,
$
, with the property that the
on the vector space , V, of possible values of
9
decomposes
V
into
G-orbits). The above notion depends of course on our choice
F . It has been shown in [12] that, for some class of complete gauge fixings, F ,
including the temporal gauge, and arbitrary "symmetry breaking" b.c.
•4 > - 0 . xF
(8)
(This follows either from a "spin-wave" argument related to the one in [22], or from the principle of "symmetry restoration via defects", such as instantons). In [12] Morchio, Strocchi and the author have therefore proposed a gauge-invariant description of the physics of a Higgs theory in the continuum limit, with the hope that this might lead to a useful notion of "spontaneous local symmetry breaking". In the example of the Georgi-Glashow model considered above, the appropriate, gauge invariant fields are ^•F
(photon), and
4>•
(Higgs particle) .
(9)
Moreover there is a gauge-covariant field
IT+F" = N[|J| 2 f -+(*•? )] (W+ and W") , cj> uv ' ' uv uv where
N
(10)
indicates normal ordering. From this field one may formally construct
fields localized on curves
(y
)
with given endpoints
(x,y) :
xy d r + f )(x)P[exp / A (5)-d5P](7r-j-f .)(y) . • uv 'xy p * The conventional, gauge-dependent picture is recovered if
(11)
<j> 1 e„ . Note that the
physical fields introduced in (9), (10) and (11) do not form any and there is no reason why the masses of the photon and the
W
SU(2) and
W
multiplets, boson ought
to be degenerate.In fact, this theory is expected to have a non-perturbative phase in which there is only one massive, neutral vector particle (a massive photon), a
203 127 "Higgson" and neutral
W -W
bound states. In that phase the electric charge would
be confined, (region I of Fig.l). At large renormalized values of
x,
, u
and
6 ,
the theory should however have a QED phase with a massless photon (coupled to the vacuum by
) , a massive (unstable) "Higgson", massive
W
and
W
vector
bosons and massive magnetic monopoles, (region II, Fig.l). When one speaks of "spontaneous breaking of
SU(2)" in this theory one is thinking of phase II. The picture
developped here can be tested in the lattice Georgi-Glashow model for which one ex2 2 pects the following phase diagram, (X,u > 0 ; X,u -»-•», as £-*•<•>):
pure
theory
U(l)
? 6 c
\
II pure \ 1$ lattice t
I
/ / / / / 1
?
c
A.
Fig. 1 pure
SU(2) theory
I : confining phase/II : QED phase. The dashed line might correspond to a line of singularities of the electric string tension, [34] . The only rigorous results concern the existence and nature of the transitions on the lines
g « <° and
£ = °°
(at
C .
g c , resp.). See [7,28,27,33]. The above considerations extend to more general Higgs theories with the property that the stability group the gauge group
G , for all
the Lie algebra of
H-*
H-* of • *
<(>
, for all
4> j* 0
x
H
of
x , are the "electromagnetic and gluon fields",
the remaining generators of the Lie algebra and should correspond to massive
is conjugate to one subgroup
x
. The gauge fields corresponding to generators of
A
of
G
are the "broken generators"
(H-neutral) bound states or massive vector bosons.
Things become problematic when there are several, inequivalent Higgs orbits, i.e. the abstract group corresponding to (Example : G = SU(3) ,
•+
$
H-jj-
depends non-trivially on
X
in the adjoint representation, . . . ) . Let
the effective Higgs potential, (including radiative corrections). Let
->the orbit on which V stability group, and -fc
Is 0 •
7 >? ^
^
V ,.(*) be erf T (J ) denote
°
(<(>) takes its minimum, H the corresponding (abstract) the Lie algebra of H . Let (f be the Higgs field ave-
204 128 raged over a ball centered at
x
of radius » M
scale (a fluctuation length scale of
, where
M
is a typical mass
f ) of the theory. Then with high probability
Jx e (tj Perturbation theorists then say that
•
G
(12)
is broken down to
fields corresponding to generators of -fn.
H
, that the gauge
(in the sense indicated above and in [12]) fy-Qfy
remain massless, while the ones corresponding to
acquire masses.
On a non-perturbative level, this prediction is probably wrong, as argued by Morchio and Strocchi [35] on the basis of ideas and results in statistical mechanics [36,37,38] <(> = $
and of [12] : Suppose
+ 5<j>
V ., (<(>) has a local minimum on an orbit
, with stability group
H. c= H
. (In general
R.
this assumption for clarity). Let M ,M~ be the curvatures of ' o 1 to ($ ) , ($ ) at ~$ ,$.. , respectively. If Aa E V
)
, then
ly. (If (13) fails it may happen that even if
V' „ ($.) > V „
(13)
~~+~
-1
a <* (?
$
,
V „ transversal eff.
eff.tfl > " V .ff. ( *o )+ * (M l" , £ )M2>> ° '
for some positive constant
($.)
, but we make
-*•
is close to
is close to
(iji )
predominant-
($ ) , predominantly,
($ )) . However, with some probability (vanishing in per-
turbation theory, but positive non-perturbatively) there appear "bubbles, B , of the false vacuum" such that
f
is close to
($..) , for
x E B . The effect of these
bubbles on the physics of such a theory can be estimated by a Peierls (action-entropy) argument [36] and a study of mass generation [38]. (I follow a presentation in [37]): First we must estimate the probability —^
p
($.) , i.e. that
belongs to a bubble
x
(e.g. the origin)
false vacuum. We choose b.c. such that
9
of the event
€($),as
E ^
that
is close to —T
B £ {y||y-x| <M
)
of
|z| -»•"•. A connected piece,
T , of the boundary of a bubble is called a contour (or phase boundary). For the event E
to occur it is necessary that there be a contour
from bubble
separating
{y||y-x| < M
}
B
such that
3B = r
is
* and of contours. The action of a x bounded below by A(|r|) , where
A(|r|) > aM"3|r| + Ao-M_1|r| Here
r
•> , as follows from our definitions of
a
is a constant
.
(14)
2 —1 )
and Irl is the volume of r . The first ren. '' term is a surface term, the second term a lower bound for a volume term. The precise
dependence of
M , a
«(cu X
and
A(|r|)|r|
a 1
on coupling constants is not known, presently. If
» l
(15)
205 129
the statistical weight of a contour fore the probability
p
for
E
r
is bounded above by
exp[-A(|r|)]
. There-
to occur is bounded by
p < I X exp[-A(|r|)]
1}
,
(16)
r X where
I
— *? ranges over all contours
surrounding
{y||y-x| < M
|r| = const.M
}
r
of volume
|r| • const.M
n , n « 1,2,3,...,
. The number of such contours with given volume,
n , is bounded by e
, (c M 0(1) is a geometrical constant) .
(17)
From (14), (16) and (17) we conclude that
p «
1
if (15) holds.
(18)
By the results of [38,35] one then expects that gauge fields corresponding to generators in
OlQtp
(in the sense of [12] ) acquire masses "l+J
corresponding to the generators of try
Qliy-.
> while gauge fields
acquire masses
« ^PIS+J.
d9)
Similar considerations apply to Fermion masses. Thus - if there are no further local minima giving rise to other bubbles than to the larger group the "breaking" from
H
H
G
is really "broken down" to
. Moreover, there is no o —
down to
H
, rather
elementary Higgs field causing
H.. , [35]. Finally, one does not expect any dyna-
mical monopoles with charges labelled by
IT (H /H ) , (ir
= k
homotopy group),
but only ones with charges labelled by
TT.(G/H ) ~ ir, (H ) , as argued in [37]. 2 o = 1 o Finally, we should mention that the applicability of conventional perturbation theory, based on the (generally incorrect) assumption that <$ >_ j* 0 , to Higgs x r gauge theories has been discussed in [12], with the result that the deviations can generally be expected to be entirely non-perturbative. This ends our discussion of the notion of "spontaneous breaking of local (gauge) symmetries" : In abstracto, it is somewhat vague and misleading. It must be
understood dynamically, and one should be aware of the fact that non-perturbative effects generally alter the conventional interpretation. Such effects, together with the requirements of renormalizability, some form of asymptotic freedom and the requirement that there exist an "unbroken"
SU(3)
x U(l)
may be useful guides for
model builders.
1)
A lower bound on
p
is more difficult to derive, see [37].
206 130 4. Renormalization group ideas. We consider a class of physical systems which can be described by a family (algebra), (X , of local "observables", e.g. Euclidean fields, in quantum field theory (QFT), or spin fields, in statistical mechanics (SM), and some space, X , of time-translation invariant states, e.g. Euclidean vacuum functionals in
QFT , equi-
librium states in
A
SM . Let
A € 01. By
A
we denote the translate of
by a
x vector
x
in space-imaginary time
(QFT) , or space
(SM) . Let
p £ X
be a state
characterizing a specific physical system. Question : How do correlations, p(A -B ) , A.B x y behave, as
in 01 ,
|x-y| •+• «> , i.e. in the (infrared) scaling limit ? (In a continuum sys-
tem one may also be interested in the behaviour of p(A -B ) , as x y
|x—y| •+ 0 : the
short distance, or ultraviolet limit. We focus our attention on the scaling limit). In order to answer that question, one tries to construct functions, a »(8) • dependir on
A C 01 and on a scale parameter
8
, such that
G A>B (x,y) E lim a A (e)a B (6){p(A ex .B ey ) - P(A ex )p(B ey )}
(20)
exists. (My discussion is slightly oversimplified at this point, since one often chooses
p
on the r.s. of (20) to depend on
9
, as well, such that
p
approaches 6
a critical state, as
a
6 •+ « ) . In order to find
,(9) » A E d , and other quantities
of interest at large distances, one tries to determine the large scale effective dynamics, by integrating out fluctuations on a sequence of increasing length scales. One popular scheme to accomplish that is the Kadanoff "block spin transformations". Abstractly, they can be described as a non-linear transformation,
x
, acting on
X x 0L: x : (p,A) ->• x(p,A) = (P T .A x ) , p (A ) - p(A) 2 )
with
T
such that each application of
x
(21)
,
T
increases the scale of effective fluctuations,
i.e. transforms a dynamics on a given scale into an effective dynamics on the next larger scale. In order to answer the question raised at the beginning by means of such a scheme one must study the manifold p* £ M c X
2)
or
x •= x e , with
iff
M m
of fixed points of
p* = p*
.
P C A ^ - B ^ ) - const.pT (A x "B y ).
x
: (22)
(21')
207 131 Under suitable hypotheses on the properties of the vicinity of some manifold,
p* 6 M m
T
, one can decompose
into a stable manifold,
X
in
M (p*) , and an unstable s
M (p*) :
U
Fig. 2
States on
M (p*)
, states on
are driven towards
p* , under the action of
x
The tangent space,
linear space spanned by eigenvectors of
Dt
corresponding to eigenvalues of modulus
> 1 .
perturbations". The space, x
by
x
I
a
. W
R
to
p*
are driven away from
M (p*)
at
, the linearization of
t
R
p* T
at
is the p*
,
is called the space of "relevant
, and the space,
M at m are computable in terms of
(...(p ) ) ...
M (p*)
, to
, of "irrelevant perturbations" is defined by replacing
in the definition of
is the tangent space to ions
R
Let AA
M
, of "marginal perturbations"
p € M o (p*) . One argues that the funct-
and of the rate of approach of and
(See (21), (21')) .
n times The point of interest to us is now the following : It may happen that the fixed point
p*
has a larger symmetry group than a state
p
on
M (p*) . This en-
tails that the scaled correlations, original correlations
G. „(x,y) , exhibit a larger symmetry than the A,ii p(A -B ) . If this happens we speak of asymptotic enhancement x y
of symmetry, or of the (dynamical) generation of asymptotic symmetries. It is quite irrelevant in this general discussion, whether the symmetry in question is internal or spatial, global or local (i.e. gauged). One might argue that the concept of symmetry enhancement is only interesting for physics if it has some stability properties. Let
G
be some (global or local,
internal or spatial) symmetry group, and let H be a subgroup of G . Consider a (x is assumed to have suitable smoothness G-invariant fixed point, p* of properties). Suppose that the H-invariant subspace of riant subspace of
M
. Then, in some vicinity, N , of
M p*
coincides with the , every
G-inva-
H-invariant
208 132
fixed point of
G-invariant. Thus, all states in ^ ranges over all H-invariant fixed points, p , of T in
U
T
is also
U M (p) , where p*/N S N , are driven towards
•p/N
G-invariant fixed points. Moreover if the H-invariant subspace of with the G-invariant subspace of of p *
, the
p*
, then for some neighborhood
N
H-invariant subspace of marginal and relevant perturbations of a
H-invariant fixed point p*
M @ R , at
M © R coincides
p £ N
is also G-invariant, (i.e. H-invariant states near
tend to approach G-invariant states under the action of
T
) . This is the de-
sired stability of our concept. The concept of symmetry enhancement has been described e.g. in [14], (see also Diirr's notes). The first rigorous study of models exhibiting this phenomenon (e.g. the ZZ -models, see Sect. 5) probably appeared in [27,39]. An abstract discussion very similar to the one presented here appeared subsequently in [16] (which inspired the present section).
209 133 5. Symmetry enhancement : Generation of asymptotic, global and local symmetries. In this final section we sketch very briefly some examples of the phenomena described at the end of Sect. 4 and point out why "symmetry enhancement" might be the right concept permitting us to decide whether a local (gauge) symmetry in some gauge theory is "spontaneously broken", or not. 2 First, we consider the 7L spin models on the square lattice, TL , N • 5,6,7, -V 2 ... . The classical spin, S , at a site x G.ZL is given by 2im !J = (cos 9 ,sin 6 ) , 8 = — — - , n = 0,...,N-1, Vx . x x x x N x The equilibrium state of the model at inverse temperature
g is given by the measure
dup°(e) = [z< N ) ] _ 1 exP[gr cos(e -e )] , 3 where
B
xy
X
(23)
y
xy is an arbitrary pair of nearest neighbors. This measure is the limit of
the measures du
h cos(NB ) x . (6) - Z~ exp[g£ cos(9 -6 )] n e de p,n g,n x y x x
,
(24)
as h -* •» ; (de - Lebesgue measure on unit circle). The classical XY- , or rotator (N) model corresponds to h = 0 . For h > 0 , the measure du„ . and du have a ' p,h p discrete, global symmetry group (generated by 2Z and reflections) while the rotator (h - 0) has a continuous, global symmetry group. In [39] Spencer and the author have shown that, for all h €[0,~]
and N > N , where N is a suffiently large integer B o o independent of h , there exists an interval [j[(h,N) , g(h,N)] of values of p which are all critical points and at which the correlation length of the spin systems described by du . is infinite. Moreover, g» n
—g(h,N) =< —g(0,N)
= B (rotator) < «. , c
'g(h,N)-».o> , as h - » 0 or N-«-» . We have constructed an infinite sequence of renormalization transformations which drive
du„ . towards a U(l)-invariant state du? , , for all h £ (0,<»] , N > N , g,h ~- _p,n = o and all g 6 (g ,g.) , with ^(h,N) < g < g 2 < B(h,N) . Thus, asymptotically, the discrete symmetry of the 7L ture that for each some
models is enhanced to a continuous symmetry. We conjec-
h € (0,~] and each
g € (j$(h,N) ,"g(h,N)) , N > N
, there exists
g' = g'(g,h) > g (rotator) such that spin correlations in du , and in «= c p,n du , have identical (long distance) scaling limits, (although this does not quite
210 134
follow from our construction). In [27] we have established similar results for the QED phases [29] of the TL
lattice gauge theory in four dimensions : Local ZL -invariance is asymptotically
enhanced to local
U(l)-invariance.
Recently, we have also examined examples of non-abelian gauge theories coupled to some Higgs fields (not transforming under the fundamental representation) for 2 which we argue that, for suitable choices of the coupling constants $,£,A,u , the theory is in the same (long distance) "universality class" as the corresponding pure Yang-Mills theory (for some
6' - g ' ( g , 5 , — ) , 5 = 0 ,
field expectations are considered. In such a case one could say that the matter fields leave the full gauge group "unbroken". (In the opposite case it would be appropriate to speak of "local symmetry breaking"). It would be interesting to study symmetry enhancement at short distances in continuum grand unified theories. More standard examples of symmetry enhancement which are, however, not very well understood mathematically are : - Restoration of full Euclidean invariance of correlations of lattice theories in the scaling limit (as
3^g
, where
0
is a critical point).
- Restoration of translation invariance above the roughening temperature in the threedimensional Ising model or in a lattice gauge theory, [34]. Problems of symmetry enhancement are typically very involved, technically, so that we cannot present any details here.
135 References. 1. H. Weyl, "Symmetry", Princeton, N.J. : Princeton University Press, 1952. 2.
S. Coleman, Secret Symmetry : An Introduction to Spontaneous Symmetry Breakdown and Gauge Fields, Erice Lectures 1973, A. Zichichi (ed.).
3. J. Frohlich, Bull. Amer. Math. Soc. 84_, 165, (1978). 4. L. Michel, Reviews of Modern Physics 52_, 617, (1980). 5. J. Goldstone, Nuovo Cimento _19_, 15, (1961); Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345, (1961), 124, 246, (1961). 6. H. Ezawa and J.A. Swieca, Commun. math. Phys. 5_, 330, (1967). 7. J. Frohlich, B. Simon and T. Spencer, Commun. math. Phys. 5_0, 79, (1976). 8. J. Frohlich and T. Spencer, in "New Developments in Quantum Field Theory and Statistical Mechanics", M. Levy and P. Mitter (eds.), New York & London : Plenum, 1977. 9.
J. Frohlich, Acta Physica Austrica Suppl. XV, 133, (1976).
10. S. Elitzur, Phys. Rev. D12, 3978, (1975). 11. G.F. De Angelis, D. De Falco and F. Guerra, Phys. Rev. D17, 1624, (1978). 12. J. Frohlich, G. Morchio and F. Strocchi, Phys. Letts. 97B, 249, (1980) ;. Nucl. Phys. JB, in press. 13. K. Wilson and J. Kogut, Physics Reports 12C, N°2, 76, (1974). K. Wilson, Rev. Mod. Phys. V7, N°4 L. Kadanoff, A. Haughton and M. Yalabik, J. Stat. Phys. ^4, N°2, 171, (1976). G. Jona-Lasinio, Nuovo Cimento 26B.99, (1975). S. Ma, Rev. Mod. Phys. 46_, N°4, 589, (1973). P. Bleher and Ja. Sinai, Commun. math. Phys. 33, 23, (1973), 45_, 247, (1975). G. Jona-Lasinio, in "New Developments..." (see ref. 8). M.E. Fisher, Rev. Mod. Phys. ^6_, N°4, 597, (1974). 14. D. Foerster, H.B. Nielsen, M. Ninomiya, Phys. Lett. 94 B, 135, (1980). J. Iliopoulos, D.V. Nanopoulos and T.N. Tomaras, Phys. Lett. 94 B, 141, (1980). Ref. 39 (Sect. 7); ref. 27; ref. 16. K. Cahill and P. Denes, Preprint, Univ. New Mexico : UNMTP-81/020. 15. Refs. 39 and 27; J. Frohlich and T. Spencer, Phase Diagrams and Critical Properties of Classical Coulomb Systems, Erice 1980. 16. C. Newman and L. Schulman,"Asymptotic Symmetry : Enhancement and Stability', submitted to Phys. Rev. Letters. 17. L. Michel and L. Radicati, Ann. Phys. (NY) 66, 758, (1971). D. Kastler et al., Commun. math. Phys. 2]_, 195, (1972). 18. J. Frohlich, G. Morchio and F. Strocchi, Ann. Phys. (N.Y.) 119, 241, (1979), Phys. Lett. 89 B, 61, (1979). 19. Ph. Martin, Preprint, EPF-Lausanne, 1981. 20. N.D. Mennin, J. Math. Phys. 8, 1061, (1967). J. Phys. Soc. Japan, Suppl. 26_, 203, (1969).
212 136 21. S. Coleman, Commun. math. Phys. 2L>
259
» (1974). See also ref. 6.
22. J. Frohlich and C. Pfister, Commun. math. Phys. (1981). 23. H. Kunz and C. Pfister, Commun. math. Phys. 4£, 245, (1976). 24. J. Frohlich, R. Israel, E.H. Lieb and B. Simon, Commun. math. Phys. 6i2_, 1, (1978). 25. F. Dyson, E.H. Lieb and B. Simon, J. Stat. Phys. j ^ , 335, (1978). 26. E.H. Lieb, in "Mathematical Problems in Theoretical Physics", G.F. Dell'Antonio, S. Doplicher and G. Jona-Lasinio (eds.), Springer Lecture Notes in Physics, BerlinHeidelberg-New York : Springer Verlag, 1978. 27. J. Frohlich and T. Spencer, "Massless Phases and Symmetry Restoration....", Commun. math. Phys., to appear. 28. A. Guth, Phys. Rev. D21, 2291, (1980). 29. S. Elitzur, R. Pearson and J. Shigemitsu, Phys. Rev. D19, 3698, (1979). 30. K. Wilson, Phys. Rev. DIP, 2445, (1974). 31. K. Osterwalder and E. Seiler, Ann. Phys. (NY) j^O, 440, (1978). 32. D. Brydges, J. Frohlich and E. Seiler, Ann. Phys. (NY) m . , 227, (1979). 33. E. Seiler, "Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics", Springer Lecture Notes in Physics, to appear. 34. H. van Beijeren, Commun. math. Phys. 40, 1, (1975), Phys. Rev. Lett. ^ 8 , 993, (1977); ref. 39, (Sect. 7); C. Itzykson, M.E. Peskin and J.-B. Zuber, Phys. Lett. 95 B, 259, (1980); A. Hasenfratz, E. Hasenfratz and P. Hasenfratz, Nucl. Phys. B180, 353, (1981); M. Liischer, DESY Preprint 1980. 35. G. Morchio and F. Strocchi, Phys. Lett. 104 B. 277, (1981). 36. J. Glimm, A. Jaffe and T. Spencer, Commun. math. Phys. 45, 203 (1975); R. Dobrushin and S. Schlosman, Preprint 1981. 37. J. Frohlich, "The Statistical Mechanics of Defect Gases", unpublished. 38. D. Brydges and P. Federbush, Commun. math. Phys. 62, 79, (1978); D. Brydges, J. Frohlich and T. Spencer, "The Random Walk Representation of Classical Spin Systems and Correlation Inequalities, Commun. math. Phys., to appear. 39. J. Frohlich and T. Spencer, "The Kosterlitz-Thouless Transition in Two-Dimensional Abelian Spin Systems and the Coulomb Gas", Commun. math. Phys. to appear.
II Non-Perturbative Quantization of Topological Solitons
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215
QUANTUM THEORY OF NON-LINEAR INVARIANT WAVE (FIELD) EQUATIONS OR:
SUPER SELECTION SECTORS IN CONSTRUCTIVE QUANTUM FIELD THEORY* t Jurg Frohlich
§
Department of Mathematics Princeton University Princeton, N. J. 08540 TABLE OF CONTENTS General Introduction to Parts 1 and 2 Part 1 : General Framework of Constructive Quantum Field Theory 1.1 Introduction 1.2 Outline of a general theory of super selection sectors in re I ativistic quantum field theory. 1.2.1 Remarks on Osterwalder-Schrader reconstruction. 1.2.2 Reconstruction of algebras of local observables. 1.2.3 Vacuum super selection sectors 1.2.4 Charged super selection sectors. ,4 Part 2 : The Super Selection Sectors of the g
C o n s t r u c t i o n of t h e vacuum s e c t o r s 4 The vacuum super s e l e c t i o n r u l e s of the g l ^ theory . The charged super s e l e c t i o n ( s o l i t o n ) s e c t o r s of g(P2
-
Research p a r t i a l l y supported by the NSF under grant MPS 75-11864 Lectures d e l i v e r e d a t the I n t e r n a t i o n a l
School o f Mathematical Physics "Ettore
Majorana", E r i c e , S i c i l y , Summer 1977 *A. Sloan Foundation Fellow
Reprinted from Invariant Wave Equations, G. Velo and A.S. Wightman (eds.), Proceedings of the "Ettore Majorana" International School of Mathematical Physics, Erice, Italy, June 27 July 9, 1977, Lecture Notes in Physics 73, Springer-Verlag Berlin, Heidelberg, 1978. © Springer-Verlag 1978
216 340 QUANTUM THEORY OF NON-LINEAR INVARIANT WAVE (FIELD) EQUATIONS OR:
SUPER SELECTION SECTORS IN CONSTRUCTIVE QUANTUM FIELD THEORY
General Introduction to Parts 1 and 2 In these lecture notes we f i r s t give a b r i e f summary of the general framework presently underlying the constructions of n o n - t r i v i a l r e l a t i v i s t i c quantum f i e l d s (r.q.f.'s).
Completely absent from previous reviews of the general framework of con-
structive quantum f i e l d theory ( c . q . f . t . ) is a discussion of a pragmatic, rigorous theory of super selection sectors useful f o r applications to specific r . q . f . models: that is to say, previous reviews have exclusively concentrated on describing the general (Euclidean) framework underlying the construction of r . q . f . ' s in the vacuum ("Wightman") representation.
The phenomena of quantum solitons and n o n - t r i v i a l t o -
pological charges in standard r . q . f . models i n two space-time dimensions and higher dimensional gauge theories have taught us that this is not enough. We hope to close the gap described here in these notes, at least to some extent. Another topic that may not have received the attention i t deserves, in some of the recent reviews, is new developments in scattering theory useful for applications to r . q . f . models.
We make a few remarks on this subject (mainly concerning
m u l t i s o l i t o n scattering theory), but we can unfortunately not cover the most interesting recent developments in the subject. The rigorous connections between the classical and the quantum theory of nonl i n e a r , invariant wave equations is only treated, in these notes, in the form of references.
This i s u n f a i r , because the problems coming up here are most i n t e r e s t i n g .
But a valuable review of this topic would go beyond the l i m i t s of these notes. However, heuristic connections between the classical and the quantum theory w i l l be mentioned at several places, in particular in the discussion of instantons and solitons.
We shall also meet some l i m i t a t i o n s to these connections.
b r i e f l y indicate what sort of heuristic connections we have in mind.
We now
(The reader is
advised to also consult Gervais 1 contribution to these proceedings): Most parts of this review are motivated by the study of r e l a t i v i s t i c Hamiltonian systems.
The classical Hamiltonian functionals of these systems provide
Hamilton equations of motion which are c l a s s i c a l , non-linear, invariant wave equations; (see Strauss' contribution to these proceedings). In order to get some q u a l i t a t i v e insight into the quantum theory of such r e l a t i v i s t i c Hamiltonian systems we shall determine the c r i t i c a l points (absolute and local minimas) of t h e i r classical Hamilton functionals:
Their structure w i l l teach
us some q u a l i t a t i v e properties of the quantum theory, in particular properties of the physical vacuum state and of the super selection rules. But i t turns out (especially in the study of gauge theories) that i t is not always enough to investigate the c r i t i c a l points of the classical Hamilton functional
217 341
in order to predict a l l q u a l i t a t i v e features of the quantum theory.
This is because
of the p o s s i b i l i t y of quantum mechanical tunnelling between d i f f e r e n t absolute minima of the classical Hamilton functional.
I f the (classical) Euclidean action has local
minima at non-constant f i e l d configurations (instantons) interpolating d i f f e r e n t , constant f i e l d configurations corresponding to absolute minima then there is tunnelling, and the structure of the quantum mechanical ground state (vacuum) is not correctly predicted by the minima of the classical Hamilton functional. A well known example is the one dimensional anharmonic o s c i l l a t o r with Hamilton function P
2
-T*
2
+
9
x 4 +
efe
where g is s t r i c t l y p o s i t i v e . One sees immediately that this Hamilton function is non-negative.
The poten-
tial - T *2
+
9 *4
+
<3g
(2)
has two degenerate minima at X = + ( 8 g ) " 2 (where i t vanishes) separated by a (for 0 < g «
1) large barrier.
The curvature of the potential at x = + (8g)
2
is = 1.
I f we did not know better these properties of the classical Hamilton function might mislead us to make the following wrong conjecture:
For 0 < g «
1 , the
groundstate and the entire spectrum of the corresponding quantum mechanical Hamilton operator are two f o l d degenerate, (and the spectrum resembles the one of two uncoupled, harmonic o s c i l l a t o r s ) . That this is wrong can be seen by considering the Euclidean action of the anharmonic, double hump o s c i l l a t o r , defined by +<» S= /
{x ( t ) 2 - \ x ( t ) 2 + g x ( t ) 4 + gl^-} dt
(3)
It has two absolute minima at
x(t)
E
±
-JTSg"
(4)
and local minima at x(t) = + — /8g~
tanh (-1-) TSg"
(5)
representing homotopy classes of ( " f i e l d " ) configurations x ( t ) d i f f e r e n t from the ones represented by (4).
The solutions (5) (of the Euler-Lagrange equations obtained by
varying S) interpolate the solutions ( 4 ) , x ( t ) = + (8g)
2
, and signalize quantum
mechanical tunnelling between the two absolute minima (classical groundstates) (4) of the classical Hamilton f u n c t i o n ( l ) . (barrier penetration).
I t is well known
t h a t , indeed, there is tunnelling
The groundstate of the quantum mechanical Hamilton operator is
218 342
unique, the spectrum is simple.
However, i t does resemble - for small g > 0 - the
one of two uncoupled, harmonic o s c i l l a t o r s : of the two degenerate minima x = + (8g) such p a i r ; see £R1~]
.
2
Eigenvalues come in pairs separated by a
)and perturbation theory in g 2 about either
small, but positive gap °= exp (-const, g
is asymptotic to both eigenvalues in one
Moreover, A. Sokal has shown [Si]
that these eigenvalues
as functions of g 2 have s i n g u l a r i t i e s along a horn-shaped region { z : Rez > 0, |arg z| < e (Rez) • Rez } ,
(6)
with e (Rez)^» 0, as Rez\ 0, accumulating at g = 0. These a n a l y t i c i t y properties are incompatible with Bore! summability |_S1J . The situation described here for the anharmonic, double hjmp o s c i l l a t o r appears to be typical for gauge theories with instantons £m two dimensional abelian Higgs model which we discuss in f F l J We summarize this somewhat vague discussion:
, Cl^]
- such as the
; see also
£C2j
.
A study of the c r i t i c a l points (absolute
and local minima) of the Euclidean action of a Hamiltonian system (and of the classical Euler-Lagrange equations obtained by varying the action) appears to be a useful guide for guessing the q u a l i t a t i v e properties of the groundstate and the low-lying spectrum of the corresponding quantum system.
Unfortunately we cannot discuss the recent
developments of this topic ("instanton physics," see in these notes for reasons of "space-time
[PI,
B l , P2, H2, C3, J1J
limitations";
Next we want to b r i e f l y indicate in what sense in a Hamiltonian system the c r i t i c a l points of the classical Hamilton functional y i e l d information about the structure of the super-selection sectors of the corresponding quantum system. that by considering a specific r . q . f . notation:
model in two space-time dimensions.
We do
Here some
Space-time points in M are denoted x = (x_,t) (with t the time coordinate).
Partial derivatives with respect to x_, resp. t are denoted 3 , resp. 3..
Furthermore
4>(x) is a r e a l , scalar c-number f i e l d on M , and TT(X) denotes the momentum ( f i e l d ) canonically conjugated to 4>. We now define the classical Hamilton density > i of the 4 well known g
- M U , *) = ~ U 0 (ir,«) + "&J (
(7)
with ~^0
M x ) , *(x)) = \
{TT ( x ) 2 + (3 x <J»(x))Z } ,
(8)
—
and
~ & ! (*(x)) = g *(x) 4 - I
(9)
The constant term in the " p o t e n t i a l " 01, is so chosen that$£ > 0.
The Hamilton equa-
tions of motion derived from (7) - (9) are TT(X)
= 3 t <j, (x)
0
}
j
219 343
For a complete existence theory for solutions of (10) see the contributions of Garding and Strauss to these proceedings, and r e f s . given there. A finite-energy solution of (10) is a solution * functional
for which the Hamilton
^ H U,
dx * M ( i r ( x , t ) , <j>(x, t ) ) > 0
(11)
— CO
takes a f i n i t e value EQ (
d x ^ ( 3 t * o ( x , 0 ) , 4>Jx, 0))
< »
These finite-energy solutions f a l l into four homotopy classes represented by the stationary solutions 4>+ (x) = ( 8 g ) - \ *_ (x) = - (Sg)"3*
*- (x) = (8q)'h
tanh ((Sg)"* 5 x ) ,
(12)
<(. (x) = - (8g)'h s
tanh ((Sg)"1"2 x)
The solutions <J> and $— are called soliton - , resp. a n t i - s o l i t o n solutions. interpolate between the constant solutions
(13) They
(Note that <)> and <J> are
solutions" (5) of the anharmonic, double hump o s c i l l a t o r .
At this point we meet the following rather general p r i n c i p l e :
A theory in d space-
time dimensions with solitons corresponds to an analogous theory in d-1 space-time dimensions which has instantons). From heuristic and rigorous work concerning quantum solitons that has been done over the past few years emerged the following p i c t u r e :
The homotopy classes of
f i n i t e energy solutions to some classical Hamiltonian equations of motion (non-linear, invariant f i e l d equations) are, f o r small values of g (<"= Planck's constant), in a one - one correspondence with inequivalent super-selection sectors of the corresponding quantum f i e l d theory, provided the homotopy classes represented by the constant solutions ( i n g (J>„, $
and <> j ) are in one - one correspondence with the c r i t i c a l points
of the Euclidean action ( i n g <> j ~ given by <(>(x) = + (8g)
2
) , i . e . , provided there is
no tunnelling leading to a more complicated structure of the physical vacuua. 4 In Part I I we shall show that this picture is correct for the g <J>2 - theory in two space-time dimensions: In this r . q . f . model the homotopy classes represented by <(> and
correspond to two inequivalent vacuum sectors, whereas the ones repre-
sented by <)> and <>j— correspond to two (inequivalent) soliton sectors with non-vanishing, opposite "topological charge" Q =
/
dx^ (a
4.) ( x ) .
220 344
By considering the two space-time dimensional, abelian Higgs model one can also show that this picture may be false in the strong coupling regime (where there may exist superselection sectors not predicted by the classical f i e l d equations)CFl 1. This is perhaps bad news for many heuristic approaches to the problem of quantizing s o l i t a r y waves which are usually based on some sort of expansion in g («= M = Planck's constant), only valid for smal1 g. So f a r we have pretended that the Euclidean action (more generally:
the
Euclidean description of r . q . f . t . ) provides information concerning the properties of the quantum mechanical ground s t a t e , in r . q . f . t . the vacuum sectors of a r . q . f . model, but that we must investigate the Hamilton functional (more generally the Hamiltonian description of r . q . f . t . ) in order to obtain insight into the structure of superselection sectors inequivalent to the vacuum sectors.
While this principle is of
considerable, heuristic value, i t i s conceptually not satisfactory. Our main contention is that with the help of somewhat more sophisticated mathematical analysis i t is possible to extract a l l information about a r . q . f . theory from i t s Euclidean description.
This contention is detailed in Part I .
In Part I we summarize the general framework underlying the construction of n o n - t r i v i a l r . q . f . 's and t h e i r super-selection sectors.
In Part I I we shall discuss
an e x p l i c i t model (the g
221 345 Part 1 The general framework of constructive quantum f i e l d theory (includinq a praqmatic theory of
super-selection sectors)
1.1.
Introduction One of the thinqs a general framework of c . q . f . t . ouqht to teach us is how
to attempt to solve a non-linear, invariant f i e l d equation of the form n*(x)
where Q
is the
bounded below
= N [ F (4»(X))1 ,
(1.1)
d'Alenbertian, F = - P' and P' is the derivative of a polynomial P
(e.q. P(4>) = q <(> - T <> t
+
"gin")'
xe
M with v = 1 , 2, 3, (4) (the number
of space-time dimensions), and N denotes "normal ordering," a prescription of how to define higher powers of 4>{x); f i n a l l y the solution c(>(x) of (1.1) is
supposed to be a
r e a l , scalar f i e l d on Mv with values in the "operator-valued d i s t r i b u t i o n s " on a "suitable Hilbert space". The followina are some of the central problems arising in attempting to solve (1.1)
under these requirements: —What is a "suitable Hilbert space"? --What is a "suitable" topological vector space of operator - valued d i s t r i -
butions w i t h i n which solutions to equation (1.1) for some (hopefully interesting) class of F's can be constructed?
What class of Cauchy data (at time t = 0, say) can be im-
posed? --Do physics or mathematics impose additional requirements on the class of F's and on the solutions <)>(x)? Clearly, the c-number solutions to (1.1) constructed in the lectures of Garding and Strauss can be viewed as operator-valued d i s t r i b u t i o n s on a one-dimensional Hilbert space and hence represent a solution to the problem posed above.
Rut they do
not have anything to do with quantum mechanics, resp. the quantum theory of non-linear, invariant f i e l d equations. In quantum mechanics observables do generally not commute with each other, and there is no reason why, quantum mechanically, \>(x), * ( v ) } =
*(x)
d>(y) - <My)
(1.2)
(in the sense of d i s t r i b u t i o n s ) . I f we impose (1.2) as a condition on the solutions of (1.1) we f a l l back into the classical theory.
Quantum mechanics t e l l s us t h a t , in general, the r . h . s . of (1.2)
ought to be f 0 and proportional to ft (Planck's constant).
222 346
Quantum physics does however impose specific restrictions on £
\jM, Here (x-y)
x
e
= (x°-y°)
- (x. - y_)
;
(*
=
(1.3)
(x°>2i) with x° the time-component of
x,
1T\ However, for ( x - y ) 2 > 0,
£<|>(x), * ( y ) J t 0.
(1.4)
I t is well known that for many years the combination of (1.1) with (1.3) (1.4) looked l i k e an i l l - d e f i n e d problem, or at least a very
d i f f i c u l t one, except
for v = 1 , (the case of the anharmom'c o s c i l l a t o r ) where (1.3) is void. This lead
to f r u s t r a t i o n , and f r u s t a t i o n lead
to (amonq other things)
axiomatic f i e l d theory which attempted to formulate general principles (axioms) r . q . f . t . , and in particular any quantum theory of ( 1 . 1 ) ,
any
ought to s a t i s f y .
One of the main purposes of formulating axioms was to show that the basic principles of r . q . f . t . are compatible with each other (and with the requirement of n o n - t r i v i a l scattering) or, more concretely, to pose the problem of developing
a
quantum theory of f i e l d equations such as (1.1) in a mathematically precise way. Most suitable for our purposes are the Wightman axioms £ S2, J2] which b r i e f l y recall here, for the convenience of the reader.
we
(At this point we note, however,
that the developments of the past two or three years, in p a r t i c u l a r the discovery of quantum s o l i t o n s , call for a s l i g h t modification of the framework l a i d down by the Wightman axioms.
A possible such modification or extension, inspired by the Haag-
Kastler axioms ["H3"l - and the author's work on quantum solitons in r . q . f . models in two dimensional space-time \_F2, in Part
F3 J - is
described later in Part 1 and exemplified
2). F i r s t we give the Hilbert space formulation of the Wightman axioms, postulates
which - i t is proQosed - ought to be obeyed by a l l admissible solutions of a r e l a t i v i s t i c f i e l d equation such as (1.1): (Wo)
The states of a physical system are the unit ravs of a separable Hilbert
space an(more aenerallv, of a familv of such Hilbert soaces:
the super-selection sec-
tors). (Wi)
To each test function f in the Schwartz space -& (Rd) there corresponds
an unbounded operator
<j>(f)*
—• 4>(f).
i n ' f l t which is in the domain of a l l the operators
There is a domain 0 dense
{(f>(f); f e « ( R )} and l e f t invariant
by them. (Wii) There is a continuous, unitary representation U: (a,A) e
group^+ U(a,A) f
on ^§(such that
(a,A) M = f (A"1 (x-a))
U(a,A)
223 347
(Wiii) The spectrum of the infinitesimal generator (H,P) (the energymomentum operator) of {U(a,l): ae R v } is contained in the forward liaht cone ¥ + , and (0,0) is an eiqenvalue of (H,P). (Wiv) On the domain n of (Wi),
C*(f). *(q)3 = 0, i f the supports of fand g are space-like separated; (see ( 1 . 3 ) ) . (Wv) Let ) be the *aloebra of polynomials in {
f e&(R d )}.
Then
there is a vector £2 in the eigenspace of (H,P) corresponding to the eigenvalue (0,0) such that Q e D and { ( ? (<|>) f2}
is dense in ^ , .
From now on we choose D = {(p(4>) n } . is
In this formulation the f i e l d equation
equivalent to the requirement that {•*(x)
- N[F
(*(X)}} a
in the sense of *J^-valued d i s t r i b u t i o n s .
= 0
(1.5)
This is a consequence of the Reeh-Schlieder
theorem £S2 , J2] . Amonn (Ho) - (Wv) we feel (Wv) is the least plausible axiom; and i t is the one that w i l l
require modifications in models with quantum
solitons.
Wightman's reconstruction theorem says that a r . q . f . t . satisfying (Wo - Wv) is completely determined by the vacuum expectation values ( v . e . v . ' s ) Wn ( x r . . . , x n ) =
2 , . . . , and equation (1.6)
is
,
(1.6)
to be understood as an equation between
tempered d i s t r i b u t i o n s , thanks to (Wi) and (Wv). butions.
iu
The W 's are called Wightman d i s t r i -
They s a t i s f y - see l_S2 > J
WQ = 1 (<=>
(WI) (W2)
(<=>
(Wi), (Wv) + nuclear theorem).
P o s i t i v i t y (<=> the scalar product < - , - > of ^Jv, is positive d e f i n i t e ) . Invariance:
The distributions W are Poincare - i n v a r i a n t (<=> W ( i i ) ,
W(v)). (W3)
Spectrum condition:
Support properties of the Fourier transform W of
W , n = 1 , 2, . . . (<=> spec (H,P) S 7 + ) . (W4)
Locality:
For (x. - x i + 1 ) 2 < 0
W (. . . x ^ x i + 1 , . . . ) = W
(...xi+r
x . . . . ) , for a l l n (<=> (Wiv)). Wightman has shown the equivalence of (Wo) - (Wv) and (W0) - (W4). One of the main problems of constructive quantum f i e l d theory is to construct models of r . q . f . ' s with n o n - t r i v i a l scattering satisfying (W0) - (W4), i f possible in a space-time of dimension 4.
Conventionally one looks f o r such models which are formally
parametrized by a non-linear, invariant f i e l d equation, e.g. of the form ( 1 . 1 ) , ( 1 . 5 ) ;
224 348 (nowadays other parametrizations, formally equivalent to a f i e l d equation, are however usually preferred). In dimension v = 4 this main problem is s t i l l unsolved, but, f o r v = 1,2,3, plenty of models with n o n - t r i v i a l scattering satisfying (WO) - (W4) and (1.5) e x i s t ; ( f o r these models u n i t a r i t y of the scattering operator at high energies is s t i l l an open problem. explicitly
Recently Zamolodchikovt Zl 3and a group of people in B e r l i n C T l "}
constructed some n o n - t r i v i a l ,
unitary scattering
models in two space-time dimensions, a t r u l y yet
operators for r . q . f .
admirable result.
proven that these scattering operators come
However i t is not
from a r . q . f . theory satisfying
(WO) - (W4), and, moreover, they are t r i v i a l in so f a r as they do not describe i n e l a s t i c processes, and a l l
momenta are conserved.
This makes these constructions
special to two space-time dimensions, in contrast to some constructive q . f . t . £S3
results
, 0 1 , El ] ) . In pursuit of the main problem of c . q . f . t .
i t has turned out to be extremely
d i f f i c u l t to attempt a d i r e c t construction o f the Wightman distributions {Wni _ of some formal r . q . f . t m o d e l , e.g. one formally defined in terms of a f i e l d equation such as ( 1 . 1 ) , ( 1 . 5 ) ; see C Gl This lead
, H4 3 .
to the following discovery r S4 • Nl"1
:
I t is always possible
to convert the hyperbolic problem o f solving a non-linear, invariant f i e l d equation such as ( 1 . 1 ) , (1.5) subject to the requirement that the solutions satisfy (WO) - (W4) into an e l l i p t i c problem + analytic continuation. Part of the discovery ( i n a sense i t s real depth) is the following recipe: Try to solve the e l l i p t i c problem by means of p r o b a b i l i s t i c potential theory, (the theory of generalized, stochastic processes). This discovery has a prehistory in quantum mechanics where i t is often easier itH to construct the unitary time evolution group e of a quantum mechanical system by f i r s t constructing the semigroup e
, t s 0 , using path space techniques and the
Feynman-Kac formula and then doing an analytic continuation in the time t .
(This is
by now a well established technique f o r proving selfadjointness of quantum mechanical Hamiltonians H). Before i l l u s t r a t i n g the discovery described here by considering some examples we explain
on what mathematical facts i t i s based: From axioms (W2) and (W3) i t follows, according to the Bargmann-Hall-Wightman
theoremfj S2 , J2 ] , that the Wightman d i s t r i b u t i o n s W ( z . , . . . ,z ) are holomorphic in a large domain of £ v n (called the extended tube) which contains the domains
ZJ
n=
{ 2
l
V
i M z j - z ^ c V
(and, by (W4), also the domains obtained by permuting the arguments) and the so called Euclidean points
£
n = {z!
V zj = %• Hj}' Zi * V
f0r
'" * j} '
(1 7)
'
225
349
where x. e R
denotes the space - and i t j . t- e R, the time-component of z.,.
On x. one defines rn S
n (xr---'xn^
= M
( n
-l'
U
l
V
it
n)>
x
j
= (
-j"
V'
These are called the Euclidean Green's or Schwinger functions.
(1-8)
For t , < t - < . . . < t ,
S ( x , , . . . , x ) is formally given by Sn (x,
x n ) = < n, _TT U (*.,
0)
e"(tj+l " V
H
) * ( x ^ 0) Q>.
(1.9)
Most of the properties of the functions S can be guessed by studying the formal eguation (1.9).
Among these properties - which follow from (WO) - (W4) - are:
The Schwinger functions S
are Euclidean invariant, (to be compared with
(W2)); they have a positivity property, called Osterwalder - Schrader positivity [OZ 2 > (a rigorous expression for the facts that, accordinq to (1.9) they can be -tH written as scalar products and that e , t * 0, is a selfadjoint contraction semigroup); they are symmetric under permutations of their arguments (a consequence of locality (W4)). The discovery C S4
, Nl J
described above can now be rephrased as follows:
A construction of generalized
functions {S } °°n satisfying the properties n n=0 stated above (and some additional technical conditions^ Nl ,02 , F4 3 ; this w i l l turn out to be an " e l l i p t i c problem") automatically yields Wightman d i s t r i b u t i o n s {Wn> " Q , by
analytic continuation in the time-variables, such that equation (1.8)
holds. The past f i v e years of c . q . f . t . and theoretical physics ("instanton-physics" t
P2 , H2 , CI
, C2])
have shown that i t is advantageous to f i r s t t r y to construct
the Schwinger functions {S }
™- of some r . q . f . t . , using the theory of functional
integrals, (renormalized Fredholm determinants) and stochastic processes. This raises the following natural question: ing the ones already
mentioned) of a sequence
{S }
What precise properties (includ« of generalized functions are
s u f f i c i e n t for the S n 's to be the Schwinger functions of W 's satisfying (WO) - (W4), in the sense of equation (1.8) (see also (1.9))? This question has been answered by Osterwalder and Schrader in a very s a t i s factory way £ 02
^ .
Here we b r i e f l y recall a somewhat weaker version of t h e i r main
theorem ("Osterwalder-Schrader reconstruction"). We define R* .
= * ( i . t ) : t > e, E >
<&(R+ _) = { • $ : $ ,S
and &
+
e =
0}
*&(R V ), supp-J c R^ " +,e ^(R+>£=0)
(1.10) }, (1.11)
226
350
For f e i & (R v ), define Exp f - { f n } J o . w i t h f 0 fn ( x r . . . . x n ) - (n!)"1
_"
Following Osterwalder and the following properties on { S > _
1, f , = f
<xj>
£
(n:)_1
Schrader \j)2 2
"« :
(EO)
f8n
(yl
x
n)
<1.12)
we postulate - as in D)l
J-
S = 1 , S e -sS' ( P v n ) ; o
n—u
for a l l f e *&(R ) for which | f | < 1 , where | - | I n=o
f...
n
tv
is some
,
norm continuous on z* (R ),
(n.')" 1 S ( f * n ) = S (Exp J)
(1.13)
converges absolutely. (This is much more than has to be assumed £02 ^; but it is convenient for the purposes of this review). (El) The S 's are Euclidean invariant, for all n = 1, 2, 3, ..., or S (Exp 5 ) - S (Exp^ B ) ,
(1.14)
for all proper Euclidean motions B of R v , where § . (x) = $(B~ X ) , and (E2) Let fe (jc.t) = For fj e-^>+, J = 1 M
|f|. < 1.
f (x,-t). N, arbitrary N = 1, 2, 3, ...
= S (Exp f1" x Exp f j ) = J (m.1)"1 ( n . T 1 S m,n=o
((T^)*" 1 • (fJ)'n)
are the matrix elements of a positive semidefinite matrix M. Remark: i t y condition, (E3)
This is a stronger version of the usual Osterwalder-Schrader p o s i t i v -
tozl. For a l l n = 1 , 2, 3, . . . . S (x 1
x ) is symmetric under permuta-
tions of i t s arguments. Remark:
Postulates (EO) - (E3), as formulated here, are suitable for the
Euclidean description of r e l a t i v i s t i c , neutral, scalar f i e l d s .
For f i e l d s with charge
and (or) spin and (or) internal degrees of freedom they have to be modified in a standard way.
For gauge f i e l d s the
modification is
not
standard, and this is s t i l l an
important research topic) Theorem 1.1, p 2 ] For generalized functions S
(x,
x n ) , n = 0, 1, 2
to be the Schwinger
functions, in the sense of equation (1.8) - (1.9), of a r.q.f.t. satisfying (WO) - (W4) it is sufficient that {Sp} n " Q
satisfy (EO) - (E3).
227 351
Remark:
Osterwalder and Schrader ^ 0 2 3 (and Glaser F G 2 3 ) prove a more
general theorem of this type, with considerably weaker
hypotheses, but identical con-
clusions. A few remarks on the proof of Theorem 1.1 - which consists in essence of continuing the Schwinger functions S - and estimates on S - analytically in the timevariables back to the (real time) Minkowski region and verifying (WO) - (W4) for boundary values so obtained - can be found in the next section.
the
This result t e l l s us
that a r . q . f . t . s a t i s f y i n g (WO) - (W4) can be described in terms of i t s Schwinger functions, i . e . there exists a purely Euclidean description of r . q . f . t . other versions of (E0) - (E3) equivalent to (WO) - (W4), [02]
(There are .
).
Next, we explain how this Euclidean description of r . q . f . t . may be related to functional integrals and stochastic processes: Postulates (E0) - (E3) ( i n particular (E3)) are obviously compatible with the existence of a p r o b a b i l i t y measure y on (the a - algebra £ generated by the Borel cylinder sets o f ) j»X = ^>' „»i (R v )of real-valued, tempered distributions such that * — reai S
n
(f
1
,...,f
n
= / for
arbitrary
f,,...,f
)H/S
n
(x
Xn) ^
1
f,
(X.)d\.
n TT (|,(f.) dp U ) , 1 i =l
(1.15)
ins*(Rv).
Here {<(>(f) : - f e i ( R v ) } are the "coordinate functions" defined by 4>(f) [ T ] = T(f),
for a l l T e V . Given (E0) and (E3), the existence of a probability measure y on 4 1 is equiv-
alent to the following i n e q u a l i t y , usually referred to as (NS)
Nelson-Symanzik p o s i t i v i t y :
For a r b i t r a r y f 1 , . . . , f" i n ^ > ( R v ) with |f J | < 1/2 ; N = 1 , 2, 3 M..
2 S (Exp $
x
Exp f j )
(1.16)
are the matrix elements of a positive (semi-) definite matrix M. Note: In
Condition (NS) does not follow from the Wightman axioms (WO) - (W4).
rare cases (NS) follows from the combination of (WO) - (W4) with a field equation,
such as (1.1), (1.5). Nevertheless, condition (NS) holds in most of the r.q.f. models thus far constructed, and it has been a powerful tool for constructing them. By Minios1 theorem [ Ml J
, (1.16), (E0) and (E3) imply that S (Exp-f ) ,
If | < 1, is the Laplace transform of a probability measure y on ^ ' : S (Exp f) = /
e + ( f ) d y U ) . |f| < 1-
(1.17)
In terms of the measure y, postulates (El) and (E2) can be reformulated as follows:
228 352
Let F be a I - measurable function o n ^ ' . Let 8 be some Euclidean motion of Rv. We define B F(i)>) = F (<S>R), where * B (f)
= * (fg). i
e •5^(R V )-
From (El) we then get /
y
6 F (<(.) dy M
= / ., F M
dy (<(,),
(1.18)
For all 8; i.e. y is Euclidean invariant. Let l+
be the smallest a - algebras on j& ' such that all the coordinate
functions {*(f) : i
e ^ ( R V ) . supp .fc { * *
0 }}
are £ + - measurable. We define the Euclidean time-reflection 6 by 6F M
= F (*e), ^
(f) =
for a r b i t r a r y I - measurable functions F on ^
e &(RV),
. Clearly 6 defines an isomorphism
between £ + and £_. Lemma 1.2.T F I Q
:
Suppose the Schwinger functions {S } °°
satisfy (EO) - (E3) and (NS). Then
there exists a probability measure y on si such that (1) Equations (1.17) and (1.15) hold, i.e. the moments of y are the Schwinger functions S . n (2) y is Euclidean invariant, in the sense of equation (1.18). (3) For arbitrary £ + - measurable functions F on v«' /
6 F(*) F(*) dy(<|>) > 0
(1.19)
(This is the p r o b a b i l i s t i c version of Osterwalder-Schrader p o s i t i v i t y ) . Remark: Every L - measurable function F eL (iX'.J.dy) can be approximated 2 in L by functions o f the form m I c, e * ( f i } , c. et. f i e ^ > ( R V ) , 1=1 1^1 <
1/2 ; seer.Fr].
So Lemma 1.2 (3) follows from (E2), (NS) and (1.17).
The class of probability measures described in Lemma 1.2 is denoted Jl/{ These measures are called quantum measures C F5 ,F6 2 •
229 353 Now we can rephrase the main problem of the
quantum theory of non-linear,
invariant f i e l d equations as follows: Consider the equation ( 1 . 1 ) , D*
(x) = N \ V ( < | > ( X ) ) 3
,
where F = - P 1 , and P is e.g. a polynomial bounded from below, and degree (P) = 4, for v = 3.
(For v = 1,2, much larger classes of functions P can be studied). Let <(>(x) be the stochastic process given by some measure u e J^\
.
Let N rF(<)>(x))~l be some normal ( I t o ) ordered version of F ( i . e . N is ^ n a prescription, depending on the measure y, of how to define powers (j>(x) of the process <J>(x) with d i s t r i b u t i o n y ) .
In the cases understood
X<j>" - models in 2 or 3 space-time dimensions) that can be determined with N.
at this time (e.g. the
the operator N depends on y in a manner
a p r i o r i , and, as a formal operation on powers, i t coincides
Let A be the v-dimensional Laplacean, and define N y
= / d v x Ny [P((4»(x) +
CP(* + f) - P (•)]
f(x)) - P (<Mx))]
Theorem 1.3: In v = 1,2,3 space-time dimensions, and for the class of functions F specified above, any solution y e AA
(a
quantum measure) satisfying the "Radon-Nikodym"
equations
dti(
(RN,
* +,f)
-
e*(Af)
-N e u
x for arbitrary
f
e J A = J&
+ * ( f . Af) [>(<(, + f ) - P(<j>)] , F =-P',
, (R v ), provides a solution 4>(x) to the non-linear,
invariant f i e l d equation (1.1) satisfying the Wightman axioms (Wo) - (Wv) (resp. (WO) (W4))and the equation ( 1 . 5 ) . Remarks: the
By Osterwalder-Schrader reconstruction, Theorem 1 . 1 , and Lemma 1.2,
moments of any measure y e J\^
are the Schwinger functions, in the sense of
equations ( 1 . 8 ) , ( 1 . 9 ) , of a unique r . q . f . t . satisfying (W4).
the Wightman axioms (WO) -
The proof of Theorem 1.3 is therefore reduced to showing that equation (RN)
implies the f i e l d equation (1.5). Let
f. e 4 ^ (R.
Here is the outline of a formal proof of (1.5):
), for some e > 0.
Using equation (RN) in infinitesimal
form ("integration by parts on function space", seeC.D2 n Av / , *(x) TT x sf' j=1
*(fj J
diift) = / . N v a/
. F7-J
) one finds
n [F(<|,(x))J TT <|,(f.) dy(4»), J j=1
(1.20)
230 354
f o r a r b i t r a r y n = 0 , 1 , 2 , . . . , in the sense of distributions on ^
(R^J s I f : f e e ^ ( R ^ ) } .
Equation (1.5) follows from equation (1.20) by Osterwalder-Schrader reconstruction (analytic continuation in the time variable,T02
T .
See also (J F7 ]
).
Theorem 1.3 t e l l s us that the main problem of the quantum theory o f equation (1.1) can be solved by constructing solutions
y e -\
to equation (RN).
In one
dimension the problem of solving equation ( 1 . 1 ) , resp. ( 1 . 5 ) , in the class of f i e l d s <> ( satisfying (WO) - (W4) is equivalent to solving equation (RN) in the class -M of quantum measures on *G'
=
^ ' r p , - i (R)-
This i s the quantum mechanical anharmonic
o s c i l l a t o r , and the equivalence follows from the Feynman-Kac formula fJN2, S 5 ^ . the case v = 2,3, F = - P ' , P(<\>) = gtt>* + | - <(>2 , 0 < g « follows from £ G3
, E2 , M2^
( f o r g > 0:
In
1 , m2 > 0, t h i s equivalence
under one a d d i t i o n a l , most natural assump-
t i o n ) , but i t is strongly believed to be true for the class of functions F specified above, without r e s t r i c t i o n s .
( I n v = 4 dimensions the situation is very
unclear),
A very formal "solution" (generally meaningless, but of great heuristic power) of equation (RN) is given by the so called Euclidean Gel!'Mann- Low formula:
, -h I {(V*)2 (x) + NJzP(Hx))]} uU dvM = Z"1 e
dvx TT fe <J>(x),
(1.21)
xeRV where Z is an i n f i n i t e normalization factor. 2
I f P(4>) = j -
meaning: In this case u is simply the Gaussian measure on « j ' with mean 0 and 2 -1 covariance (-A + m ) ; <> f is then called the " f r e e , Euclidean f i e l d " ; see £ N3]] . The moments of y are the Schwinger functions of the r e l a t i v i s t i c . n e u t r a l , scalar f i e l d 4> of mass m.
( I t is easy to show that y is indeed
a quantum measure, e t c . ; see e.g.(T6J
for a discussion of the free Euclidean f i e l d (and some interacting f i e l d s in v = 1 and 2 dimensions) from the point of view adopted in these notes). Further solutions u e / \
of equation (RN) have been constructed for the
following choices of P, setting F = - P 1 ; (we omit the discussion of the case v = 1 which is standard quantum mechanics). 1)
v = 2, P an a r b i t r a r y polynomial bounded from below. and refs. given there.
2)
v = 2 , P(
ro,2ir); seerF8
See [|G3 ,G4
<(>2, m2 > 0 , X r e a l , e
< 4ir, 6 e
, F5) and r e f s . given t h e r e .
v = 2 , P(<(>) = X Cosh (e
3)
,S5 ]
~
, FICj
<> f , m > 0,
and r e f s .
v = 3, P(<(>) = XQ* + y- <(>2, X > 0, m2 r e a l ; see £ G5, M3
, FllJ and refs.
given there. In some of these cases ( e . g . 3) with m2 = - 1 , X > 0 s u f f i c i e n t l y small) i t is known that there are at least two solutions y + mutually singular.
and u_ to equations (RN) which are
On the other hand, in case 1) uniqueness theorems are known f o r
231 355 2
P((J>) = Part
XQ(c|>) + ?r- <()2, m2 > 0 ( f i x e d ) , Q a polynomial bounded from below, and 0<X « 1.
2 of these notes is devoted to a discussion of examples, where the solutions of
equation (RN) are non-unique, and of the consequences of such non-uniqueness. The most substantial results concerning the construction of measures u e^M q.m. satisfying (RN) (resp., h e u r i s t i c a l l y , (1.21)) f o r a large class of functions F are due to Glimm,
Jaffe and Spencer (see £
and Simon (see \^_G4
G3 , G5
, S6
, G8 J ) and to Nelson (see £ N4 ^
• G6 ,G7j) to Guerra, Rosen ).
In these references, however, only the construction of solutions u e -M, to equation (RN) and, as a consequence, of a quantum f i e l d <(>(x) satisfying (WO) - (W4) and equation (1.5)
is studied.
In other words, in these references only the "vacuum
representation" of a quantum f i e l d <|>(x) satisfying (1.1) is constructed.
In some cases
(two dimensional r . q . f . models in the multiple phase region and gauge theories in two, three or four dimensions) this does not appear to y i e l d a complete quantum theory of the non-linear, invariant f i e l d equation in question, because of the existence of n o n - t r i v i a l Poincarg -covariant super-selection sectors orthogonal to the vacuum sector
^C4,F2,F3J
(see the heuristic explanations given in the introduction to Parts 1 and 2).
A general
theory of such super-selection sectors is presented in Section 1.2.4.
232 356
1.2.
Outline of a general theory of super selection sectors in r e l a t i v i s t i c quantum f i e l d theory
1.2.1. Remarks on Osterwalder-Schrader reconstruction For the purposes of developing a general theory of super selection sectors s t a r t i n g from the Euclidean description of r . q . f . t . which is concrete enough to be useful in the study of special models (e.g. the \
J than the one formulated in postulates (EO) - (E3).
Thus we f i r s t pre-
sent a reformulation of these postulates. Let {h } , v be a C°° p a r t i t i o n of the i d e n t i t y on R with the following a ac Z properties: For a l l aeZ , 0 < h < 1 , and h is C" with supp h contained in a cube centered at a with faces parallel to the coordinate hyperplanes and sides of length 3/2. Let || • || be a norm continuous on -*3(R V ) with the following properties: 1)
| | • | | i s translation invariant.
2)
I f x is the characteristic function of a compact rectangle in R , | | x l l < °°-
3)
For f ( x ) c - »
(Rv~ ) and xrn x-i
the characteristic function of the s t r i p
{x : x = (.x.t), 0 < t < T} , II f » X [ 0 J J
U 2 i o (T) | f | 2 ,
(2.1)
where | • | is some norm continuous on i ( R
) .
Examples: = ||f|| ~= ( / | f ( x ) | P d v x ) 1 / P , P < Z . - P > (b) | | f | | 2 = / dp (m2) ( | f | , (- A+ m 2 )" 1 | f | ) , 0 2 where (•,•) is the L scalar product, A is the v dimensional Laplacean and p is a (a) | | f | |
(2-2)
positive measure on \j) ,<*>"] with dp
f 0
(™ ) m
< oo ;
(2.3)
see t Dl " ] • We now state a stronger version of the regularity condition (EO). (EO')
Exponential bound ( " S t a b i l i t y under linear perturbations of the dynamics") Let
f c ^ ( R v ) ; set
f
=f • h , a e
Zv.
Suppose that || f | | < 1 , for a l l a e Z
v
.
Then S(Expf) =1 n=o
(n:)'1 S ( f8n)
233 357 converges absolutely, and |S (Exp f ) | < K, exp TlC,
f
I || ae 7»
J|~\ -J
.
(2.4)
for some f i n i t e constants K, and K-. The remaining postulates (El) - (E3) are unchanged, i . e . (El)
Euclidean invariance S(Exp f ) = S(Exp f )
(E2)
(2.5)
Osterwalder-Schrader p o s i t i v i t y For f • e J 3 + , j = l , . . . , N , N = l , 2 , 3 , . . . S (Exp f l x Exp-p J ) are the matrix elements of a positive semi-definite matrix.
(E3)
Symmetry S
Remark:
( x , , . . . , x ) is symmetric under permutations
of i t s arguments.
By (E3) S (Exp f x Exp g) = S (Exp ( f + g ) ) ,
(2.6)
where, by (EO1). both sides are well defined for f and g in-iX(R ) with | | f || <
Let «X be the linear space of f i n i t e sequences F of test functions with
f
o
1/2,
1/2, for a l l a e Zv .
llgjl <
e t and f
f
ejJ(R
)
= 0, for a l l n > n„ ( F ) , for some f i n i t e n ( F ) . n o o
Let tSj. be the subspace of sequences F e Wwith the property that
where
supp f n <= Rv" , for a l l n = 1 , 2 , . . . ,n Q (F), R " = { x . j , . . . , x n ; x , = ( x _ . , t . ) , t , > 0, j = l , . . . , n } .
(2.7)
v
We set S(F) =
\ Sn ( f n ) , n=o
S(FxExpgxH)E Standard arguments
(2.8) \ m,j,n
(jl )
A
^ F7 . Dl ~\ show that
S^
(f
8 g
8j
8 h)
(2.9)
(EO1) implies that the r . h . s . of (2.9) con-
verges absolutely for F and H in & and g ess (Rv) with ||g || < 1 , for a l l For a r b i t r a r y Euclidean motions B of
R we define
a e Zv.
234 358
n 0 (F) FD = {f J g n,g n=o f
0,B
=
f
with
0'
(2.10)
f
n.3
g = 6 denotes r e f l e c t i o n at t = 0, and g = t n (F) We set F = ( f n > n ° 0
translation by (0_,t) ( time-translations). (2.11)
From (El) we then obtain S(FB) = S(F),
(2.12)
S (Fg x F) > 0,
(2.13)
and from (E2) for a l l F e
&+.
By (2.9) and (2.13), S (F 0 x G) defines a positive semi-definite inner product on M .
Let N be i t s kernel and consider ceri D
V/ N
(2-14)
Then S defines a scalar product on D . Given F e J S + , we denote by W(F) the equivalence class of F modulo N. We define <W(F), W(G)> = S(Fe x G),
•
(2.15)
for F and G i n u + . Completing D in the norm separable Hilbert space'Jf,.
| | - | | given by the scalar product <•,•> yields a
By construction D is dense in *)^ w ; *^ w turns out to be
the physical (Wightman) Hilbert space of some r . q . f . t . s a t i s f y i n g (WO) - (W4) with Schwinger functions given by S p , n = 0 , l , 2 , . . . £2 = W(l), where 1 5 {f n >
We set
, (2.16)
f a
o
= 1, f
m
= 0, for a l l m > 1 ;
turns out to be the physical vacuum. Using (E01) and (2.9) i t is easy to show - see f p4 , ni 1
- that the map W
can be extended to sequences of test functions of the form Exp f x F = { g n } n : o n , k=o
, k
(2.17)
235 359
n
(E\
where F = { y n f 0 n n=o such a way that
e. ^
W(Exp f x F) e ^
w
and f e aS+ with | I f J | < 1/2, for a l l a e z V , in
+
.
(2.18)
This defines a dense subspace of??,, containing D. If F e ^ T
t
then F e.&
+
: F
t
""""" F t '
defines a semigroup o n "
+
. for a l l t > 0; hence
= °'
+
For a l l F and G i n w S(Fe x G t )
+
(ED =
_
S((F e )_ t x G)
= S UT^~).e x G).
(2.19)
Thus, f o r G e N, |S (F e x G t )| = |S ( ( F p e x G)| < S ((T^ ) e x F t ) 1 / 2 S (Ge x G ) 1 / 2 0, i . e . G t e N, so that N is invariant under T . , and T. can be l i f t e d
t oJ ff+/N.
This permits us to define a semigroup P., t > 0, on D by P t W(F) = W(TtF) = W(F t ), t > 0 .
(2.20)
Lemma 2 . 1 : (1) Pf . t > 0, is a densely defined, symmetric semigroup on (2) For a l l <|) e D, s-lim P. i|i = i|i. t+o (3) | | P t * | | < || for a l l i, e D . Proof: (1) Since D is dense in^^v,, (2.20) shows that P t> t > 0, is densely defined. Furthermore <w(F), P t W (G)> = <W(F), W(Gt)> =
S(FQ x G t )
=
S ((F^g x G)
236 360
= <W(F t ), W (G)> = . (2) Clearly f
•+ f , as t ^
t
0, in J & ( R v n ) , for a l l n = 0 , 1 , 2 , . . .
.
vn
Since S ( x , , . . . ,x ) e s i ' (R ), for a l l n, and sS + consists of f i n i t e sequences of test functions, we conclude that S (Gg x F
) + S (G9 x F), as t * 0,
for a l l F and G in »
.
From this and the d e f i n i t i o n s of W and <•,•> (2) follows.
(3) Let ii = W (F) e D. Then | | P t * M 2 =