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An asymptotic loop extension for the effective potential in the p(ø)₂ quantum field theory Slade, Gordon Douglas 1984

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AN ASYMPTOTIC LOOP EXPANSION FOR THE EFFECTIVE POTENTIAL IN THE P((f))2 QUANTUM FIELD THEORY By GORDON DOUGLAS SLADE B.A.Sc, The University of Toronto, 1977 M.Sc, The University of Toronto, 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF J THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1984 © Gordon Douglas Slade, 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. -Department of ^A-TK^AM.TIC.'S . The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 E-6 (3/81) i i Thesis Supervisors; Dr. J o e l Feldman and Dr. Lon Rosen ABSTRACT: The e f f e c t i v e p o t e n t i a l V(ft,a) f o r the Euclidean P(<J>) ^ quantum f i e l d theory i s defined to be the Fenchel transform (convex conjugate) of the pressure i n an external f i e l d , and i s shown to be f i n i t e . The parameter Ti i s Planck's constant divided by 2ir . The c l a s s i c a l l i m i t (ti+O) of the e f f e c t i v e p o t e n t i a l i s shown to be the convex h u l l of the c l a s s i c a l 1 2 2 p o t e n t i a l P(a) + -m a . For values of a f o r which the c l a s s i c a l p o t e n t i a l i s equal to i t s convex h u l l and has a nonvanishing second d e r i v a t i v e , the usual one-particle i r r e d u c i b l e loop expansion f o r the e f f e c t i v e p o t e n t i a l i s shown to be asymptotic as Ti 4- 0 , using a uniformly convergent (as Ti 4- 0) high temperature c l u s t e r expansion and i r r e d u c i b i l i t y properties of the Legendre transform. For the same values of a , V i s shown to be a n a l y t i c i n a f o r s u f f i c i e n t l y small ti . F i n a l l y an example i s given f o r a double well c l a s s i c a l p o t e n t i a l where the one-particle i r r e d u c i b l e loop expansion f a i l s to be asymptotic, and the true asymptotics are obtained. i i i TABLE OF CONTENTS Abstract i i Table of Contents ^.. .. i i i L i s t of Figures i v Acknowledgement v Chapter 1: Introduction and Main Results 1 §1. The p(4>) 2 Quantum F i e l d Theory 1 §2. Phase Transitions 6 §3. Graph Notation 1 0 §4. Main Results ^ 3 Chapter 2: Preliminaries 1 7 §1. Convex Functions I 7 §2. Some Useful Transformations 23 §3. The C l a s s i c a l P o t e n t i a l 28 Chapter 3: The Main Estimates 3 6 §1. The Tr a n s l a t i o n 36 §2. The Cluster Expansion 3 9 §3. Convergence of the Cluster Expansion 4 4 §4. A n a l y t i c i t y of the Pressure 57 Chapter 4: Smoothness of the E f f e c t i v e P o t e n t i a l 64 §1. Proof of Theorem 1.4.1 64 §2. Proof of Theorem 1.4.2 68 §3. Proof of Theorem 1.4.3(a) 6 9 §4. Proof of Theorem 1.4.4 71 Chapter 5: The Loop Expansion 76 N §1. -V^y(0,a) i s a sum of Graphs 76 §2. The Test f o r I r r e d u c i b i l i t y 85 §3. The L a t t i c e Theory 88 §4. Regularity of the L a t t i c e Legendre Transform 9 4 §5. I r r e d u c i b i l i t y 1 0 3 Chapter 6: F a i l u r e of the 1-PI Loop Expansion 118 §1. An Asymptotic Connected Loop Expansion 118 Bibliography I 2 4 L i s t o f F i g u r e s F i g u r e 1 : The e f f e c t i v e p o t e n t i a l a n d p h a s e t r a n s i t i o n s F i g u r e 2. The r e l a t i o n s h i p b e t w e e n D f ( u ) a n d u(a) f o r f e C V Acknowledgement I t i s a p l e a s u r e t o thank P r o f e s s o r s J o e l F e l d m a n and L o n Rosen f o r what t h e y have t a u g h t me, f o r h e l p f u l s u g g e s t i o n s and f o r t h e i r e n c o u r a g e m e n t . I a l s o w i s h t o thank B r u c e Sharpe and P e t e r Sharpe f o r t h e i r company i n l e a r n i n g quantum f i e l d t h e o r y . I am g r a t e f u l t o Joanne Nakonechny f o r e n c o u r a g e m e n t , and t o the N a t i o n a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada f o r f i n a n c i a l s u p p o r t . F i n a l l y I w i s h t o thank C a r o l Samson f o r a f a s t and a c c u r a t e j o b o f t y p i n g t h e t h e s i s . 1 Chapter 1: INTRODUCTION AND MAIN RESULTS §1. The P((f))2 Quantum F i e l d Theory The subjects of axiomatic and constructive quantum f i e l d theory arose as an attempt to put quantum f i e l d theory on a sound mathematical foundation. The idea, which took shape i n the 1950's, was to write down a p h y s i c a l l y motivated l i s t of properties or axioms that a mathematically well-defined quantum f i e l d theory would be expected to s a t i s f y , and then look f o r examples s a t i s f y i n g the p r o p e r t i e s . One of the f i r s t sets of axioms [WG 64], the Garding-Wightman axioms, involves operator-valued tempered d i s t r i b u t i o n s (the f i e l d s ) a c t i n g on a H i l b e r t space, which transform i n an appropriate way under Lorentz transformations. An equivalent set of axioms i s formulated i n terms of vacuum expectation values of the f i e l d [SW 78]. When the axioms were f i r s t introduced the only models known to s a t i s f y them were free f i e l d s . I t was r e a l i z e d that the t e c h n i c a l problems involved i n the construction of i n t e r a c t i n g models were les s imposing when the dimension of space-time was reduced from four to three or two. Work began on the construction of two-dimensional models, with the hope that a technology could be developed that would be useful i n attacking the p h y s i c a l l y relevant case of four dimensions. A b r i e f account of early progress i n the construction of i n t e r a c t i n g two-dimensional models can be found i n the Appendix to [SW 78]. In the early 1970's a new strategy began to emerge, that of a n a l y t i c a l l y continuing the vacuum expectation values from r e a l to imaginary time. This new strategy was c a l l e d the Euclidean strategy because the replacement of t by i t changes the Minkowski metric to the Euclidean metric. In [OS 73] 2 conditions were given on the Schwinger functions (the a n a l y t i c continuations of the vacuum expectation values) which are equivalent to the Garding-Wightman axioms, i n the sense that existence of a family of d i s t r i b u t i o n s s a t i s f y i n g the Osterwalder-Schrader axioms guarantees the existence of a unique f i e l d theory s a t i s f y i n g the Garding-Wightman axioms, and vice-versa. Motivated by ideas of Feynman [Feyn H 65] the attempt since t h i s time to construct d-dimensional s c a l a r boson f i e l d theories has been centred on obtaining Schwinger functions s a t i s f y i n g the Osterwalder-Schrader axioms as the moments of a measure on the space S ( R )- of tempered d i s t r i b u t i o n s . In [F 74] a set of axioms (the POS, or p r o b a b i l i s t i c Osterwalder-Schrader axioms) was introduced f o r p r o b a b i l i t y measures on S'( R d) which guarantees that the moments of such a measure s a t i s f y the Osterwalder-Schrader axioms. The converse need not be true: the Osterwalder-Schrader axioms do not imply that the Schwinger functions are the moments of a measure. A theory whose Schwinger functions are the moments of a measure i s said to be Nelson-Symanzik p o s i t i v e . So f a r each of the several models constructed i n two and three dimensions s a t i s f y the stronger axioms. No i n t e r a c t i n g models have been constructed yet i n four dimensions. To state the POS axioms we introduce some d e f i n i t i o n s and notation. d d Denote the Schwartz space of real-valued t e s t functions on R by S( R ) , the space of continuous l i n e a r functionals on S( R d) by S'( R d) , and the action of <$> e S' ( R d) on f e S( R d) by <Mf) • Although not every i d <f> e S' ( R ) i s a function, we follow common usage and oc c a s i o n a l l y write 4>(f) = /<t> (x)f (x)dx • Let I denote the a-algebra of subsets of S'( R d) generated by the Borel c y l i n d e r sets, i . e . , sets of the form {<(. e 5' ( R d) : (<Mf.) , . ..f<Mf )) e ^} where f. e S( R d) ( i = l,...,n) , I n i 3 A is a Borel subset of R , and n can be any positive integer. Denote by £ + the o-algebra generated by sets of the same form but with the f supported in {x £ Rd : x^ > 0} , and l e t E denote the Euclidean group of translations, rotations and reflections of R E a c t s on S" as follows. For y e. E and f e S l e t (yf)(x) = f(y~ ^ x ) and -1, Define 6 e E by 6 ^ , — = (x^ ' X d-1' X d ) (Y+) (f) = +(Y f) x u x u—X <-l Following [FS 77] the POS Axioms for probability measures v on (S '( R d), are the following: POS 1: v i s invariant under E . POS 2: Fo6 F dv > 0 for a l l Y -measureable functions F on S '( R ) L+ POS 3: There i s a norm ||J • || continuous on S ' ( Rd) such that e ^ ^ d v i s uniformly bounded and continuous in the norm on {f e Si Rd) : |||f 1} POS 4: The action of the time-translation subgroup T is ergodic on (S ' ( Rd) , I, dv) . That i s , for a l l A e L ' (dv) and <j> e S ' ( Rd) , t lim >^ t - x » A(x 4>)ds = s A dv POS 1 embodies the requirement that the real time f i e l d theory be Lorentz covariant. POS 2 gives a positive definite inner product on the Hilbert space of states of the theory. The third axiom is a regularity condition and guarantees that the moments of dv exist. Finally, POS 4 is equivalent to uniqueness of the vacuum. The question of whether or not POS 4 is satisfied is closely connected with the study of phase transitions. The simplest example of a measure satisfying a l l the POS axioms is the 2 _ 1 Gaussian measure with mean zero and covariance C = (-A + m ) , where A 4 i s the Laplacian on R and m > 0 . We s h a l l denote t h i s measure by du^ or du „ depending on the context. I t i s the unique measure on ( S ' ( R d ) , I) whose Fourier transform i s i ^ t f ) ^ 2 d e d u c = e L (R ) ; the existence of t h i s measure i n guaranteed by Minlos' Theorem [GV IV 64, p. 3 5 0 ] . The Gaussian measure du^ i s the s t a r t i n g p o i n t f o r the construction of measures corresponding to two-dimensional i n t e r a c t i n g s c a l a r boson f i e l d s , namely the P (<}>) models. These models describe a s c a l a r boson f i e l d i n two dimensions with mass m and a polynomial s e l f - i n t e r a c t i o n , and give the quantization of the c l a s s i c a l f i e l d theory with (Euclidean) Lagrangian density L(<f>(x), V<J> (x)) = j(V<J> (x) ) 2 + j m2<}> (x) 2 + P (<)> (x)) . Suppose P i s a polynomial on R which i s semibounded (bounded below) and l e t A be a square centred at the o r i g i n with sides p a r a l l e l to the coordinate axes. Set :P(* (x) ) : Ddx B e d y c B d v A f P = • S - ( i . i ) :P (<J>(x) ) : dx C B B B 2 ~ ^" B where C = (-A + m ) and A i s the Laplacian with B-boundary conditions on 3A . B may be p e r i o d i c , free, or D i r i c h l e t f o r example; a p a r t i c u l a r choice i s often preferred for t e c h n i c a l reasons. The Wick dots : : n i n d i c a t e that P has been normal ordered with respect to C C a B B and are defined i n §2.2. One then attempts to show that dv approaches a A , P 2 l i m i t as A + R and that the l i m i t i n g measure s a t i s f i e s POS 1-4. To state the r e s u l t of [FS 77] concerning the existence of these l i m i t s i t i s necessary to introduce the pressure i n an external f i e l d y , defined by a (u) = l i m -r-jk- &n A+R :P (<j))-y<J>: A C B dy (1.2) C B Here |A| i s the volume of A and we have dropped the dummy va r i a b l e x from the i n t e r a c t i o n . I t i s shown i n [GRS 76] that the l i m i t i n equation (1.2) e x i s t s and i s independent of the choice of B f o r a wide cla s s of boundary conditions. I t i s an easy consequence of Holder's i n e q u a l i t y that a i s convex. In [FS 77] i t i s shown that a i s i n f a c t s t r i c t l y convex. From convexity i t follows that the d e r i v a t i v e Da(y) e x i s t s f o r a l l but countably many u , and the r i g h t and l e f t d e r i v a t i v e s 0*01 (y) e x i s t f o r a l l y • Let P y ( x ) = P(x) - yx . F r o h l i c h and Simon prove the following theorem i n [FS 77] . + - B Theorem 1.1: I f D a(y) = D a(y) then the measures dv given by —————— I\ * XT 2 V eqn. (1.1) converge to the same l i m i t dv p as A + R (B = free, p e r i o d i c , y D i r i c h l e t ) i n the sense of convergence of Fourier transforms, and the measure dv s a t i s f i e s POS 1-4. • P y + More generally, f o r any y they construct two measures dv p y corresponding to the i n t e r a c t i o n P^ which s a t i s f y POS 1-4, and are equal i f + - ± and only i f D a(y) = D a(y) , i n which case dv p = dv p . The existence of y y a y f o r which dv p ^ dv p corresponds to the existence of a phase y y t r a n s i t i o n f o r the theory. Phase t r a n s i t i o n s are discussed i n the following s e c t i o n . 6 §2. Phase Tr a n s i t i o n s In [FS 77] i t i s shown that D +a(y) <j> ( 0)dv p for a l l y e R y (2.1) where the one-point 4>(0)dVj, i s the number s a t i s f y i n g y <Mf)dv_ (f)(0) dv" f(x)dx for a l l f e S( R ) . Such a number u J e x i s t s by t r a n s l a t i o n invariance of dv p . When a i s d i f f e r e n t i a b l e at y eqn. (2.1) i s what i s obtained by formally d i f f e r e n t i a t i n g eqn (1.1). I f a i s not d i f f e r e n t i a b l e at a point y^ then <t>(0)dvp ? y 0 <M0)dv and so (|>(0)dvp i s discontinuous at y Q . When t h i s happens i t i s s a i d that there i s a phase t r a n s i t i o n at y^ , because a continuous change i n the parameter y r e s u l t s i n a discontinuous change i n the theory. A number of r e s u l t s have been obtained i n the l a s t ten years concerning the existence of phase t r a n s i t i o n s f o r various polynomials. In [SG 73] i t was shown that f o r P(x) = ax^ - bx 2 with a > 0 and b e R , a(y) has an a n a l y t i c extension to the complement of the imaginary axis and hence a phase t r a n s i t i o n can only occur f o r y = 0 . I t was shown i n [GJS 75] that f o r some values of a and b a phase t r a n s i t i o n does occur at y = 0 , and the i n d i v i d u a l phases were studied i n [GJS 76] using a low temperature c l u s t e r expansion. Low temperature expansions were used i n Cl 81] to obtain d e t a i l e d information about multi-phase theories, where d i f f e r e n t phases are obtained by varying not only the external f i e l d but also the other c o e f f i c i e n t s of P . 7 There i s a c l a s s i c a l i n t u i t i o n at work i n [GJS 76] and [ i 81] which we 1 2 2 now describe. Let (x) = P(x) + — m x be the c l a s s i c a l p o t e n t i a l , and 1 -1 2 consider U Q ( t i , x ) = tl U (Ti x) , f o r h small and p o s i t i v e . Here Ti i s Planck's constant divided by 2TT , and Ti 4- 0 i s the c l a s s i c a l l i m i t . I f U Q a t t a i n s i t s global minimum more than once, decreasing h has the e f f e c t of separating the minima and r a i s i n g the b a r r i e r ( s ) between them. For deep and widely separated minima of p o s i t i v e curvature (small ti ) the existence of more than one phase i s expected, because of the contribution of each minimum to the exponent i n the fun c t i o n a l i n t e g r a l d e f i n i n g the pressure. I f one the other hand U ^ has a uniquely attained g l o b a l minimum of p o s i t i v e curvature then U Q ("h; •) w i l l have that minimum magnified with respect to the others so that f o r small enough ti a unique phase i s expected -only the global minimum contributes s i g n i f i c a n t l y to the fun c t i o n a l i n t e g r a l . P o s i t i v e curvature i s required to ensure a p o s i t i v e mass f o r each phase. The work of [GJS 76] and [I 81] provides some j u s t i f i c a t i o n for t h i s i n t u i t i o n i n the case where U ^ a t t a i n s i t s global minimum more than once. In Lemma 4.4.1 we show that the i n t u i t i o n i s j u s t i f i e d i n the case of a uniquely attained global minimum. While the c l a s s i c a l p o t e n t i a l U Q provides a ru l e of thumb t e s t f o r the occurrence of phase t r a n s i t i o n s , a rigorous t e s t i s provided by the e f f e c t i v e p o t e n t i a l (which as we s h a l l see i s a quantum analogue of U ^ ) . The e f f e c t i v e p o t e n t i a l was f i r s t introduced i n [GSW 62]. Based on ideas of Jona-Lasinio [J-L 64], i t i s defined i n [CW 73] e s s e n t i a l l y to be the Legendre transform of the pressure i n an external f i e l d . Since the Legendre transform does not always e x i s t we f i n d i t convenient to use instead the unique convex extension of the Legendre transform, i . e . , the convex 8 conjugate or Fenchel transform [Fen 49] of the pressure. We define the e f f e c t i v e p o t e n t i a l V for the P(<f>)0 model by V(a) = sup [ua - a(y)] , yc R a e R (2.2) As we s h a l l see i n Theorem 4.1, V(a) i s always f i n i t e . By d e f i n i t i o n V i s convex and thus cannot have a double well structure. The v a r i a b l e a i s known as the c l a s s i c a l f i e l d , since the supremum i n eqn. (2.2) i s attained at a value of y s a t i s f y i n g provided such a y e x i s t s . The importance of V for the study of phase t r a n s i t i o n s i s a consequence of the f a c t that points of n o n d i f f e r e n t i a b i l i t y of a convex function are i n a one-one correspondence with l i n e a r portions of i t s convex conjugate. (See §2.1 f o r a b r i e f review of convex function theory). That i s , l i n e a r portions of V are i n a one-one correspondence with the occurrence of phase t r a n s i t i o n s . Two examples of the r e l a t i o n s h i p between the one-point function, the pressure, and the e f f e c t i v e p o t e n t i a l are shown i n Figure 1. The most common method for the c a l c u l a t i o n of the e f f e c t i v e p o t e n t i a l i s the loop^expansion [CW 73], [Jack 74], which provides a power se r i e s expansion i n h . U n t i l now the dependence of V on "h has been suppressed. Putting the ti's i n e x p l i c i t l y , the pressure i s given by y 1 -1 : P(t))) - y<j>: a (h ,y) = lim In e (2.3) 2 A+ R' and the e f f e c t i v e p o t e n t i a l by 9 Figure 1: The e f f e c t i v e p o t e n t i a l and phase t r a n s i t i o n s Figure 1 depicts the r e l a t i o n s h i p between the e f f e c t i v e p o t e n t i a l , the pressure and the one-point function i n two examples of an even c l a s s i c a l p o t e n t i a l . In (a) the absence of a l i n e a r p o r t i o n i n V implies that a i s differentiable" everywhere and hence the one point function D a ( y ) = cj>(0)dVp i s continuous. y In (b) the i n t e r v a l C - l , l ] on which V i s l i n e a r with slope zero corresponds to the f a c t that the pressure a i s not d i f f e r e n t i a b l e at y = 0 with l e f t and r i g h t d e r i v a t i v e s -1 and +1 r e s p e c t i v e l y , and hence the one point function jumps from -1 at y = 0~ to +1 at y = 0 + . 10 Vfh,a) = sup [ya - a(h,p)] . (2.4) ye R V i s then approximated by expanding i n a power s e r i e s i n "h and keeping N the f i r s t few terms: V(h,a) - T v (a)Ti n . T y p i c a l l y N i s one or two. „ n n=0 In four dimensions, the approximation with N = 1 has been used i n cosmology [Bran 82]. As we s h a l l see i n Theorem 4.2, v.(a) = U (a) for 0 o most values of a , so U Q ( h ) 1 S t n e c l a s s i c a l l i m i t of V(h,a) . In the physics l i t e r a t u r e i t i s argued that v n ^ a ) 1 S given by a c e r t a i n sum of one-particle i r r e d u c i b l e n-loop Feynman graphs. (We describe the graph notation i n the next s e c t i o n ) . The main r e s u l t of t h i s t h e s i s i s a proof that f o r most values of a the expansion of V as a power s e r i e s i n "n i s asymptotic i n the P(<f>)2 theory, with a proof that the c o e f f i c i e n t s of the expansion are given by the appropriate sum of graphs. We also give an example (in Theorem 4.5) where the one-particle i r r e d u c i b l e loop expansion f a i l s f o r c e r t a i n values of a . § 3. Graph Notation Feyman graphs provide a convenient notation for representing i n t e g r a l s of a form that a r i s e s frequently i n quantum f i e l d theory. In t h i s section the graph notation used i n t h i s thesis i s explained. To begin with an example and a fixed t r a n s l a t i o n i n v a r i a n t covariance C(x,y) = C(x-y), the X graph / \ i s by d e f i n i t i o n equal to 2 \ & = X 1 X 2 X 3 ^ 1 d x 2 C ( 0 i x i ) C ( 0 l X 2 ) C ( x l X 2 ) 2 ( 3 ' 1 ) A2 S The r i g h t side of eqn. (3.1) i s obtained from the l e f t side by i d e n t i f y i n g 1 1 2 . . . any one vertex as the o r i g i n i n R and ass o c i a t i n g with the remaining v e r t i c e s the v a r i a b l e s x^ and x^ . To every l i n e there corresponds a fac t o r of C evaluated at the endpoints of the l i n e . These factors are 2 m u l t i p l i e d together, integrated over R with respect to x^ , and the r e s u l t i s m u l t i p l i e d by the vertex factors A. . This procedure i s followed to obtain the value of any graph. Usually the vertex factors depend only on the number of l i n e s emanating from a vertex and are under-stood to be part of the graph without w r i t i n g them e x p l i c i t l y . These graphs a r i s e v i a Wick's Theorem [GJ 81] , which gives the expectation with respect to the Gaussian measure of covariance C (denoted ^ * ^ c ) of a product of Wick-ordered monomials as a sum of graphs. In p a r t i c u l a r , introducing the semi-colon notation defined by < F 1 ( * ) ; P 2 ( * ) > C = < F 1 W ) F 2 ( * ) > C C<V*)> c (3.2) ^ ( + );... ;F ( + ) % = I (-1) ' * ' + 1 ( | II | -1) '. II n < C Fi ( < t > )^> C  N TTCP j=l ieir . / n D where P i s the set of p a r t i t i o n s of {l,2,...,n} and ir = {TT , . .. ,ir i i } , n |TT| i t i s a standard f a c t that f or any t r a n s l a t i o n i n v a r i a n t covariance 2 ~1 P 2 ~1 that approaches (-A+m ) as A + R 2 , (for example = (-A^ +m ) with p A^ the Laplacian with p e r i o d i c boundary conditions on 3A), 1 / ^1 ^ \ lim TTT (A) :<() m ( A ) : ) i s given by the sum of a l l connected At R 2 l A l N X CA graphs with no s e l f - l i n e s , that can be made up of m v e r t i c e s with k. 2 _ 1 legs (i=l,...,m) and l i n e s of covariance (-A+m ) . Here a s e l f - l i n e i s a l i n e that connects a vertex to i t s e l f and a connected graph i s one 12 for which any two v e r t i c e s are path-connected by l i n e s i n the graph. Graphs are u s u a l l y taken to include c e r t a i n combinatorial f a c t o r s ; we explain our convention f o r combinatorial factors i n the next se c t i o n . Expectations of the form (3.2) occur most often as d e r i v a t i v e s , as n v k follows. Let S(A,<j>) = £ a (A) :<J> (A): . Then using induction i t can be 1 k=l * shown that f o r c e r t a i n p o s i t i v e integers C dA' <<n -S(A,<|>) , e dy, < | TT | I 1 1 I I I V -D S(A,<>);...; 1* ' S(A,cj>f> g TTeP. ' k . where • e e - S ( A ' * } d u C (3.3) I f a^(A) ->- 0 as A -»- 0 , then dA £n f e S ( A ' < f > ) d y o = I C <f-D ' 1 ' S (0 , * ) ; . . . ; - D ' ^ 1 7 1 S (0, <f>)\ C TJ TT > 1 1 /C 1 TreP, ' 2 "I Note that a graph with a s e l f - l i n e i s i n f i n i t e because (-A+m ) (x,x) 2 _ 1 IS i n f i n i t e . In f a c t a graph with (-A+m ) l i n e s i s f i n i t e i f and only i f i t i s connected and has no s e l f - l i n e s . A graph i s said to be 1-PI (one-particle i r r e d u c i b l e ) i f i t i s connected and i f the removal of any one l i n e leaves a connected graph. A graph i s sa i d to be 1-PR (one-particle reducible) i f i t i s not 1-PI . For example, Q and ^ are 1-PR, while (^ )^ a n d O a r e 1-PI. A graph G with L l i n e s and V v e r t i c e s i s said to be an n-loop graph, where n = L - V + 1 . For example, ^ ) Q ^ ^ a s n = 3, ^ a s n = 2 . , We close t h i s section with another d e f i n i t i o n . D e f i n i t i o n 3.1: Given a graph G and d e R , the d-renormalized graph G, i s the graph obtained by removing a l l s e l f - l i n e s from G , introducing d \ a vertex f a c t o r d for each removed s e l f - l i n e , and introducing a factor C . for every k-legged vertex of G having j s e l f - l i n e s , where k, D k! i s the number of ways of choosing j p a i r s from k , j 2 D j ! ( k-2j) ! k o b j e c t s . D For example, G = = > ^ = C4 i d ^} G = GL\) => G d = G -A -As we w i l l see i n Theorem 4.3, the 0 (TL 1) c o n t r i b u t i o n to V(fi,a) i s given by a sum of d-renormalized graphs, for a c e r t a i n d = d(a) . The d-renormalized graphs a r i s e from a Wick re-ordering procedure that introduces the combinational factors C i n a natural way Csee eqn. (5.5.8)). §4. Main Results Before s t a t i n g the main r e s u l t s , we introduce some notation. For a r e a l function f we denote i t s convex h u l l ( i . e . , the greatest convex function majorized by f) by conv f . Recall the d e f i n i t i o n of the 1 2 2 c l a s s i c a l p o t e n t i a l : (x) = P(x) + — m x . Let and B = B u B The remainder of the thesis i s devoted to the proof of the following theorems for the P (<{>) ^  model. The f i r s t theorem states that the supremum i n the d e f i n i t i o n of V(Ti,a) i s f i n i t e . Theorem 4.1: V(?i,a) < 0 0 for a l l Ti > 0 , a e R . The second theorem gives the r e s u l t f o r the c l a s s i c a l l i m i t of the e f f e c t i v e p o t e n t i a l that was ant i c i p a t e d i n §1.2. Theorem 4.2: l i m V(ft,a) = (conv U )(a) for every a e R . TUO 0 For a \ B , the following theorem provides a rigorous proof for the P(<j>)2 model of a standard (non-rigorous) r e s u l t i n quantum f i e l d theory [IZ 80] [Ram 81], namely that V(Ti,a) can be approximated by the f i r s t terms of the one-particle i r r e d u c i b l e loop expansion. Theorem 4.3: (a) Let a 4 B . Then there e x i s t s a y > 0 such that 00 V(R,a) i s a n a l y t i c i n -ft f o r Ti e (0,y) • Moreover, V(7A,a) i s C at Ti = 0 + ( i . e . , a l l right-hand derivates e x i s t at "h = 0) , and so 00 the expansion VCh,a) ^ £ v (a)^ 1 1 i s asymptotic, where n=0 n v n(a) = D^VC0 +,a)/nI . (b) Let a I B . Then v Q ( a ) = U Q(a) and v 1 ( a ) = -Y ( a ) = - lim j i j - Jin A+R2 P" (a) 2 ^ : (P : A 15 For n > 2 , -v Ca) i s the Cf ini te) sum of a l l d(a)-renormalized — n 1-PI n-loop diagrams with k-legged v e r t i c e s taking factors (k) - P (a)/k! (3 <_ k <_ deg P) and l i n e s corresponding to the free U"(a) covariance of mass A l " (A) , where d(a) = - — log — = — . A combinatorial U 4TT 2. m factor i s associated with each graph - see Remark 1 below. Remark 1: The renormalized graphs i n -v (a) are to be understood to n include combinatorial f a c t o r s . Given a renormalized graph, l e t V . be kD the number of v e r t i c e s that o r i g i n a l l y had k legs and have been renormal-ized with the removal of j s e l f - l i n e s . The combinatorial f a c t o r f o r the graph i s the f a c t o r associated with the graph by Wick's theorem divided by n V 1 . For example, the combinatorial f a c t o r of (/\) i s 1728 = 288. j,k 3 k 3-Note that since each vertex has at l e a s t 3 l i n e s , the number of graphs contributing to -v (a) i s f i n i t e because the number of loops n = L - V + 1 > | V - V + 1 = | v + 1 . As an example of Theorem 4.3 we obtain a renormalized (and rigorous) version of a r e s u l t of [Jack 74]. Let U Q ( X ) = + -|- x 2 and P(x) = x^ . Since U Q i s convex and U Q(a) = 12a 2 + 1 > 0 f o r a l l a , B i s the 1 2 empty set. Then Theorem 4.3 implies that with d(a) = - — - log(l+12a ) , 4TT -v 2(a) = [® + Q ] d ( a ) = ® + 3 d ( a ) 2 * , and -v 3(a) = [(Q)+CXD+ © + © + ® + O O O ] d ( a ) = © + 6d(a) <D + © + © + ® + 6 2 d ( a ) 2 O Lines are (-A+1) ^ l i n e s and 3- and 4-legged v e r t i c e s take factors 4a and 1 r e s p e c t i v e l y . Amputated legs have been p a r t l y drawn to keep c l e a r what the vertex factors should be. A f t e r t h i s research was completed work of Eckmann [E 77] was brought to the author's attention, i n which the loop expansion for the Schwinger functions of the P (<j>) model i s shown to be asymptotic. Theorem 4.3, which gives an asymptotic loop expansion f o r the e f f e c t i v e p o t e n t i a l , i s a natural follow-up to Eckmann's work. We comment i n Chapter 3 where some of our estimates mirror those of Eckmann. 4 In [F 76], i t i s shown that f o r <j> models V i s a n a l y t i c i n c e r t a i n a , with Ti f i x e d . The next theorem gives s u f f i c i e n t conditions for a n a l y t i c i t y of V i n a f o r general polynomials. Theorem 4.4: Let K c B c be compact. There e x i s t s a y > 0 and an open set 0 => K such that V(Ti_, •) has an a n a l y t i c extension to 0 for every "h e (0,y) . There are three main ingredients to the proofs of these theorems. The f i r s t step i s to reduce a n a l y t i c i t y properties of V(Ti_,a) to a n a l y t i c i t y properties of Tia(ri,u) by some elementary convex a n a l y s i s . This i s done i n §2.1. The proof that the pressure has the required a n a l y t i c i t y i s v i a a high temperature c l u s t e r expansion [GJS 73], and appears i n Chapter 3. The proof of Theorem 4.3(b) uses the t h i r d ingredient: an i r r e d u c i b i l i t y analysis i n the s p i r i t °f tCFR 81], which i s the subject of Chapter 5. F i n a l l y , i n Chapter 6 we prove the following r e s u l t which gives an asymptotic expansion for V(Ti,a) when a i s the bad set, f o r the 2 1 2 c l a s s i c a l p o t e n t i a l U (a) = (a - —) 0 o Theorem 4.5: Let V(n,a) denote the e f f e c t i v e p o t e n t i a l f or m = 1 and 17 PCx) = Cx2 - h _ i - x 2 . Than f o r |a| < — , Dv(0,a) = -y (—) = 0 , 8 2 /8 1 /8 and f o r n >_ 2 , - —j- D^V(0,a) i s given by the sum of a l l n-loop connected graphs with no s e l f - l i n e s , with three- and four-legged v e r t i c e s taking factors ^ P ( 3 ) ( — ) = -^2 and 7 7 P ( 4 ) ( — ) = -1 r e s p e c t i v e l y , and l i n e s corresponding to the free covariance of mass 1 . Graphs take combinatorial factors as per Remark 1. A number of authors [FOR 83], [BC 83], [CF 83] have recently calculated the 0 (Ti) contribution to the e f f e c t i v e p o t e n t i a l corresponding to the c l a s s i c a l p o t e n t i a l considered i n Theorem 4.5. They f i n d that the correct 0 (TL) approximation to the e f f e c t i v e p o t e n t i a l i s the s t r a i g h t l i n e i n t e r p o l a t i o n of the 0 (TL) approximation given f o r | a | > —— by Theorem 4.3. Theorem 4.5 gives a rigorous j u s t i f i c a t i o n of t h i s f a c t ; the proof i s an easy consequence of using the Fenchel transform to define V(Ti,a) and the known f a c t that there i s a phase t r a n s i t i o n i n t h i s model i f Ti i s s u f f i c i e n t l y small [GJS 76]. The observation that the n order contribution f o r |a| < — i s the connected graphs rather than the 1-PI /8 graphs appears to be new. Chapter 2: PRELIMINARIES § 1. Convex Functions We begin t h i s section by s t a t i n g some well-known properties of convex functions. Proofs can be found i n [Rock 70] or [RV 73]. Theorems 1.1 and 1.2 below w i l l be used i n the proofs of Theorems 1.4.2 and 1.4.3 r e s p e c t i v e l y . 18 Given a convex function f : R R , i t s convex conjugate i s defined to be f*(a) = sup [ua - f ( y ) ] , a e R . (1.1) ye R Since f i s convex i t i s continuous and the r i g h t and l e f t d e r i v a t i v e s ± + D f e x i s t everywhere and are nondecreasing. Let M = sup D f(y) and y m = i n f D f(y) . Then f or a e (m,M) the supremum i n equation (1.1) i s y f i n i t e and i s attained at any y for which D f (y) <_ a <^  D +f (y) . I f f i s s t r i c t l y convex then there i s one and only one such y , which we denote y (a) . I f a < m or a > M then the supremum i s +00 . In Figure 2 the r e l a t i o n s h i p between Df and y (a) i s depicted g r a p h i c a l l y . We denote by ^(C^) the cla s s of convex ( s t r i c t l y convex) functions ± 0 * f on R f o r which lim D f (y) = ±°° . For f e C , f (a) < 0 0 for a l l a . y-»-±oo A property of the convex conjugate we have already mentioned i n §1.2 i s that points of n o n d i f f e r e n t i a b i l i t y of f are i n a one-one correspondence with l i n e a r portions of f (see Figure 2). The pre c i s e correspondence i s that D~f(y) 7* D + f ( y ) i f and only i f f* i s l i n e a r with slope y on the i n t e r v a l [D~f(y), D + f ( y ) ] . ** For any convex function f , f = f . I t i s pos s i b l e to define the conjugate of an a r b i t r a r y function f by the formula (1.1) but i n general ** ** f ^ f • Something can be said however about the r e l a t i o n s h i p of f to f ; to avoid s u b t l e t i e s associated with i n f i n i t e - v a l u e d functions we ** only mention that f o r Q a semi-bounded polynomial, Q = convQ . Here convQ i s the convex h u l l of Q , i . e . , the greatest convex function majorized by Q . 1 9 Figure 2: The r e l a t i o n s h i p between Df(u) and u(a) for f e C£ (i) f (V) ( i i ) Df(y) ( i i i ) D +f (V (a x)) (i) Given a convex function f e C , ( i i ) the point u(a) at which s _ + sup[ya - f ( y ) ] i s attained i s the unique y for which a e [D f ( y ) , D f ( y ) ] y ( i i i ) I f Df has a jump d i s c o n t i n u i t y a t y Q , then for a e [ D _ f ( y ), D + f ( y Q ) ] f*(a) = y Q a - f ( y Q ) , so f i s l i n e a r on [D f ( y Q ) / D + f ( y Q ) ] with slope y Q . 20 Convex functions are well-behaved with respect to convergence prop e r t i e s . For example, i f f(ti,«) i s convex for a l l Ti > 0 and f(u) = lim f(Ti,u) iHO e x i s t s f o r a l l y i n a dense subset of R , then f(y) = l i m f (h,y) 1 H 0 f o r a l l y e R , the convergence i s uniform on compact subsets of R , and the l i m i t function f i s convex. I t also follows that D~f(y) < lim D~f(ft,y) <_ lim D*f(ti,y) £ D + f ( y ) , for a l l y e R . In 7A4-0 714-0 p a r t i c u l a r , i f f i s d i f f e r e n t i a b l e at y then li m D~f(n,y) = lim D*f(h,y) = Df (y) . ( 1 . 2 ) Ti+0 h+0 We now give conditions on a family f (Ti, •) of convex functions which imply smoothness of f ui,a) , beginning with the following theorem. Theorem 1 . 1 : Suppose f (h,») and f are i n C f o r a l l * i > 0 , and s suppose lim f = f(y) for a l l y i n a dense subset of R . Denote "h + 0 by y(h,a) and y (a) the unique values of y where ya - f(h,\i) and ya - f(y) a t t a i n t h e i r suprema. Then lim yftl,a) = y(a) and "h + 0 * * lim f (n, a) = f (a) . •h+o Proof: We f i r s t prove that l i m y(h,a) = y(a) . F i x a e R and e > 0 . Ti + 0 Choose p e ( 0 , e ) such that Df(y(a) ± p) e x i s t . Let a = -j min{Df (y (a)+p) - D + f ( y ( a ) ) , D _ f ( y ( a ) ) - Df(y(a) - p)} . Since f i s s t r i c t l y convex, a > 0 . Then Df(y(a)+p) > D + f ( y ( a ) ) + a >_ a + a and s i m i l a r l y Df(y(a ) - p ) < a - a . By eqn. ( 1 . 2 ) there i s a 6 > 0 such that |D ±f (h,y (a )±p) - Df(y(a ) ± p ) | < ^ for a l l "h < 6 . 21 Therefore D~f(ft , y(a) - p) < a < D^fOXyCa) + p) f o r a l l "h < 6 and so y(n,a) e Cy (a) - p, y (a) + p] for a l l "h < 6 , and hence l i m y(n,a) = y (a) Now |f (ti,a) - f * ( a ) | = | sup Cya - f(Ti , y ) ] - sup Cya - f ( y ) ] | . u y But i f sup a(x) and sup b(x) are attained at x & and x^ re s p e c t i v e l y , x x then | sup a(x) - s u p b ( x ) | <_max{|a(x ) - b (x ) | , |a(x ) - b(x b)|} X X < sup |a(x) - b(x)| , assuming x, < x. . — r 1 a — b x e C x a / X b ] Therefore for any Ti < 6 , |f*(Ti,a) - f* (a) | <_ sup |f(ti , y ) - f ( y ) | yeCy(a)-p,y(a)+p] Since f(Ti ,y) -v f(y) uniformly on compact i n t e r v a l s , the r i g h t side goes to zero as ti + 0 . • Theorem 1.2: Suppose f(ti,«) and f belong to the set C , with s lim f (ti,y) = f(y) f o r a l l y e R . Let A = {a e R : there i s no y Ti+O with D f(y) = D f(y) = a} . F i x a & A and suppose that f o r some y > 0 there i s an open i n t e r v a l I containing y Ca) , such that f i s a n a l y t i c i n (h,y) e (0,y) x I c c 2 and |D 2f(Tl , y ) | >_ C > 0 for every (ti,v) e (0,y) x I • d-3) Then f o r some y' > 0 , f*(h_/a) i s a n a l y t i c i n ti e (0,y') • I f i n addi t i o n the mixed p a r t i a l d e r i v a t i v e s of f are uniformly bounded i n Cn,y) , i . e . , there are constants M such that m,n iD^D^f (Ti,y) I < M f o r every (Ti,y) e (0,y) x i ; m , n = 0,1,2,... (1.4) 1 1 2 — m,n 22 then f (ft,a) i s C°° at TL = 0 + with D n(0 +,a) = l i m D n f * ( f t , a ) , T H O n = 0/1,2,... Remark: I f i t i s assumed that f i s C rather than a n a l y t i c i n * 00 (0,y) x I , the same proof gives that f (•,a) i s C i n [0,y') • Proof: By Theorem 1.1 we can choose y' < y such that yCh,a) e l i f Ti < y' . Also, i t follows from a n a l y t i c i t y of f and the bound (1.3) 2 C that there i s a neighbourhood 0 =» (0,y) * I on which |D f ( h , y j | > — . Y z z Let g(ft,y) = — [ya - f(ft , y ) ] = a - D 2f(ft , y ) f o r (ft,y) e V where we set V = 0^ n {1i,y) e C : 0 < Re "ft < y' } . Then y(Ti,a) i s uniquely defined by g(ft ,y(ft,a)) = 0 , for TL < y' . By the f a c t that |D2f(Ti,y) | >^  — on 0 and the i m p l i c i t function theorem [Horm 73] i t Z Z y follows that y(Ti,a) i s a n a l y t i c i n Ti i n an open neighbourhood uy . 3 (0,y') , with (Ti,y(h,a)) e V for a l l Ti e , . Therefore * f (h,a) = y(Ti,a) a - f (Ti,y ("h,a)) i s a n a l y t i c i n ti e U , . Suppose now that the bounds (1.4) hold. We show t h i s gives upper bounds on the absolute values of d e r i v a t i v e s Dny(Tiya) uniform i n ft e (0, y') . In f a c t y(Ti,a) i s defined by the equation g(Ti,y (ft,a) ) = a - D 2 f (ft,y (ft,a)) = 0 . (1.5) D i f f e r e n t i a t i n g eqn. (1.5) with respect to Ti gives 2 - D 1D 2f (ft,y (ft,a) ) - D 2 f (ft,y (ft,a) ) D ^ (ft,a) = 0 , - D D f (ft,y (h,a) ) i . e . , D y ffi, a) = . (1.6) D^f (h ,y(ft,a)) I t follows from the bounds (1.3) and (1.4) that |D^u(n,a)| i s uniformly bounded i n ti e (0,Y') • Repeated d i f f e r e n t i a t i o n of eqn. (1.6) together with (1.3) and (1.4) gives uniform bounds on the higher order d e r i v a t i v e s . These bounds on |D^y("h,a)| and the bound (1.4) imply that ID^f* (h,a) I < M < °° uniformly i n ft e (0,y') • But t h i s implies that 1 — n * 00 + f (ti,a) i s C at ti = 0 . To see t h i s , note that iD^f ( X,a) - Djf (y,a)| = x D ^ + 1 f * (s,a)ds| IM^^^Iyj + |x|) for a l l n+1 n * i y e (0,y') . Therefore ^D^f (n,a) }^>o i s Cauchy and so d = lim D nf (Ti,a) e x i s t s , n > 0 . But for n > 1 , ti+O D? 1 f*(Ti,a) - d 1 n-1 - d 1_ Ti (D"f (s,a) - d )ds 1 n i n * i < sup Dn f (s,a) - d | — „ «. 1 n 0<s<Ti Since the r i g h t side goes to zero as n * n equals lim D^f (ti,a) , n >_ 1 . • tiio n * + n 4- 0 , D^f (0 ,a) e x i s t s and §2. Some Useful Transformations This section contains some standard facts about Wick ordering and fu n c t i o n a l i n t e g r a l s that w i l l be needed l a t e r . We begin by d e f i n i n g Wick order. Let C be a covariance operator, and l e t h e C Q ( R ) be p o s i t i v e with h(x)dx = 1 . Define the approximate 6 function at x e R by 6 (y) = r h(r(x-y)) , r > 1 . The u l t r a v i o l e t cutoff f i e l d <j> i s r,x — r 24 given by <f (x) = <j>(6 ) and the cu t o f f Wick powers by 3T IT a X Cn/2] ; (l> r(x) n: c = I <-l) 3c ( ^ ( x ) ^ <x)n 2 3 (2.1) j=0 where c . = — r and cr (x) = n l (n-2j):j-.2 D 6 (y)C(y,z)6 (z)dydz r,x r,x ,P 2 2 - 1 For V c A , m, > 0 , and C = (-A +111^ x 1^11 X^ v) i t i s easy to see that there i s an M such that |cr^ _ ( x) | <_ M log r for large r . (2.2) I f f has compact support and i s an element of L^( R 2) for some p > 1 then :<|>n(f):c = lim : <J>"(f): e x i s t s i n L 2 ( d u c ) , where :<f>n(f): = r :tf)"(x) :f (x)dx. This defines the Wick monomials. The following lemma provides a Wick re-ordering formula. •Lemma 2.1. [GRS 75], [Sp 74]. For V a f i n i t e union of l a t t i c e squares i n A and m, m^  > 0 , l e t 2 ~~ I 2 2 ~^ C = (-A+m ) and = (-A+rt^Xy+m X^y^) with p e r i o d i c boundary conditions on dA . Then for any h > 0 and x e A , Cn/2] : < f, n(x): T^= I c (Fld(V,A,x)) k:<(,(x) n" 2 k: , K: k = Q nK TiC 1 2 -1 m i 2 where d(A,A,x) = — log — + K(A) with K(A) •+ 0 as A + R , and 4TT 2 m |d(V,A,x)I £ |d(A,A,x) Proof: By a standard r e s u l t [GJ 81, p. 168], [n/2] : ( ( > n ( x ) : h C = Jn C n k C r i 6 c V ( x ) } ^ ^ - h r k=0 1 where <5cv(x) = l i m [C (x,y) - C(x,y)] . y-*-x F P Denote by A the Laplacian with free boundary conditions and by A the Laplacian with p e r i o d i c boundary conditions on dh . Then f o r a > 0 (-AF+a2) 1 ( x ,y ) = (2TT) 2 2 p 2+a 2 (2.3) Writing nL = (n^L,n 2L) f o r n e Z and L the side length of A , (-A P+a 2) 1(x,y) = I (-A F+a 2) 1(x-y+nL) . neZ 2 For V = A , <5c (x) = li m [(-A P+m 2) 1(x,y) - (-AP+m2) """(x^)] y->x = lim [(-A F+m 2) 1(x-y) - (-AF+m2) """(x-y) + y->x I ((-AF+m2) 1( x-y+nL) - (-AF+m2) 1(x-y+nL))] nez 2 \ ( 0 } F. 2-1,.. ... , . F. 2." 1. % " 1 . m l By eqn. (2.3), l i m C (-A +xa~) (x-y) - (-A +m ) (x-y) = — log ~ . ^ 1 4TT 2 y->-x m Since (-AF+a2) (z) <_ const-e a ' z | for any |z| >_ 1 , i t follows that m2 d(A,A,x) = — log - i - + K(A) with K(A) ->- 0 as A + R 2 . 4TT 2 m -1 To handle the case when V ^  A we use the following Wiener i n t e g r a l representation f o r C [GJ 81]: C 1(x,y) = dt r 2 2 -, ds[m 1 x v(w(s)) + m x A\ y(w(s))] dW (w) e x,y where dW i s Wiener measure on the torus A f o r paths s t a r t i n g at x x,y and e n d i n g a t y a t t ime t . S i n c e the e x p o n e n t i a l f a c t o r o f t he - m 2 t _ m 2 f c i n t e g r a n d a lways l i e s between e and e , C^(x,y) a lways l i e s between KP. 2 -1 P 2 v _ 1 (-A +m1) (x,y) and (-A +m ) (x,y) , and hence |c i (x ,y ) - C ( x , y ) | < |(-A P +m 2 ) 1 ( x , y ) - ( - A P + m 2 ) _ 1 ( x , y ) T h e r e f o r e 16c (x) | <_ | 6 c A ( x ) | . • The f o l l o w i n g f o u r lemmas can a l l be seen on a f o r m a l l e v e l by w r i t i n g - 1 2 e [(V<f>(x))2 + m 2<()(x) 2]dx n d * (x) xeA - 1 r 2J e [ (V<(>(x) ) 2 + m2<|>(x)2]dx n d(j)(x) xeA Lemma 2 .2 : [Sp 7 4 ] , [GRS 76] ~ m k. 2 2 i L e t w. e L ( R ) have compact s u p p o r t and A($) = JJ s<f> (w.) 1 i = l 1 P. 2 -1 L e t a e R , P be a semibounded p o l y n o m i a l and C = (-A +m ) w i t h 1 2 2 p e r i o d i c boundary c o n d i t i o n s on dh . Then f o r (x) = P (x ) + -^m x , :P(<|>) : A(<f>) e r dyc(<J>) = JA(ip+a) e V U ^ k ) (a) ,k 1 2 . 2 A k=0 k i d y c ( ^ ) P r o o f : The lemma f o l l o w s by t r a n s l a t i o n by a . • P. 2 -1 Lemma 2 .3 : F o r V c A , b + m > 0 , and C = (-A +m ) w i t h p e r i o d i c BC on 3A , 27 j2 : cp : V - b d y c = d V > 2 L l 2 2 -1 where ^ = (-A+(m +b)xv + m X^ y^) with p e r i o d i c BC on 3A Proof: See [GJ 81, §9.3], Lemma 2.4: For A , P and C as i n Lemma 2.2, and f o r any h > 0 , = P(<f>) : A (if)) e dy, :Pfh <|>): A(Ti 40 e V d y c . The Wick dots i n each integrand match the corresponding measure. Proof: This lemma follows by s c a l i n g the f i e l d [GRS 76]. • Lemma 2.5: For A and P as i n Lemma 2.2 and a > 0 , :P(40 A (40 e dy C(A,m ) _1 2 A (4i) e a :P(40 oA dy -2 2 ' C(aA,a m ) 2 P 2 -1 where C(A,m ) = (-A +m ) with p e r i o d i c BC on 8A , and m i (a) (a) A (40 = II :4> (w. ): , with w. (x) = w. (ax) i = l Proof: See [GJ 81]. • 28 § 3. The C l a s s i c a l P o t e n t i a l 1 2 2 In Chapter 1 we defined the c l a s s i c a l p o t e n t i a l U Q ( x ) = p ( x ) + x and the bad set B = u , where B 2 = {a e R : U£(a) = 0} and = {a e R : U Q(a) ^  (conv U Q)(a)} . I t i s clear from t h e i r d e f i n i t i o n s that B^ consists of at most n -2 points, where n = deg P , while B^ consists of a union of at most ^— - 1 f i n i t e closed i n t e r v a l s . Let U u(x) = U Q(x) - yx . Let = {y e R : has a uniquely attained g l o b a l minimum} , and f o r y e G^ denote the l o c a t i o n of the minimum by ?(y) . Define F = {y e G 1 : UQ' (£(y)) = 0} and G = G^F . I t i s c l e a r that E, i s s t r i c t l y i ncreasing on G , and hence F i s f i n i t e . Let m(y) = min U (x) . x » Then f or y e G 1 , m(y) = U (£ (y)) • In t h i s section we prove the following f a c t s about B, G, E, and y . The set G c i s f i n i t e . The sets B and G are r e l a t e d by E, : B c = E, (G) . The functions E, and m are a n a l y t i c on G , with m'(y) = -£(y) and E,' (y) = rj" ( u) ) f ° r y e G . I t i s not hard to see from the d e f i n i t i o n of E, that E, i s s t r i c t l y increasing and continuous on G^ , and discontinuous on G^ . We show that l i m £(y) = ±°° . This, together with y-V+oo the f a c t s that -m'(y) = £(y) and E, i s s t r i c t l y increasing, implies that -m e C s F i n a l l y we prove a t e c h n i c a l lemma that w i l l be needed i n proving a n a l y t i c i t y of the pressure i n § 3 . 4 . 29 In p r e p a r a t i o n f o r p r o v i n g t h a t G ° i s f i n i t e we p r o v e t h e f o l l o w i n g lemma. n v k Lemma 3 . 1 : Suppose T (x ) = I t x a t t a i n s i t s g l o b a l minimum a t x = 0 k=2m o n l y , where m, t , t^_ > 0 . Then t h e r e i s a 6 > 0 s u c h t h a t n 2m T ( x ) > 6 ( x n + x 2 m ) f o r a l l x e R . P r o o f : S i n c e T i s bounded b e l o w , n i s e v e n . T h e r e f o r e t n _ 1 T ( x ) - | t n x n = - ^ x n + I t k x k + ~ as | x | - » , k=2m and t h e r e i s a K > 1 s u c h t h a t , % 1 n 1 . 2m I I T ( x ) ^ - t n x >_- t ^ x f o r |x|>.K (3 .1) To d e a l w i t h s m a l l | x | , o b s e r v e t h a t » t 2 m« 2" - I | t t | - t 2 m x 2 " c i - I ^ M*"2": k=2m+l k=2m+l 2m 1 v . I L e t e = m i n { l , — ( \ ——) } . Then f o r | x | <_ e , k=2m+l 2m T(x) > t x 2 m C l - e I - ^ - ] > t , x 2 m [ l - k = 4 x 2 m > 0 x n . (3 .2) — 2m £ t__ — 2m 2 2 2m — 2 2m k=2m+l 2m F i n a l l y , l e t a = m i n T(x ) > 0 . F o r e <_ [x| <_ K , e<_|x|<K , . a n a 2m a , 2m, n . ._ T (x ) > a > x + — — x > (x +x ) (3.3) ~ ~ 2 K n 2 K 2 m ~ 2 K n L e t 6 = min{--t , - r t__ , -a—} . By e q u a t i o n s ( 3 . 1 ) - ( 3 . 3 ) , 4 n 4 2m n 30 T(x) >_ 6 (x +x ) for every x £ R . • Lemma 3.2; G° i s f i n i t e . c c Proof; F i r s t , since F i s f i n i t e and G = G^ u F , i t s u f f i c e s to show that G^ i s f i n i t e . Note that (x) = 0 i f and only i f U Q(x) = u . For |y| s u f f i c i e n t l y large u Q ( X ) = u has a unique root and hence has a uniquely attained global minimum. I t follows that there i s an N > 0 such that G^ c [-N, N] . We claim that f o r a l l y e C-N, N] , there i s a deleted neighbourhood 0 of u such that 0 n Gf = <b . Given the claim, u V 1 l e t 0' = 0 u {y} . There i s a f i n i t e subcover {0 1 ,...,0' } of C-N, N] , 1 1 y y l % and therefore G^ c {y^, . . . ,y } . We now prove the claim, considering the cases y e G^ n C-N, N] and y e G1 n C-N, N ] separately. Suppose y e G x , and l e t W(x) = U u(x+5(y)) - U^(tl(y)) . By Lemma 3.1 there i s a 6 > 0 such that W(x) >_ 6 x n for a l l x e R . (3.4) The claim i s proved i n t h i s case provided i t can be shown that there i s a p > 0 such that Z (x) = U (x+S(y)) - U (Uy) = W(x) - ex (3.5) e y+e y+e has a uniquely attained global minimum for a l l e with | e | < p . By eqn. (3.4) Z (x) > 6 x n - ex for a l l x e R . (3.6) e — s We consider e > 0 ; the case of e < 0 i s s i m i l a r . C l e a r l y f o r e > 0 the global minimum of Z^ i s negative, and occurs i n { x e R : x > 0 } . Now Z (x) < 0 only i f x e (0, ( e 6 _ : L ) 1 / n _ 1 ) c (o, ( p 6 _ 1 ) 1 / n - 1 ) . Thus i t e suffices to show that there exists a p > 0 such that Z £(x) = 0 has only one root in (0, (pcS ^-)^/n f o r a n e e (o,p) . Note that Z 1(x) = 0 i f and only i f w'(x) = e . e Let a = min{l, min{x > 0: W"(x) = 0}} . Then a > 0 and W is one-one on (0,a) . Let p = 6a11 1 . Then (0,(p6 1 ) 1 / n ^) = (o,a) . Since W i s 1-1 on (0,a) , W* (x) = e has at mc5st one root in ( C C p S - 1 ) 1 7 1 1 " 1 ) . To prove the claim for p e , again consider the case e > 0 . Let £ = max{x e R: U attains i t s global minimum at x} , and l e t p W(x) = U (£+x) - U (£) . The proof here follows the previous case, using the fact that there is a 6 > 0 such that W(x) >_ 6x n , for every x > 0 which is clear from the proof of Lemma 3.1. For the E < 0 case, s h i f t U by 5 = min{x e R: U attains i t s global minimum at x} . • y y Lemma 3.3: The functions m and £ are analytic on G , with m' (p) = -£(p) and (p) = ^n ^ f ] 1 ^ • Furthermore, £ is s t r i c t l y increasing on G ^ , continuous on G ^ , and discontinuous on G ^ ; lim £(p) = ±°° ; and -m e C p-y+oo S Proof: The derivative U Q is an entire function, and for p e G , UQ(5(p)) = p and U Q ( £ ( P ) ) > 0 . By the Inverse Function Theorem [Rudin 74, p. 231], there are open neighbourhoods 0 containing p and V containing £ ( P ) such that U ' is invertible and the inverse i s 0 v analytic on 0 . This inverse is an extension of £ . Since for p e G , m(p) = U^(£(p)) = UQ(5(p)) -p£(p) , m is also analytic on G , with m'Cp) = U^UCvm'OO " P5'(p) - £(p) = -5(p) . To calculate £'(p) , differentiate the equation tM(£(p)) = p with respect to p to obtain c The f a c t that £ i s s t r i c t l y increasing and discontinuous on G 1 i s c l e a r from the d e f i n i t i o n of E, . I t i s also easy to see that £ i s continuous on F , and hence on . For large y , E,(y) i s the unique root of u Q ( x ) = y • As y -*• ± 0 0 that root diverges to ±°° , so lim £ (y) = ± » . This l a s t f a c t , together with the s t r i c t monotonicity y->±oo of E, and the equation -m1(y) = £ (y) , implies that -m £ C • s The following lemma shows how £ r e l a t e s B and G . Lemma 3.4: B C = £(G) . Proof: Suppose a £ £ (G) . Then there i s a y £ G such that E, (y ) = a a a Since U" (a) = U" (£ (y ) ) > 0 , a 4 B . We now show a j B . Now 0 0 3. £ X * * * (conv U Q)(a) = U Q (a) = sup Cya - UQ(y)] . Since y U*(y) = supCyx - UQ(x)] = -min U (x) = -m(y) , (3.7) x x (conv U Q)(a) = supCya + m(y) ] . y But -m i s d i f f e r entiable a t y and D(-m) (y ) = E, (y ) = a . Since a a a -m £ C , t h i s implies that s (conv U n) (a) = y a + m(y ) = y a + U (E,(\i)) = U.(?(y a)) = U.(a) . u a a a y a u « u a (3.8) Since G i s a union of open i n t e r v a l s and E, i s s t r i c t l y i n creasing and continuous on G , E, (G) i s a union of open i n t e r v a l s . Together with eqn. (3 .8) , t h i s implies that a { B^ . Hence £(G) c B C . On the other hand, l e t a £ B° . Suppose, contrary to the statement of the lemma, that a k K (G) . Then a e E, (F) or a £ £ ( G 1 ) C . I f a e 5(F) then U^(a) = 0 so a e B 2 . Therefore a e . By C r — *T* C Lemma 3.3 there must be a y Q e G^ for which a e L£ (y^) , 5 (y Q) ] c £(G^) The i n t e r v a l [5(y Q), 5 y^)] i s n o n t r i v i a l since y Q e G^ i s a p o i n t + + where £ undergoes a jump d i s c o n t i n u i t y . Since ^ ^ Q ) = D (-m) (y^) by Lemma 3.3, a e [D~(-m)(y 0), D +(-m)(y Q)] c £ (G^) C • I t follows from the f a c t that a e B c , eqn. (3.7), and the correspondence depicted i n Figure 2 that ** * _ U Q(a) = U Q (a) = (-m) (a) = y Q a + m(y Q) for a l l a e [D (-m)(y Q), D+(-m) ( y Q ) ] . But t h i s i s impossible because U Q cannot have a l i n e a r segment. • We close t h i s section with a lemma that w i l l be used i n the proof of a n a l y t i c i t y of the pressure i n §3.4. D e f i n i t i o n 3.5: For 6,L > 0 denote by T, the set of a l l polynomials o ,L T(x) = I t.x with |t | < L (k=2,...,n) and T(x) _> 6 (xn+x^) k=2 k K for a l l x . For T e T , small perturbations of the c o e f f i c i e n t s of T and a 0 , L small l i n e a r perturbation T(x) ->- T(x) - yx do not change the f a c t that the polynomial has a unique g l o b a l minimum, located say at £ . Transla-t i o n of the perturbed polynomial so that i t s global minimum s i t s a t the o r i g i n w i l l give a polynomial i n T , , for some 6 1 > 0 s l i g h t l y 0 , L smaller than 6 and L 1 s l i g h t l y l a r g e r than L . I f the perturbations are smooth then £ w i l l also be smooth. These elementary f a c t s are proved i n the following lemma. oo + Lemma 3.6: Let a be a n a l y t i c i n ( 0 , Y N ) and C at 0 , and l e t x 1 n T(Tl,x) = y a. (n)x . Suppose T(0 , O e T . Then there e x i s t k=2 ^ 6 , L 6',L',y,p > 0 such that T u(n,x) = T(n,x) - yx has a uniquely attained global minimum at say £(h,y) f o r a l l (n,y) e [0,Y) x ( _P»P) i with Sfti,y;x) = T (ti,E, (ti,y) +x) - T (ti,£ (n,y)) e T, , T , f o r a l l y y o , L (n,y) e Co,y) x (-P,P) . CO Moreover, 5 i s a n a l y t i c on (0,y) x (~P,P) and C on LO,y) x (~P»P) Proof: The proof that £(n,y) e x i s t s i s much l i k e p a r t of the proof of Lemma 3.2. F i r s t , note that by choosing ti s u f f i c i e n t l y small we can arrange that T (n,«) e T f o r a l l Ti e CO,y) / and hence 0 ^-,L+£ 2 2 T (h,x) >_ |- x 2 - yx for a l l (n,y,x) e [0,y) x R x R . (3.9) Consider the case y > 0 , for which the minimum of T^(n,-) i s s t r i c t l y negative and occurs when x > 0 . I t s u f f i c e s to show that D ^ T ^ ^ x ) = 0 has at most one root when T (n,x) < 0 , i . e . , when x e (0, 7^-) by y o eqn. (3.9). For "fc e [0,y) , l e t a(ti) = min{l, min{x>0:D 2T Q (ti,x) = 0}} . Then D T (n,•) i s one-one on (0,a(Ti)) . Let a = i n f a(ti) . To 2 0 0<h<Y 2 2 ~ V arrange that a > 0 note that D ^ T (?i,x) = D oT N (0,x) + I k ( k - l ) ( a Cti) _ z o z u k = 2 x a, ( 0))x k~ 2 and that a(0) > 0 (since T(0,«) e T. ) and l e t k o ,L 2 c = min D T_(0,x) , so c > 0 . Choose y smaller i f necessary a(0) 2 ° 0<x< — n 2 c to arrange | £ k(k-l)(a^(h) - a (0)x | <_— for a l l n e Co,y) and k=2 k k 0 < x < a(0) . Then D 2 T Q ( h , x ) = D 2 T Q ( 0 , x ) + I k (k-1) (a^ (ti) -a^ (0)) x k ~ 2 k = 2 > c - f - f for a l l h e [0 , y ) , 0 < x < 2 l 2 L and hence a >_ ^ ^ 0 ) . Since D 2 T 0 ( T i , x ) i s one-one on (0,a) f o r a l l Ti e [ 0 , y ) , there i s a t most one r o o t o f D 2 T Q ( h , x ) = y i n the i n t e r v a l ( 0 , 2 y 6 _ 1 ) p r o v i d e d 2y6 1 < a , i . e . , y < y a6 . Thus we take p = — a6 . 2 2 Now the l o c a t i o n £(h ,y) o f the g l o b a l minimum o f T ^ ( t i , ' ) l i e s i n the i n t e r v a l (0 ,a) , where a = 2p6 . By t a k i n g p and y s m a l l e r , I l k k i we can make — |D T 0 (tl,£(Tl ,y)) - D 2 T Q ( 0 , 0 ) | as s m a l l as d e s i r e d , u n i f o r m l y i n tl e [ 0 , y ) , | y | < p and k = 2 , . . . , n . T h e r e f o r e , s i n c e n D k T(h , t :(t i ,y ) ) SCh ,y ;x) = I - £ — x* , k=2 k -we can arrange that S ( n , y ; « ) e Tj. f o r a l l ti e [0 , y ) , | y | < p . CO I t remains to prove t h a t £ i s a n a l y t i c on (0,y) x (-p,p) and C on C0 ,y) x (-p,p) . L e t f ( t l , y ; x ) = D 2 T Q ( h , x ) - y . Then f i s a n a l y t i c on 0^ x c x C , f o r some complex open s e t 0^ => (0 ,y) . Since f (n ,y ; t;(ti ,y) ) = 0 , and D 3 f (tl ,y;?(ti ,y) ) = D 2 T 0 ( n,t ; ( n , y ) ) >_ | - > 0 f o r (h ,y) e (0,y) x (-p,p) , the I m p l i c i t F u n c t i o n Theorem [Horm 73] i m p l i e s t h a t there i s a neighbourhood U => (0 ,y) x (-p,p) on which £ i s a n a l y t i c . By d i f f e r e n t i a t i n g the equation f (ti ,y ; £ (h, y)) = 0 with r e s p e c t to Ti or y and u s i n g the uniform lower bound on D^f i t i s easy to see t h a t d e r i v a t i v e s o f £ are u n i f o r m l y bounded i n ti e (0 ,y) , and hence £ i s smooth a t tl = 0 + • • 36 CHAPTER 3: THE MAIN ESTIMATES §1. •The Transl a t i o n To prove a n a l y t i c i t y i n ti f o r the e f f e c t i v e p o t e n t i a l and obtain the desired form f or the de r i v a t i v e s at Ti = 0 i t i s convenient to perform a change of v a r i a b l e , so as to e x p l i c i t l y i s o l a t e the leading term. Let 2 _ 1 C = (-A+m ) with p e r i o d i c B.C. on 8A and r e c a l l that U (x) = u (x) - ux y 0 1 2 2 where U Q ( X ) = P ( X ) + 2 m X " B^ L e m m a 2 . 2 . 2 , f o r any f i x e d a e R 1 * J d y h c = 6 n U .00 [:P(H--y(fr ] - i|A|u (a) - j C I P. , (a):<t>k:-3m2:cj,2:J dy, tiC (1.1) By d e f i n i t i o n of the pressure i n eqn. (1.2.3), eqn. (1.1) implies Tia(ti,y) = -U (a) + Tia. (h,y-U' (a)) . y 1 0 (1.2) where a^CTijj) lim_ AtR 2 In 1 A k=2 k i - j * 3 dy, tic (1.3) I n s e r t i n g eqn. (1.2) i n the d e f i n i t i o n of V i n eqn. (1.2.4) gives V(ti,a) = sup[ya - Tia(Ti,y)] y supCya + U (a) - tic ^ (Ti ,y-U Q (a)) ] yeR y = U Q(a) + supC-Tio^ (h,y) ] yeR (1.4) 37 Next, we perform a mass s h i f t so as to e x p l i c i t l y i s o l a t e the 0(h) cont r i b u t i o n to the e f f e c t i v e p o t e n t i a l . Let m2 = U^'(a) = P" (a) + m2 . 2 For a 4 B , m^  > 0 . For the remainder of t h i s section we assume a 4 B . o -1 Let = (-A+mp with p e r i o d i c BC on 8 A . By using Lemma 2.2.3 i t follows from eqn. (1.3) that o (ti,y) l i m A in 1 [ I p ( k ) ( a ) : , k : - y 9 ] A K = 3 K L * C dy. TiC, + lim A 1 P (a) .2 dy. TiC (1.5) Here and throughout t h i s t hesis the Wick dots appearing i n an integrand are with respect to the covariance of the measure unless otherwise i n d i c a t e d . Introducing y(a) = l i m in 1 Ti dU. TiC (1.6) i t i s c l e a r from Lemma 2.2.4 that y i s independent of Ti > 0 . n r,(k) I 1 „ " J A k=3 Let a2Ch,y) = li m ^ in 11 ^ f ^ * k t i C dy. tiC, (1.7) Then by eqns. (1.4) and (1.5), V(h,a) = U Q(a) - tiy (a) + sup C-tio 2 (Ti,y) ] . yeR (1.8) The next step i s to Wick re-order the i n t e r a c t i o n i n to match the 38 (k) covariance C . Writing a = P ( a ) A• (k=3, ,n) and using Lemma 2.2.1, X K the i n t e r a c t i o n i n a 2(ti,0) can be rewritten as I a k : + k : f t c = ^ q i - ( ^ ) : < ( ) k k=3 k=o k ~ * C 1 (1.9) n-l-k - 1 m l where each q, i s a polynomial of degree C — - — ] i n tid = TlT ( — log ——) k 2 4u 2 m 2 plus an Ti-independent term that goes to zero as A i R ] . To si m p l i f y the notation we drop the A-dependent term (which i s i n s i g n i f i c a n t f o r large A 2 and disappears i n the A + R l i m i t ) . In view of Lemma 2.2.1 (see eqn. (5.5.8)), q Qm) = 0 ( t i 2 ) , q^Ti) = 0(h), q 2(h) = 0 (Ti) , q f c(n) = a f c + 0(n) (3<k<n) . (1.10) [n/2] E x p l i c i t l y , q om) = I C 2 k ^ a 2 k ( t i d r . k— 2 ( L I D Inserting eqn. (1.9) i n eqn. (1.7) gives tia 2(ti,y) = -q 0(h) + ti lim jjn In 1 Ti n A k=2 1 % C l ( 1 - 1 2 ) / Let a(Ti,j) = lim A 1 (Al Jin 1 ti C I q, :* k: " j*] A k=2 tiC, (1.13) so that ha 2(Ti,u) = -q Q(ti) +ha(h, y-q^Ti)) (1.14) Inserting eqn. (1.14) into eqn. (1.8) gives V(Ti,a) = U Q(a) - Try (a) + q Q(n) + sup[-Tia (Ti,y) ] , a | B y (1.15) D k q o ( 0 ) k Observe that — — = C v a o v d gives the value of the d k • ^k ,K ZK renormalized k loop graph with a s i n g l e 2k legged vertex a and legs joined up i n p a i r s . To show that the tran s l a t e d e f f e c t i v e p o t e n t i a l E(h) = sup[-Tia(h,y)] (1.16) y i s a n a l y t i c i n Ti with d e r i v a t i v e s at h = 0 given by the appropriate sum of graphs, we w i l l use Theorem 2.1.2 to reduce the problem to the study of Tia(Tl,y) . This pressure i s studied using a high temperature c l u s t e r expansion. § 2. The C l u s t e r Expansion The main d i f f i c u l t y i n proving a n a l y t i c i t y of the pressure Tia(Ti,y) and the p o t e n t i a l E(h) i s the i n f i n i t e volume l i m i t . The high temperature c l u s t e r expansion [GJS 73] i s often a useful t o o l i n dealing with the i n f i n i t e volume l i m i t i n a weakly coupled theory. (The terminology "high temperature" comes from the f a c t that weak coupling i n quantum f i e l d theory i s analogous to high temperature i n s t a t i s t i c a l mechanics). By Lemma 2.2.4 we can write a as -1 C I q Tl :<j) : - uT> cj>] i dy . (2.1) 1 a(Ti,y) = li m -ry-p Jin By eqn. (1.10), f o r k >_ 2 the c o e f f i c i e n t of :<J> : i s at l e a s t 0 (Ti ) 1 -1 2 2 so for small h and small yh we are i n a weak coupling s i t u a t i o n . However the s i t u a t i o n i s more complicated than the weak coupling case r -. h treated i n LGJS 73J since Ti occurs to d i f f e r e n t powers i n the d i f f e r e n t terms of the i n t e r a c t i o n . In t h i s section we write down the c l u s t e r expansion and give conditions f o r and consequences of i t s convergence. In the next section we prove bounds on the p a r t i t i o n function i n eqn. ( 2 . 1 ) (for y = 0) that guarantee convergence uniform i n "h . To begin, we introduce some notation. Since we are using p e r i o d i c BC , we i d e n t i f y opposite sides of A to obtain a torus. Let 8^ denote the set of a l l l a t t i c e bonds j o i n i n g nearest neighbour s i t e s i n the 2 p e r i o d i c l a t t i c e A n Z . For each b e we introduce a parameter S, e [0, 1 ] . Let b C ( S ) = I TT S TT ( 1 - S )C r c BA b e r C r r c 2 - i r c where C = (-A^  +m ) with the Laplacian with p e r i o d i c BC on 8A c r c and D i r i c h l e t BC on T , ( i . e . , A^ i s the F r i e d r i c h ' s extension [Kato 66] of A r e s t r i c t e d to C°°(A\rC)) . When s,_ = 0 each nonzero u b term i n C(s) has D i r i c h l e t BC on the bond b , whereas f o r s = 1 each term does not have D i r i c h l e t BC on b but i s f u l l y coupled across b . The parameter gives a measure of the amount of coupling i n C(s) r d across the bond b . For T c B , l e t 8 = TT and A ber d s b s ( D b = s^ i f b e T b 0 i f b | r Let dy(s) be Gaussian measure on S'(R 2) with covariance C(s) , and l e t Z(A,A ( S) = k - V ( A , X , s ) d u ( s ) where V(A,X,s) = I a. (X) k=0 k !* -C(s) 41 and the a are functions of X e 0 <= C m . We sometimes write Z (A,X) f o r Z(A,X,x ) , the p a r t i t i o n function with D i r i c h l e t BC on T . Let r c O x , A , s = Z ^ ' S > -1 -V(A,X,s), e dy(s) Then the following expansion holds k., • • • k 1 r , S A (x k k .,...,x ) = <":<)> (x ):...:<(. r ( x ) :\ 1 r \ 1 r / X,A,1 I x,r 1 r f r k-n :<(> ^"(x. ) :e i = l 1 - V ( X , X , s ( D ) dy ( s ( r ) ) d s ( D Z9x(A\x,X) Z(A,X,1) (2.2) k .. .k I T r(X,r,A,X ;x ,...,x ) , x,r r where i n the summation X ranges over a l l unions of closed l a t t i c e squares i n A containing {x ,...,x r) while T ranges over subsets of n i n t X o such that each component of X\T has nonempty i n t e r s e c t i o n with {x,,...,x } . The formal d e r i v a t i o n of eqn. (2.2) i s r e l a t i v e l y s t r a i g h t -1 r forward i n v o l v i n g l i t t l e more than the fundamental theorem of c a l c u l u s . The hard work goes i n t o bounding the r i g h t side of eqn. (2.2) with bounds independent of A and X . The proof of the following theorem i s i m p l i c i t i n [GJS 73]. Theorem 2.1: Suppose |a (X)| <_ L for a l l X e 0 , k e {0,1,2,...,n} , and that the following bounds hold: For some p > 1 , If -PV(A,X,s) | K|A| | e dp (s) I <_ e 1 , f o r a l l X e 0 , A , s (2.3) 42 and _^ — , for a l l A e 0 , and f o r any u n i t l a t t i c e square A . (2.4) o A ' Then for any C > 0 there i s an M > 0 depending only on K and L (and C) 2 r such that for m > M(K,L) and we L (A ) , and for a l l A and X e 0 , T r(X,r,A,X;x 1, . .. ,x r)w(x 1 # ... ,x r)dx| <_ |w|e~ (2.5) {X , r : X >D} where |•| i s a t r a n s l a t i o n i n v a r i a n t norm on L 2(A r) . • There are three main ideas i n the proof of Theorem 2.1. The f i r s t i s that the number of terms i n the sum over X and Y i n eqn. (2.2) having a c 1 j X | f i x e d value of |X| can be bounded above by e where i s a constant depending only on the geometry. The second key r e s u l t i s that fo r any constant C > 0 there i s a constant > 0 , depending on the K of eqn. (2.3), such that for m s u f f i c i e n t l y large (depending on C) , J 1 0 ( r kH -vrx x s ( m i - c l r l + c J x l n :<|> ""(xJre v t x ' A ' s u >> dy (s (r)) ds (D w (x ..., x ) dx |<_ e - - 2 |w| i = l -clr l The decay fa c t o r e 1 comes from estimating the d e r i v a t i v e s of C(s) r produced by applying the d e r i v a t i v e s 8 to the Wick dots or the measure dy(s (r ) ) on the l e f t side. See [GJS 73] for d e t a i l s . The bound (2.3) i s used with Holder's i n e q u a l i t y to control the exponential of the i n t e r a c t i o n . F i n a l l y , using the lower bound (2.4) i t can be shown using ideas from s t a t i s t i c a l mechanics and the two estimates j u s t stated that f o r m s u f f i c i e n t l y large (again depending on K) there i s a constant C 3 such that 43 Z 3 x(A\X,X) Z(A,X,1) < e J Combining the above three estimates with the fact that | r | ^_"j(|x'|-r) -c I r I because of the constraints on T , the factor e 1 1 gives convergence for C (hence m) sufficiently large. k i 2 Now l e t i>. = :<t> (w.) : where w. e L (A) , and l e t . be the 1 1 1 1/X 2 translation of 1 by x e R . In [GJS 73] i t is shown how to use Theorem 2.1 to prove that there are positive constants and m' such that -m' |x-y| 1 <C*1 x ;*2 1 -\ l,x 2,y/ X f A f l K 1 S with depending on ^>i'^ >2 " T ^ e m o s t important consequence of Theorem 2.1 for our purposes i s the following generalization of the above bound to higher order truncated expectation values, which i s due to Dimock [Dim 74]. (See [EMS 75] for related estimates). Theorem 2.2: Suppose the hypotheses of Theorem 2.1 hold. Then there are positive constants and m* such that sup sup I <V \ I 1 A XeO X 1 , X 1 N , XN/X,A,1 -m*<5 (x ,... ,x )/N I N K2 6 where 6 (x , ...,x ) = sup |x. - x.| , K depends on the ii, , and sup l l i f j f N 1 D 1 A ranges over squares A . • An immediate Corollary of Theorem 2.2 is the following result, which i s the main result we need from this section. Corollary 2.3: Suppose the hypothesis of Theorem 2.1 hold. Then there i s a positive constant M such that r 44 1 / 1 \ sup sup - r - T - l <:(() X(A) :;...;:<() r ( A ) : > \±K (2.6) A XeO | A | X X X,A,1 r where sup ranges over squares A of integer side-length. A 2 Proof: Denoting the u n i t l a t t i c e square centred at j e Z by A_. , we have k k -rjr | <^<j> 1 (A) :;...;:<(, r(A):^> | X, A, 1 k. k < 4 r y 2 j , . . . ,j eZ nA 1 r | 1 (A ):;...;:<(, r (A. ) \ | N 31 3 r X,A,1 (2.7) 1 / 1 0 \ [TT I | <j<f> (A ) :;:<(> (A ) : ; . . . ; : * r (A ) :> | 1 1 . • „2 . X J2 J l 3 r 31 ' X,A,1 j , ..., j eZ nA J l r 1 r - mjh 2-D 1l + ...+ | j r - J 1 | ] / r 2 1 T T T I K e (2.8) J 1 , . . . , J r e Z nA K -m [|jl |+...+|£ | ] / r 2 r V * 21 1 r J 1 e Z nA Jl , . . . , £ r e Z 2 <^  K2* const. where (2.7) follows from t r a n s l a t i o n invariance of the p e r i o d i c theory and (2.8) follows by Theorem 2.2. • § 3. Convergence of the Cluster Expansion To prove a n a l y t i c i t y of t i a ( t i , u ) i n Ti and u , the f i r s t step i s to perform a t r a n s l a t i o n i n the f u n c t i o n a l i n t e g r a l d e f i n i n g a to remove the 45 l i n e a r term from the i n t e r a c t i o n . For a = K(U ) ^  B , the c l a s s i c a l a p o t e n t i a l occurring i n a(h,y) i n eqn. (1.13) s a t i s f i e s (for Ti = y = 0) n n (k) I <*k<0>*k + I V 2 - I + = U y ( 5 (V + X> " U y «<V 6 T6,L k=2 k=2 a a fo r some 6,L > 0 , by Lemmas 2.3.4 and 2.3.1. For small Ti and y the t r a n s l a t i o n w i l l replace the ^ ( O ) by a n c ^ u dependent c o e f f i c i e n t s that are close to the q (0) , so by Lemma 2.3.6 the c l a s s i c a l p o t e n t i a l w i l l remain i n some T_ .We prove convergence of the c l u s t e r o ,L expansion i n j u s t t h i s context. The idea f or the proof of Lemma" 3.1 below o r i g i n a t e d i n work of Spencer [Sp 74]. A f t e r t h i s research was completed the author learned of a paper by Eckmann C E 77] where an estimate b a s i c a l l y the same as eqn. (3.1) i s proved by e s s e n t i a l l y the same method and used to prove convergence of a c l u s t e r expansion uniform i n small r e a l l l . Let S = { z e C : 0 < R e z < y , 0 < |Arg z| < Q) . 6,y 1 n — r k 2 Lemma 3.1: Let T(h,x) = I a ( t l)x and a ( t i ) = 0 ( t i ) where the a are k=2 k 1 * continuous i n some S . Suppose ReT(0,*) e T . for some 6,L > 0 . H iY o /L Then there e x i s t 6,y > 0 such that J [h :T(Ti,h2<|>) :+a (tl)*- \ m2:<(>2:] V dy 2 ( s ) m < e K ' V l (3.1) for everv t l e S„ and for every s , and for every f i n i t e union V of l a t t i c e squares i n A . The constant K depends on 6 and L . Before proving Lemma 3.1 we o u t l i n e the proof. F i r s t we reduce the problem to r e a l t l using an elementary argument and use conditioning 46 [GRS 76] to reduce to the case s = 1 . We then s h i f t mass from the 1 2 measure to the i n t e r a c t i o n , leaving a mass term — 6<p i n the volume V i n the measure. The r e s u l t i n g new i n t e r a c t i o n (Wick re-ordered to match the measure) evaluated at the u l t r a v i o l e t c utoff f i e l d <b i s bounded r n/2 below by -const (log r) , uniformly i n "h , using the f a c t that f o r "h 1 2 s u f f i c i e n t l y small T(?i,x) >_ — 6 (x n+x ) . An appeal to a r e s u l t of [DG 74] completes the proof. Proof of Lemma 3.1: The f i r s t step i s to reduce the problem to the case of r e a l Ti and a . Note that [^-:T(ri,ti2<j)) :+a (h)<f>- \ m 2 : 9 2 : ] v dy 2 ( s ) m (:Re[ir(ti,ti 2<j))]:+Rea 1(Ti)i})- ^m 2:f 2:) V dy 2 ( s ) m 2 2 For small 9 and y > Rea^ti) = 0 (Reti ) . Let t = ti = p +iq , and for 2 <_ k <_ n , l e t b, (t) = p 2 _ k R e (a ( t 2 ) t k ~ 2 ) for t ? 0 and b, (0) = Rea, (0) . Then k k ReETi -^T (Ti ,Ti2cp) ] I R e [ a v ( t 2 ) t k _ 2 ] 0 > k = I b t ( t ) p k V k=2 k k=2 K (3.2) But |b k(t) - Rea k(0) | < Re (a, ( t 2 ) t k 2) I " 1 _k-2 R e ( a v ( 0 ) t k 2) Re(a, ( 0 ) t k - 2 ) 1 + I — h K e % l 0 » Since t " = £ (™) p™" ( i q ) £ = p m [ l + £ (™) {^) ] , i t follows that 1=0 £=1 P t m | — - l | can be made a r b i t r a r i l y small by taking 6 (and hence |^|) P P small. Therefore , tk"2 k -2 |b k(t) - Rea k(0) | < | a k ( t ^ ) - a k ( 0 ) | |-^.| + 1^(0) | \ ^ - l | P P can be made as small as desired by taking 0 i y s u f f i c i e n t l y small, so there e x i s t 6,y such that T b, (t)(t>k e T„ „ for a l l t 2 e S„ k=2 k e ' Y 2 2 In view of eqn. (3.2), i t s u f f i c e s to consider tl r e a l and replace aj^ by b e R . k The second step i s to use conditioning to reduce the estimate (3.1) T 2 2 to the case s = 1 . To see t h i s , note that as forms -A, + m c - A , +m A A r 2 ~ i 2 ~ i [Kato 66, Thm. 2.10, p. 326] and hence (-A^ +m ) <_ (-A^ +m ) [Kato 66, Thm. 2.2.1, p. 330]. Since C(s) i s a convex combination of covariances Y 2 -1 2 ~1 of the form (-A^ +m ) , C(s) (-A^ +m ) and hence by the Conditioning Comparison Theorem [GRS 76, Thm. I I I . l , p. 256] i t follows that f J — 2 [~T(h,ti2<|>) :+a. (tl)*- ^ f ) ) 2 : ] . n l z e dy 2 ( s ) m 1 L^ -:T(ti,ti2)<fr) :+a. (ti)*- T m2:*2:] 71 1 2 v m where dy = dy 2(1) . m m The t h i r d step i s to perform a mass s h i f t i n the covariance. By Lemma 2.2.3 V [^:T(Ti,?i2<))) :+an ( t i ) * - ^ m2:(|)2:] n 1 2 dy m [i:T(ti,h2<f,) : „+an(ri)9 - ^ m 2:^: ] -V 1 1 m2 1 2 m2 .1 2 6, .2 6 (^ m — ) : 9 : V ^ 6 dy (3 A 2 cS. ,2 c5 J v {2 I n " I ) : < ) > : dy 6 2 "I where dy i s Gaussian measure with p e r i o d i c covariance (-A+6x v + m X ^ V ' The two quadratic terms ^-m2:92: e s s e n t i a l l y cancel; the term - ^t)> 2 1 6 r 1 n 6 2 w i l l be cancelled using Ti T(h,Ti <\>) 9 + ~ 9 • Wick order with 6 6 respect to dy i s denoted : : . Applying Jensen's i n e q u a l i t y to the denominator of the r i g h t side of eqn. (3.3) we obtain C^:T(ti,ti 29) :+ a ; L(ri)9-im 2:(() 2:] :A(9) dy 2 < m V dy 1 (S —1 2 1 ? 2 1 2 6 2 5 where :A(cf>): = ri : T ( t i , h <|>):2 +a1 (h) 9"-m : 9 : 2 + (^  m - -) : 9 : . m m By Lemma 2.2.1 A has the form 1 „ 9 K A(x) = h T (T I , T I X) - ^ x + I a, (h)h x K (3.4) k=0 where the a are bounded i n absolute value by a constant depending only k on 6 and L . The fourth step i s to provide a lower bound on : A ( < j > r ) : , where 9 i s the u l t r a v i o l e t c u t o f f f i e l d defined i n §2.2. By eqns. (3.4), (2.2.1) and (2.2.2) 1 n _ 2 k : A ( 9 ) : 6 = h ^ r T t t i ^ ) : 6 - f : ^ : 6 + T a ^ t l ) * 2 : 9* : 6  T r r 2 r u k r k=0 - n-2 - — = h _ 1 T ( f i , h 2 9 r ) - I 9^ + I c (Ti,r)Ti 2 a r 2 9* k=0 with Ic Ch,r)I uniformly bounded i n small ti and large r . The key to 1 k b x e T ^ , i t k=2 - , L + 2 follows that k 2L _! ifWftV >\t? 9^ + |*2 and so --1 n-2 - n ~ k : A ( 9 ) : 6 > ^ T l 2 9" + \ n ft2 0 2 9* r — 2 r , „ k r Y r k=0 2R6 -1 n r k-, = 0 L-ti x + > c, x J r 2 k=0 k 1 _1 2 2 where x = Ii 0 9 . For 0 < Ti < 1 , i t follows that r r — Since the c, are bounded uniformly i n h and r , t h i s implies :A((f> ) : 6 > -r — n_ 2 const*a r (3.5) n 2 >_ -const* (log r) by eqn. (2.2.2), where the constant i s independent of Ti and r . The f i n a l step i s to appeal to Proposition 2.9 of [DG 74], which uses a 2 decomposition of S' (R ) to show that a bound of the form (3.5), together with standard estimates for Gaussian expectations of Wick ordered products, imply the upper bound Lemma 3.1 corresponds to the bound (2.3) needed f o r convergence of the c l u s t e r expansion. However i t must be improved for the"following reason. In general the mass m^  w i l l not be large, and a s c a l i n g trans-formation must be performed to increase i t . However t h i s s c a l i n g a f f e c t s K . Since the s i z e of the required mass depends on K , there i s a problem. The bound (3.7) below i s an adequate modification of (3.1); see C o r o l l a r y 3.3. \ Theorem 3.2: Let T(ti,x) = V a, (h)x k and a, (ti) = 0 (ti ) where the a, k=2 k 1 k k are continuous i n some Sfl . Suppose Re T(0,x) e T . f o r some 6,L > 0 , :A(* ) : 6 dy < e' K v and hence (3.1). • e and f i x m,e > 0 Then there e x i s t 6,y,b > 0 such that i f : T(Ti,h2(j)) :+a, (n ) ( ( ) - im 2 : < j ) 2 : ] v tl 1 2 dy 2 ( s ) m < e s|v| (3.7) for every ti € Sfl , for every s , and f o r every f i n i t e union V of un i t l a t t i c e squares i n A . Moreover, _1 2 1 2 2 T(tl,h <(>) r+a^^Cti)*--!!! :* :] dy 2 ( s ) m (3.8) for every h e S Q , for every s , and for every u n i t l a t t i c e square A o /Y Proof: The proof follows [Sp 74] . For A c v we define v C : i T(tl,h2<j>) :+a1(7l)<j>-]|m2:4.2:] *A = 6 - 1 (3.9) Then C:^ r(h,h2<|>) :+a.. (ti)<J>-^>2:cf>2] V * 1 2 dy 2 ( s ) = m n (^ A+l)dy 2 ( s ) Acv m (3.10) • I X c V n iji dy 2 ( s ) . Acx m We claim that there i s a y = y(e , 6,L) such that f or h < y , n ip dy (s) | < e 1 Acx A m (3.11) We w i l l show how (3.11) implies the theorem and then prove (3.11) Given (3.11), i t follows from eqn. (3.10) that 1 T(Ti,Ti2<j>) :+a (Ii) if-\m2 : <f>2: ] V dy 2 ( s ) m « i .1*1 XcV l\W)* m< ( l + e ) ' V | < e*|v| „ m — m=0 which proves (3.7). The bound (3.8) follows from (3.11) with X = A . I t remains only to prove the in e q u a l i t y (3.11) . To s i m p l i f y the 1 notation, l e t :S(V): = [i-:T(tl,ti2<j)) :+an (h)4>-^m2:<()2:] . By the y Ti 1 2 Fundamental Theorem of Calculus, dA :S (A ±) :e -A . :S (A.) : l 1 (3.12) By eqn. (3.12) and Holder's i n e q u a l i t y n * du (s) | <. II n:S(A.) : Acx fl " • m I  ™PJle 1 -I X.:S(A.) sup p o<A.<r — 1— (3.13) where p > 1 w i l l be chosen below to be near one. The norm || • || i s the norm i n L (dy 2 ( s ) ) • 1 2 By assumption the c o e f f i c i e n t s of S are 0 (Ti ) or 0(b) • For Ti < y , i t follows from standard estimates on Gaussian i n t e g r a l s [GJ 81] that for given f i x e d p' , n :S (A.) 2 X < (max{y ,b}«M) 1 (3.14) 53 fo r some constant M independent of fl and s . To bound the other f a c t o r on the r i g h t side of eqn. (3.13), we cannot use Lemma 3.1 d i r e c t l y , because when A^ = 0 the c l a s s i c a l p o t e n t i a l w i l l not be i n any T . However the proof of Lemma 3.1 0 ,L can be modified to overcome t h i s d i f f i c u l t y , as we w i l l now show. As i n the proof of Lemma 3.1 we assume that ti and T are r e a l and that 2 s = 1 . Note that f o r p > 1 and y e (0,m ) , -pA.:S(V): e 1 du -pX :S(V) : 2 ~(m 2-Y) J y : * ' m , y e e du m 1 , 2 i - 5(m -y) : * : dy -p\±:S(V) : 2-i(m 2-y) m :* V dyY by Lemma 2.2.3 and Jensen's i n e q u a l i t y . But by Lemma 2.2.1 pX.Crti ^ ( t i , * 2 * ): _+a (h)* - im 2 : * 2 : ] + i (m2-y) : * 2 : Y m m 1 -1 2 >_ pA i[tl :T(7l,n <j>r) : 2 +a i(tl)' m im 2 : * 2 : ,] + i ( m 2 - y ) : * 2 : 2 - C m m 1 -1 2 A.[ph :T(n,Ti * ): 9 +pa (Ti)* 1 £ X * m 6 2 , 2 s-V 2 ] m + [ X . ( | " f m 2 > + |(m2"Y)]:<i>2: 2 - C m (3.15) 54 I f T(0, •) e T then f o r p e (1,2) , pT(0 , O e T. „ , so the o ,L o,2L estimates of the fourth step of the proof of Lemma 3.1 show that 1 n ~ 1 2 6 2 2 ph" :T(n,ri (|)r) : 2 +pa1(Tl)<}>r - - :9 : 2 >_ - M (log r) (3.16) m m (Compare the l e f t side of the above i n e q u a l i t y with the expression f o r :A(<))) : given above eqn. (3.4)). As for the second term on the r i g h t side of eqn. (3.15), we choose p and y such that 0 £ min [X.(6-|m2) + ^ (m2-Y) ] = min CX.(6-? m 2) + \(m 2-y)l 0<X.<1 1 2 2 A e{0,l} 1 2 2 = min{i(m 2- Y), 6 - ( ^ m 2 - \ y) (3.17) 6 m2 6 C l e a r l y p = 1 + — and y = min{—,—} s a t i s f y (3.17). By eqns. (3.15), m (3.16) and (3.17), we have 1 pXiCri"1:T(ri,n2().r: 2 +a ( f t ) ^ - |m2:<j)2: ] + \ (m 2- Y) : <f2 : Y m m n n 2 2 _^ - M (log r) - c log r - c >_ M (log r) I t now follows as i n the l a s t step of the proof of Lemma 3.1 that - Y X. : S ( A . ) : . 1 x 0<X.<1 — i — sup II e 1 < e K ' X l . (3.18) P _ Using the bounds (3.18) and (3.14), eqn. (3.11) follows from eqn. (3.13) by taking b and y s u f f i c i e n t l y small. • 5 5 The main consequence of Theorem 3.2 i s the following. C o r o l l a r y 3.3: For an i n t e r a c t i o n T and a function a^(h) = 0(h ) as i n Theorem 3.2 there e x i s t 6,y,b > 0 such that the c l u s t e r expansio'n f o r 1 the i n t e r a c t i o n i - T (h ,h2<}>)+a, (h)<j> - im 2* and mass m converges with n 1 2 bounds depending only on m,6 and L , independent of A and of n e S e,y In p a r t i c u l a r , eqn. (2.6) holds f o r t h i s i n t e r a c t i o n , with independent of h e S e,y Proof: Theorem 2.1 and Co r o l l a r y 2.3 cannot be immediately applied because the mass may not be large. To overcome t h i s problem we consider the theory obtained by replacing the given theory, abbreviated (T,a^,m) , by a theory -2 -2 -1 (a T,a a^,a m) where a > 0 i s chosen s u f f i c i e n t l y small that a "Sn > MC1,1) , where M(K,L) i s the lower bound on the mass f o r convergence of the c l u s t e r expansion given i n Theorem 2.1. For ReT(0,') e T. with 6 ,L 1 2 2 2 2 ^ 2 |a 2(0) - -m | < b we have a~ ReTCO,•) e T _ 2 _ 2 and \a a 2(0)-o m | < a b. a 6 ,a L -2 -2 -1 By Theorem 3.2 applied to the theory (a T,a a^,a m) there are b ,9,y > 0 such that 1 CTi~1:T(Ti,ti2<|») :+a (h) *-im 2: <(>2 : ] dy 2(s) <5>. < e 1 (3.19) and CTi" 1 :TCTi,Ti2<j>) :+a 1(Ti)*-im 2:* 2:] dy ,(s) a > I - 2 (3.20) 56 for Ti e S and \a 2a„(0) -ho 2m21 < a 2b . By taking y and b i o ,Y ^ 2 1 — 1 2 1 2 2 to be smaller the c o e f f i c i e n t s of ti T (fi,ti 9) + a^ (fl) 9 - -m 9 can be made less than one i n absolute value. The bounds (3.19) and (3.20) correspond to the bounds (2.3) and (2.4) of Theorem 2.1. (Clearly the - j m u l t i p l y i n g the i n t e g r a l a on A p the l e f t side of eqn. (3.19) can be replaced by — f o r p s u f f i c i e n t l y a close to one by fur t h e r decreasing y and b ). By Theorem 2.1 we obtain a uniformly convergent c l u s t e r expansion f o r the theory -2 -2 -1 (a T , a a^, m) . But by Lemma 2.2.5, a generalized Schwinger function -2 -2 -1 fo r the theory (a T , a a^,a m) i s equal to the corresponding generalized Schwinger function f o r the theory (T,a ,m) provided we also replace A 1 _ 1 -1 by a A and w by w . Since a i s j u s t a constant the C o r o l l a r y i s proved. • §4. A n a l y t i c i t y of the Pressure We are now i n a p o s i t i o n to prove that f o r some e,y > 0 and open set Oy 3 (0,y) the pressure 7ia(Ti,y) of eqn. (1.13) i s j o i n t l y a n a l y t i c i n (Ti,y) e 0^ x D £ , where = {z e C : |z| <e} , and i s C°° a t Ti = 0 + . n The strategy of the proof i s the following. Given T(Ti,x) = £ a (Ti)x k k=2 k with a^ a n a l y t i c i n (0,y) and T(0,*) e L , l e t tiT(Ti,u) = lim -TTT- £n A 1 1 1 1 2 2 - — [ :T(Ti,<|>) :--m :<)> :-yc|>] * J A 2 e d . (4.1) y h C By Lemma 2.3.6, i f ti and |p | are s u f f i c i e n t l y small then T(ti,x) - yx has a uniquely attained global minimum, at say £(ti,y) , with E, a n a l y t i c i n (h,y ) e (0,y') x ( _ e ' , e « ) a n d c°° at h = 0 + . By Lemmas 2.3.6 and 2.2.4, t r a n s l a t i n g the f i e l d i n eqn. (4.1) by £(ti,y) gives r i s e to a new pressure whose c l a s s i c a l p o t e n t i a l l i e s i n some T . uniformly i n small o ,L 2 Ti and |y| . For la2(°) " ~\ s u f f i c i e n t l y small we can appeal to Cor o l l a r y 3.3 to conclude that TIT (h,y) i s a n a l y t i c i n 1 1 2 0 0 2 + 0 0 (71 ,y) e (0,y") x (-e",e") and C at Ti =0 . To improve t h i s to^ C at Ti = 0 + we w i l l show that odd order d e r i v a t i v e s with respect to t i 2 1 2 vanish a t Ti = 0 . In Lemma 4.2 below i t i s shown that the vanishing of the odd order d e r i v a t i v e s implies the required smoothness at Ti = 0 . The following Lemma w i l l be used to show that the above-mentioned odd 1 2 order d e r i v a t i v e s a t Ti = 0 vanish. For the a n a l y t i c i t y of the pressure we only need f = 0 i n Lemma 4.1. However we prove the more general r e s u l t 58 because i t w i l l be needed i n Theorem 5.1.3. The parameter t corresponds to t i . -2 2 1 2 2 Lemma 4.1: Let B^fx) = t T ( t /tx) - -m x - f ( t ) x , n 2 V Z JC 0 0 where T ( t ,x) = \ a (t )x with a. a n a l y t i c i n (0,y) and C i n k=2 k * 1 2 [0,Y) , Re T(0,*) e T . t ,|a fo) - -m I < b where b i s s p e c i f i e d by o,L ' 2 2 i 2 C o r o l l a r y 3.3, and where f ( t ) = t g ( t ) with g a n a l y t i c i s (0,y) and C°° i n [0,y) . Let <•> J :P(<f>) A dy m P,A :P(<i>) dy m Then tto W < : * k l ( A ) : ; - - - ; : * k r ( A ) > B t > A = TAt < :* k l ( < t > ) :'---- ; :* k r ( A )>>B 0 ,A (4.2) uniformly i n A . In p a r t i c u l a r , i f k^+...+k^ i s odd the l i m i t i n (4.2) i s zero, since B^ i s quadratic. Proof: I t follows from C o r o l l a r y 3.3 and the Fundamental Theory of Calculus that ^ 1 ( A ) : ; . . . ; : ? K R ( A ) > > B - ^ ( A ) : ; . . . ; : 0> R ( A ) >> f t k JN. ]AT i j df <C:tt> 1 ( A ) T ^ ^ > ds s,A 59 T^T I | ^ < : * k ^ A ) : ; . . . ; : A ( A ) : ; } A T | : B s ( * ) > > B s ^ d s | <_ Mt uniformly i n A . • Lemma 4.2; Let 0 be an open neighbourhood of (0,/y) on which a function f i s a n a l y t i c . Suppose f i s C at 0 with f (0 ) = 0 , k = 0,1,2,... . Let f (x) = f (x^) , x e U = 0 n(C\{x e R : x <_ 0}) . OO j . Then f ^ i s a n a l y t i c on U and C at 0 . CO + Proof: The only thing to check i s that f^ i s C at 0 But t h i s i s obvious since f. (x) = f (/x) - f (0) + - f f ( 2 ) (0)x + ^ f ( 4 ) (0)x 2 + - f f ( 6 ) (0)-- 3 X + . • V k Theorem 4.3; Let T(71,x) = ). a (h)x where the a are a n a l y t i c i n k=2 k k an i n t e r v a l (0 ,p) and C°° at 0 + , and T(0,«) e T . Then f o r 0 ,L 1 2 la^CO) - -m | s u f f i c i e n t l y small there e x i s t y > 0 and complex open neighbourhoods 0^ 3 (0,y) and D containing 0 such that - i [:T ( t i , $ ) :- i m 2 :* 2 :-y<j>] d y h c " h T ( t l,y) = t l l i m T . ( t i , y ) = t l l i m J J T - £n A A ' ' oo + i s j o i n t l y a n a l y t i c i n (h,y) e 0^ x D and C at h = 0 , with uniformly bounded d e r i v a t i v e s . Moreover, there i s a c > 0 such that |D 2 t l T(h,y)| >_ c for a l l (ti,y) e 0 x D. (4.3) Proof; By Lemma 2.3.6 there e x i s t y',e' > 0 such that for y e (-£',£') and h e [0; y') T^(n,x) = T(Ti,x) - yx has a uniquely attained global minimum, 60 at say £(ri,y), with S(tl,y;x) = T (n,x+£(ti,y) ) - T , ( h ,Uh,V) ) e T,, T l f o r a l l y y o , L (fi,y) e CO, y') x (-£',£') . (4.4) Moreover 5 i s a n a l y t i c i n V , x D £, , where D^, = { z e C : |z| < e' } and V i s an open neighbourhood of (0,y') , and C°° at Ti = 0 + . Using Lemma 2.2.2, t r a n s l a t i n g i n by E, and using Lemma 2.2.3 gives Tix A(Ti,y) = " T ym , 5(rx,y)) In 1 C:S(tl,y;9) :~m 2:(j) 2:] <3y. tic " T pm , 5(n,y)) + -nn Jin Ci:S(ri,y;ti 2(J)) i m 2:? 2] i n 2 dy. (4.5) 1 2 1 2 for a l l Ch,y) e x D£, . Since - ^ S (0 ,y; 0) = - D^iO,^ (0,y)) we can. 1 2 1 2 make - D 2S(0,y;0) as close as desired to -m by taking e 1 and |a 2(0) - ^m2| s u f f i c i e n t l y small. Then by Co r o l l a r y 3.3 and eqn. (4.4) 1 / 1 r \ expectations of the form -r^-r- <^:<t> (A) :;...;: 9 (A) iy are bounded i n absolute value independent of A, t l , y , where 2 - 1 2 S (ti ,y;x) = ti S(tl,y;tl x) 1 2 2 - -m x . The f i r s t term on the r i g h t side of eqn. (4.5) i s a n a l y t i c i n oo . (ti,y) e V . X D and C at Ti = 0 , and does not depend on A . Y e Its d e r i v a t i v e s are uniformly bounded. To see that tix(ti,y) i s a n a l y t i c , and C at t i = 0 , we show that the i n f i n i t e volume l i m i t of the second term on the r i g h t side of eqn. (4.5) i s a n a l y t i c i n (h ,y) using V i t a l i ' s theorem, then use Lemma 4.1 to show that odd de r i v a t i v e s with respect 1 2 to t = Ti vanish a t t = 0 and appeal to Lemma 4.2. Let £ A(t,y) = -rjr in : S(t,y;<f>) : A dy , (t,y) e ( V n S ) x D C y T  e where 0 < y y' and 0 < z <_e' are such that the logarithm i s well defined. Since l D l ? A ( t ' y ) l = |I| K :D S(t,y;(j)) A y S,A and l D 2 C A (t,y) | = |<^JA:D2S(t,y;<(.):>^ ^  are uniformly bounded i n A,t,y , the same i s true of |?^ ( t , y ) | so by V i t a l i ' s theorem C(t,y) = li m S^(t,y) i s a n a l y t i c i n (t,y) e (V , n S ) X D . Moreover a l l d e r i v a t i v e s of C y T  e 2k+l i are bounded uniformly i n (t,y) . We now show that D2^A^ t , y^ ^ as t + 0 , uniformly i n A , for k,i = 0,1,2,... . In f a c t , by eqn. (1.3.3), :D S(t,y,•<))): 2k+l la_ . 1 v / D I ?A(t'y)=]ir i CTT< 2k+l : D S ( t , y ; * ) : ; . . . ; A ^ S , A (4.6) where P i s the set of p a r t i t i o n s of { l , . . . , n } , TT. are the elements n i of a p a r t i t i o n TT , and are p o s i t i v e integers. D i f f e r e n t i a t i o n of eqn. (4.6) with respect to v gives f , 7 r | T r | l l ^ u l 1 - f l a | T r | + l ' ~ f h o l ' - V J : ^ D2 I I S:,j:D2 S:, . . . ,J:D 2 > ' S where we allow | \ , ...,|a|^ j | to be zero. (4.7) n D ^ T ( t 2 , 5 ( t 2 , y ) ) k _ 2 k Since S(t,y;<J)) = £ — t 9 - -m 9 , the t = 0 k=2 k - 2 contribution to D ^ D 2 ^ ^ s 3 l i n e a r combination of terms of the form r c(y)<j> (A) where r i s odd i f j i s odd and r i s even i f j i s even. By eqn. (4.4) and Lemma 4.1 (with f = 0) , as t 0 the r i g h t side of eqn. (4.7) approaches uniformly i n A a sum of terms of the form C (U )/:<(. 1 (A) '°I(A)N (4.8) \ X S(0 , V ; O , A where r 1 , . . . , r | ^  | have the same p a r i t y as | TT^ | , . . . , | TT | ^  | | , and r i 1 r I 1 are a l l even (in f a c t equal 2). Since |TT|+1 I cr j I TT 1 +.. .+I TT 1 1 I = 2k+l i s odd, r + +r. 1 i s also odd. The expectation ' 1 ' I17! 1 l a l i n eqn. (4.8) i s i n v a r i a n t under 9 -v - 9 since S(0,u;*) i s quadratic, and hence equals zero. I t now follows from Lemma 4.2 that TiD2T(h,y) i s 00 + C at Tl = 0 . I t remains to prove the lower bound (4.3). Using the notation (F. (<j>) ;F_ (<)>)) dv = J 1 2 F ((fr)F2(<fr)dv - F1((f))dv F2(c|))dv , we have tl A 1 tl C:T(tl,<()): -(<J)(A) ;<j)(A) ) e A 1 2.2 , - i -111:9: - y<j> ] dy. tiC 1 ti [:T(tl,9): - |m2:<(>2: - U9 ] dy. tiC h A 1 ti 1 2 2 [ :S (h,y ;<j>) :--m :<j> :] ((9(A)+5(ti,y));(9(A)+C(ti,y))) e dy. tic _1 "ti 1 2 2 [ :S (ti,y ;<J>) :--m :<j> :] dy. tic ti A 1 tl [:S(ti,y,•(()): -1 2.2 , -m:9 :J (9(A);9(A)) e dy. TiC 1 ti C:S(ti,y;9): -1 V 1 2^:9 :J dy. t iC r j j < 9^(A) ;9(A))> S,A where we used Lemmas 2.2.3 and 2.2.4. By Lemma 4.1, lim tiD 2x(?i,y) = l i m -ryr <f<f> (A) ;9 (A)S _ ti+O A 1 1 S(0,y;-),A (4.9) The r i g h t side of eqn. (4.9) i s continuous i n y and equals (-A+2a (0)) 1 ( x ) d x > 0 f o r y = 0 . Therefore, taking e and y 2 R j smaller i f necessary, the lower bound (4.3) holds. • CHAPTER 4: SMOOTHNESS OF THE EFFECTIVE POTENTIAL § 1. Proof of Theorem 1.4.1 Theorem 1.4.1: V(h,a) < °° f o r a l l ti > 0 , a e R . Proof: To show that V(Ti,a) = sup Cya - tm(ti , y ) ] i s f i n i t e , i t s u f f i c e s to V + show that l i m tl D 2 ot(ti , y ) = ±°° ( r e c a l l Figure 2 ( i i ) ) . Since h plays no y->-±°° important r o l e i n t h i s discussion we drop i t from the notation. We write :P(4>): ' A <•> = I P , A,m 2 - I :P(<)>) dy c ± r i g h t and ^ ^ 2 for the l e f t continuous i n f i n i t e volume expectation [FS 77], P ,m Then by eqn. (1.2.1) D +a(y) = <% (OU * , 2 J ^ \ / P(x)-yx,m Note that i t s u f f i c e s to prove that l i m D +a(y) = °° f o r a l l semibounded P because t h i s implies y-x» that < <t>(0)\ " 2 = lim <^-cj)(0)^ + y->-°o y-*—00 P(x)-yx,m y->— °° P(-x)+yx,m" lim D a (y) = l i m x Y v - , / „ — - x T 2 (0 )^ > + - - l i m ^ T v- ,yr 2 y-*>° P(-x)-yx,m Also since a i s d i f f e r e n t i a b l e except on a countable set i t s u f f i c e s to show that D +a(y) i s unbounded on the set of p o s i t i v e y's for which Da(y) e x i s t s . For such y , lim <J>(oh = < J > ( 0 ) ) + = ^ ( 0 ) ) ' A + R 2 N ' P(x)-yx,A',m N ' P(x)-yx,y N / P(x)-yx,m For large y , l e t £(y) denote the unique poi n t at which 1 2 2 (x) = P(x) + -m x - yx att a i n s i t s global minimum. By Lemma 2.3.3 E, i s increasing and l i m £(y) = 0 0 . Therefore i t s u f f i c e s to show that y-H» | \to (0)y 2 - S(y) | i s bounded above uniformly i n A and large ' P(x)-yx,A,m y . This upper bound i s a consequence of the c l u s t e r expansion of [Sp 74]. We s p e l l out the d e t a i l s . By Lemma 2.2.2, w r i t i n g P^(x) = P(x) - yx , <^(0)^ 2 = 5(u> + <^(0^> x 2 P ,A,m ' S(y,«),A,m y where S(y,x) = U(x+5(y)) - U(£(y)) - ^ m2x2 = \ a (y)x k - ^ m2x2 , with y y k=2 a, (y) = U '(E(y))/k! . Therefore i t s u f f i c e s to show that k y | v M O W I i s bounded uniformly i n A and large y . X S(y,«),A,m 2 Applying s c a l i n g (Lemma 2.2.5) and mass-shift (Lemma 2.2.3) transform-ations , we obtain < * < 0 3 > S ( U , W < M O , > 2 N 2 / < * ( 0 » ^ 2_, ?2 _ nS ( y , . ) , C 2 A, ? 2" nm 2 Q , 5 2 A , ? 2 ' (1.1) 66 Here Q i s defined by 5 a 2 k=3 £ m k=3 5 m k-n where we have introduced b^(y) = £ ' N o t e that l i m b (y) y-x» e x i s t s and i s f i n i t e . 2-n £ A 2 n-2 By Lemma 2.2.1 and the f a c t that | log — - A = 0(log £ ) , 5 m n-k n n-2 _ 2 Q(x) = £ b r x + I 0(5 k ( l o g ? n 2) ) x k . k=3 k=0 The constant term i n Q can be cancelled i n eqn. (1.1 ) , so the c l a s s i c a l p o t e n t i a l occurring on the r i g h t side of eqn. (1.1) i s n-k ,ii s V , i-2-k k n r 2 „,^-k,, ^n-2 X 2 k 1 ,.2-n 2 W(x) = I h.K x + I 0(5 (log? ) )x + - £ a 0x k=3 k k=l 2 . n = I (b +X ) ? 2 _ k x k + A x k=2 K K n-k n-1 2 _ 2 where A R = 0 U ~ 2 (log£ n~ 2) ) , k >_ 2 ; ^ = 0(? - 1(log£ n ) ) . n n Since I b x K = 5~n I a ( ? x ) k = £ n [ u (5x+E) - U (5)3 has a uniquely k=2 k=2 y y attained global minimum at zero, f o r large y W w i l l have a uniquely attained g l o b a l minimum, at say n(y) , with n(y) -»- 0 as y -> 0 0 , Transla-t i o n by n (Lemma 2.2.2) gives (1.2) ^ f r ( O ) ^ n = n(y) + ^ ( o j ^ ) ^ r> r 2 ' » ^2-n _ r2 . R2-n Q»S A,5 a 2 Q l fC A,5 a 2 1 2 - n 2 where Q±(x) = W(x+n) - W(n) - ~K a.^ . The expectation on the r i g h t side of eqn. (1.2) has the c l a s s i c a l p o t e n t i a l TT , s r\ t \ A- lc2-n 2 V w ( k ) (1) k W^x) = Q l ( x ) + - 5 a 2x = I — x . Let T(x) = g~2W (Sx) . (1.3) 1 2 - 1 - 1 2 Then W 1(x) = 5 T(g x) . Here 5 plays the r o l e of h mi i V W ( K ) ( n ) r k _ 2 k , Now T(x) = 2, — 5 x , and k=2 k -r4w (k )(n)5k-2 = 7 T 5 k - 2 I ( b , + X . ) ? 2 - j j ( j - i ) . . . ( j - k + i ) n j - k X. k. j=k = I (b.+X . ) S k ~ j ( j ) T , j " k j=k k ^ b k for large y , k >_ 2 . (1.4) Reca l l Lemma 4.1 from [Sp 74]: v k Lemma 1.1: Let U (x) = I t x be bounded below, U (x) = U (x) - yx , 0 k=2 k y 0 and f o r large y l e t £ (y ) be the l o c a t i o n of the uniquely attained global minimum of U . Then there e x i s t M,c > 0 such that f o r a l l y y > M n U (x+£(V)) - U (5(10) 1 c I Sty)" | x | k for a l l x e R . • y y k=2 By t h i s lemma, I \xk = £ n[Uu(5x+5) - U | (£) ] k=2 l T n c I 5 n - k | 5x | k = c I |x| k . (1.5) k=2 k=2 68 By eqns. (1.4) and (1.5) i t follows that f o r y s u f f i c i e n t l y large T(x) > £ f |x| k . k=2 Since the c o e f f i c i e n t s of T are uniformly bounded for large y by eqn. (1.4), T e T f o r some L > 0 . I t follows from eqn. (1.3) and Co r o l l a r y 3.3.3 that | ^ ( 0 ^ n I i s bounded uniformly i n r, r 2 " - 1 , 2-n Q±,K A , £ a 2 A and large y . Since n(y) ->- 0 as y -> °° i t follows from eqns. (1.2) and (1.1) that | O ( 0 ) y | i s bounded uniformly i n A and X ' S(y,-) ,A,m 2 large y . • §2. Proof of Theorem 1.4.2. Theorem 1.4.2: For every a e R , l i m V(n,a) = (convU )(a) . h+0 0 Proof: Because of the f a c t that tia(ti,«) i s s t r i c t l y convex [FS 77] and hD 2 aCh,y) -*- ±°° as y -»• ±°° (as shown i n §4.1), i t follows that h a ( h , « ) e C . In Theorem 2.1 below we w i l l show that lim "ha (ti,y) = -m(y) S h+0 fo r a l l y . Using t h i s , and the f a c t that -m e C by Lemma 2.3.3, i t s follows from Theorem 2.1.1 and eqn. (2.3.7) that * ** l i m V(Ti,a) = -m (a) = U (a) = (convU ) (a) , for a l l a e R . TlyO We now prove the promised Theorem, which i s a Laplace's method type r e s u l t 2 for f u n c t i o n a l i n t e g r a l s on S'(R ) . For r e l a t e d r e s u l t s i n the context of Gaussian i n t e g r a l s on C[0,1] , see [Sim 79], [ER 82] . 69 Theorem 2.1: l i m ftct(ft,y) = -m(y) , f o r a l l y e R . ti+O Proof: Let a. Ch,y) = -rfj- £n 1 "ft [:P(9)'.-y<J)] A dy^ c , and f i x y e G Let T(x) = U (x+£(y)) - U (5(y)) . By Lemma 2.3.1, T e T. for some y y •* 6 ,L 6,L > 0 . By Lemma 2.2.2, ft°<A(h,y) = -U ( S (U ) ) + ft In 1 "ft t:T((f)): -A 1 2^2 -m:(|, : ] dy. ftC (2.1) By Jensen's i n e q u a l i t y the argument of the logarithm on the r i g h t side of eqn. (2.1) i s bounded below by one, and by Lemma 2.2.4 and Lemma 3.3.1 i t i s bounded above by e K ^ i f Ti i s s u f f i c i e n t l y small. These bounds and eqn. (2.1) show that |ftaA(Ti,y) + m(y) | -v 0 uniformly i n A , as ft -1- 0 , for y e G . But by Lemma 2.3.2 G i s dense i n R and hence limTia(Ti,y) = -m(y) for a l l y e R by convexity. • ft+O §3. Proof of Theorem 1.4.3(a): Theorem 1.4.3(a): Let a | 6. There e x i s t s a y > 0 such that V(h,a) i s a n a l y t i c i n ft for ft e (0,Y) • Moreover V(ft,a) i s C at ft = 0 , and so the D^V(0 +,a) expansion V(h,a) ^ £ v ( a)"ft n 1 S asymptotic, where v n ( a ) ~ 1 • n=0 n Proof: R e c a l l eqn. (3.1.15): V(ft,a). = U Q(a) - Try (a) + q Q (ft) + supC-fta (ft,y) ] , a | B , y •where q and a are functions of a . F i x a I B . Since q i s a 70 polynomial we need only show that E (h) = sup C-ha(Ti,u)] i s a n a l y t i c on + U (0,y) and C at Ti = 0 . We show t h i s using Theorem 2.1.2 . Note that i t s u f f i c e s to show that l i m ?io-(h,u) = -m (y) , for a l l y e R , (3.1) h+0 ° 5 k 1 2 2 where m n ( y ) = mint I q, (0)x + - m x - y x ] . In f a c t , w r i t i n g (as i n U K 2 1 x k=3 Theorem 2 . 1 . 1 ) y(0) f o r the l o c a t i o n of the supremum i n sup [+m ( y ) ] , y 0 i t follows from Lemma 2.3.3 that yCO) i s the unique root of -mQ(x) = 0 . V k 1 2 2 By Lemma 2.3.3 t h i s root i s the unique y for which I q (0)x + -m x - yx k=3 k 2 a t t a i n s i t s global minimum at zero. Since a k B , there are 6,L > 0 such that I a (0 ) x k + i m j x 2 e T . , (3.2) , - k 2 1 6 ,L k=3 and so y(0) = 0 . Now given eqn. (3.1), i t follows from (3.2) and Theorem 3.4.3 that Tia(h,y) s a t i s f i e s the a n a l y t i c i t y requirements of Theorem 2.1.2, as well as the necessary bounds on the d e r i v a t i v e s , and hence E i s a n a l y t i c i n (0,y) and C°° at h = 0 + . I t remains to prove eqn. (3.1). We show that (3.1) holds for y e GCO), where f o r A > 0 n k 1 2 2 G(A) = { y e R: T q (A)x + -m,x - yx has a uniquely attained global k=3 k 2 1 minimum and has p o s i t i v e curvature at that minimum} . The set G(0) i s dense i n R by Lemma 2.3.2, so (3.1) holds f o r a l l y i f i t holds f o r y e G(0) , by convexity. { Let o A(h,X,y) = -rrr £n , (• n JA k=2 K and l e t a A(Ti,y) = o"A(Ti,Ti,y) , so a(h,y) = lim o"A(Ti,y) . By the Fundamental A Theorem of Calculus, |ha A(li,u) +m Q(y)| <_ |TiaA(Ti,0,y) +m Q(y)| + h | V£K (h,X ,y) | dX . (3.3) By Theorem 2.1, the i n f i n i t e volume l i m i t of the f i r s t term on the r i g h t side of (3.2) goes to zero as Ti 4- 0 . As f o r the second term, f i x y e G(0) and y > 0 s u f f i c i e n t l y small that y e G(X) f o r X e (0,y) . In the expectation hD2a A(h,X,y) , t r a n s l a t e the f i e l d by the l o c a t i o n n k 1 2 2 £(X,y) of the global minimum of J q (X)* + -m A - y<J> , scale the f i e l d ± k=3 k 2 X 2 d> -»- Ti d> , s h i f t the quadratic term of the i n t e r a c t i o n over to the measure, and Wick re-order the i n t e r a c t i o n to match the new measure. Then by Co r o l l a r y 3.3.3, ti ! D2° A(h,X,y) | i s bounded uniformly i n A and i n small Ti and X , and therefore the second term on the r i g h t side of (3.3) i s O(Ti) uniformly i n A . • Note that i t was also proven i n Theorem 2.1.2 that the poi n t y (Ti) at which sup C-Tia(Ti,y)] i s attained i s a n a l y t i c and bounded on (0,y) and y hence C at h = 0 . In p a r t i c u l a r lim y(Ti) = y(0) = 0 (3.4) ti+0 §4. Proof of Theorem 1.4.4. Theorem 1.4.4: Let K c B c be compact. Then there i s a y > 0 and an 72 open set 0 => K such that V ( h , * ) has an a n a l y t i c extension to 0 for every Ti < y . Proof: F i x a j B. Since "ha(ti,*) i s s t r i c t l y convex, i t follows from § 4 . 1 that t iaCTi,*) e C , and hence there i s a unique y(h,a) f or which s V(Ti,a) = y(Ti,a)a - ha (Ti,y (Ti,a)) . S i m i l a r l y by Lemma 2 .3 .3 -m e C g so there i s a unique y(a) at which sup [ya + m(y)] i s attained. I t follows from Theorems 2.1 and 2 .1.1 y that |yCh,a) - y(a)| can be made a r b i t r a r i l y small by taking h s u f f i c i e n t l y close to zero. Let K c B be compact. We show V(?i,a) = y(Ti,a)a - Tia (Ti,y (h,a)) i s a n a l y t i c i n a neighbourhood of K by showing the following. Lemma 4 . 1 : For a e K there i s a neighbourhood 0 containing y(a) a and y > 0 such that ha(ft,*) has an a n a l y t i c extension to 0 f o r a a every Ti < y . That i s , for a l l fi < y there i s no phase t r a n s i t i o n i n a a the neighbourhood 0 of y(a) . Si Lemma 4 . 2 : There i s an open disk V containing a to which y(Ti,*) has — ^ — c l an a n a l y t i c extension for every Ti < Y , with y(Ti,V ) c o a a a Since K i s compact, u V has an open subcover {V ,...,V } , a€K 3 a i ^ N and V(h,«) i s a n a l y t i c on U V for a l l Ti < min y . I t remains i = l a i I f A lN a i to prove Lemmas 4.1 and 4 . 2 . Proof of Lemma 4 . 1 : F i x y e G . Making the a dependence of e x p l i c i t by w r i t i n g a^Ch,y;a) f o r o^(?if]i) , i t follows from eqn. (3 .1 .2 ) that Tia(ri,y) =-U (S(y)) + ho^ (ft , 0 ; £ (y)) . (4.1) 73 By Lemma 2.3.3 and the above equation, i t s u f f i c e s to show that f or f i x e d a^ j I B there i s a f i x e d neighbourhood 0 o f a^ and a y^ > 0 such t h a t ha^(h,0 ;a) i s a n a l y t i c i n a e 0 f o r a l l h < y ^ . Let o l f A ( h , 0 , a ) - - j ^ £n A k=2 k: du i so that C a (ti,0;a) = lim a (h,0,-a) . We w i l l give bounds on \a . (h,0;a)| 1 A 1 / A I,'' uniform i n a, h and A f o r ti < y and l a - a 0 l K e > with e'Y Q > 0 The Lemma w i l l then follow by V i t a l i ' s Theorem. 2 Now P"(a Q) + m > 0 since a Q 4 B , so by Lemma 2.2.3 a1 A(h,0;a) = In ? P ( k ) ( a ) r 1 k F " { A 0 ] 2 A k=2 c 2 C Hn r :<P : (4.2) where CQ i s the p e r i o d i c covariance of mass m + P"( ap) • The second term on the r i g h t side of eqn. (4.2) i s bounded uniformly i n A . The f i r s t term w i l l be bounded by using C o r o l l a r y 3.3.3 to give a uniform bound on i t s d e r i v a t i v e with respect to a . To apply C o r o l l a r y 3.3.3 the Wick order of the i n t e r a c t i o n must match the measure. Using Lemma 2.2.1 to Wick re-reorder we obtain a new i n t e r a c t i o n n . (k) |-1 :S(h,a; r) :_ = £ s,(n,a):d> : , where s = [ ^ a ;+0(h)]h , 3 £ k £ n , C 0 k=0 k C 0 k k. s 2 = i ( P " (a) - P"(a 0)) + 0(h) , s± = 0(h 2) and s Q = 0(1) . Now 74 n P ( k ) ( a Q ) 1 2 ) x + -m x e T (for some 6, L) since a e B , so for . _ k! 2 o,L 0 k=2 l - k k x |a-a | and ti s u f f i c i e n t l y small, Re £ s tl x + - (m +P"(a n)x e T * , k=2 |,L +-and C o r o l l a r y 3.3.3 can be applied. • Note that the convergence of the c l u s t e r expansion obtained i n the proof of Lemma 4.1 shows that f o r some y > 0 and open disk U containing V (a) , 2 - d 2 | D tia(ft,y) | = | — - u UtU)) + 0(ti)| '2 1 1 2 y 1 dy 1 + OCh) I > c > 0 for "h < y , y £ 0 (4.3) U£(E(V)) by eqn. (4.1) and Lemma 2.3.3. Lemma 4.2 i s a consequence of the following g e n e r a l i z a t i o n of the inverse function theorem [Rudin 74], [Rudin 76]. Lemma 4.3: Suppose f(ti,') , ti > 0 are a n a l y t i c functions i n a neighbourhood 0 of a point y Q , and that |D f Ch,u) | > c > 0 f o r a l l ti < y , y e 0 (4.4) and |D 2f(ti,y)| <_ M for a l l Ti < y , y e 0 . (4.5) Suppose also that there e x i s t u with l i m y^ = y„ , and f(ti,y. ) = an ti • ,« ti 0 T i O ti+O independent of ti . Then there i s a y > 0 and an open neighbourhood 75 V of a Q such that f o r a l l 0 < Ti < y f ( t i , * ) has an a n a l y t i c inverse function on V , with f _ 1 ( t i,V) c o . Proof: Denote the open disk of radius r centred at y by D(y,r) , and choose r < — such that D(y r) c o . Choose y > 0 such that — 2M 0 VL e D(U n- ^) for a l l h < y . Then = D(u , \) c D(y n,r) i f tl < y Tl 0 2 tl Tl 2 0 2 Fi x a e D(a » — ) and l e t 0 8M g(ti,y) = y + [D f (ti,y Q) ] _ 1 ( a - f (ti ,u ) ) . Note that for any given fl , g(ti,*) has a f i x e d point i n B^ i f and only i f a = f(tl,y) has a s o l u t i o n y e B . We w i l l show that f o r Ti n s u f f i c i e n t l y small g(h,*) has a unique f i x e d p o i n t i n B^ , and therefore 2 f(ti,-) has an inverse on V = D(a , r~") . The inverse function must be 0 8M a n a l y t i c i n view of (4.4) and the a n a l y t i c Inverse Function Theorem. Note that |D 2gCtl,y)| = | l - [ D 2 f ( T i / y 0 ) ] " 1 D 2 f (h,y) | <_.! |D 2f(tl,y) - D 2 f (ti,y 0) | £ ^Mr < | for y e D(y Q,r) , by eqn. (4.5) and the f a c t that r <_ — . Therefore |g(ti,y 1) - g(ti,y 2) | <_ i |y 1 - y | i f Ti < y , M2 e B^ . Moreover i f y e B_ then n |g(Ti,y) - M^I £ |g(li,v) - gflwu^) | + IgCTi,^) - y j - ^ Y " V T J + l D 2 f ( h ' V I 1 , a " f (h'V i 2 ^ 1 r , 1 C / C • s: ^  ^ < - - + - — - — • i f ti < y , - 2 2 c 8M 4M 76 so g(tl,y) e B. i f TI < y . But since g(h,*) i s a contraction from B Tl n to i t s e l f , i t has a unique f i x e d point, for a l l tl < y [Rudin 76]. • Proof of Lemma 4.2: Let f(ti,y) = tiD 2a (?i,y) . Then f(ti,*) i s a n a l y t i c i n 0 by Lemma 4.1. By eqn. (4.3), |D f ( t l , y ) | >_ c > 0 f o r ti < y , y e 0 . By the uniform convergence of the c l u s t e r expansion obtained i n a the proof of Lemma 4.1 i t also follows that | D f c n , y ) | £ M i f t i < Y / y e O a . By Theorems 2.1 and 2.1.1 li m V (ti,a) = y (a) . Since f(ti,y(ti,a)) = a , tl+O a l l hypotheses of Lemma 4.1 are s a t i s f i e d , and the r e s u l t follows. • CHAPTER 5: THE LOOP EXPANSION N §1. -D^V(0,a) i s a sum of Graphs N In t h i s section we f i x a e B and prove that f o r N ^  2 , -D^v(0,a) i s equal to a f i n i t e sum of graphs with l i n e s corresponding to the free 2 » 2 " covariance of mass m^(a) , where ir^fa) = P (a) + m = U Q (a) . The proof of Theorem 1.4.3(b) w i l l then be completed by i d e n t i f y i n g the graphs t o p o l o g i c a l l y . R e c a l l eqn. (3.1.15) : V(ti,a) = U (a) - tia(a) + q n(ti) + sup[-tl (h,y)] , a 4 B (1.1) 0 U y where y , q^ and a are given by eqns. (3.1.6), (3.1.11) and (3.1.13) r e s p e c t i v e l y . By eqn. (3.1.11) , „ k n !k,k a2k d k = 2 2 ' (1.2) k T D k q o ( 0 ) =i 0 otherwise. 77 1 2 i . e . , i n the notation of D e f i n i t i o n 1.3.1 -D q^(0) = a^ -^D 3q (0)=a m and so on, where d = •—• loq —— \ 0 \ 4 7 T m2 Let E ( T i ) = supC-ha (h,y) ] = - h a ( f i , y ( h ) ) . y In section 4.3 i t was shown that E i s a n a l y t i c i n (0;y) and C at 0 By L e i b n i t z ' Rule N N-1 D NE(0) = l i m D NE(h) = - l imCh — o(h,y (h) ) + N - o(ti,y(h))] . (1.3) N N— 1 THO Ti+O dh dh , N We w i l l show that E(0) = DE(O) = 0 and that dh N a(?i,0(Ti)) i s a f i n i t e sum of graphs with l i n e s of mass m^  . Re c a l l i n g the notation v*,t a) = 7^ " D^V(0,a) from Theorem 1.4.3, t h i s shows that v (a) = II (a) , N N : 1 0 0 v 1(a) = - Y ( a ) , and that -D^V(0,a) i s a sum of graphs. The t o p o l o g i c a l structure of the graphs c o n t r i b u t i n g to - v N ^ a ) ^ o r N — 2 w i l l be shown i n the remainder of Chapter 5 to be as stated i n Theorem 1.4.3(b). The f i r s t step i s the following lemma. Lemma 1.1: For some y > 0 , a(h,y(h)) i s C i n h e [ 0 , y ) , with a(0,y(0)) = 0 . r Proof: As was j u s t mentioned, E i s C i n L0,Y) I and therefore the same i s true of Ra(h,y(h)) = -E(ri) . We now show that l i m o"Ch,y(Ti)) = 0 , Ti+O which w i l l prove the lemma. 7 8 By eqn. (3.1.13), a(Ti,y) = l i m A in 1 n [ I q. (n) :* k: A k=2 K - y<t>] dy. he. r k 1 2 2 Let Q (Ti,x) = 2, q v ^ ^ x + - m x - yx . By Lemma 2.3.6, Q (h,«) has y k=2 2 1 y a uniquely attained global minimum, at say £(h,y) with £ smooth, provided Ti and y are s u f f i c i e n t l y small. By Lemma 2.2.2, c(h ,y) = - cQ (h,£(h,y) + l im J - J - £n H V A I I 1 h 1 2 2 [:T(<j>):--m :<(> :] A 2 dy. he, (1.4) where T(x) = Q (Ti,£ (h,y)+x) - 0 (Ti,£(Ti,y)) e T„ uniformly i n small Ti y y 6 , L and y . Evaluating eqn. (1.4) at y = y(Ti) and using Lemma 2.2.4 gives a(h,u(h)) = ~Q ^ (h,£(h,y (Ti)) + l i m y - , h y(h) A IA j in J . - | [ i : T ( t l 2 Y ) :-im 2: T 2:] dv. (1.5) But since y(0) = £(0,y(0)) = 0 , r e g u l a r i t y of y and £ imply that y(h) = 0(Ti) and £(h,y(h)) = 0 (Ti) . I t follows by s u b s t i t u t i n g i n t o Q y C h ) (Ti,£(h,y (Ti))) that (Ti, £ (h,y (Ti))) = 0 (Ti 2) , and therefore -^Q u ( T l ) (ti,£(Ti,y(Ti))) 0 as Ti + 0 To show that the second term on the r i g h t side of eqn. (1.5) goes to zero as Ti + 0 , we c a l l i t B(Ti) H l i m 3. (Ti) and note that the c l u s t e r A A expansion converges f or B^Cti) uniformly i n small Ti , by Co r o l l a r y 3.3.3. 79 In p a r t i c u l a r , there i s a constant M such that ^-r 3 01) 8h < M for a l l small h ; and therefore 1 2 1 |B A(ti) - B A ( 0 ) | = |6 A(Ti) - o| < Ig^ - B A ( x 2 ) |dx < Mh 2 • 0 Cor o l l a r y 1.2: E(0) = DE(0) = 0 . Proof: Since E (h) = -no (h,y (h)) , E(0) = 0 i s an immediate consequence of Lemma 1.1. Also, DE (h) = -a(h,y(h)) + ti — a("h,y(Ti)) goes to zero as dh Ti + 0 by Lemma 1.1. • Before s t a t i n g the main r e s u l t of t h i s section, we introduce some notation. Let o^j = TT D I lq k(0) (j = 0,.,...,-) (1.6) n 2 so that q (TO = £ q .h11 . k j=o k : Recall from Theorem 1.4.3 the notation v N(a) = D^V(0,a)/N! . Theorem 1.3: For a & B , v Q(a) = U Q (a) and v^(a) = -y (a) . For N >_ 2 , -D^v(0,a) i s equal to -D Nq Q(0) plus - D NE(0) . The d e r i v a t i v e -D NE(0) i s given by a l i n e a r combination of graphs with no s e l f l i n e s , with p o s i t i v e or negative c o e f f i c i e n t s , made up of l i n e s of mass m^  and v e r t i c e s n N -q , k = 2, 3,...,n; j = 0, 1,...,- . The graph corresponding to -D q n(0) K"2 2 K) i s given under eqn. (1.2). Proof: By eqns. (1.1) and (1.3) and C o r o l l a r y 1.2 the only thing to check 80 i s that f o r N > 2 , -D NE(0) = N — N-l dri N-l a(ti,y(Ti)) i s a sum of graphs with v e r t i c e s -q , . , k jt o k i By d e f i n i t i o n of a and Lemma 2.2.4, o(Ti,y) = l i m a (h,y) , where A ' ? r 1 k -i £ [q Ch)h :<|) : - yh <j,] a ACh,y) = -A-r- in e ' A k = 2 , dy. 2 2 oo Let f(Ti ) = y(h)h . By eqn. (4.3.4) there i s a function g , C on 1 1 2 2 [0, Y) / such that f (Ti ) = ti g(ri) (1.7) Also (ri,y(h)) = in llq. (Ti)h 2 : 9 k : A k=2 K - f(Ti )+] For x e R , l e t C(t,x) = l i m ? (t,x) , where ^ A ( T ' X ) = TKJin I [q ( t 2 ) t k 2 : < ) , K : - X<j>J A k=2 dy. Then C ( t , f ( t ) ) = a ( t 2 , y ( t 2 ) ) . 2 We see by s u b s t i t u t i n g t for h i n the asymptotic expansion f o r a(Ti,y(h)) that dh' a(Ti,y(ri)) = nl d 2n (2n) ! 2n at C ( t , f ( t ) ) , so i t s u f f i c e s ,2n to show that dt 2n C ( t , f ( t ) ) i s given by a l i n e a r combination of appropriate graphs. To show t h i s , we begin by introducing some notation. Let 81 S(t,<|>,A ) = I q k ( t 2 ) t k 2:<},k(A): - f ( t ) 9 ( A ) . Then by eqn. (1.3.3), k=2 d t k C A ( t , f ( t ) ) = I - r - j < -D 1 1 S ( t , 9 / A );...; -Dx | i r | S(t,9,A )> TreP, ' ' t , i (1.8) where ^ ' ^  t A """s ^ B e x P e c t a t i ° n with respect to the i n t e r a c t i o n S ( t , 9 / A ) and the notation P, , TT, c i s as i n eqn. (1.3.3). The expectations on the k TT r i g h t side of eqn. (1.8) are f i n i t e sums of p o s i t i v e integers times p o s i t i v e powers of t times expressions of the form ( r e c a l l eqn. (1.6)) . k k (A ) a ) "[AT \ q k j :<f> (A)-•••'•-<5k j :<j> r ( A ) - f (t)<f.(A);...;f S (t)<MA)^ t f A , (1.9) with k_^  e {2,...,n} , e {0,1,..., '|} , r > _ 0 , s >_ 0 , ^ 1 • We denote the i n f i n i t e volume l i m i t of the expression (1.9) g r a p h i c a l l y by K V / ^ — ^ - f 5 (t) ( 1 - 1 0 ) We now show that the vertex factors f (0) are a c t u a l l y graphs which hood onto the corresponding legs. To s i m p l i f y the notation we use Z^JLL  TO denote a l i n e a r combination of terms of the form (1.10) with vertex factors 1 U ± ) instead of f (t) ; which l i n e a r combination w i l l be apparent from the context. The c o e f f i c i e n t s of the l i n e a r combination w i l l include combinatorial factors and powers of t . 82 Derivatives of f are ca l c u l a t e d as follows. Since D 2a(h,u(h)) = 0 by d e f i n i t i o n of u (Ti) , i t follows from the f a c t that D 2 C ( t , f ( t ) ) = t D 2 a ( t 2 , y ( t 2 ) ) that D 2 C ( t , f ( t ) ) = 0 . (1.11) By eqn. (1.7), f i s C i n t . D i f f e r e n t i a t i n g eqn. (1.11) with respect to t gives Df(t) = - D ^ C t ^ f (t)) D 2 C ( t , f ( t ) ) (1.12) Using the graph notation described i n the l a s t paragraph, eqn. (1.12) can be written Df(t) = (-1) - O - o -As explained below, d i f f e r e n t i a t i o n of eqn. (1.13) gives D*f(t) = (-1) - P + <$> - o - - o -- 6 - ( - i K > H > - / 2 -6-(-D -o (1.13) (1.14) The terms on the r i g h t side of eqn. (1.14) a r i s e as follows. The f i r s t three terms come from d i f f e r e n t i a t i n g the numerator -O* °f eqn. (1.13): the f i r s t term comes from d i f f e r e n t i a t i n g t's appearing as c o e f f i c i e n t s of r 2 k—2 k ; the second term from d i f f e r e n t i a t i n g the £ ) f c :<J> : p a r t k=2 k of the i n t e r a c t i o n ; the t h i r d term from d i f f e r e n t i a t i n g the f(t)<f> part of the i n t e r a c t i o n and using eqn. (1.13). The l a s t term on the r i g h t side of 83 eqn. (1.14) comes from d i f f e r e n t i a t i n g the f a c t o r . Since there i s no t dependent c o e f f i c i e n t as a f a c t o r i n —O" » there are only two terms i n the d e r i v a t i v e of —O - • Dropping minus signs we can rewrite eqn. (1.14) as D 2 f ( t ) = J&L. + _I^L + - 0 - - 0 , <>--<> + O . ( i . i 5 ) -O - - 0 - ( H » 2 (-O-) 2 i - 0 - ) 3 In the l a s t three numerators of (1.15) note how a l l but one of the s i n g l e legged v e r t i c e s can be matched i n p a i r s , and that the power of —Q^- i n the denominator exceeds the number of matched p a i r s by one. We w i l l now show how eqn. (1.15) generalizes to higher order d e r i v a t i v e s . By the same reasoning used to d i f f e r e n t i a t e - 0 > above, _d dt Q /k\ £\ ~~"'-o-Using the formula (1.16) i t follows from eqn. (1.13) and induction that D f ( t ) i s a l i n e a r combination of quotients of the form (1.17) l -0 - ) M where the diagram eventually terminates; M - 1 i s the t o t a l number of matched p a i r s of legs, i . e . , M = m +m +m ; and there i s only one j- ^ *J k-1 unmatched l e g . To see t h i s , suppose D f ( t ) i s of the form (1.17) and note that d i f f e r e n t i a t i o n of any factor of the numerator (using (1.16)) produces a sum of terms of the form (1.17). Also, using the quotient r u l e to d i f f e r e n t i a t e the denominator gives terms of the form (1.17) by matching one leg of ^ - Q - to the unmatched leg of the numerator. (There w i l l s t i l l 84 be one unmatched l e g l e f t over). In the l i m i t t 0 the measure i n (1.9) becomes dy by Lemma 3.4.1. c u l Hence by Wick's Theorem D f(0) i s a l i n e a r combination of products of connected graphs without s e l f - l i n e s , with v e r t i c e s and l i n e s as i n the statement of the Theorem as well as one-legged v e r t i c e s which match up i n M - 1 p a i r s as depicted i n (1.17), divided by I • • ) . Thus there i s one power of • » f o r each matched p a i r of legs, with one power l e f t over. The unmatched l e g i n (1.17) should be thought of as being matched to the corresponding l e g of (1.10), and the extra power of *• « i n the denominator as corresponding to these legs. As we w i l l now show, at t = 0 each f a c t o r of • » i n the denominator serves to l i n k together one matched p a i r of legs to create a connected graph. We w i l l now show that at t = 0 (1.18) Ik » Ik 1 2 where each c i r c l e denotes a connected graph with no v e r t i c e s other than those e x p l i c i t l y drawn. In f a c t , each of the l i n e s L^ and must be connected to a multi-legged vertex; choose these to be the v e r t i c e s f i x e d at zero when evaluating the graphs. Then the numerator can be written dxC.lO,x)l" T ^ 2 " 1 (1.19) £ _ i T ^9 where the dashed l i n e s i n d i c a t e the absence of L^ and L^ . One of the dxC^(0,x) on the r i g h t side of eqn. (1.19) cancels the factors denominator on the l e f t side of eqn. (1.18). The remaining factor serves 85 to l i n k up the two graphs on the r i g h t side of eqn. (1.19). To see t h i s , take one of the graphs under the i n t e g r a l dxC^(0,x) and use t r a n s l a t i o n invariance to f i x the f i x e d vertex of that graph at x instead of at the o r i g i n . Since the remaining graph has one vertex f i x e d at zero, C^(0,x) l i n k s the two graphs together. This proves eqn. (1.18). Theorem 1.3 now follows by repeated a p p l i c a t i o n of eqn. (1.18) to see that at t = 0 the M - 1 matched p a i r s of legs i n (1.17) can be joined by c a n c e l l i n g M - 1 factors of • . • • , i n the denominator, and that the sing l e unmatched l e g of (1.17) can be joined to the appropriate unmatched l e g of (1.10) by c a n c e l l i n g the remaining f a c t o r of »••' • » i n the denominator, r e s u l t i n g i n a connected graph. • § 2. The Test f o r I r r e d u c i b i l i t y I r r e d u c i b i l i t y properties of a graph depend only on the t o p o l o g i c a l structure of the graph and not on the rules f o r evaluating the graph. In th i s section we define the notion of a t o p o l o g i c a l graph and show how a function can be assigned to a t o p o l o g i c a l graph i n such a way as to provide a t e s t f o r whether or not the graph i s 1-PI . D e f i n i t i o n 2.1; A t o p o l o g i c a l graph i s a c o l l e c t i o n of f i n i t e l y many v e r t i c e s , each having a f i n i t e number of legs ( h a l f - l i n e s joined at one end to the vertex), such that every l eg of every vertex i s paired with some other l eg to form a l i n e . ^ P l e s ; Q / Q__Q , 0 0 , • [> . As i n §1.3, a t o p o l o g i c a l graph i s s a i d to be connected i f i t s v e r t i c e s are path connected by i t s l i n e s . A t o p o l o g i c a l graph i s 1-PI i f the removal 86 of any one l i n e from the graph leaves a connected graph. There are many ways to assign a function to a t o p o l o g i c a l graph i n such a way as to be able to use the function to t e s t the graph f o r i r r e d u c i b i l i t y p r o p e r t i e s . The choice we make i s guided by our strategy f o r i d e n t i f y i n g the t o p o l o g i c a l structure of the graphs contributing to -D^V(0,0) . That strategy i s to introduce a l a t t i c e theory and an " e f f e c t i v e p o t e n t i a l " f o r the l a t t i c e theory which generates exactly the same t o p o l o g i c a l graphs as the continuum e f f e c t i v e p o t e n t i a l , but with d i f f e r e n t rules of evaluation. These rules of evaluation make the i r r e d u c i b i l i t y t e s t straightforward. We now explain the method of CSp 75] f o r t e s t i n g a graph f o r one-p a r t i c l e i r r e d u c i b i l i t y i n the context we need. For a f i x e d p o s i t i v e integer m , we consider the l a t t i c e L„ of 2m points {x,,...,x } , thought of 2m 1 2m as c o n s i s t i n g of the two s u b l a t t i c e s {x,,...,x } and {x ,...,x„ } . 1 m m+1 2m Write m2 = U Q (a) as usual and l e t where CQ) = m -4 R 1 XR ( R l ) . . m. 1 ' i = D r. . , i / j I 3-D XR , X e CO, 1] , 2 (2.1) (R_). . = < 2 ij m 1 ' i = D , and m+i,m+j R. . = r for a l l i , j . The matrices R, R, , and R„ are a l l m x m , 13 1 2 the r . . are s t r i c t l y p o s i t i v e with r . . < r , r . . = r . . for a l l i and 1: ID — ID D i j , and r > 0 i s chosen s u f f i c i e n t l y small that C(X) i s p o s i t i v e d e f i n i t e f o r a l l X e CO, l ] and a l l r . . e (0,r) . (It i s poss i b l e to ID _2 so choose r since for r = 0 , C(X) = m^  I. See Lemma 4.1). The f a c t that C(X) i s p o s i t i v e d e f i n i t e i s not relevant f o r the i r r e d u c i b i l i t y t e s t ; p o s i t i v e d e f i n i t e n e s s i s required because C(X) w i l l be the covariance 87 of a Gaussian measure on R 2 m i n the next se c t i o n . The v a r i a b l e X measures the coupling between the sets {x„,x„,...,x } and {x ,,x „,...,x„ }. 1 2 m m+1 m+2 2m - 4 "° R _ Observe that DC(O) = m^  has nonnegative e n t r i e s . D e f i n i t i o n 2.1: Let L„ (the l a t t i c e of 2m points) c o n s i s t of the 2m 2m points labeled { X / • • • / X } . A t o p o l o g i c a l graph G i s imposed on L 1 2m zm by assigning each vertex of G to a d i f f e r e n t p o i n t i n L„ . Such an 2m assignment i s c a l l e d an imposition of G on L . A n admissible 2m imposition (AI) i s an imposition for which at l e a s t one vertex i s assigned to each of the s u b l a t t i c e s {x .. ,x } and { i t , . ,...,x„ } . • 1 m m+i ziri Now consider a graph with 2m v e r t i c e s or l e s s that has been imposed x. x. 1 . 1 -on L„ . For example, G = \J 1/ , where the i . are d i f f e r e n t elements 2m * j 2 i of {l,...,2m} . The rul e for evaluating such a graph i s to form the product with one fac t o r of C(X), '. f o r each l i n e j o i n i n g x to x. . The l . i i i 3 k j 3 2 2 graph G depicted above has the value G(A) = C(X). . C(X). . C(X). . C(X). . . V 2 V 3 V4 V 4 The t e s t f or i r r e d u c i b i l i t y i s the following CSp 75]. Lemma 2.2: A t o p o l o g i c a l graph G with V v e r t i c e s i s 1-PI i f and only i f DG(O) = G(0) = 0 for every AI of G on L„ , for some m > V . 2m — Proof: Note that 0 ( 0 ) ^ _^ 0 for a l l i and j with CfOK.. = 0 i f and only i f i e {l,2,...,m} and j e {m+1,...,2m} or vice-versa. Also, DC(0) . . > 0 fo r a l l i and j . ID -We f i r s t consider connectedness. Since G(X) i s a product of C(X). . , G(0) = 0 for every AI i f and only i f at l e a s t one C(0). . = 0 Vk 1 j 1 k 88 for every AI . This happens i f and only i f at l e a s t one l i n e joins {l,...,m} to {m+1,...,2m} for every AI , i . e . , i f and only i f G i s connected. Now we consider one-particle i r r e d u c i b i l i t y . Note that DG(O') i s a sum of products, each of which consists of one DC(O). . m u l t i p l i e d by 1 j 1 k the remaining C(0). . • Each such product i s greater than or equal to zero, so DG(O) = 0 for every AI i f and only i f each such product i s zero f o r every AI . This happens i f and only i f a t l e a s t two factors C(X)^_. occur i n G(X) with i e {l,...,m} and j e {m+1,...,2m} or vice-versa f o r every AI , i . e . i f and only i f G i s 1-PI . • Note that the above proof goes through i f we take m = 1 and do not require that d i f f e r e n t v e r t i c e s be assigned to d i f f e r e n t l a t t i c e p o i n t s . The requirement that d i f f e r e n t v e r t i c e s be assigned to d i f f e r e n t l a t t i c e points w i l l be needed i n Theorem 5.6. §3. The L a t t i c e Theory In t h i s s e c t i o n we introduce a l a t t i c e analogue to the e f f e c t i v e p o t e n t i a l E(h) which generates exactly the same t o p o l o g i c a l graphs as E but which assigns values to the graphs i n such a way that the i r r e d u c i b i l i t y t e s t of §5.2 can be applied. In the l a t t i c e theory we include space-time dependent coupling constants g .. , which w i l l be used to reduce the k] i analysis of graphs with v e r t i c e s summed over the l a t t i c e to the analysis of graphs with f i x e d v e r t i c e s . Because of these space-time dependent coupling constants, i t i s necessary (although i t i s not obvious at f i r s t glance) to make the external f i e l d space-time dependent to preserve the i r r e d u c i b i l i t y of the e f f e c t i v e p o t e n t i a l . I t i s because of the space-time dependent coupling constants and external f i e l d that i t i s more convenient to work on a l a t t i c e theory than a continuum theory. (See the remark a f t e r Theorem 4.3). The l a t t i c e i n t e r a c t i o n i n an external f i e l d y e R M i s given by n 2m n 2 . I (Ti,g,x) = I [ Y q, .TiV ..x. - y. x. ] (3.1) y i i i k=2 j=o k:> k : ) 1 1 1 1 where y=(y,,...,y_ )e R 2 M , x = (x ....,x„ )e R 2 M , and the q, . are defined 1 zm 1 zm kj i k i n eqn. (1.6). The v a r i a b l e serves to l a b e l the quantity h x^ i n I . The vector g has components g . (k=2,...,n;j=0, ,^;i=l,...,2m) — Nm and i s r e s t r i c t e d to l i e i n the subset C c R N = 2m(-+l)(n-1) , E m 2 defined as follows. The p o s i t i v e constant e w i l l be f i x e d below. N D e f i n i t i o n 3.1: For E > 0 , c R i s the open cone with vertex at the N o r i g i n , axis the l i n e segment { ( t , t , t , . . . , t ) e R : 0 < t < 1} , and radius £ at i t s wide end. • By taking £ small and any coordinate g, .. near 1 , we can ensure k ] i n n 2 k that the c o e f f i c i e n t s of I y q, .h^g, ..x. are close to those of the k £ 2 j i o q ^ k ^ -n _ polynomial £ q ( h ) x . , for a l l g e C k=2 k 1 E We now prove a useful f a c t about C .Let P : C -*- [0, l ] denote E - £ the mapping which takes a vector i n to the f i r s t component of i t s orthogonal p r o j e c t i o n on the axis of C £ Lemma 3.2: For any g e C and any component g of g , £ k j i | g k j i - Eg I l £ eg Proof: Let E,g denote the p r o j e c t i o n of g e C on the axis of C 1 £ e 90 By the t r i a n g l e i n e q u a l i t y |g .. - Eg | <_ |g - £ g| . But by the cone JC ^  1 -J-l £ l g l geometry, |g - E g| <_ E ——^ = £ Pg . • / N m The import of t h i s lemma i s that by choosing £ small, we can make n n j k n k the c o e f f i c i e n t s of \ \ q /h g x, near to those of Pg \ q (h)x. . k=2 j=0 1 3 1 1 k=2 k 1 The analogue of the pressure i n the l a t t i c e theory i s given by T 2 m(Ti,g,y ,X) = in (fc,g,x): _ e n M dy,.,. / Cli,g,y,X) e (0,») x C x R x [ o , l ] , > TiC (A) E (3.2) where dy^ i s Gaussian measure on R 2 i n with covariance D , i . e . , 1 T,-1 --xD x e 2 dx D . 1 - 1 --XD X 2 , e dx and the Wick dots are with respect to the covariance TiC(X) , i . e . , : x k : = I a .(-D I I ChC(X) . .l3xk'23 . j=0 ^ 1 1 1 Because T 2 m has not been normalized by d i v i d i n g by the volume i t generates l a t t i c e graphs that have a l l v e r t i c e s summed over rather than having one vertex f i x e d as with the continuum pressure. The l a t t i c e analogue of E(?i) i s the Legendre transform T (evaluated 2m with the c l a s s i c a l f i e l d equal to zero) given by r_ (h,g,X) = sup C-TiT. (h,y,g ,X)] , (Ti,g,X) e (0,°°) * C * [0,111. (3.3) 2m „ 2m E 2m yeR The following lemma w i l l be used i n the proof that T i s f i n i t e . 2m Lemma 3 . 3 : Let dv = g(x)dx be a f i n i t e p o s i t i v e measure on R , with g > 0 and e " D X e L'(dv) . Let dv. = - — — : r e 3 X d v Then l i m j-*±o° J xdv . = ±°° D Proof: I t s u f f i c e s to prove that lim xdv. = + «= , since j-KO J -1 xdv -3 _ jx xdv where dv = SAIiSi f a n d dv = g(-x)dx s a t i s f i e s 3 j f , , . e g(-x)dx the hypotheses of the Lemma. To prove the j + 0 0 case, we begin by showing that given any a < 1 and y > 0 there i s a J(y) such that dv. > a for every j > J(y) 3 ~ ~ In f a c t , l e t E > 0 and choose x^ < y such that dv < E ." Choose J, such that e j ( x Q - y ) < E for j 21 J Q • Then Dx n e J dv = Dx, e J dv + DX e dv + DX, e J dv < e DX n f 0 ° dv + DX, e J dv = e 3 y C e dv + E + e~3Y e D X d v ] JY so dv. > Ce -DY e j Xdv] C E dv + E + e •DY -1 e D X d v 2 l a for E s u f f i c i e n t l y small But for y > 0 and j ^_ J(y) , xdv = o 3 J xdv . + —oo -J - -y xdv . + 3 J xdv . > -y 3 - J_oo 3 xdv. - y(1-a) + ya -y xdv. + (2a-l)y J-co ^ (3.4) And i f y > 0 and j > 0 then xdv xdv . < "3 ~ J xdv -3 — jx x e g(-x)dx e J g(-x)dx x g(-x)dx = c g(-x)dx (3.5) By eqns. (3.4) and ( 3 . 5 ) , xdv. > - c + (2a-l)y i f j > J(y) . The ;_oo 3 -Lemma then follows by taking a = - , since y can be taken to be arbi t r a r i l y large. • Theorem 3 .4 : The lat t i c e Legendre transform T (Ti,g,X) i s f i n i t e for 2m (ri,g,X) e (0,°°) x C £ x CO, 1] , and the supremum in i t s definition i s attained at a unique point u(?i,g,A) Proof: The variables Ti,g,X,m play no role in the proof so we drop them from the notation and simply write 93 r = sup C-T(y)] = - i n f T (y) (3.4) yeR' 2m yeR 2m Now T(y) = in - : I Q ( x ) : x e e y x dy^x) , so for 6 e [0, l ] and y ^ v , T(6y +(l-9)v) = in e(-:I (x):+yx) (1-0)(-:I (x):+Vx) e e dy c(x) -:I (x):+yx 0 < in [ (| e u dyn) ( -:I (x):+VX 1-0 e 0 a v c . ] = 0T(y) + (l-9)T(v) , (3.5) by Holder's i n e q u a l i t y . By the conditions f o r s t r i c t i n e q u a l i t y i n Holder's i n e q u a l i t y given i n [Rudin 74, p. 66], Holder's i n e q u a l i t y i s s t r i c t here provided there are no constants a, B (not both zero) such that a e ~ : I Q ( x ) : + y x -:I (x):+vx e a.e.(dy) . Since there are no such a, i f y ^ v the i n e q u a l i t y i n (3.5) i s s t r i c t : T(6y + (l-6)v) < 0T(y) + (l-0)T(v) That i s , T i s s t r i c t l y convex. I t follows that i f T i s bounded below then the supremum i n eqn. (3.4) i s f i n i t e and i s attained at a unique point. By a standard theorem [Rock 70, Thm. 27.2], T i s bounded below i f lim -r— T(ty) > 0 f o r every y ^ 0 . We use Lemma 3.3 to show more, that ot t-x» i n f a c t lim -r~ T(ty) = +°° . By d e f i n i t i o n of T , ot t-x=° 94 ^ T ( t y ) = C -:IQ ( X ):+tyx yx e dy - :I (x) :+tyx e dy. - i x c ^ x Expand the Wick dots, write d y c = const e dx , and choose an i f o r which y . ^ 0 . Let z = y • x and y = (x.. x x ) £ R 2 m 1 1 1 i 2m Then for some polynomial P i n 2m v a r i a b l e s , t z . z e ( -P(z,y) e dy)dz 3t T(ty) = e ( -P(z,y) e dy)dz which goes to +°° as t -»- 0 0 by Lemma 3.3. • §4. Regularity of the L a t t i c e Legendre Transform 00 In t h i s s ection i t i s shown that IV i s C as a function of 2m Ch,g,X) £ [o, y) x C x [o, 1] . Because the l a t t i c e i n t e r a c t i o n involves E n ^ 1 2 2 the basic polynomial £ q x + -m x , the l a t t i c e and continuum theories k=2 K 2 1 have s i m i l a r s t r u c t u r e s . The proofs i n t h i s section are based on the same ideas as the proofs of smoothness of the continuum pressure and e f f e c t i v e p o t e n t i a l , and can be omitted i n a f i r s t reading. We begin with a lemma about C(X) ^ . Recall from the d e f i n i t i o n of C(X) (eqn. (2.1)) that the nondiagonal elements of C(X) are required to be l e s s than or equal to r . Lemma 4.1: For any £ > 0 there i s an r ^ > 0 such that C(X) ^ e x i s t s for a l l r < r Q , and xC(X) "^ x >_ (m 2 -E) | X | 2 for a l l x e R 2 m , r < r Q , X £ [0, 1] . -2 -1 2 Proof: For r = 0 , C(A) = m^  I , so C(A) = m^l . By choosing r s u f f i c i e n t l y small the spectrum of C(A) can be confined to a neighbourhood -2 -1 of m^  small enough to guarantee that the spectrum of C(A) i s within o e of . • The following Lemma provides a multi-dimensional analogue to Lemma 2.3.6, and i s the key to proving smoothness of T„ . A f t e r T„ 2m 2m has been shown to be smooth Lemma 2.1.2 w i l l be applied to obtain smoothness of r . 2m Lemma 4.2: There e x i s t r Q , e, p, y > 0 such that the polynomial i n „2m x e R (h,g,A ;x) = :I Q(Ti,g,x) • h c ^ X ) - yx + i x C ( A ) _ 1 x (4.1) (after undoing the Wick ordering) has a uniquely attained global minimum, at say £(h,g,y,A) , f o r a l l (Ti,g,y,A,r) e [0, y) x C £ x Dp * Co, l ] * [0, r Q ] , where D = {x e R 2 M: I x l < p} . Also, % e C°°(Co, y) x c x D x [o, 1]) . p e p Moreover, there e x i s t s a c > 0 such that 2 K(h ,g,y ,A;x) = J y (Ti,g,A ;x+£ (h,g,y ,A)) - J (n,g,A;£(h,g,y,A)) >_ c|x| (4.2) 2in for every (Ti,g,y,A,x) e [0, y) x c x D x ^0, 1] x R E P Proof: R e c a l l from eqn. (4.3.2) that f or a 4 B there i s a 6 > 0 such that n k l 2 2 n k 1 I q k Q y + 2m]_y 1 6 I lyl f o r e v e r y y e R . (4.3) ^ k=2 k=2 By Lemma 4.1 r ^ > 0 can be chosen s u f f i c i e n t l y small that i x C ( A ) _ 1 x >_ (im 2 - |) |x| 2 f o r a l l (A,r,x) e [0, l ] x [o, r Q ] x R 2 m . (4.4) 96 By Lemma 3.2 i t i s pos s i b l e to choose y and e small enough that the c o e f f i c i e n t of x. i n :I (h,g,x): (after undoing the Wick ordering) 1 0 TiC (A) ^ i s within -Pg of ( E g ) q k Q f o r a l l (h,g,A) e [0, y) x x [o, l ] and i £ {1,2,...,2m} , k £ {2,...,n} . This can be done because the c o e f f i c i e n t i s a sum of terms each of which i s l i n e a r i n one component of g and which are at l e a s t 0(Ti) for a l l terms except the g .a. x. term. Using kOi TcO l t h i s , together with eqn. (4.4), we have 2m n £ v l ? J 0(Ti,g,A;x) > I I C(Pg)q k 0x^ - (Eg) ||x.| ] + [±m± - |) |x| i = l k=2 + C ^ t i ^ A J x + C(ti,g,X) where and C Q come from the Wick ordering, are 0(h) , and are a sum of monomials i n components of g . Therefore, using eqn. (4.3), 2m r n i [ a i = l k=2 a0Ch,g,X;x) > ( E g ) J [ ( I q k o x k + - V x 2 ) - -J|x. | k - | | x . | 2 ] + (1-Eg)(|m 2 - | ) | x | 2 + C xx + C Q 1 (Pg)^|x| 2 + (1-Eg) (imj - |) |x| 2 + c^x + C Q But ( E g ) | + (1-Eg) (|m2 - «, - i»2 _ 6 + g g ( 3 | _ 1 ^ , 1 2 6 36 1 2 6 - i m i " 4 + T " 2 m l = 2 ' 1 2 since -m^  >_ 6 by eqn. (4.3) (and the f a c t that q 2 Q = 0) . Therefore J 0(Ti,g,X;x) >_ | |x| 2 + C l X + C Q, f o r (Ti,g,X,x) e [0, y) x C~£ x [0, 1] x R 2 m (4.5) 97 and hence J y ( n , g , X ; x ) - C Q ^ | | x | 2 + (C^ - u ) x > 11 x | 2 - ( |y| + \c±\)\x\ I t fo l lows that J ( t i , g , X ; x ) " CQ 1 0 i f |x| > 4 6 " 1 ( | y | + |cj) ( 4 . 6 ) Since J (Ti,g,X;0) - C = 0 we have min J Ch,g,X;x) - C < 0 for every y 0 y 0 — x 2iti 9 y e R and i t su f f i ce s to show that - — J (h,g ,X;x) = y . ( i = l , . . . , 2 m ) dx^ 0 i has at most one root i n {x: |x| <_ 46 1 ( | y | + IC.J)} . By eqn. (4.5) , det 9x.3x. • i D J Q (h ,g ,X;x) x=0 6 2m >. (-) , f o r (h,g,X) e CO, y) x C £ x Co, 1] Using t h i s and the fac t that d e r i v a t i v e s o f J Q with respect to x are uniformly bounded i n Ch,g,A) e CO, y) x C £ x CO, 1] and choosing y and e smal ler i f necessary, there i s an a ' > 0 such that det 9x.8x. 0 J „ (h ,g ,X;x) 1 5 2m _ l - ( - ) for a l l (h ,g ,X,x) e Co,y) x c x C o , l ] x D , 2 4 s SL (4.7) One can argue us ing (4 .7) , the fact that d e r i v a t i v e s of J Q are uniformly bounded and an adaptat ion o f the proof o f the inverse funct ion theorem, that ,2m there i s an a > 0 such that VJ ( T i , g , X ; « ) : D R~ U cl i s one-one and 2m hence VJ (h ,g ,X;x) = y has at most one s o l u t i o n i n D , for any y e R 0 a Let 2p = a6/4 . Then for y e D P and y smal l enough that |c.J < p , {x : |x| £ 4 6 - 1 ( | y | + |c |)} c D and so J has a uniquely a t t a ined g l o b a l l a y 98 minimum. oo C a l l the l o c a t i o n of the glo b a l minimum £(ti,g,y,X) . Then 5 i s C i n (ti,g,y,X) e LO,y) * C x D x [ o , l ] by eqn. (4.7) and the i m p l i c i t £ p function theorem [Warn 71] . F i n a l l y , eqn. (4.2) follows from the f a c t that £(ri,g,y,X) can be made a r b i t r a r i l y close to zero by choosing y and p s u f f i c i e n t l y small by eqn. (4.6 ) , and hence the c o e f f i c i e n t s of K can be made a r b i t r a r i l y close to those of J . • From now on we take r = r . Theorem 4.3: r i T 2 m (ri, g, y ,X) i s C i n (ri,g,y,X) e [0,y) * C £ * D p x [ o , l ] 1 2 Proof: Translating x by £ i n eqn. (3.2) and then s c a l i n g by ti gives TiT 2 m(Ti,g,y ,X) = - j Ch,g,A ;5(?i,g,y,A)) + ti Jtn f -^K(Ti,g,y,X;n 2x) e * dx -ixC(X) Xx e dx (4.8) where J and K are given by (4.1) and (4.2) r e s p e c t i v e l y . y y The f i r s t term on the r i g h t side of eqn. (4.8) has the required smoothness, by Lemma 4.2. Using the bound of eqn. (4.2) and Lebesgue's Dominated Convergence Theorem, the second term can b e ^ d i f f e r e n t i a t e d under the i n t e g r a l sign with 2 2 — respect to (t=Ti ,g,y,X) e [0,y ) * c e x D p x ^ • T h e o n l Y thing to check i s that odd t de r i v a t i v e s vanish at t = 0 , i . e . k & - V DCXDi3 — C t 2 T ( t 2 , g,y,X)] -v 0 as t 4- 0 , i f k i s odd. 3 t k g y 3X* 2 m (4.9) To see t h i s , note that by eqn. (4.2) 1 1 4j- K(t 2,g,y,X;tx) l c | x | 2 for a l l (t,g,y,X,x) e (-y^Vc x D x [ o , l ] x R 2 i n , 2. £ P so that i n f a c t the second term on the r i g h t side of eqn. (4.8) i s C 1 1 2 2 — i n (t,g,y,X) e (-y , Y ) x C x D x [ 0 , 1) . But by s c a l i n g —|K(t 2,g,y,X;tx) t , -2m e dx = t 1 2 —2-K(t ,g,y,X;x) dx Therefore the second term on the r i g h t side of eqn. (4.8) i s i n v a r i a n t under t -*• - t , and eqn. (4.9) follows. • Remark: A continuum version of Theorem 4.3 would be more d i f f i c u l t to prove than the l a t t i c e version, because i n a continuum theory with space time dependent coupling constants the function £ which removes the l i n e a r term from the i n t e r a c t i o n s a t i s f i e s a non-linear p a r t i a l d i f f e r e n t i a l equation. Also, smoothness properties of T with respect to g and y 2m would be i n terms of f u n c t i o n a l d i f f e r e n t i a t i o n i n the continuum theory rather than the p a r t i a l d i f f e r e n t i a t i o n i n the l a t t i c e theory. In the next lemma, we use the notation <•> • e dy, TIC (A) h , g , y , x --:I (h,g,x): n y e dy, TiC(X) Lemma 4 . 4 : The following l i m i t s are uniform i n g, y, X e C^ * Dp * [0, l ] Ci) l i m Ti T Oi,g #u,X) = Ti+O - J CO,g,X;£(0,g,y,X)) Cii) l i m Ti+O < * i > . Ti,g,y ,X = C i (0,g ,y,X) 100 ( i i i ) l i m h 1 / x . ; x . \ = (M x ) . . , where M h+0 1 D / h , g , y , X 1 3 ab 8x a9Xj_ ) K ( 0 , g , y , X ; x ) x=0 i s i n v e r t i b l e ( i n f a c t , p o s i t i v e d e f i n i t e ) by e q n . (4.2) . -1 I n p a r t i c u l a r , l i m h / x . J x v ti+0 1 3 / T i , g , 0 , ; = C(X) . . . I D P r o o f : ( i ) The r e s u l t f o l l o w s f rom e q n s . (4.8) and ( 4 . 2 ) . ( i i ) D i f f e r e n t i a t i n g e q n . (4.8) w i t h r e s p e c t t o y ^ , t h e l e f t s i d e g i v e s <*i>. w h i l e T i , g ,y ,X - — J (T i , g ,X ; £ ( h , g , y ,X)) = - M ; J (Ti ,g, X ; £ (h,g,y ,X)) - y £ C h , g , y ,X ) ] 9y. y 9y. 0 I i 2m - I i f " J 0 » . g . x . 5 ) ^ " «i - .1, ^ ^ = - 6 . . 2m 9£ j - 1 j 9 y i The r e s u l t t h e n f o l l o w s s i n c e r a p p l i e d t o t he s e c o n d term on the r i g h t 3 y i s i d e o f e q n . (4.8) i s 0(h) by e q n . ( 4 . 2 ) . C i i i ) Ti 1 / x ,x \ N 1 3 / T i , g , y , ; --J (h,g,X;x) • , . T I y , ( x . , x . ) e dx i ,D ( --J ( h , g , y ; x ) Ti u , e dx - -K (h ,g ,y ,X;x) ( ( x . + S . ) ; ( x . + 5 , ) ) e dx i i D D - -K (h ,g ,y ,X;x) n -, e dx 101 --K(n,g,y ,X;x) (x,;x.)e dx f -^K(h,g,y,X;x) j e dx -iK(h,g,y,\;h 2x) (x.;x.)e " dx i 1 1 2 --Ktfi,g,y,X;h x) h e dx -ixMx 2 (x.;x.)e dx r —xMx 2 e dx using eqn. (4.2) i n the l a s t step. Since M i s a symmetric p o s i t i v e d e f i n i t e quadratic form, the l a s t i n t e g r a l i s a Gaussian i n t e g r a l "that can be evaluated e x p l i c i t l y to give (M ^ ) 3-D • Lemma 4.5: lim y(ti,g,X) = 0 uniformly i n (g,X) e C * [0, 1] . h+0 G Proof: To s i m p l i f y the notation, l e t f Ch,y) = Ti 7^Ch,g,]i,X) and f(y) = - J (0,g,X,-£ (0,g,y ,X)) . By Lemma 4. 4 ( i ) , l i m f(7i,y) = f(y) V 7i+0 uniformly i n g and X . * By eqn. (3.5) f(h,') i s convex, so the same i s true of f . Also, f i s smooth f o r small |y| and f(y) >_ 0 with f(y) = 0 only i f y = 0 . Let e e (0,p) and set a = min | — f (sy) | . Then a > 0 and for any |y| = 1 , . o S s=±e \v\=i 102 T2- f ( S P ) < 9s < - a s = - E > a s = +£ 3 A. y But — f (Ti, sy) = y <TX > . - i and 3s / h , g , s y , X 3 ^ - 3 * — f(sy) = ; ^ £ J o ( 0 , g , ; \ ; £ ( 0 , g , s y , X ) - s y £ (0,g,sy ,X) ] = y £ ( 0 , g , s y , X ) Therefore 3s" :(Ti,sy) - — f ( s y ) | = |y <^ x)> ^ - yC(0 ,g , sy ,X) s Ti ,g,sy,X I <x> ^ - ? (0 ,g , sy ,X) | f l ,g ,sy ,X (4.10) The r i g h t hand side of eqn. (4.10) goes to zero as h + 0 uniformly i n g,X , s = ±e and | y | = 1 by Lemma 4.4 ( i i ) . 3 ~ 3 ^ Therefore there e x i s t s a 6 > 0 such that I— f (Ti,sy) - — f (sy) I < — 3s 3s 2 for a l l (Ti,g,s,y,X) e (0,6) x C £ x { - £ , + £ > x {y : |y| = 1} x [o, 1] , and so — f (Tl, sy) • < - - , s = -e - 2 > - , s = +e - 2 for a l l Ti < 6 , y = 1 . I t fo l lows that the minimum of f (h ,y ) i s a t t a ined at some p o i n t s(Ti)y(Ti) with |y(Ti) | = 1 and sCh) < £ . • Theorem 4 .6 : r (h,g,X) i s C i n (Ti,g,X) e [0,y) x C x [o, l ] . 2m e Proof : We f i r s t show smoothness of r_ (ti,g,X) = -Ti T_ (Ti,g,y (h,g,X) ,X) 2m 2m i n the open set (0,y) x C x (0,1) . By Lemma 4 .5 , y(Ti,g,X) e D for c P Ti < Y s u f f i c i e n t l y small. Therefore by Lemma 4.4 ( i i i ) , det S y ^ y ft T 2 m ( f t,g , y , X ) = det -1 y ( f t ,g ,X) < V x j > > c > o ft,g,y ,x_ (4.11) uniformly i n Ti, g and X i f E and Y a r © s u f f i c i e n t l y small. By eqn. (4.11) and the i m p l i c i t function theorem [Warn 71] , y(h,g ,X ) i s C°° i n (h,g ,X) e (0 , y ) x C £ x (o,l) . The extension of smoothness to [0,Y) x c e * ^0, l ] poses no d i f f i c u l t y since d e r i v a t i v e s of T can be seen to be uniformly bounded 2m i n (ft,g ,X) e (0,Y) x c e x (0,1) using eqn. (4.11) and the f a c t that d e r i v a t i v e s of T are uniformly bounded (by Theorem 4.3) . • 2m § 5. I r r e d u c i b i l i t y In t h i s section we give the proof of Theorem 1.4.3(b). The f i r s t theorem of t h i s s ection allows us to analyze the graphs occurring i n N i r 2 N ( " 0 ' g ' A ) instead of those i n D^EtO) N Theorem 5.1: For N >_ 2 , - D 1 r 2N l' 0' g'^ ^ S < 3 ^ v e n b y a f i n i t e l i n e a r combination of graphs which i s t o p o l o g i c a l l y i d e n t i c a l to the sum of graphs N equal to -D^E(O) (as given i n Theorem 1.3), with the following rules of evaluation: 1. Whereas a vertex i n -D^E(O) takes a fa c t o r -q, .:<bk(R2): , a vertex 2N k k = ) i n -D r(0,g , X ) takes a factor - £ q g .:x.: . X , ^  Kj K j l 1 2. No vertex i s f i x e d — a l l are summed over the l a t t i c e . 3. A l i n e j o i n i n g x. to x. contributes C ( X ) . . . I D I D 104 Proof: Since r 2 N ( h , g , A ) = -Ti Jin ™ o j*-k-l "I n 2 J - . k 2 - I I I I g k i i : x i : _ T l yiCh-g.^).*.: ( i = l k=2 j=0 K D k D i i i i <3Y C(X) and E (Ti) = - T I T T T L I M £ N A A n n 2 A k=2 j=0 k 3 j+k-1 . 2 xk r :-"h u(Ti)<|>] d i f f e r e n t i a t i o n of T with respect to t = ti i s formally very s i m i l a r to d i f f e r e n t i a t i o n of E with respect to t = Ti , and with the rules 1-3 above, y i e l d s graphs of the form (1.10) with f replaced by -1 2 b(t,g,A) = t y ( t ,g,A) . However a d i f f e r e n t mechanism i s responsible for hooking the graphs D^h(t,g,X) onto the corresponding legs, as we 2N now explain. For b e R , l e t zCt,g,b,A) = TCt ,g,tb,A) = £n n 2N 2 - X C X q„.t ' i - l k=2 ^ 2j+k-2 k ^ , g, . . : x. : -b . x. ] k j i i l i d y c ( X ) . (5.1) Then b(t,g,A) i s characterized by r — z (t,g,b (t,g, A) , A) = 0 , (i=l,...,2N) db. 1 S i m p l i f y i n g the notation by denoting d i f f e r e n t i a t i o n by subscripts and using an implied summation convention, d i f f e r e n t i a t i n g eqn. (5.1) with respect to t gives -1 "bt + z b b b t = 0 , and so b f c = - z ^ z ^ 2 2 Since z, , = t T , i t follows from Lemma 4.4 ( i i i ) with t = h bb yu that z,_ i s i n v e r t i b l e i f t i s s u f f i c i e n t l y small. In f a c t , bb l i m z \ = C(X) 1. . . The matrix inverse z, ^  plays the r o l e here of t+o b i b j 1 3 b b 2 the denominator D 2 C ( t , f ( t ) ) i n Theorem 1.3. To see that z ^ hooks things up i n the r i g h t way, consider for example z. ..z, ?"z, . at t = 0 , i . e . , a l i n e a r combination of terms of t t b bb b t t t the form j i 2N 11> 2 J3 where a vertex denoted k i s f i x e d at *k Suppose the l i n e — — « i i s connected to vertex i . Then 1,3=1 i 2 j 3=1 1 2 1 i , 3 D3 2 33 This shows that at t = 0 , 1 1 j , xi i1 2N rVc<M;--rl, 32 2 X2 j3 J 3 • C o r o l l a r y 5.2: ^ 1 ^ ( 0 , 1 , X ) = \ ^ D2 D1 F2N ( ° ' ° ' A 5 ' I a I <N where a i s a multi-index with 2N( n+l)(n-l) components. N Proof: By Theorem 5.1, D i r 2 N ^ ° , g ' ^ ^ S a P o x y n o m i a l i n 9 o f degree N so the C o r o l l a r y follows by Taylor's Theorem. • The following Lemma shows that when g = 0 the i n t e r a c t i o n (defined i n (3.1)) occurring i n the l a t t i c e pressure T(Ti,g,u (Ti,g,X) ,X) vanishes. 106 Lemma 5.3: y(Ti,0,A) = 0 f o r (h,X) e [0,y) x [0,1] Proof: I t was shown i n the proof of Theorem 3.4 that Ti T,.„Ch,0,y ,A) = Ti £n 2N 1 -yx e dy i s s t r i c t l y convex as a function o f y TlC (A) Since Ti T 2 N Ch,0 ,-y, A) = Ti T 2 N(h,0,y , A ) , i t follows that i n f Ti T„ (h,0,y,A) occurs a t yCh,0,A) = 0 . • W2N 2 N yeR To s i m p l i f y the notation for d e r i v a t i v e s with respect to components of g , given i n d i c e s k , j , i we write g. = g . . , and denote £-'£ 1 £ derivatives with respect to g^ with a subscript £ , e.g., T , „ = T ?; r and we drop the subscript 2N from T and T„„ . 12...N 3 3 2N 2N The following lemma i s the f i r s t step i n i d e n t i f y i n g the graphs c o n t r i b u t i n g to -r fh,0,A) . 12. . .N Lemma 5.4: For Ti < y , _ r ^ 2 N(n,0 ,A ) i s a f i n i t e sum of graphs with the V K " 1 K £ N v e r t i c e s -q, . "h :x. : (£=1,...,N) and l i n e s C(A) . No V £ Xl s e l f - l i n e s can appear. Graphs enter the sum with e i t h e r a plus sign or a minus sign, but a l l those with minus signs are 1-PR . Furthermore, every 1-PI graph with the mentioned v e r t i c e s enters the sum with a plus sign. The combinatorial f a c t o r of a 1-PI graph i s the same as for T M(Ti,0,0 ,A) . 12. . .N Lemma 5.4 w i l l be improved i n Theorem 5.6 where i t w i l l be shown that a l l the 1-PR graphs i n _ r 1 2 N(h,0 , A ) cancel, leaving only the 1-PI graphs. 107 Proof of Lemma 5.4: The variables Ti and X play no s i g n i f i c a n t r o l e i n the proof so we drop them from the notation. Derivatives are denoted by subscripts and an i m p l i c i t summation convention i s used. In the following, a l l d e r i v a t i v e s of T are evaluated at (g,y(g)) . D i f f e r e n t i a t i n g the equation - r(g) = T(g,u(g)) with respect to g 1 gives -I" = i + T y = T. , since T = 0 . Note that i n T, the g 1 1 y 1 1 y 1 3 dependence of y i s not d i f f e r e n t i a t e d . D i f f e r e n t i a t i n g -T^ = T^ ., with respect to g^ gives - F12 " T12 + T l y y 2 ' To compute y^ , d i f f e r e n t i a t e the equation T^ = 0 with respect to g^ to obtain T . + T y. = 0 , i . e . , y i yy I y. = - T _ 1 T . , (5.2) i yy y i where the inverse on the r i g h t side i s a matrix inverse. Therefore -r._ = T._ - T T - : LT „ . (5.3) 12 12 l y yy y2 Note that when g = 0 (g,y(g)) = (0,0) by Lemma 5.3 and we have a free theory. Using the l a t t i c e analogue of the formula (1.3.3) for the der i v a t i v e of the logarithm of a p a r t i t i o n function and the d e f i n i t i o n of T i n eqn. (3.2), a d e r i v a t i v e of the form T.. , at g = 0 i s the i 3 . . . k y ^ 4 J M sum of a l l connected graphs with f i x e d v e r t i c e s as s p e c i f i e d by the 9^'s » and M f i x e d one-legged v e r t i c e s because of the y d e r i v a t i v e s . As shown i n the proof of Theorem 5.1, T ^ serves to l i n k up graphs i n a free theory. yy We use a graph notation for the de r i v a t i v e s as follows. Denote 108 i j k y ^ j j by D M and y ± by (-1) ^ y>i , where the dot on the y. = -T "*"T . graph indicates that a T ^ has amputated a leg that 1 yy y i yy was brought down by d i f f e r e n t i a t i o n with respect to y . When g = 0 or M <^^>-i i s given by a sum of connected l a t t i c e graphs without s e l f - l i n e s . (In p a r t i c u l a r , a t g = 0 ^^P*" ~ ^ * I n th-'-s notation, eqn- (5.3) becomes -Y 12 • 0 » - O - O The theorem now follows by repeated d i f f e r e n t i a t i o n of eqn. (5.3) using the following f a c t s : 3 —1 —1 —1 T = -T (T „ + T y ) T 3g £ yy yy yyX, yyy I yy 9 g „ i j . .ky ... y M 39, C l e a r l y a l l graphs occurring i n ~ r 1 2 w i a m i n u s s i g n a r e 1-PR , because a minus sign i s introduced with every factor of T 1 (and yy - l i n no other way) and a factor of T corresponds to a l i n e whose removal yy disconnects the graph. Furthermore contains the term + T 1 2 M (0,0) which i s the sum of a l l connected graphs (with combinatorial factors) having v e r t i c e s as i n the statement of the Lemma, and hence contains as a subset a l l 1-PI graphs. • The following theorem i s the key to obtaining the c a n c e l l a t i o n of a l l 1-PR graphs i n - r i 2 N C h,0 , A ) . Is i s i n s p i r e d by [CFR 81]. Theorem 5.5: Given g„ = g, . . ,. . . - ^ ., i k j ^ 1 ^ d = l » — F N ) 1 i f at l e a s t one i ^ i s an element of {l,...,N} and at l e a s t one i ^ i s an element of {N+1,...,2N} then f o r a l l Ti e CO,Y) D s r . . ( f t , 0 , 0 ) = 0 , s = 0, 1 12. . .N Proof: Since "ft plays no r o l e i n the proof i t i s omitted. -1 4 Beginning with the case s = 0 , since C ( 0 ) = m^  r 1 1 0 -1 does not mix {x ,...,x } and {x„,.,...,x„ > we can write 1 N N+l 2N T(g , y,0) = S ( 1 ) (gtl) ,y (1)) + S, 0 1 (g(2) ,y (2)) (2) where y(1) and g(l) (respectively y(2) and g(2)) co n s i s t of those y. and g, .. with i e {l,...,N} (respectively i e {N+l,...,2N}) , and 1 JC J 1 s ( 1 ) (g(D ,y (l)) = In J n N n 2 ,  _ 1 t I 1 1^ • 9i . . : x. : -y . x. ] r i - l k=2 j=0 k D k 3 1 1 1 1 , e d-Y _ 4 ( x ^ . . . , ^ ) m i R l n 2N n 2 " I C I T q. .g. . . :x. :-y .x. ] S ( 2 ) (g(2) , y(2)) = £n i=N+l k=2 j=0 d Y -4 ( X N + 1 X 2 N > m l R2 Therefore T(g ,y,0) = • 3y . 3 — S ( l ) ( g ( 1 ) i c {1,...,N} i S,_. (g(2),y(2)) i e {N+1,...,2N} d\i± (2) 110 I t follows that y i(g,0) = " y | 1 } ( g ( l ) , 0 ) i e {l,...,N} y f 2 ) ( g ( 2 ) , 0 ) i e {N+l,...,2N} and hence T(g,0) = -T(g,y(g,0),0) = - S ( 1 ) ( g ( l ) , y ( 1 ) ( g ( l ) , 0 ) , 0 ) - S ( 2 ) ( g ( 2 ) , y ( 2 ) ( g ( 2 ) , 0 ) , 0 ) and the theorem follows i n the case s = 0 . To prove the theorem i n the case s = 1 , we begin by noting that D 2 r(g,0) = d dX X=0 T(g,y(g,X),X) = -D 2T(g,y(g,0),0)D 2y(g,0) - DjT(g,y(g,0),0) = -D 3T(g,y(g,0),0) , (5.4) since D 2T(g,y(g,X),X) = 0 . Denoting expectations with respect to dy -:I (g,x): by and expectations with respect to e dy A C (A) C(X) -:I (g,x): e M dy C(X) bv C •^ , , we have D 3T(g,y,0) = — -:I (g,x): -:I (g,x): . -:I (g,x) Ce y ]X 0 = Ce M 0 3X -:I (g,x): ( - : I o ( g ' x ) : c ( X ) ) e ]o 2N 1,3=1 (5.5) 3X ( : I o ( g ' x ) : c ( X ) } 2N g,y,o i/3=i g ,y»o I l l By eqns. (5.4) and ( 5 . 5 ) , D 2r(g,0) = < ^ 2N + \ . I ^ ^ S K V J ) , M " ( 5 - 6 ) i/D=i ' g.y(g#o),o J Now d i f f e r e n t i a t e eqn. (5.6) with respect to and g^ where i e {l,2,...,N} and i , € {N+l,N+2,...,2N} . Since a b [ k ] 2 :x k: = I c A - l ^ C a ) 3 . * * ' 2 3 , 1 c(X) j=0 k 3 _9_ 9A k - = I c (-i) jjC(o):T1DC(o)..xk-2j :x. : 9 Therefore o 1 c(A) j^ o k j 1 1 1 1 1 : I n (g,x) .* , i s a sum of two polynomials: one i n o x^,...,x depending only on g^ with i ^ e {l,...,N} , and one i n x N + 1 , . . . , x 2 N depending only on g^ with i ^ e {N+1,...,2N> . Since as was seen i n the proof of the s = 0 case the measure <f * ) /g,y(g,0),0 factors into a product of p r o b a b i l i t y measures i n x^,..., x^ and X N + 1 , . . . , X 2 n depending only on g^ with i ^ e {1, ,N} and g^ with 1 6 { N + 1 2 N } r e s p e c t i v e l y , ^ - | — < ^ | :I (g,x) .- V = 0 ^a ^b '0 '/ g,y(g,0),0 Next, observe that the term i n v o l v i n g [x.x.] on the r i g h t side of 1 D 0 eqn. (5.6) does not depend on g at a l l and hence vanishes a f t e r taking g d e r i v a t i v e s . I t remains only to show that 112 a 2 ^ x i x j y  = °'  ( 5 " 7 ) 3 q a 9 g b N - J ' g , y ( g , 0 ) , 0 Consider the case where both i and j are i n { l , — , N } . Then by f a c t o r i z a t i o n of the measure ^x.x.\ depends only on the 1 3 g,y(g ,o ) ,o with i ^ e {l,...,N} and eqn. (5.7) holds since i ^ e {N+l, .,..,2N} . The case where both i and j are i n {N+1,...,2N} i s s i m i l a r . Now consider the case where exactly one of i , j l i e s i n {l,...,N} . Then by f a c t o r i z a t i o n of the measure, < V x . \ = < X i > ' ( X-\' 1 2 ' g,V (g,0) ,0 1 "g,y (g,0) ,0 D g,y(g,0),0 Each fa c t o r on the r i g h t side of the above equation vanishes by d e f i n i t i o n of y(g,0) . This completes the proof of eqn. (5.7) and hence of the Lemma. • We now show that a l l 1-PR graphs occurring i n ~ r 1 2 N(fi,0,A) cancel, and i d e n t i f y e x p l i c i t l y the remaining 1-PI graphs. As i n the - i u o ( a ) statement of Theorem 1.4.3 we write d(a) = — log — = — . 4ir z m Theorem 5.6: The d e r i v a t i v e _ r 1 2 „(n,0,X) i s a polynomial i n ti where the c o e f f i c i e n t of f i m i s the sum of a l l d (a)-renormalized m (k £) loop 1-PI graphs with v e r t i c e s -(P (a ) / k 0 l ) x . (£=1,...,N) and C(A) l i n e s with s e l f - l i n e s allowed. Note that the v e r t i c e s are f i x e d . Each graph takes the same combinatorial factor that i t has i n T 1 2 . . . N ( * ' ° ' ° ' X ) ' 113 Proof: We f i r s t show that ~^i2 ^(n,0,\) can be written as a sum of 1-PI graphs having Ti dependent v e r t i c e s . Part of the work was done i n Theorem 5.4, from which i t follows that we can write " r i 2 . . . N = X^*'" + l V * ' X ) ~ JAm'M ' (5-3) k=l m=l £=1 where the three sums on the r i g h t side of eqn. (5.3) are re s p e c t i v e l y the sum of a l l 1-PI graphs made of C(X)-lines and v e r t i c e s V i V 1 k £ -q, . h :x. : (having the same combinatorial f a c t o r as i n "^ 12 . N ( n ' 0 ' 0 ' A ) ) > the sum of a l l 1-PR graphs occurring i n the expansion of Theorem 5.4 with a plus sign, and the sum of a l l 1-PR graphs occurring i n the expansion with a minus sign. We now use Theorem 5.5 to show that the l a s t two sums cancel. In f a c t , t r e a t i n g i , . . . , i N as free v a r i a b l e s , i t follows from Theorem 5.5 that D 2 T 1 2 NCh,0,0) = 0 , s = 0, 1 for any admissible imposition of i , . . . i on the l a t t i c e L„„, of 2N points. On the other hand I N 2N K £ D^I (h,0) = 0 , s = 0, 1 f o r any AI , by Lemma 2.2. I t follows from k=l 2 k eqn. (5.3) that M L I D^R (n,0) = I D ^ C h j O ) , s = 0, 1, for any A.I. (5.4) m=l m £=1 M We now show that t h i s implies that ][ R (h,X) consists of exactly the same L m = 1 m graphs as £ N.(h,X) . £=1 * For a graph G with v e r t i c e s as i n R or N 0 , denote by G the graph obtained from G by c a n c e l l i n g a l l factors 1 1 4 : + i - i 2 -q, .n . Since R , i s reducible and has N v e r t i c e s , i t can be kj 1 imposed on L 2 N by choosing i , . . . , i N i n such a way that a l i n e of r e d u c i b i l i t y of R ^ ( i . e . , a l i n e whose removal disconnects R ^ ) j o i n s x, to x„,, , and no other l i n e joins {x,,...,x } to 1 N+l 1 N {xN+-j_»• • • , X 2 N ^ " This imposition of R ^ on L 2 N of course also imposes the other R 's and N ' s ori I. T . Since a l l these graphs are connected, m % 2 N at l e a s t one l i n e crosses from {x.,...,x } to {x , x„„} for each 1 N N+l 2 N graph. But — R ( 0 ) or -^r N . ( 0 ) i s zero i f and only i f more dA m dA x, than one l i n e makes the crossing from {x 1,...,x N} to ^ X J J + I ' * * ' , X 2 N ^ " Hence for the above imposition \ d f V 0 ) = I if V°> • I if V°> (5-5) m=l one l i n e one l i n e V d — where ) — - G. denotes the sum over those i for which G. has a s i n g l e , . dA i l one l i n e l i n e j o i n i n g {x, , . .. ,x } to {x.T. ,x_„}. But because of the form of C(A) 1 N N+l . 2 N (eqn. ( 2 . 1 ) , f o r a graph on R^ with exactly one l i n e j o i n i n g { X ^ , . . . , X n } to {x . x l , -=?N ( 0 ) on - T T - R ( 0 ) i s r m u l t i p l i e d by a product of r..'s N+l . 2N uA si, Q A m 13 (l<_i,j<N or N+l<i,j< 2 N ) , because i t i s only when the l i n e j o i n i n g {x.,...,x } to {x ,,...,x„„} i s d i f f e r e n t i a t e d that the r e s u l t i s non-1 N N+l 2 N zero. I t follows that the second e q u a l i t y i n eqn. (5.5) i s an equality of polynomials i n the r^_. (l<i,j<N or N+l<i,j< 2 N ) , and so the c o e f f i c i e n t s of these polynomials must agree. However these c o e f f i c i e n t s characterize the graphs t o p o l o g i c a l l y . To see t h i s , note that the r^_. are i n a one-one correspondence with l i n e s j o i n i n g x^ to x^ . Thus a product of r i j ' s characterizes the parts of the graph s i t t i n g i n each of the s u b l a t t i c e s {x,,...,x„} and {x x_„} . Because there w i l l be only one vertex 1 N N+l 2 N 115 x. i n each s u b l a t t i c e that does not have i t s f u l l quota k of l i n e s provided by the s u b l a t t i c e graphs, there i s one and only one way that the l i n e crossing from {x^,...,^} to {XM+I' • • • , X2N^ c a n ^ ° ^ n t h e t w o s u b l a t t i c e s , and the graph i s uniquely determined. Therefore I K = I No ( 5 ' 6 ) one l i n e one l i n e with exactly the same graphs occurring on each side of the equation. Now d i s c a r d the graphs contributing to eqn. (5.6) from eqn. (5.4) and repeat the above procedure u n t i l none of the R remain. We now show that no m graphs remain, arguing by c o n t r a d i c t i o n . Discarding a l l R^ graphs and the corresponding N graphs from eqn. (5.4) leaves 0 = £'D;!N0(h,0) Jo Ct JO s = 0,1 , for every A I , where denotes the sum over the remaining graphs. Therefore 0 = — ^ N . ( O ) , s = 0, 1 , for every A I . Each dA & v1 d — — term i n I ~ j r N £ ( ° ) ^ s nonnegative, and since N^ i s 1-PR, for a given the i , , . . . , i can be chosen i n such a way as to make N (0) > 0 . 0 1 N dA JCQ ~ 1 d — But t h i s contradicts 0 = ) - r r - N. (0) and hence there can be no N 0 dA JG * M L remaining. The end r e s u l t i s that 7 R (Ti,A) = T N„ (h,A) , with exactly m=l SL=1 the same graphs on each side of the equation, and hence K " r i 2 . . . N a i ' ° ' A ) = - I k ( ' h ' X ) * ( 5 , 7 ) k=l To i d e n t i f y the graphs contr i b u t i n g to the r i g h t side of eqn. (5.7) as those stated i n the theorem, we begin by obtaining an e x p l i c i t formula for q . . By d e f i n i t i o n (eqn. (1.6)), q, . = -rvD^q (0) , where q, i s n n k defined i n eqn. (3.1.9) by the requirement £ a,:* : = £ q (h) :<j> :_ k=3 * C k=0 * n C l 116 Let *k = I a . 3 < k < n k — — 0 , otherwise , and extend the d e f i n i t i o n of k 1 k c . = —: 1 by s e t t i n g c = 0 i f j > [-] . Then by Lemma 2 . 2 . 1 , k 3 2 3 j ! ( k - 2 j ) I K 3 2 l V * \ c ' X V * \ c " ln \ X ^tM)J|*k'23|«:1 k=0 3=0 J 1 k=3 k=0 n 2 n j=0 k=0 n n j 0 cI W^.^-'S (5.8) Therefore q . = a. . c . .d , and a vertex k 3 K + 2 3 K + z3,3 . k , . k , 3 +—1 3+—1 * 2 k ~ , J 2 -q, . n :x. : = -a, . Ti c, ,„. ._ -.-Tc} l k+23 k+23,3 1 dj:x kcan be in t e r p r e t e d as j k where each closed loop takes a fa c t o r of d , the combinatoric factor c, „. . counts the number of ways of choosing k+23,3 p/ 2 3 p a i r s from k+2j l i n e s , each h a l f - l i n e takes a factor h , and the vertex takes the factor - a . . This means that there i s a one-one fi k+23 correspondence between I-PI graphs having v e r t i c e s j+-k-l , 2 k -q .Ti :x.: and no s e l f - l i n e s , and d-renormalized 1-PI graphs having kj 1 117 v e r t i c e s - a, , x K + 2 ^ with s e l f - l i n e s allowed and each l i n e taking a h Tc+2j i f a c t o r h . I t remains only to i d e n t i f y the o v e r a l l power of "h of a graph. An unrenormalized graph has a power of h given by P = I - V + 1 , where I i s the number of l i n e s of the graph, V i s the number of v e r t i c e s , and the extra +1 comes from the h i n - r = hT . But I - V + 1 i s exactly the number of loops i n the unrenormalized graph. • In conclusion we combine the r e s u l t s of t h i s chapter to prove Theorem 1.4.3(b). By eqn. (1.1) and C o r o l l a r y 1.2 we need only show that for N >_ 2 , -V a ) = NT Di v<°' a> - " i t D \ ( 0 ) " NT D I E ( 0 ) ( 5 - 9 ) i s the appropriate sum of graphs. The f i r s t term on the r i g h t side of eqn. (5.9) was i d e n t i f i e d i n eqn. (1.2) to be the d(a)-renormalized s i n g l e vertex N-loop diagram. By Theorem 5.1 the second term i s a sum of graphs 1 N t o p o l o g i c a l l y i d e n t i c a l to the L 2 N-graphs whose sum i s - ^TT-^r (0,1,1) / where V i s the L 2 N Legendre transform (evaluated at the c l a s s i c a l f i e l d equals zero)- By C o r o l l a r y 5.2, I a I <N -1 d N dh' N I - T D ^ ( h , 0 , l ) . _1 t! la j <N (5.10) But by Theorem 5.6 the r i g h t side of eqn. (5.10) i s exactly the desired sum of graphs: the d i f f e r e n t terms i n the sum over a give the N-loop graphs with d i f f e r e n t kinds of v e r t i c e s . F i n a l l y we show that the combinatorial factors are as indi c a t e d i n 118 Remark 1 under Theorem 1.4.3. By Theorem 5.6 the combinatorial f a c t o r of a graph contr i b u t i n g to D°(fi,0,l) i s the same as for D^T(h,0,0,1) , namely the factor associated with the graph by Wick's Theorem. The f a c t o r o f — 7 occurring on the r i g h t side of eqn. (5.10) provides the factor ot» — - appearing i n Remark 1. Since the ^y- on the r i g h t side of (5.10) d N i s cancelled by an NI brought down by — — , the combinatorial f a c t o r of dh 1 N a graph i n - — D.T(0,1,1) , and hence i n -v (a) , i s as stated i n N i l N Remark 1. CHAPTER 6: FAILURE OF THE 1-PI LOOP EXPANSION §1. An Asymptotic Connected Loop Expansion U n t i l now we have discussed the asymptotics for the e f f e c t i v e p o t e n t i a l when a I B . The set B^ c B i s not very i n t e r e s t i n g because i t corresponds to a massless theory which w i l l be divergent when ti = 0 . The set B^ c B i s more i n t e r e s t i n g and has recently received some atte n t i o n i n the l i t e r a t u r e [FOS 83], [BC 83], [CF 83] for the case of a double-well 2 1 2 p o t e n t i a l . Consider the c l a s s i c a l p o t e n t i a l U (a) = (a - -) , with m = 1 0 8 2 1 2 1 2 1 1 (so P(a) = (a - -) - - a ) , for which B = B = C , ] . 8 2 1 /~8 /~8 (The constant ^ appearing i n U i s a r b i t r a r y and can be replaced by any 8 0 p o s i t i v e constant provided that m i s also changed so as to agree with the curvature of U^ at i t s minima). I t i s c l e a r that the loop expansion must break down at l e a s t for the i n t e r v a l Ial < — where u " (a) < 0 , 1 1 12 0 since i n that i n t e r v a l y(a) and the graphs contributing to v n ( a ) a s 119 given by Theorem 1.4.3 are divergent. In the above references the authors take both minima of the c l a s s i c a l p o t e n t i a l i n t o account for I a l < — /8 and conclude that for |a| < —— the 0(h) approximation to the e f f e c t i v e /8 p o t e n t i a l i s the s t r a i g h t l i n e i n t e r p o l a t i o n of U Q ( - ) ~ hy(a), |a| > — • In t h i s section we give a rigorous proof that t h i s p i c t u r e i s correct, showing how i t i s a consequence of the d e f i n i t i o n of the e f f e c t i v e p o t e n t i a l using the Fenchel transform and the occurrence of a phase t r a n s i t i o n [GJS 76]. Moreover we f i n d that the N1"*1 order c o n t r i b u t i o n i s the sum of a l l N-loop connected graphs, for N >_ 2 . Theorem 1.4.5: Let V(h,a) denote the e f f e c t i v e p o t e n t i a l for m = 1 and P(a) = ( a 2 - i ) 2 - ;=a2 . Then for |a| < — D.V(0,a) = -y (—) = 0 , 8 2 /£" 1 /8 - I N and f o r N >_ 2 , — D^VtCa) i s given by the sum of a l l N-loop connected graphs with no s e l f - l i n e s , with three- and four-legged v e r t i c e s taking factors "-7-P ( 3 ) (—) = - /2 and 7 - p ( 4 ) ( — ) = -1 r e s p e c t i v e l y , and l i n e s corresponding to the free covariance of mass one. Graphs take combinatorial factors as per Remark 1 under Theorem 1.4.3. Proof: Let P(x) = U Q(x) - |x 2 = x 4 - |x 2 + , and l e t p(h,y) = h lim -rK- in A+R W J _1 ( ~h [:p(<t>) :-y r] A d H h C ' where C = (-A+1) ^ . The boundary conditions on C can be chosen as a matter of convenience since p i s independent of a wide v a r i e t y of boundary 120 conditions [GRS 7 6 ] . By taking c to have p e r i o d i c boundary conditions _1 /8 and appealing to Lemma 2.2.2, t r a n s l a t i o n of T by — gives p(h,y) = h lim A+R «,n r. :* 4:±v /2:* 3:-y (<J>±—)] A SB dy, hC By Lemma 2.2.4 we obtain p(h,y) = A+R' 2 W L N J 1 _1 £h-A 4:±S2 h 2 : Y 3 - y ( h 2 T ± f i " 1 — ) ] A /8 -y. = ± — y + Ti lim AtR A in 1 _1 Ch:4 4:±/2-h 2: Y 3:-yri 2 T ] -y, (1.1) We w i l l apply r e s u l t s of [GJS 76] which use free BC , so for the remainder of t h i s s ection we take C to have free boundary conditions. As usual, Wick dots i n an integrand match the measure. In Theorem 2.2 of [GJS 76], f o r s u f f i c i e n t l y small h and y the one-point function corresponding to the pressure p(h,y) i s c o n t r o l l e d using a low temperature c l u s t e r expansion. I t follows from t h e i r r e s u l t s that |D;p(h,0) - (± — ) | = 0 (h 2) , 2 V s as perturbation theory and eqn. (1.1) would suggest. (In the notation of [GJS 76] the^ + vers i o n of the above equation i s X ^<j> ( x ) ^ = A £ + + 0(X 4) 2 ~ 2 where X = h and £ + = (8h) ^ i s the l o c a t i o n of the minimum occurring Therefore f o r - 1 2 4 1 2 -1 on the p o s i t i v e axis of h Ug ( h a L = ha ~ ^ a + (64h) ) i l l — + any a < — there i s a 5(a) > 0 such that a e [D p(Ti,0) , D„p(h,0)] ft 2 2 for a l l Ti < 6 (a) , and hence (see Figure 2 on p. 19) 121 V(Ti,a) = supCya - p(Ti,y)] = -p(Ti,0) , Ti < 6 (a) . y (1.2) In [GJS 76] an i n f i n i t e volume theory corresponding to the i n t e r a c t i o n 1 Tix 4 + / z "h2 x 3 and covariance C i s obtained. In §6 of [GJS 76] i t i s 1 2 shown that the perturbation ser i e s i n h for a generalized Schwinger function of t h i s theory i s asymptotic. The pressure i s not discussed, but i t i s straightforward to use the estimates of [GJS 76] to show that perturbation theory i s also asymptotic f o r p(Ti,0) , as we now show. Let t = Ti 1 2 and £ A( f c) = In [t 2:(f) 4:+/2t:<j) 3:] dy. Then p(t 2 , 0 ) = t 2£(t) , where t, (t) = l i m £.(t) . By the Fundamental Theorem of Calculus, C.Ct) - C A(s) D£. (x)dx = T7T [ <^2x:<t>4(A) : + /2: (j)3 (A) :\ dx , s,t > 0 A l A l Js / x,A (1.3) [x2:())4:+/2~ X : ? ' 3 : ] where <•> x,A [x2:<f>4:+/2 x:?'3:] dy. 122 By Theorem 6.2 of [GJS 76] , Tjr|<^2x:(j)4 (A) :+/2 : Y 3 ( A ) : ^ x ^ I i s 2 bounded uniformly i n A and small x >_ 0 . Taking the l i m i t A + R i n eqn. (1.3) and using Lebesgue's Dominated Convergence Theorem gives £(t) - c(s) = f <^2x:(j)4(0) :+/2: Y 3(0) :^> x dx , s , t > 0 j e , t small, J s where < • )>x = l i m (• >^ . Since ^2x: Y 4 (0)+/2:<j>3 (0) :^> x i s C°° i n A X ' small x > 0 by Theorem 6.1 of [GJS 76], i t follows that D£(t) = <2t:<D4(0) + /2: Y 3(0) :^> t t >_ 0 , t s m a l l . (1.4) In Theorem 6.1 of [GJS 76] i t i s also shown that the d e r i v a t i v e s of the r i g h t side of (1.4) at t = 0 are given by perturbation theory. Since 2 2 2 pCt ,0) - t £(t), the same i s true of p ( t ,0) . The odd order d e r i v a t i v e s 2 of p ( t ,0) vanish at t = 0 (because of the t - t symmetry of t,) , and the d e r i v a t i v e s with respect to t of order 2n at t = 0 correspond to d e r i v a t i v e s with respect to h of order n at h = 0 . I t follows that 0 0 2 + p(Ti,0) i s C i n small Ti = t , i n c l u d i n g h = 0 Since p(Ti,0) = hC(h 2) , D^ptCO) = C(0) = 0 . But by d e f i n i t i o n of y i n eqn. (3.1.6), y (—) = 0 , and hence -D..V(0,a) = -y (—) = 0 by i/i" 1 /8" (1.2). Moreover for N > 2 , - i.D^v(0,a) = - T D ^ P ^ 0 ' 0 ) i s given by — Nl 1 Nl 1 perturbation theory to be the sum of a l l connected n-loop graphs with 3 and 4 legged v e r t i c e s taking factors - / J and -1 r e s p e c t i v e l y and with legs corresponding to (-A+1) 1 , with the usual combinatoric factors and no closed loops. • 123 Theorem 1.4.5 shows that the asymptotic behaviour of the e f f e c t i v e 2 1 2 p o t e n t i a l corresponding to the double-well c l a s s i c a l p o t e n t i a l U (a) = (a --) U 8 i s quite d i f f e r e n t f o r |a| < — and |a| > — . For |a| > — Theorem /Hi J/8 / i " 1.4.3 gives an asymptotic one-particle i r r e d u c i b l e loop expansion f o r "V("h,a) with graphs having three- and four-legged v e r t i c e s taking factors -4a and -1 2 1 1 r e s p e c t i v e l y and l i n e s of mass 12a - - . However for |a| < — Theorem ^8 1.4.5 gives an asymptotic connected loop expansion for VCh,a) graphs having three and four-legged v e r t i c e s taking factors - 4 ( — ) = -^2 and -1 r e s p e c t i v e l y 1 2 1 and l i n e s of mass 12 (—) - - = 1 , i . e . , the vertex factors and l i n e s are ' r e -c a l c u l a t e d at an endpoint of B = [• , — ] . The asymptotics for V(h,a) are independent of a when I a l < — because for l a I < — and Th /I" ° ft s u f f i c i e n t l y small VCh,*) i s l i n e a r with slope zero near a^ and approaches the l i n e a r p o r t i o n of convU Q at an a Q-independent rate. The mechanism responsible f o r the f a c t that connected graphs rather than 1-PI graphs occur for |a| < — i s clear from the proof of Theorem /8 1.4.5: the supremum i n the d e f i n i t i o n of V i s attained at a point independent of a and small "h and the c a n c e l l a t i o n of reducible graphs c • provided by u(Ti,a) when a e B does not take place. Thus i n some sense Theorem 1.4.5 shows what i s l o s t by d e f i n i n g the e f f e c t i v e p o t e n t i a l to be l i n e a r when there i s a phase t r a n s i t i o n . I t i s an i n t e r e s t i n g open question whether V("n,a) can be defined i n B by an a n a l y t i c continuation from B , so that V i t s e l f might have a double-well s t r u c t u r e . 124 Bibliography [BC 83] Bender, C , Cooper, F., F a i l u r e of the naive loop expansion f o r the e f f e c t i v e p o t e n t i a l i n T ^ f i e l d theory when there i s "broken symmetry", Nucl. Phys. B 224, 403-426 (1983). [Brand 82] Brandenberger, R.H., Quantum F i e l d Theory Methods i n Cosmology, Harvard Un i v e r s i t y P r e p r i n t , HUTMP 82/B(22) (1982). [CW 73] Coleman, S., and Weinberg, E., Radiative Corrections as the Orig i n of Spontaneous Symmetry Breaking, Phys. Rev. D 1_, 1888-1910 (1973). 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