APPENDIX TO PART I
Hilbert Space Operators and Functional Integrals
Mathematical background material assumed throughout Part I is presented here. A basic knowledge of Hilbert space and self-adjoint operators through the spectral theorem is assumed; this material is summarized in the opening sections. The notions of trace class operators and nuclear spaces are devel• oped, leading to Gaussian measures over the Schwartz space fl" of tempered distributions. Basic properties of measures on Y' are presented, including the Bochner theorem and the Feynman-Kac formula.
A.I Bounded and Unbounded Operators on Hilbert Space
A Hilbert space Yf is a vector space over the complex numbers having a positive definite inner product < , ), and which is complete in the corre• sponding norm 11811 = <8,8)1/2. The inner product < , ) is a map from ff- x Yf to the scalars which is linear in its second argument and antilinear in its first argument. Positive definiteness is the statement 0::; <8,8) with equality only for 8 = o. Some useful elementary formulas relating the norm and inner product are:
118 + xl12 + 118 - xl12 = 211811 2 + 211xl12 (Parallelogram law), (A.U) <8, X) = H 118 + xl12 - 118 - xl12 (Polarization), (A. 1.2) + illx + Wll 2 - illx - i8112}
122 A.l Bounded and Unbounded Operators on Hilbert Space 123
<0, X) ~ 11011 Ilxll (Schwarz inequality), (A. 1.3) 110 + xii ~ 11011 + Ilxll (Triangle inequality), (A. 1.4) Completeness of Yf means that Cauchy sequences have limits. Thus if
then there is a 0 E Yf for which On ~ O. To give examples and fix notation we list as Hilbert spaces 12 (square summable sequences), L2 (square summable functions), and en (complex n-space). Vectors 0 and X are perpendicular (orthogonal) if <0, X) = 0, also written as 0.1 X. A set of vectors {O;}7=1 is orthonormal if
<0. 0.) = J.. = {O for i -# j, " J 'J 1 for i = j.
Proposition A.I.}. Let Xl' ... , Xn be orthonormal and let X E Yf. Then
(Bessel's inequality).
A Hilbert space Yf is separable if it has a countable dense subset. An orthonormal sequence (or net) {eJ7=1' n ~ 00 is a basis provided the finite linear combinations Lf=1 Aiei> N < 00 are dense in Yf. The number of basis elements n is the dimension of Yf, and is independent of the basis {e i }7=1. Also Yf is separable if and only if it has a finite or denumerable basis. Most Hilbert spaces encountered in mathematics and physics are separable. The notion of a net generalizes that of a sequence, and is necessary only for limiting processes in nonseparable Hilbert spaces.
Proposition A.1.2. Let {ei} be an orthonormal basis. If Ai is a sequence (or net) of scalars with L IAil2 < 00, then
(J = LAieiEYf
and every 8 E Yf is uniquely representable by such a series.
Proposition A.l.3. Let 0 E Yf and let % be a closed linear subspace of Yf. There is a unique vector XO E % which minimizes the distance from % to Yf.
For g £; Yf, let
g.1 = {8: Proposition A.l.4. Let % be a closed linear subspace of a Hilbert space Yf. Then each 8 E Yf can be uniquely written as 8 = 81 + 82 124 Appendix to Part I Hilbert Space Operators and Functional Integrals with 81 E %,82 E %~. Let [I' be any subset of Yf. Then [l'H is the smallest closed linear subspace of Yf containing [1'. In particular, % = %H. Let EE> denote a direct sum of Hilbert spaces. With the obvious identifica• tions, the proposition states that Yf = % EE> %~. A linear operator T from a Hilbert space Yf to a Hilbert space % is bounded if for some constant c, IIT811 :::;; c11811, all 8EYf. Then II T II is defined to be the smallest such constant, so that IIT811 :::;; II TIl 11811. The property that Tis bounded is equivalent to Tbeing continuous in the topology defined by the norm on Yf. Elementary properties of the norm are II T + 2SI1 :::;; II Til + 12111SII, IITSII:::;; IITIIIISII· Now use the notation of Proposition A. 1.4 and define an operator E = E,AI' by the formula E8 = 81, From the uniqueness part of Proposition A1.4, E is linear. Because 81 J.. 82, 1181 II :::;; 11811 and so II E II :::;; 1. However, also by uniqueness of the decomposi• tion, E81 = 81 so IIEII = 1, and E2 = E. E is called the orthogonal projection onto %. A linear operator mapping from a Hilbert space Yf to the scalars, C, is called a linear form. For any 8 E Yf, the map x ---+ (e, X> is a linear form, and it is continuous by the Schwarz inequality. The converse to this example is the Riesz representation theorem. Proposition A.I.5. Let f be a continuous linear form on a Hilbert space Yf. Then there is a unique 8 E Yf such that f(x) = (8, X>, The adjoint T* of a bounded operator T: Yf ---+ %, is defined by the formula (e, TX> = (T*8, X>· Note that T*: % ---+ Yf. Elementary properties of the adjoint of bounded operators are (ST)* = T*S*, A.1 Bounded and Unbounded Operators on Hilbert Space 125 (S + AT)* = S* + IT*, II TIl = II T*II = II T* Til 1/2, T** = T. An operator T is self-adjoint if T* = T. It is normal if T* T= TT*. It is isometric if II T811 = 11811 for all 8 E.Yt', which is equivalent to the relation T* T = I. Tis unitary if it is isometric and its range is the full Hilbert space, which is equivalent to T* T = TT* = I. Tis an orthogonal projection if and only if T= T* = T2. The spectral theorem states that a commuting family of normal operators can be simultaneously diagonalized. Theorem A.1.6. Let r be a commuting family of bounded normal operators on.Yt'. There is a measure space (X, dv) and a unitary map U: L 2(X, dv) -+.Yt' such that U* TU is a multiplication operator by a complex valued measurable function for each T E To In the theorem, eachf E Loo(X, dv) defines a bounded operator on L 2(X, dv) by the formula 8 -+ f8, where (f8)(x) = f(x)(J(x). The spectral theorem is often formulated in terms of spectral projections. Consider a single self-adjoint operator T and the corresponding real valued f ELoo(X, dv). Let e). be the characteristic function of the set {x:f(x) ~ A} and let E). = Ue).U*. Theorem A.l.7. In the above notation, T= SA dE). where the integral is a Riemann-Stieljes integral. Other types of convergence besides norm convergence are useful in the study of operators. Let 8n -+ 8 be a sequence of vectors in .Yt' and let An be a sequence of bounded operators. Then 8n -+ 8, weak convergence, if Also An -+ A, weak operator convergence, if For nonseparable Hilbert spaces, sequential convergence as above should be replaced by convergence through nets. An elementary but very useful fact is Proposition A.1.S. The unit sphere in .Yt' is compact in the weak topology. 126 Appendix to Part I Hilbert Space Operators and Functional Integrals A main tool in using the spectral theorem is the functional calculus. Theorem A.1.9. Let F be a bounded Borel measurable function of a real variable and let T be self-adjoint. We can define an operator F(T) so that T = f A dE). => F(T) = fF(A) dE)., U* TU = fELoo => U* F(T)U = F(f)ELoo. The mapping T --+ F(T) respects polynomial operations, composite functions and uniform limits of the function F. The most interesting operators are unbounded. However, bounded oper• ators are used to study unbounded ones. Let an operator A be defined on a linear subspace £»(A) (called the domain of A) of a Hilbert space Yf, and let A map £»(A) into Yf. In general, £»(A) is not closed and A is not bounded, but we suppose that £»(A) is dense in Yf, as this occurs in most examples. The graph ~(A) is the linear subspace of Yf E8 Yf defined by ~(A) = {(e, Ae): eE£»(A)} s; £» E8 Yf. A is closed if ~(A) is closed. A is closable if ~(A)- is the graph of an operator A -. In this case and A - is called the closure of A. A is an extension of B, written B s; A, if ~(B) s; ~(A). Thus A-is an extension of A. A criterion for A to be closable is that if I/In E .0J(A). and I/In --+ 0 and also Al/ln --+ e E Yf, then e = o. The adjoint A * of an unbounded operator A is defined by the formula Proposition A.t.tO. A * is closed. A is closable if and only if A * is densely defined. In this case, A - = A **. Definition. A is symmetric if As; A*, A is essentially self-adjoint if A- = A*, or equivalently A* = A**, and A is self-adjoint if A = A*. EXAMPLE. Let Yf = L2(X, dll) for some measure space (X, dll). Let f be a real valued measurable function on X, and let ~ = {gEL 2 : flf(xWlg(xW dll(X) < oo}. A.l Bounded and Unbounded Operators on Hilbert Space 127 Then the operator M = Mf defined by Mg(x) = f(x)g(x) (i.e., multiplication by f) with domain !'fi £ .if is self-adjoint. The spectral theorems A. 1.6, A.1.7, and A.1.9 extend to unbounded opera• tors. For unbounded operators, commutativity is defined in terms of the corresponding bounded functions, for example, the spectral projections. For a symmetric operator A, we define the defect space .%(z) as the eigen• space for A * for the eigenvalue z for any complex z. The dimension v(z) of .%(z) is called the defect index . .% and v are basic to the study of self• adjointness. Let ~(z) be the range of A - zI. Proposition A.1.II. The dimension v(x) takes on constant values in the upper half plane and constant values in the lower half plane. Theorem A.I.12. Let A be a symmetric operator. The following are equi• valent statements: (i) A is essentially self-adjoint. (ii) v(i) = v( - i) = O. (iii) ~(i) and ~( - i) are dense in Yf. Let A be a closed symmetric operator. The following are equivalent statements: (i) A = A*. (ii) v(i) = v( - i) = O. (iii) ~(i) = ~( - i) = Yf. Theorem A.1.l3. If the defect indices v( ± i) are both strictly positive then A has proper symmetric extensions. A necessary and sufficient condition for the existence of self-adjoint extensions is equality of the defect indices v(i) = v( - i). If A commutes with a complex conjugation on Yf or if A is semibounded, then v(i) = v( - i). In most concrete problems, the distinction between symmetric and self• adjoint operators is a question of boundary conditions. If A ~ A *, then A has " too many" boundary conditions and A * has too few. One example is the operator A = i(d/dx) with the domain of smooth func• tions compactly supported on the interval [0, 1], acting on Yf = L2[0, 1]. The defect indices are v( ± i) = 1, and the self-adjoint extensions of A can be parametrized by a real number (). Each choice of boundary conditions f(O) = eiOf(l) restricts A* to a self-adjoint operator. For the interval [0, 00], the defect indices are v(i) = 0, v( - i) = 1. Hence i(d/dx) has no self-adjoint extension on that interval. A third example is the transformation A = - A - k 2 r3. Here a classical particle moving in the potential - k 2 r 3 is repelled to infinity in a finite time. 128 Appendix to Part I Hilbert Space Operators and Functional Integrals The transformation A defined on smooth, compactly supported functions also requires a boundary condition at r = 00 for A* to specify whether A* is (or is not) self-adjoint. Lemma A.I.14. If A is an operator with range ~(A) = .Yf and A -1 is bounded, thenfor 161 sufficiently small, A + 61 has a bounded inverse also. PROOF. 00 (A + e1)-1 = A-V + eA-1)-1 = A-1 L (-eA-1)". n=O For e sufficiently small, so the series converges in the operator norm, and defines (A + e1)-1 as a bounded operator. The series of Lemma A.l.14 is called the Neumann series. We see that the set of complex numbers z {z: (A + Z)-1 exists as bounded operator} is an open subset of the complex plane C. The complement, denoted O'(A), is a closed set, called the spectrum of A. Now we put ourselves in the situation of the spectral theorem, with A normal and unitarily equivalent to (and without loss of generality identified with) a multiplication operator A=Mf , then IIAII = IlfllLoo' O'(A) = ess range f Also A is self-adjoint iff f is real valued; A is unitary iff f has values in the unit circle Izl = 1; and A is a projection iff f takes on the values 0, 1 only. Now let A be self-adjoint. By the functional calculus and the above remarks, eitA is unitary for t real. The map t --+ eitA == Ii defines a one-parameter group representation, which means that (a) t --+ 7; is continuous from the reals to the strong operator topology and (b) t --+ 7; is a group homomorphism: Proposition A.I.15. Let A be a self-adjoint operator with domain '@(A). Then eitA.@ = .@(A). For (}E.@(A), d 'A 'A -ell (} = iAelt (} dt ' with the difference quotient converging in norm in .Yf. A.2 Positive Operators and Bilinear Forms 129 Theorem A.l.16 (Stone). Let t ~ 7; be a one-parameter unitary group representation. Then there is a self-adjoint operator A (its generator) such that 7; = eilA. A.2 Positive Operators and Bilinear Forms An operator A is positive if °~ ~(A) x ~(A): t/I, e ~ Definition. A bilinear form t is a map t: yt x yt ~ C which is linear in the second factor and conjugate linear in the first factor. An example is t(e, X) = t*(e, x) = t(x, e)-, (t1 + IXt 2)(e, X) = t1(e, X) + IXt 2(e, X) for scalar IX. Also t2 is an extension of t1, written t1 c t2 if ~(td c ~(t2) and t 1 (e, X) = t2(0, X), for all e, xE~(td. A positive bilinear form t is closed if whenever both ~(t)3en ~ e and t(en - em, en - em) ~ 0, it then follows that eE~(t) and t(en - e, en - e) ~ o. t is closable if it has a closed extension. The closure of t is the smallest closed extensipn of t. Proposition. A.2.t. Let the bilinear form t be defined by a positive operator T, as in (A.2.1). Then t is closable. PROOF. Because 0 :-:; T and consequently 0 :-:; t, we obtain a Hilbert space .it; by completing fl&(t) = fl&(T) in the metric defined by the inner product (t + 1)(8, X) = t(8, X) + (8, X> = (8, X>Jt', 130 Appendix to Part I Hilbert Space Operators and Functional Integrals Since Ilell ~ IleIIJl"" J'l; convergence implies.Yl' convergence, and we identify J'l; with a linear subspace of.Yl'. Let t- be the bilinear form t-(e, X) = (e, X)JI", - (e, X) with domain £&(t-) = J'l;. Then t- is an extension of t. Completeness of J'l; implies that t- is closed, and so t is closable. We note that t- is the closure of t. Theorem A.2.2. Let t be a positive closed bilinear form with a dense domain ~(t) c Je. There is a positive self-adjoint operator T with domain ~(T) = {8E~(t): It(x, 8)1 ~ constllxll for all XE~(t)}, (A.2.2) and such that the bilinear form to, defined by to(x, 8) = on the domain ~(T) x ~(T) is a restriction oft. In other words, for all x, 8 E ~(T). Moreover, T is uniquely determined by (A.2.2) and (A.2.4). PROOF. We start by defining the bounded operator A = (T + 1)-1; from A we recover T. Define J'l; as in Proposition A.2.1, and for r/J E.Yl', we define the linear functional X..... (r/J, X)· This functional is continuous, for XE.Yl' and is thus also continuous as a functional on XEJ't,. Thus for some bounded operator A: .Yl' ..... J'f, ~ .Yl'. In fact, I(r/J, x)1 ~ 1Ir/JllllxlI ~ 1Ir/JIIIIXIlJl", so that IIAII ~ 1 as an operator on .Yl'. For r/JE.Yl', o ~ so that I ~ A-I, and A-I - I is positive. Also for XE.Yl', X..... (X, e)JI", = (X, Ar/J)JI", = (X, ¢i) is continuous in the .Yl' norm. Thus e E £&(T) and £&(T) ::;:) £&(A -1). Conversely, let r/J, 1/1 E £&(T) c J'l;. Then (r/J, 1/I)JI", = <¢i, (T + 1)1/1) = <¢i, A(T + I)1/I)JI", by the definition of A. Thus 1/1 E Range A = £&(A -1 ) and A-I = T + I. A.2 Positive Operators and Bilinear Forms 131 Since the domain ~(T) defined by (A.2.2) is dense (as it equals ~(A -1)), it follows that the bilinear form (A.2A) uniquely determines T on this domain, proving uniqueness. Definition. Let To be a positive densely defined operator on a Hilbert space Yt. Let to be the corresponding bilinear form. By Proposition A.2.1, to is closable. Let t be the closure of to. Let T be the self-adjoint operator constructed by Theorem A.2.2. Then T is called the Friedrichs extension of To. We see from (A.2.1), (A.2.2), and (A.2.4) that T actually is an extension of To. Notice that ~(t) =f. ~(T) However, it is possible to identify ~(t) as the domain of an operator. Proposition A.2.3. I n the above notation, ~(t) = ~(Tl/2). Definition. A bilinear form v with domain ~(v) is relatively form bounded with respect to a positive bilinear form t if ~(v) :::J ~(t) and there exist constants a and b such that Iv(8, 8)1 :s; at(8, 8) + bl18112 for all 8 E ~(t). The smallest such a is called the relative form bound of v. Proposition A.2.4. If v is relatively form bounded with respect to a positive bilinear form t, with a relative form bound a < 1, then t + v is closed and bounded from below. Remark. That t + v is bounded from below means that t + v + c is positive for some constant c (i.e., c is a constant multiple of the form 8, X ~ <8, X> defined by the identity operator on .YC'). In this case, t + v defines a self-adjoint operator. Suppose there are operators T and V defining t and v. In general, ~(T) n £&(V) = 0 and so T + V has a trivial domain of definition. Still, t + v defines a self-adjoint operator, which we would like to call T + V. If we do so, there is in general no 0 =f. 0 for which (T + V)8 = (TO) + (VO). Proposition A.2.S. Let HI and H 2 be self-adjoint operators with 0 :s; HI :s; H2. Then (H2 + a)-I :s; (HI + a)-I for any a> o. PROOF. For eEYf, set tPl = (Hl + a)-le, tP2 = (H2 + a)-leo Then «(H2 + a)-le, e) = (tP2, (Hl + a)tPl) = «(Hl + a)1/2tP2' (Hl + a) 1/2 tPl ) S (tP2, (Hl + a)tP2)1/2(tPt> (H1 + a)tP1)1/2. Here ¢2E~(H2) c ~((H2 + a)1/2) and because H1 ::; H2, ¢2E'lfl((H1 + a)1/2) also. 132 Appendix to Part I Hilbert Space Operators and Functional Integrals Substituting HI S H2 in the first factor above yields «(H2 + a)-Ie, e) s «(H2 + a)-Ie, e)I/2 «(HI + a)-Ie, e)I/2, to complete the proof. A.3 Trace Class Operators and Nuclear Spaces An important class of operators on Hilbert space Jf are those operators A which are in some sense finite dimensional. A has finite rank ifthe range 9t(A) is finite dimensional, or equivalently if the null space %(A) has finite codi• mension (i.e., Jf e %(A) = %~ is finite dimensional). A is compact (or com• pletely continuous) if either of the following equivalent conditions are satisfied: (i) A maps the unit ball in Jf into a subset of Jf with a compact closure. (ii) For any bounded, weakly convergent sequence (or net, for nonseparable Jf) of vectors en' the sequence Aen is norm convergent. The equivalence of (i) and (ii) follows from the weak compactness of the unit ball in .1f. The finite rank operators are compact, but not conversely. Between these two classes of "approximately finite dimensional" operators, there is a range of intermediate cases, similar to the Lp spaces of measure theory. Theorem A.3.t. For a positive self-adjoint operator A, the quantity 00 Tr A = L (ei , Ae), (A.3.1) i=1 where {eJ~l is an orthonormal basis, is independent of the basis {ei}~l. PROOF. Clearly Tr A < 00 only for bounded A. By the spectral theorem, if {ij} 1=1 is another orthonormal basis, then (ei , Ae;) = L (A 1/2 ei , jj) (jj, A 1/2 ei )· j The right side is a sum of positive terms, so a reordering of the summation gives Tr A = L \(A 1/2 ei , jj)\2 = L \(ei, A 1/2jj) \2 i,j i,i = L (jj, Ajj), j as claimed. Evidently Tr A s Tr B for A s B. We define the norm IIAllp = (Tr(A*A)P/2)1/P. The special cases p = I and p = 2 are called the trace and Hilbert-Schmidt norms, respectively. The p = 00 norm, II A 1100, is defined to be the operator norm of A. Because (A.3.I) is independent of the choice of basis, A.3 Trace Class Operators and Nuclear Spaces 133 Tr A = Tr U*AU (A.3.2) for any unitary operator U. Through linear combinations, Tr is extended to general (nonpositive) trace class operators A. Completion in the norm II' lip defines the class rp of P integrable trace operators. roo is the class of compact operators, and r pi ~ rp2 if PI :::;; P2' rp is a Banach space with the norm II' lip, and for P = 2, the Hilbert -Schmidt operators r 2 form a Hilbert space with the inner product :::; IIBlloo IIA 1/211~ = IIBlloo IIAII I . A general A has the polar decomposition A = IAI U, where 0:::;; IAI = (A*A)I/2 defines U. Then UU* = J, U*U = Proj (Kernal A) and lUI:::;; 1. Thus U B replaces B in the argument above. The inequality then follows for general p and q by interpolation, as with the ordinary Holder inequality. Taking linear combinations of the U's in (A.3.2), one can show that Tr AB = Tr BA for A E rp, BE rq, p-I + q-I = 1. For self-adjoint A, the spectral theorem allows an identification of these r p norms II A II p' In fact, a self-adjoint operator A is compact if and only if A has a complete set of eigenvectors ei with eigenvalues Ai --+ O. The spectral theorem maps A by a unitary equivalence onto the multiplication operators {AJ acting on 12, Then IIAllp = IIAillp = (1:IA;lP)I/P for p < 00 and IIAlloo = IIAilloo = sup IAJ i To illustrate these concepts, consider the harmonic oscillator operator d H = L t(Pv2 + Q;) - t v=l in d dimensions. In a basis ... , i } of Hermite functions, His diagonalized, {ei " d and d Lemma A.3.2. For any e > 0, (H + l)-(d+e) is trace class and (H + l)-(d+e)/2 is Hilbert-Schmidt. PROOF. We use the inequality 00 L (a + j)-b ~ const a-(b-l), j=O 134 Appendix to Part I Hilbert Space Operators and Functional Integrals valid for 1 < a, 1 < b, to bound successively each of the sums over i., 1 ::;; v ::;; d in the expression 00 00 ( d )-(d+e) Tr(H + I)-(d+e) = i~O ••• i~O V~1 (iv + 1) . Now we introduce a nested sequence of Hilbert spaces Yf,., Yf,.+1 C Yf,. C Yf,.-I' -00 < n < 00, with corresponding norms II' lin and inner products (', . )n' For e, t/J finite linear combinations of Hermite polynomials, (t/J, e)n = (t/J, (H + It e). This inner product and the corresponding norm Ileli n = (e, e)~/2 extend by completion to define a Hilbert space Yf,.. Because 0 ::;; H, Ileli n < Ilell n+1 and Yf,.+1 can be identified with a subset of Yf,.. Also £0 = L 2(Rd, dx). Finally, 00 00 Y = n Yf,. and Y' = U Yf,. n= -00 n= -co are the Schwartz spaces of rapidly decreasing smooth (test) functions and of slowly (at most polynomially) increasing distributions respectively. Let i = in+l be the identity map, considered as a map from Yf,.+1 to Yf,.. Lemma A.3.3. (H + 1)1/2 in+l is unitary, as a map from Yf,.+1 to Yf,.. PROOF. Let 8 E J'f,,+1' Then II (H + I)I/2i.+1 ell; = Proposition A.3.4. For D > 2d, the canonical injection map is trace class. PROOF. By definition, i8 = 8. Then i = (H + 1fD/2(H + I)D/2i. By Lemma A.3.3, the second factor, (H + It/2i, is unitary from J'f,,+D+1 to J'f", and by Lemma A.3.2 (or rather by the proof of Lemma A.3.2, applied to J'f" rather than to L2 = £0)' (H + I)-D/2 is trace class. The proposition shows that Y is a nuclear space and Y' is the dual of a nuclear space, in the sense of the following definition. Let £1 c £0 C £-1 be a nested triple of Hilbert spaces with an injection operator i, and suppose that A.3 Trace Class Operators and Nuclear Spaces 135 are trace class. Then i is also bounded, so that Thus is a bounded quadratic form on .Ye. The polarization identity combined with the Riesz representation theorem allows one to show that for a bounded operator A. We check that A is self-adjoint and positive. Because i is trace class as an operator from .Yeo to .Ye- 1 , and because by the above formula, A 1/2: .Ye- 1 -4.Yeo is unitary, A 1/2 is trace class as an operator on .Yeo. We define norms to give Hilbert spaces Yt", and then n= -00 is said to be nuclear, and its dual space is n= -00 Nested sequences of Hilbert spaces Yt" arise naturally in many problems. For example, if H is a positive self-adjoint operator, let with 118111 = II(H + 1)8110' Setting A = (H + I)-I we have .Yeoo = n !?fi(Hn) = Coo(H) n is the space of COO vectors for H. We conclude this section with a comparison of the trace p norms, and of a related L1 - Loo norm, to the operator norm. Proposition A.3.S. For p < q, IIAII = IIAlloo ::; IIAllq::; IIAllp, where II A II is the operator norm of A. PROOF. From the definition of IIAllp, it is no loss of generality to suppose that A is positive, with pure discrete spectrum. In this case, the proposition reduces to 136 Appendix to Part I Hilbert Space Operators and Functional Integrals elementary inequalities for Lp norms in the case of a discrete measure space each point of which has unit mass. Definition A.3.6. Let A be an operator on an L2 space, defined by a kernel a = a(x, y). Then we define IIA111,00 = sup f la(x, y)1 dy + sup f la(x, y)1 dx. x y Proposition A.3.7. With A as in the above definition, IIAII :-:; IIAlll,oo' PROOF. 1<1, Ag>1 = if f(xf a(x, y)g(y) dx dy I :-:; (J If(xWla(x, y)1 dx dy y/2 (J la(x, y)llg(xW dx dy y/2 :-:; IIAI11,001Ifllllgll· A.4 Gaussian Measures There is no (J finite Lebesgue (i.e., translation invariant) measure on an infinite dimensional space. A simple way to understand this phenomenon is to consider product measures 00 dv = TI dvi(XJ i=l Then convergence of the infinite product requires Vi(X;) = f dVi --+ 1 Xi sufficiently rapidly (so thatI IIn (Vi (X;)) I is finite). In the contrary case, dv is not (J finite (i.e., X is not a countable union of sets of finite dv measure) and dv is rather pathological. Thus if each Xi is the real line, at most a finite number of factors dVi can be translation invariant. We take Gaussian measures as the starting point for integration over infinite dimensional spaces. Other, non-Gaussian, measures are then obtained by perturbation, e.g., through the Feynman-Kac formula. The dual of a nuclear space (i.e., JIl'-oo) provides a convenient framework for studying Gaussian measures over infinite dimensional spaces. In this section, our Hilbert spaces ~, etc. are real. Each f E JIl'oo defines a linear (coordinate) functional 8 --+ 1(8) == 8(f) = on Yf- ro . For F a Borel measurable function on Rn, B a Borel set in Rn and fl' ... ,fn E Yfro , is called a Borel cylinder set in Yf- ro and is a Borel cylinder function on Yf- ro . If :F is a finite dimensional subspace of Yfro containing fl' ... ,f", then we say that the above cylinder sets and func• tions are based on :F, or are :F cylinder sets and :F cylinder functions, respectively. In the usual fashion of measure theory (closure under repeated monotone limits), the cylinder sets and functions define the classes of Borel measurable sets and functions for Yf- ro . For any measure dv on Yf- ro , the bilinear form 9 -+ f1(8)g(8) dv(8) is called the covariance of dv. It is defined on Yfro x Yfro and is posItive semidefinite, but in general can be infinite. We will impose as a condition on dv that the covariance is finite, continuous and (strictly) positive. Thus we define a covariance operator C to be a continuous positive linear map from Yfro to Yf- ro , or equivalently, a continuous bilinear form f, 9 -+ on Yfro x Yfro' and suppose that Cis nondegenerate «f, Cf)o = 0 only for 1 = 0). A measure dv defined on a nuclear space Yf-ro is said to be Gaussian if for each finite dimensional subspace:F c Yfro' the restriction of dv to :F cylinder sets is Gaussian. Theorem A.4.1. Let C be a covariance operator defined on a nuclear space Yfro . Then there is a unique Gaussian measure, which we denote d8c, defined on the dual space Yf- oo , and having C as its covariance operator. EXAMPLES. For C = (- L'1 + 1)-1 acting on L2 (~1), dOc is the Ornstein• Uhlenbeck measure defined on 9"(~1). We will use the same terminology with ~1 replaced by ~n. There are also conditional Ornstein-Uhlenbeck measures on 9"(~1) formed by restricting the values of the path (() E 9"(~1) at one or two points: These restrictions are well defined because the Ornstein-Uhlenbeck paths are almost everywhere continuous, as we show in Theorem A.4.4. The Ornstein-Uhlen• beck measure can also be defined in terms oftransition probabilities. This definition and its equivalence to the definition in terms of a covariance operator are discussed in Sections 3.1 and 3.2. Similarly, C = (_L'1f1 defines Wiener measure on 9"(~n) for 3 ::;:; n. For n = 1,2, however, (-L'1)-1 is unbounded as a bilinear form on 9' x 9' because k- 2 is not locally integrable, and so the approach must be modified. Let us 138 Appendix to Part I Hilbert Space Operators and Functional Integrals consider n = 1 and conditional Wiener measure with w(O) = O. This measure is defined by the statement that its covariance C has a kernel min(ltl, lsI), sign t = sign s, kernel C = { . (A.4.1) 0, otherWise, as an operator on L 2 . We note that C = (-~orl where ~o is the Friedrichs exten• sion of the Laplacian with vanishing Dirichlet data w(O) = 0 at the origin. Now C is a bounded bilinear form on g x g so Theorem A.4.1 applies. To define Wiener measure on paths taking values in a Hilbert space tS'o, we replace C above by C EB A, where A is a trace class operator on tS'o. With this definition, Theorem A.4.S below on the Holder continuity of paths remains valid for Hilbert space valued Wiener measure. The uniqueness is a consequence of our ability to compute Gaussian inte• grals of polynomials in terms of Feynman diagrams (i.e., integration by parts). Thus, writing p(e) = :Q(e): in the Wick ordered form, f p(e) dec = f :Q(e): dec = const term in :Q: = L vacuum diagrams of P. From this formula and the density of the polynomials in L2(dv) in the finite dimensional case (see Section 1.5), we see that two Gaussian measures with the same covariance agree on cylinder sets. However, since the latter generate all Borel sets under repeated monotone limits, we see that the measures must coincide, as claimed. Assuming the existence of Lebesgue measure de on £-00' then dec is defined by the formula None of the above three factors are well defined, so the proof uses a finite dimensional approximation, where the formula is meaningful, and a limit procedure to recover a measure on Yf- oo . Let:F be a finite dimensional subspace of Yfoo and let C:F be the restriction of the bilinear form C to:F x :F. Then C:F is the covariance of a unique finite dimensional Gaussian measure, d t C-I/2 e:F 1-1 dxcj' = (2n)dim:F/2 exp( -z Schmidt process gives rise to a unique Jfo-orthogonal decomposition Yf- ro = /F' EB /F.l. In this decomposition, the /F cylinder functions depend on /F only since these functions are defined in terms of the Jfo inner product between /F and Yf-ro • Thus integration over /F' is equivalent to integration of /F cylinder sets over Yf-ro • Next suppose that we have two finite dimensional subsets g c /F c Yfw Then the g cylinder sets are also /F cylinder sets. We must show that the measures dxc., and dxcgo agree on such cylinder sets. This is equivalent to the following lemma. Lemma A.4.2. Let g c /F be finite dimensional Hilbert spaces of dimen• sion m and n, respectively. Let C be a covariance operator on /F and let dxc = (2nrn/2 det C-l /2 exp( -! dXB = (2nrm/2 det C"jl/2 exp( -! PROOF. Both dXe restricted to C-cylinder sets define Gaussian measures on C'. A Gaussian measure is uniquely determined by its covariance. Both measures have covariance B = c r(C x C) == C, . In fact, B is the covariance of dX B by definition, while the restriction of dXe has as its covariance the restriction of C, i.e., B. Thus we have proved that (A.4.2) is a well-defined measure on cylinder sets. It is finitely additive and on restriction to /F, it is equivalent to Lebesgue measure. We now show that (A.4.2) is countably additive. Let S(r,j) = {O: IIOllj ~ r} be the sphere of radius r in Ytj. A measure dv is regular if v(E) = inf {v(Z): Z => E, Z weakly open}. Evidently (AA.2) is regular. Definition. Let dv be a finitely additive measure on Borel cylinder sets in Yf-w dv has vanishing measure at 00 if for each e > 0 there is a j = j(e) and an r = r(e) such that for any Borel cylinder set Y disjoint from S(r,j) we have v(Y) ~ e. If j can be chosen independently of e, then we say that dv has vanishing me'asure at 00 in Ytj. The definition contains the crucial hypothesis of the next lemma. Lemma A.4.3. Let dv be a finitely additive regular measure defined on Borel cylinder sets in Yf-ro • Suppose that dv has vanishing measure at infinity in Ytj. Then dv defines a countably additive measure on Ytj. 140 Appendix to Part I Hilbert Space Operators and Functional Integrals PROOF. Let Y = ur=l ¥ie be a Borel cylinder set, expressed as a disjoint union of Borel cylinder sets ¥ie. Let Yo = Jf-oo - Y, and then we must show that 00 L v(¥ie) = 1. k=O By finite additivity, k~O v(¥ie) = l~~ ktO v(¥ie) = l~ v(~o ¥ie) s 1. Because v is regular, the proof is completed by showing that whenever Zk is a weakly open cylinder set containing ¥ie. Let B > 0 be given. We use the fact that S(r,j) is a ball in a Hilbert space, and so is weakly compact. By weak compactness, there is a finite number of these sets, Zo, ... , Zv which form a cover for S(r,j). Let L Z = Jf-oo - U Zk' k=O Then Z is a cylinder set, disjoint from S(r,j) and so L) L B 2 v(Z) = v (Jf- oo ~ kVO Zk 2 1 - Jo v(Zd 2 1 - Jo v(Zd· Hence 00 L V(Zk) 2 1 - B, k=O and since this is true for all B ;:::: 0, dv is countably additive. If dv has vanishing measure at 00 in .1lj, define a measure dfl on .1lj by the formula dfl(Y) = dvext(Y) where dVext is the unique countably additive extension of dv to all Borel sets in H- oo . Note that dvext (.1lj) = 1 and dvext(Jf- oo ~ .1lj) = 0 so Jf-oo ~ .1lj is a set of measure zero. Then dfl is a countably additive measure on .1lj. PROOF OF THEOREM A.4.1. By general results of measure theory, we must show that (AA.2) is countably additive on Borel cylinder sets; it then has a unique countably additive extension to the class of all Borel sets. We verify the hypothesis of Lemma AA.3. For C to the continuous means that (AA.3) for some j, or equivalently C: .1lj ..... Jf_j boundedly. We choose j below to be slightly larger than that required for (AA.3) to hold. Let Z be a cylinder set based on § c Jfoo and suppose Z 1'1 S(r, -j) = 0. Since Z = Z + §l we have Z 1'1 (S(r, -j) + §l = 0 also. Thus we let S,fF = S(r, -j) + §l. S,fF is a cylinder set based on §. We assert v(Z) s f dec S B. Jff_~-S§ A.4 Gaussian Measures 141 To transfer the integral over £'-00 '" SfF to $", we write P~SfF = P~S(r,j), where P~ is the L 2-orthogonal projection of £'-00 onto $". Let Then I dec =(27trdim~/2 det Cil/2 I exp( -t = (2 7t) -dimfFI2 f exp (1IYI12)--- dy. y 2 For ye Y, Cil/2 ye$" '" P~S(r, -j) c £'-00 '" P~S(r, -j) c £'-00 '" S(r, -j), where we have used the identification $" = $' c £'-00. Moreover, S(r, -j) = (H + 1)1/2 S(r, 0), and so (H + Ifl /2 CJ)/y e £'-00 '" S(r, 0), so that Thus v(Z)::;; (27trdim~/2 Iy exp( _11~12) dy ::;; (27trdim~/2 I We evaluate this integral as a product of one-dimensional integrals, by choosing as a basis for$' eigenfunctions of the operator Thus with Tr~ denoting the ,no trace, restricted to $', v(Z) ::;; r-2 Tr~ A~. Now we show that Tr~ A~ is bounded independently of $', to complete the proof. From (A.4.3), there is a constant independent of$' such that This inequality extends to IjI, ee,no and so Cf(H + Iril2 is a bounded operator on ,no, with norm independent of $'. By the nuclear property of the Jfj norms, we increase j so that Clj(H + 1)-iI2 is Hilbert-Schmidt, with norm independent of $'. The required bounded on Tr~ A~ follows. 142 Appendix to Part I Hilbert Space Operators and Functional Integrals The space £'-00 is very large and for some purposes, such as for the proof of the Feynman-Kac formula, inconveniently large. In simple examples one finds a relation between continuity of the covariance operator (as an integral operator on L 2 (W)), and continuity of the paths. To be more explicit, let C = ( - ~ + I)-I. Then the kernel of C as an operator on L 2 (W) has 2 - n - e continuous derivatives, and dOc is supported in distributions with 1 - nl2 - e continuous derivatives. For n = 1, this is the rx. < 1 Holder continuity of the Wiener and Ornstein-Uhlenbeck paths. We now derive this n = 1 result, but the proof applies in principle for general n, to the result mentioned above. Theorem A.4.4 Let C = (-~ + I)-Ion L 2 (R 1 ). Then for any rx. < 1, dOc is supported in the space of Holder continuous paths of exponent rx.. Remark. To be slightly pedantic, the theorem states that almost everywhere distribution 0 E [Il' is equal as a distribution to a Holder continuous function. Such distributions are given by L\oC functions containing a unique Holder continuous function in their measure class; we then choose this unique rep• resentative and identify it with the path 0 E [Il'. Then omitting the comple• mentary set of measure zero, we have the Ornstein-Uhlenbeck measure supported on the set of Holder continuous paths. PROOF. Because the measure dec is defined on [/', it is necessary to regularize the paths. For fE[/, eE[/' is a smooth path. The idea is prove Holder continuity of f * e with bounds which are almost everywhere independent of f and convergent as f ---+,). Let 11!II:l = (f, Cf) as in Section A.3. By integration by parts (expansion into Wick ordered terms) the identity holds for any positive integer p and for some constant cpo Let };.(s) = f(h - s) denote reflection and translation by h. For any constant M, let X == X(M, t, h,f) be the event X(M, t, h,f) = {e: If * e(t) - f * e(t + h)1 > Mha}. Since f * e(t) = (e,J.), we can bound the measure Pr(X) == Ix dec of X as follows Pr(X) :-:; M-2Ph-2ap f (e,J. - h+h)2p dec :-:; CpM-2Ph-2apllJ. - J.+hll:~. Now (for the case C = ( - ~ + I)-I) A.4 Gaussian Measures 143 where O(h 1-£1) is independent of f In fact, we choose c(f) == + k2f£d3I-(k) IlL ~ IIIIIL . (A.4.5) 11(1 ~ 1 Therefore c(f) is bounded and convergent as 1--+15. Choose e1 so that 1 - e1 - 2IX == e2 > 0 and choose p so that pe 2 == 1 + e > O. We estimate Holder continuity of I * e at dyadic integers in (0, 1), getting Pr(X(M, kr", r",J)) ~ cpM-2Pc(f)2prn(1+£), 00 2"-1 L L Pr(X(M, kr", r",f)) ~ CpM-2Pc(f)2PC£. "=1 k=O Because 1* w is smooth, this bound applies at all points in (0, 1). Thus the Holder norm of 1* e is bounded by M on (0,1), except for paths e in a set of measure cpc.(c(f)/MfP. Now we let 1--+15 through a sequence fj chosen so that L~o c(fj+1 - fj)2P22jp is convergent. Then we can achieve Holder continuity of the paths t --+ (fj+l - fj) * w with Holder norm Mrj , for j large, again on the complement of a set of small measure O(M-2P). Thus for these paths, the IX Holder norms Ilfj*wll. ~ M are uniformly bounded and convergent as fj --+ 15. It follows that these paths ware themselves HOlder continuous with Ilwll. ~ M. Now a union over M gives Holder continuity almost everywhere. Theorem A.4.S. Let C be given by (A.4.1). Then for any IX PROOF. We modify the proof of Theorem A.4.4. To prove Holder continuity, it is only necessary to replace (A.4.4) and (A.4.5). For kernel C = c(s, t) given by (A.4.1), observe that where t5hC(O', t) = c(O', t) - c(O', 'C + h). Now and so IIC(ft - ft+h)k ~ hllIk, II ddO' C(ft - ft+h) L ~ II I k . These bounds imply 144 Appendix to Part I Hilbert Space Operators and Functional Integrals We choose a sequence fj = jg(jx) -+ 15, g a nonnegative compactly supported func• tion in Y; with integral 1. Then IIC(fj - t5)IIL~::;; 0(r1) and llad()' C(fj - 8) t ~ 0(1), and so Ilfj, - fjJ~l ::;; OUi1) + 0(jl1). Combining these bounds, we have 11ft - ft+hll~l ::;; c(f)20W-"), where c(f) is bounded and convergent as f -+ 15. This completes the proof ofH6lder continuity. Similarly, one shows Ilfj.tll~l ::;; O(ltl + r 1 ), from which it can be shown that w(O) = 0 almost everywhere. Measures dv on !/' are characterized by their inverse Fourier transform or generating functional as we will see in Section A.6. For the Gaussian case this result is the conse• quence of an explicit formula for S. Theorem A.4.6. The generating function S defined by a Gaussian measure dec is given by S = exp( -t PROOF. S(f) is cylinder function based on the one-dimensional space fF. Thus by definition the calculation of S(f) reduces to a one-dimensional Gaussian integral (2nA,f1/2 fexp( -! which is evaluated by power series expansion of the exponential eipx• A.5 The Lie Product Theorem The Lie product theorem states that e-(A+B) = lim (e-Alne-Bln)n. (A.S.1) n--+oo We present a version of this result which is sufficient to prove the Feynman• Kac formula. A.5 The Lie Product Theorem 145 Theorem A.S.1. Let A and B be lower semibounded operators on a Hilbert space£' and suppose that A + B is essentially self-adjoint on .0J(A) (\ .0J(B). Then (A.S.1) holds with convergence in the strong operator topology. Without loss of generality we suppose that A and B are positive. There is a semigroup t --+ e-t(A+lI) associated with the left-hand side of(A.5.1), defined for °~ t. The semigroup operators are contractions, i.e., Ile-/(A+B) II ~ I since °~ A + Band °~ t. The main steps in the proof are to construct a sequence of semigroups of contractions associated with the right-hand side of (A.S.I), and prove convergence of their generators to A + B. Then general arguments of functional analysis show that the semigroups also converge, which gives (A.5.1). Let and Definition. A contraction semigroup is a strongly continuous homomor• phism t --+ 1i from the positive reals [0, (0) to the contraction operators (i.e., operators of norm 1 or less) on it Hilbert space £', or more generally on a Banach space. Because of the semigroup multiplication property, 1'.7; = 1'.+1' it is sufficient to prove strong continuity at t = 0, which is equivalent to the condition lim/-+o 7;e = e for all e E £'. The strong derivative d - dt 7;11=0 = X is called the generator of 7;. By definition the domain of the strong derivative is {eE£': lim t-1 (7; - J)e exists}. 1-+0 Lemma A.S.2. F(t) is a strongly continuous function from [0, (0) to the contraction operators, F(O) = J and the strong derivative F'(O) contains -(A + B). PROOF. Let 8 E £&(A) n £&(B) = £&(A + B). Then F(t) - 18 = t-l[e-tA(e-tB8 _ 8) + (e- tA 8 - 8)] -+ -(A + B)8. t Lemma A.5.3. t --+ e-1Cn is a contraction semigroup, norm continuous in t. 146 Appendix to Part I Hilbert Space Operators and Functional Integrals PROOF. By power series expansion, Ile-tCn II ~ e-tn k~O (t:tli FG) II ~ 1, to prove the contraction property. The proof of continuity is similar. Theorem A.5.4. Let {Cn}::"=l be a sequence of bounded contraction semi• group generators. Let C be a positive operator, essentially self-adjoint on a domain £2 and suppose that CnO --+ CO for each 0 E£2. Then (A + Cnfl --+ (A + C)-l in the strong operator topology, for A > O. PROOF. The resolvents (A + Cnrl are defined by the Laplace transform(A + Cnrl = SO' e-t(Cn+J-) dt. They are uniformly bounded in norm, and so it is sufficient to prove convergence on a dense set of vectors. The orthogonal complement to (A + C)£& is a defect space for C ~ £&, and thus null. We choose (A + C)£& as the dense set. For '" = (A + C)8, 8E£&, we have II(A + Cnrl ", - (A + C)-I", II = II(A + Cnrl(A + Cn)8 + (A + Cnrl(C - Cn)8 - 811 = II(A + Cnrl(C -Cn )811 ~ rill (C -Cn)811, which tends to zero by hypothesis. Theorem A.5.5. Under the hypothesis of Theorem A.5.4, e-tCn --+ e-tC in the strong operator topology, with uniform convergence on bounded t intervals. PROOF. The unit sphere S c .Yf is weakly compact. Fix 8 E £&, and let G(I/J, t) = (I/J, e-tc(J), Gn(I/J, t) = (I/J, e-tCn 8). Because 8E £& and Cn8 -> C8 in norm, the functions Gn are equicontinuous in I/J, t. Thus by Ascoli's theorem, there is a subsequence nk (or subnet) with Gnk converging uniformly on S x {bounded t intervals} to some limit H(I/J, t). Because the Laplace transform of the semigroup is the resolvent, we have ( e-J-t H(I/J, t) dt = lim ( e-AtGnk(I/J, t) dt o k-oo 0 = lim (I/J, (A + Cny (8) k~oo = (I/J, (A + C)-18) = foo e-At G(8, t) dt. o By the uniqueness of Laplace transforms, H = G, implying the convergence of G. to G. Since the semigroups are uniformly bounded in norm, we also have conver- A.5 The Lie Product Theorem 147 gence for any IJ E;if. It follows that we have proved convergence of the semigroups in the weak operator topology, uniformly in bounded t intervals. By the semigroup property, the norms are also convergent. Then converges also. This proves strong convergence and completes the proof. PROOF OF THEOREM A.S.l. By Lemmas A.5.2 and A.5.3 the operators Cn are generators of contraction semigroups and converge strongly to A + B on EiJ = EiJ(A) n EiJ(B). By Theorems A.4.4 and A.4.5, the semigroups converge to e-t(A+B). We still must show that e-cn approximates the right-hand side of (A.5.l). Let IJEEiJ. Then by power series expansion We assert (A.5.2) Substituting this bound gives us the upper bound and the proof is complete. To establish (A. 5.2), we apply the Schwarz inequality to the left side obtaining The final identity results from expanding (k - n)2 = k(k - 1) + k - 2kn + n2. Each of the four terms can be summed explicitly after cancellation of k and k - 1 factors. 148 Appendix to Part I Hilbert Space Operators and Functional Integrals A.6 The Bochner-Minlos Theorem The Bochner-Minlos theorem characterizes the Fourier transform of measures. Let dv be a regular Borel measure on the Schwartz distribution spaces Y' and suppose Sdv = 1. The inverse Fourier transform of dv is by definition for fEY. S is also called the generating or characteristic functional of dv. Three basic properties of S are: (i) Continuity (in the topology of S). (ii) Positive definiteness: 0::::; IL=l ci cjS(f - jj) for all Ci E C, /; E Y, i = 1,2, ... , N, (iii) Normalization: S(0) = 1. Theorem A.6.1. If dv is a regular Borel measure on Y' with total weight 1, then its generating functional S satisfies (i)-(iii) above. Conversely, given a functional S defined on Y and satisfying (i)-(iii) above, then S is the inverse Fourier transform of a unique regular Borel measure dv with normalization Sdv = 1. PROOF. We prove the converse, assuming Bochner's theorem, which is the finite dimensional version of Theorem A.6.1. For any finite dimensional subspace ~ of Y, let S.9' = S~ ff. By Bochner's theorem, there is a normalized regular Borel measure dv.9' defined on ff, with inverse Fourier transform equal to S.9" As in Section A.4, we can also regard dv.9' as a measure defined on ff cylinder sets. Given two finite dimensional subspaces ffl c ff2 , we note that S.9'2 is an extension of S.9',. From this it follows that the measures dv.9' are consistent, in that dv.9', = dV.9'2 on ffl cylinder sets. It follows that a unique cylinder set measure dv is defined, agreeing with all dv.9" Moreover, dv has S as its inverse Fourier transform, by construction. It remains to show that dv can be extended to be a countably additive measure. To do this, it is sufficient to show that dv, as a cylinder set measure, is countably additive. To do this, we verify the hypothesis of Lemma A.4.3, using the continuity property (i). As in Section A.4, let S(r,j) be the sphere ofradiusj in J'tj. Let Z be an ff cylinder set with Z (\ S(r, - j) = 0, where rand j are subject to restrictions given below. Then v(Z) s f dv .9"-P§S(r,-j) s2 t,(l-exp ( -~IIP.9'ell~))dV.9'(e), where a = 2r- 2 In 2. Note that the Gaussian majorization is chosen because it has A.6 The Bochner-Minlos Theorem 149 a computable Fourier transform, as we now see. Let A = (P ,F(H + If) P,F fl. Then t, exp ( -~IIP,FOII:)) dv,F(O) = t exp ( -~ = c I I eXP(i v(Z) < 2c t (1 - S(f))exp ( - Let e be given. By the continuity of S, we can choose j and 15 so that if II fII:j S; 15 then IS(f) - 11 < e. If IlfII:j Z 15 then we use the bound IS(f)1 S; I dv = 1 which implies 11 - S(f)1 S; 2. Substituting these two inequalities in the two corre• sponding complementary regions in ff gives f v(Z) S; 2e + 4CD- 1 I,F IlfII:) exp ( - = 2e + 415- 1 o (a), where I,F II! II :jexp ( - = a Trj'"(pj'" A -1/2(H + I)-j A -1/2 P,F) = a Tr(P,F(H + Itipj'"(H + n-ip,F) after rearrangement (Tr AB = Tr BA) and substitution of the definition of A -1. Further rearrangement and the inequality P,F S; I yield the inequalities O(a) S; a Tr(P,F(H + If2j P,F) = a Tr(P,F(H + n-2)) = a Tr((H + IfjP,F(H + Iti) S; a Tr((H + 1)-2)), and so with j large enough, O(a) is bounded by a times a universal constant Tr((H + 1)-2)) which is independent of ff. Now choose r large enough so that O(a) = O(r-2 ) S; De, and then v(Z) S; 6e to complete the proof. 150 Appendix to Part I Hilbert Space Operators and Functional Integrals A.7 Stochastic Integrals Non-Gaussian measures arise as the solution of stochastic problems which are nonlinear on a classical level. The Feynman-Kac formula, developed in Section A.S and Chapter 3, provides one construction of such measures. Here we begin a second and distinct construction, based on the Ito theory of stochastic integrals and stochastic differential equations. This theory has very close connections with the theory of quantum fields, as we will see. In fact, from this point of view, there are two ideas on this section. One extends the unitary map f -+ w (I) n from .1t'-1 to the single particle subspace of Fock space, $' = Lz(dwe). In the extension, f = f(t) is replaced by a function f = f(t, w) depending on the path w in the past only. The other idea is that Leibnitz' rule for the derivative of products applies to Wick ordered (rather than to ordinary products), even in this more general context. Let dwc be the conditional Wiener measure with a covariance C given by (A.4.1), and let "fY' be the space of Holder continuous paths of exponent 0( < t for some fixed 0(. Let .I{ be the (J ring of Borel measurable sets in "fY' and let .I{,,;;,t be the (J ring generated by w(s), 0 :::; s :::; t. Definition. A nonanticipating function f=f(t,W)ELp(R+ x "fY') is jointly measurable in t and wand for each t, w -+ f(t, .) is .I{,,;;,/ measurable. Lp,na C Lp(R+ X "fY') is the space of non anticipating functions. In case w has values in a Hilbert space go, we define a nonanticipating operator to be an f as above with values in the Hilbert-Schmidt operators on go and a nonanticipating vector to be an f as above with values in go. We denote the spaces of such f as Lp,nao and Lp,naV' respectively. It is also convenient to consider the spaces L~~~ao and L;'~av offunctions whose restriction to bounded t intervals belong to Lp,nao or Lp,nav, respectively. The space .1t'-1, with norm IlfII-l = 11f'11~1 = IlfIIL using the definition C = ( - L\ot1 from (A.4.1). Thus wU') = f w(t)f'(t) dt = - f f(t) dw(t) (A.7.1) is defined as a unitary map from from f E Lz to the single particle subspace of Lz("fY', dw). The Ito stochastic integral is the same unitary map, generalized so thatfELz is replaced by fELz,na' Theorem A.7.1. Expression (A.7.1), as defined in the proof below, is an isometric map from Lz,na into Lz("fY', dw). PROOF. Iff is s step function, then the definition ofthe integral is elementary. We show that (A. 7.1) is isometricfor f a step function, and since such f are dense in L 2 , n., the theorem follows. A.7 Stochastic Integrals 151 Let 0 = to ~ tl ~ ... ~ tK+l = Tand suppose thatfis constant on each interval [tk' tHd and vanishes outside the interval [0, T). Then K ff(t, w) dw(t) = k~O f(tk, W)(W(tk+1) - w(td)· (A.7.2) Note that f(tk , w) is .I{:S;t. measurable and so (A.7.2) is a single particle vector in the interval [tk' tHl). It is for this reason that (A.7.2) defines an isometry. Similarly, it is essential that f is evaluated at the lower end in the Riemann sum; this requirement is part ofthe definition of the integral in (A.7.2). Now f f(t, w) dw(t) 112 II L2(if) K = L Ilf(tk, W)(W(tk+l) - w(tk))II£'(if") k=O K + 2 L (f(tj, w)(w(tj+1) - w(tj)),f(tk, W)(W(tk+l) - W(tk)))L2(if). O:s;j The second sum is zero because W(tk+l) - w(tk) is orthogonal to the.l{ :S;t. measur• able functions. Similarly, in the first sum, the first factor f(tk, w) is independent of W(tHl) - w(tk). The integral ofthe second factor can be evaluated, giving ffdw(t) 112 = f Ilf(tk,w)II£,(if)(tk+l- tk) II L2(if') k=O = f Ilf(t, w)IILif") dt = IlfIIL2 ••• • This completes the proof. With f E Lf,"nao, 9 E L~o,cnav and Xo E 80 , we can define the stochastic indefinite integral t t x(t, w) = Xo + f f dw(t) + f 9 dt. o 0 The corresponding differential dx is defined by the formula x = Jdx, or dx(t, w) = f(t, w) dw(t) + g(t, w) dt. (A.7.3) Under the above hypothesis, x E L~o,cnav. If 9 E Lf,cnav then x E Lf,cnao also. Com• posite functions are also defined. If u is a measurable function defined on [0, 00) x 8 0 with at most linear growth at infinity in 80 , uniformly on bounded t intervals, then loc ·f loc U (t ,x) E Lp,nav 1 XE Lp,nav. In order to work with stochastic integrals, we need to extend the elementary operations of calculus to them. The chain rule for derivatives is known as Ito's lemma. Its unusual form, and specifically the occurrence of a V 2 U term, reflect the fact that Wick ordered products rather than ordinary products satisfy Leibnitz' rule for derivatives: d:ab: = :(da) b: + :adb:. See Section 9.1. 152 Appendix to Part I Hilbert Space Operators and Functional Integrals Theorem A.7.2. Let x be the stochastic integral as defined by (A.7.3). Let x take values in ,go and let u be a function defined on [0, (0) x ,go taking values in a Hilbert space iF. Suppose that u and its derivatives up to order two in t and order three in go are continuous and bounded globally as functions on [0, (0) x go. Then t, OJ -4 u(t, x(t, OJ)) is an iF valued stochastic integral and its differential is du = Remark. The meaning of (A. 7.4) is that u(t, OJ) - u(O, OJ) = f~ OJ(m) = OJ(m) 2" ' The next lemma is basic. Lemma A.7.3. With convergence in the norm of Lz ('if/', dOJd, zn lim I 1<5;;-OJ(m)IP = 0 for 2 < p, n-oo m=l 2 n lim L 1t5;;-OJ(m)I Z = 1. n-co m=l PROOF. For 2 < p, t~ll<5,;-w(m)lpL s 2"111<5,;-w(1)IPIILz by the triangle inequality in L2(1f/", dwcl. However, since the scaling w(a) --+ tl/2w(a) does not change the covariance and hence the measure dwc, we can evaluate the right-hand side above as cp 2"2-P"/2 --+ 0, where cp is independent of n. For p = 2, one can evaluate the Gaussian integral Using the fact that the increments <5- w(m) of a Brownian path are independent, we have 2n I II 1<5';-W(m) 12 - T" III, = 2"11<5; w(1)1 2 - TnIIL· m=l A.S Stochastic Differential Equations 153 Using S(f - Sf)2 dx ::; SP dx, we bound the above by n 2 Ilb;w(W IIi2 -> 0 as in the case 2 < p considered above. PROOF OF THEOREM A.7.2 FOR A SPECIAL CASE. We only consider the case Co = :Jji = !?l, as we suppose x(t, w) == w(t). For simplicity, we take t = 1 in (A.7.5). Then u(t, w) - u(O, w) = An + Bn + Cn, where 2" 1 ) A = L Vu (m-- - w(m-l) b-wm • n m=l 2"' n 2" 1 ) 1 )) Bn = ( m~l U (m~' - w(m) - U (m~' - w(m - 1) - An, C-= L2" u (m- w(m) ) -u (m-- - 1 w(m) ) . n m=l 2n' 2n ' Then 1 An -> f 10U Cn -> f - dt. o at Now Bn does not converge to zero, as might be expected. Expanding u in a Taylor series to second order, with a remainder O(b nw(m))3, we see from Lemma A.7.3 that 1 Bn -> 2 fl TrsJ*V2uJ o to complete the proof. A.S Stochastic Differential Equations We consider equations of the form dx(t, w) = u(t, x(t, w)) dw(t) + v(t, x(t, w)) dt, (A.S.1) where u and v are Lipschitz continuous and the solution x = x(t, w)EL2•na• Such equations arise as perturbations of classical (nonstochastic) ordinary differential equations x = v(t, x) by coupling to a random driving term u dw(t). The existence of solutions is proved as in the classical case by Picard iterations. The solution x(t, w) defines a Markov process with continuous sample paths. The Markov process 154 Appendix to Part I Hilbert Space Operators and Functional Integrals defines a semigroup and an infinitesimal generator X, which is a variable coefficient Laplacian d2 X = lu(xW dx 2 + lower order. This theory can be generalized by allowing Wiener measure to have values in a Hilbert space 0"0' Then u and v map 0"0 to itself and xEL2,nav' Consider the case 0"0 = gr. Then X becomes a variable coefficient Laplacian on gr. By gluing together coordinate patches, one can also solve stochastic differential equations on a manifold An and study the corresponding generators X, which are Laplacians acting on functions on An. For an extensive treatment of stochastic integrals and differential equations, see McKean [1969] and Fried• man [1975]. As is evident from the rest of this book, one can pose the question of stochastic partial differential equations. Such theories would require infinite renormalization, and in view of their evident importance, are in need of considerable development. Theorem A.S.I. Let u and v be measurable functions on R+ x 0"0 which satisfy lu(t, x) - u(t, x/)1 ::; constlx - x'I, Iv(t, x) - v(t, x/)1 ::; constlx - x'I, lu(t, x)1 + Iv(t, x)1 ::; const(l + lxI), uniformly on bounded t intervals. Then (A.8.1) has a solution x E L~,cnav which is determined uniquely by its initial data x(O, w) == Xo E 0"0' PROOF. Since the existence and uniqueness proofs are similar and both follow the standard Picard proof for ordinary differential equations, we present the existence proof only. For 1 ::; n, we defined inductively t t xn(t, w) = Xo + f u(t, Xn-l (t, w) dw(t) + f v(t, Xn- 1 (t, w)) dt o 0 and assert xn E L~o.cnav' and 2 (const t)n Ilxn(t, .) - X - 1 (t, .) IIL (1f') ::; , ' (A.8.2) n 2 n. uniformly on bounded t intervals. For n = 1, we have T Ilxl (T) - Xo 11i,(if') ::; 211 f u(t, xo) dw 11i,(if')' o By Theorem A.7.1, the first term is dominated by T 2 f Ilu(t, xoW dt ::; const T Ilxo II~o' o A.8 Stochastic Differential Equations 155 The second term has a trivial w dependence and is bounded by const T211xo IliD :::;; const Tllxo IliD with constants which are T dependent, but bounded uniformly for bounded T. Assuming (A.8.2) for n - 1 with 2 :::;; n, we have T Ilxn(T) - Xn - I (T) IIL1f") :::;; const II f IXn - 1 (t) - xn- 2(t)1 dw(t) 11I,(1f") o T + const II f IXn- 1 (t) - xn-2(t)1 dt Ili 2(1f")' o The first term is bounded by T (const t)n-I (const T)n const f dt = . o (n - 1)! n! by the inductive assumption and Theorem A.7.1. By a Schwarz inequality in the t integration, the second term is bounded by const II (( IXn - 1 (t) - xn- 2(tW dtY2 [(1f") = const Ilxn - 1 - Xn 11I,([o.Tl x 1f") T (const t)n-l :::;; const f0 (n _ 1)! dt (const T)n n! again with constants uniform in bounded Tintervals. These estimates establish the inductive assumption (A.8.2) for all n, and show that Xn E L~.cnav' In particular, xn -+ xEL~.cnav· We set dy = u(t, x) dw + v(t, x) dt. }Jy estimates similar to those given above, lIy(T) - xn(T)IIL'ir) ~ const Ilx - xnllh:.,,-+O so that y == x. This completes the proof.