On the Preservation of Quasilocality by the Integration-Out Transformation
aa r X i v : . [ m a t h - ph ] D ec On the Preservation of Quasilocality by theIntegration-Out Transformation
T. Tlas
Abstract
We demonstrate that the integration-out step of the renormalizationgroup transformation preserves the quasilocality of the effective action.This is shown in the case of a single, real, scalar field on a torus, butthe proof holds more generally. The main result can be thought ofas showing the flow invariance of the quasilocal subset under the flowgenerated by the Polchinski equation.
One of the deepest ideas in physics is the renormalization group. This isthe concept that one can study a complicated system by analyzing it onescale at a time. Intuitively, it is clear that for this procedure to be practical,one should deal with systems which are at least approximately local. Intechnical terms, the effective actions must admit some form of derivativeexpansion. In other words, when one implements the renormalization group,a standard requirement is that the effective actions considered are requiredto be “quasilocal” using the terminology of [1]. Roughly speaking, (we willgive precise definitions later) these are functionals of the form S [ φ ] = ∞ X n =0 Z dp . . . dp n δ ( p + · · · + p n ) G ( p , . . . , p n ) φ ( p ) . . . φ ( p n ) . The goal of this paper is to prove rigorously that the integration-out stepof the renormalization group preserves quasilocality. We shall demonstratethis fact in a specific realization, notably for a theory of a single real scalarfield. However, it will be clear that the proofs can be easily generalized toother set-ups.As will be apparent below, one way to view the main result of this work isas a fact in the theory of viability [2, 3]. This is a vast subject in whicha great amount of work has been expended. However, the greatest major-ity of the established results pertain to conditions one puts on the full flowcorresponding to the evolution equation (typically, a form of tangency ofthe flow to the set is considered). There are only a few results [4, 5] whichupply sufficient conditions on the generator, and even then, they requireinformation on the behaviour of the generator away from the set which isto be invariant. Below, a viability fact is proven under the assumption thatthe linear part of the equation preserves the set, while the nonlinear partof the generator maps the set to itself (see the discussion at the end of thepaper for more details).As mentioned above, we will deal with the theory of a real scalar field φ .To make things rigorous, we need both an infrared and an ultraviolet cut-off. We shall implement the infrared cutoff by putting periodic boundaryconditions, i.e. we will consider our space of fields to be L functions on the d -dimensional torus T d = S × S × · · · × S | {z } d times . Of course, we have the usualFourier correspondence φ ( x ) ⇐⇒ a n = Z T dL φ ( x ) e − πinx dx { a n } n ∈ Z d ⇐⇒ φ ( x ) = X n ∈ Z d a n e πinx , where nx = n x + · · · + n d x d , and the measure on T d is normalized so thatit has unit volume. Note that the fact that φ is real implies that a − n = a n for every n ∈ Z d . Moreover, φ ∈ L ( T d ) ⇐⇒ P n ∈ Z d | a n | < ∞ . We shalldenote the space of the Fourier coefficients by l ( Z d ).Let Λ , Λ ′ be positive numbers with Λ > Λ ′ . Consider the following familiesof bilinear forms on the space of fields K Λ ( { a n } , { b n } ) = 12 X n ∈ Z d e − n n + 1 a n b n K Λ , Λ ′ ( { a n } , { b n } ) = 12 X n ∈ Z d e − n − e − n ′ n + 1 a n b n . The exponentials in the formulas above implement the ultraviolet cutoffs.The exact expressions chosen here are not important. Instead of e − x , wecould have chosen any other even, nonnegative function α ( x ) on R , such that α (0) = 1, α is monotone decreasing for x ≥ ∞ , as well as α is differentiable at 0 (this last condition will be neededlater). Moreover, we’ve selected the propagator of the un-cutoff theory tobe that of the massive field n +1 . We could have easily replaced it with any They will be contained in a much smaller space after we impose the ultraviolet cutoffs. onnegative sequence which doesn’t grow too fast.The two bilinear forms above are trivially seen to be positive definite. More-over, these two forms correspond to two families of diagonal operators on l ( Z d ) which are trace-class. Therefore, by standard results [6], the bilinearforms above define two families of Gaussian measures on l ( Z d ) whose co-variances are the given expressions. We shall denote these measures by µ Λ and µ Λ , Λ ′ . Now, while we have defined these to be measures on the spaceof L ( T d ) functions, they are in fact supported on a much smaller space.Again, by standard results, we can easily see that all these measures aresupported on the subspace of elements { a n } n ∈ Z d ∈ l ( Z d ) for which ||| a n ||| ≡ X n ∈ Z d e √ n + ··· + n d | a n | < ∞ . In particular, note that this implies that the measures are supported on acertain subspace of smooth functions on the torus. This subspace of l ( Z d )topologized by the norm ||| · ||| shall be denoted by V . It follows that all ourmeasures can be thought to have a common domain which is V .We can now describe our space of effective actions. Consider first the space L ( µ Λ ). As is well-known, any element S of this space can be written as S = X n ,...,n k ∈ Z d ,k =0 , ,... G ( n , . . . , n k ) : a n . . . a n k : Λ (1)where : a n . . . a n k : Λ is the Wick product of a n , . . . , a n k (with respect tothe measure µ Λ ) and the sum converges in L ( µ Λ ). Incidentally, note thatwhile the Wick products are an orthogonal basis of L ( µ Λ ) they are notnormalized and thus the L norm of the element above is given by X n ,...,n k ,k =0 , ,... k ! e − n n + 1 . . . e − n k Λ n k + 1 | G ( n , . . . , n k ) | . (2)Note that any functional of the form Z T d M Y m =0 (cid:18) Y i ,...,i m =1 ,...,d (cid:0) ∂ mx i ,...,x im φ (cid:1) α i ,...,im (cid:19) ( x ) dx, (3) In fact, the supports are significantly smaller still, see e.g. example 2.3.6 in [6].Moreover, the supports actually depend on the parameters Λ. However, the space thatwe chose is sufficient for our purposes. For the reader’s convenience, the definition of the Wick product, as well as a fewuseful formulae, are given in Appendix A. One needs to be careful here, as the a n ’s are not all independent, since a n = a − n .However, the formula for the L norm remains valid. here all α i ,...,i n ∈ N is in fact a well-defined element of L ( µ Λ ). This is aconsequence of the inequality | ∂ mx i ,...,x im φ ( x ) | . X n ∈ Z d ( n + . . . n d ) m | a n | . ||| a n ||| , (4)where . means less than a constant multiple of. It thus follows that thesquare of the expression above is bounded from above by a constant multipleof ||| a n ||| A for some integer A . As follows by Fernique’s theorem, each suchexpression is integrable, and we have our claim.We can now make an important
Definition.
A finite linear combination of functionals of the form (3) iscalled a local functional. An element in the closure in L ( µ Λ ) of the set oflocal functionals is said to be quasilocal. The set of all quasilocal functionalswill be denoted by Q (Λ) . The next proposition shows that we could have defined the notion of quasilo-cality differently.
Proposition.
An element of L ( µ Λ ) is quasilocal if and only if in (1) wehave that n + · · · + n k = 0 = ⇒ G ( n , . . . , n k ) = 0 , ∀ n , . . . n k ∈ Z d , k ∈ N (5) Proof.
The only if direction is a consequence of the fact that every localfunctional satisfies (5). This is done by a direct computation using the factsthat all the fields and their derivatives are multiplied at the same point in alocal functional, and that the propagator between a n and a m (which entersin the Wick product) vanishes unless n = − m . The fact that a limit of func-tionals satisfying (5), itself satisfies (5), follows at once from the expressionfor the L norm given in (2).To see the other direction, first note that for a fixed k , if G ( n , . . . , n k ) is apolynomial (in n , . . . , n k ) then, a direct computation shows that X n ,...,n k ∈ Z d G ( n , . . . , n k ) : a n . . . a n k : Λ is a local functional. Thus, we will be done if we show that polynomialsin n , . . . , n k are dense in the weighted l ( Z d × · · · × Z d | {z } k times ), with the weightgiven by the expression before | G | in (2). However, this is a well-known factabout the density in L of polynomials for measures with sub-exponentialtails (see e.g. [7]). Hence, the proof is complete. Twice the sum of all the α i ,...,i m works. ow, we want to introduce the integration-out operator of the renormaliza-tion group. This is meant to to act on an element of L ( µ Λ ), produce anelement of L ( µ Λ ′ ), and be given by the following formula I Λ , Λ ′ ( S )[ ψ ] = − ln (cid:18) Z e − S [ φ + ψ ] dµ Λ , Λ ′ [ φ ] (cid:19) . (6)Clearly the expression above cannot be defined on all of Q (Λ) and one needsto restrict the space somewhat. Let b ∈ R and denote by L b ( µ Λ ) the subsetof all elements in L ( µ Λ ) which are bounded from below by b . A naturalrestriction to make (6) well-defined would be to consider only elements in Q ∩ L b ( µ Λ ). However, for technical reasons, it turns out to be convenient torestrict the space a little further as is given in the following Definition.
Let Q b (Λ) be the closure in L ( µ Λ ) of the bounded elements in Q ∩ L b ( µ Λ ) . We show in Appendix B that Q b is rich enough to include every local func-tional whose integrand (defined there) is bounded from below by b .We can now state our main result which is the following Theorem. I Λ , Λ ′ is a continuous map from Q b (Λ) to Q b (Λ ′ ) .Proof. The fact that (6) is well defined on Q b (Λ) (in fact on L b ( µ Λ )) andthat the lower bound is preserved by I Λ , Λ ′ is immediate. Let I ′ Λ , Λ ′ ( S )[ ψ ] begiven by Z S [ φ + ψ ] dµ Λ , Λ ′ [ φ ] . By Jensen’s inequality, we have that I Λ , Λ ′ ( S )[ ψ ] ≤ I ′ Λ , Λ ′ ( S )[ ψ ] . Let f + stand for the positive part of a function. We then have that sZ (cid:18) I Λ , Λ ′ ( S )[ ψ ] (cid:19) dµ Λ ′ [ ψ ] ≤ | b | + sZ (cid:18)(cid:16) I Λ , Λ ′ ( S )[ ψ ] (cid:17) + (cid:19) dµ Λ ′ [ ψ ] ≤| b | + sZ (cid:18)(cid:16) I ′ Λ , Λ ′ ( S )[ ψ ] (cid:17) + (cid:19) dµ Λ ′ [ ψ ] ≤ | b | + sZ (cid:18) I ′ Λ , Λ ′ ( S )[ ψ ] (cid:19) dµ Λ ′ [ ψ ] . Since K Λ ′ ( { a n } , { b n } ) + K Λ , Λ ′ ( { a n } , { b n } ) = K Λ ( { a n } , { b n } ), we have atonce that for any L ( µ Λ ) function F the following equality The notation is due to the fact that I ′ Λ , Λ ′ is a formal linearization of the I Λ , Λ ′ . F [ χ ] dµ Λ [ χ ] = Z F [ φ + ψ ] dµ Λ , Λ ′ [ φ ] dµ Λ ′ [ ψ ]holds. Therefore, again by Jensen, we have the well-known inequality
Z (cid:18) I ′ Λ , Λ ′ ( S )[ ψ ] (cid:19) dµ Λ ′ [ ψ ] ≤ Z Z (cid:16) S [ φ + ψ ] (cid:17) dµ Λ , Λ ′ [ φ ] dµ Λ ′ [ ψ ]= Z S [ χ ] dµ Λ [ χ ] . (7)Putting everything together, we get that sZ (cid:18) I Λ , Λ ′ ( S )[ ψ ] (cid:19) dµ Λ ′ [ ψ ] ≤ | b | + sZ S [ χ ] dµ Λ [ χ ] . It follows that I Λ , Λ ′ maps L b ( µ Λ ) into L b ( µ Λ ′ ). To show that it does socontinuously, let { S n } ∞ n =1 be a sequence in L b ( µ Λ ) converging to S . First,note that for any constant c , we have I Λ , Λ ′ ( S + c )[ ψ ] = I Λ , Λ ′ ( S )[ ψ ] + c. Using this, and the equivalences I Λ , Λ ′ ( S n ) → I Λ , Λ ′ ( S ) ⇐⇒ I Λ , Λ ′ ( S n ) − b → I Λ , Λ ′ ( S ) − b ⇐⇒ I Λ , Λ ′ ( S n − b ) → I Λ , Λ ′ ( S − b ) , we can assume that b = 0 in what follows, i.e. that all our functions are non-negative. Now, using (7) we have immediately that I ′ Λ , Λ ′ ( S n ) → I ′ Λ , Λ ′ ( S ) in L b ( µ Λ ′ ). Also, using (7), the fact that | e − x − e − y | ≤ | x − y | for nonnegative x and y , and Jensen yet again we have that Z (cid:18) e −I Λ , Λ ′ ( S n )[ ψ ] − e −I Λ , Λ ′ ( S )[ ψ ] (cid:19) dµ Λ ′ [ ψ ] ≤ Z Z (cid:18) e − S n [ φ + ψ ] − e − S [ φ + ψ ] (cid:19) dµ Λ , Λ ′ [ φ ] dµ Λ ′ [ ψ ] ≤ Z (cid:16) S n [ χ ] − S [ χ ] (cid:17) dµ Λ [ χ ] , from which we conclude that e −I Λ , Λ ′ ( S n ) → e −I Λ , Λ ′ ( S ) in L ( µ Λ ′ ). Passingto subsequences, we get a subsequence { S n l } ∞ l =1 such that I Λ , Λ ′ ( S n l ) and This is of course the fundamental equation behind the renormalization group. ′ Λ , Λ ′ ( S n l ) converge almost everywhere to I Λ , Λ ′ ( S ) and I ′ Λ , Λ ′ ( S ) respec-tively. Combining this with the fact that I Λ , Λ ′ ( S n l ) ≤ I ′ Λ , Λ ′ ( S n l ) and that I ′ Λ , Λ ′ ( S n l ) → I ′ Λ , Λ ′ ( S ) in L ( µ Λ ′ ), we get that I Λ , Λ ′ ( S n l ) → I Λ , Λ ′ ( S ) in L ( µ Λ ′ ). Thus, we get a subsequence of {I Λ , Λ ′ ( S n ) } ∞ n =1 which converges to I Λ , Λ ′ ( S ). Since we can repeat this argument for any subsequence of theoriginal sequence, it follows that I Λ , Λ ′ ( S n ) → I Λ , Λ ′ ( S ), and we have conti-nuity.It remains to show that a quasilocal S is mapped to a quasilocal one. In viewof the definition of Q b (Λ) and the above continuity result, we can assumethat S is bounded. Now let F stand for a finite subset of Z d such that( n , . . . , n d ) ∈ F ⇐⇒ − ( n , . . . , n d ) ∈ F . Denote by P F the projectionof V onto those components corresponding to F (i.e. the map which setsall the a n ’s to zero unless n ∈ F ). For an L ( µ Λ ) function g , denote by g F the cylindrical approximation obtained from g by integrating out thecomponents “not in F ”. More precisely, g F [ φ ] = Z g [ P F φ + ( I − P F ) ψ ] dµ Λ [ ψ ] , where I stands for the identity map. As is well-known, g F → g in L ( µ Λ )as F ↑ Z d . Denote by µ Λ ,F the pushforward of µ Λ by P F . It follows thatif g ( n ) → g in L ( µ Λ ), then g ( n ) F → g F in L ( µ Λ ,F ). Needless to say, g F canbe considered to be a function of only finitely many variables (those in theimage of P F ).Applying this to S , we have that S F is bounded (with the lower bound being b ), that S F → S in L ( µ Λ ), and, recalling that S is quasilocal, that if S = X n + ··· + n k =0 ,k =0 , ,... G ( n , . . . , n k ) : a n . . . a n k : Λ , then S F = X n + ··· + n k =0 ,k =0 , ,... G ( n , . . . , n k ) (cid:16) : a n . . . a n k : Λ (cid:17) F , with the latter sum converging in L ( µ Λ ,F ). Note that if we consdier S F to be a function of finitely many variables then, by virtue of it being in L ( µ Λ ,F ), it should also have an expansion of the form S F = X n ,...,n k ∈ F,k =0 , ,... ˜ G ( n , . . . , n k ) : a n . . . a n k : Λ ,F , where : · : Λ ,F stands for the Wick ordering with respect to the measure µ Λ ,F . We claim that in this latter expansion, ˜ G ( n , . . . , n k ) = 0 if n + · · · + k = 0. For obvious reasons, we shall call L ( µ Λ ,F ) functions which satisfythis property quasilocal as well. Since the space of quasilocal functionsis clearly closed, we will be done if we show that (cid:16) : a n . . . a n k : Λ (cid:17) F isquasilocal. Consider one term in the definition of : a n . . . a n k : Λ . It isa product of a subcollection of a n , . . . , a n k ’s multiplied by a collection of( K Λ ) n,m ’s pairing the remaining indices (see formula (9) in Appendix A).Since ( K Λ ) n,m is proportional to δ n, − m , it follows that indices which appearin the subcollection of a n ’s are obtained from n , . . . , n k by omitting pairsof opposite ones. Since n + · · · + n k = 0, this equation is still true for thesubcollection. In view of the above, we will be done if we show ( a m . . . a m l ) F is quasilocal if m + · · · + m l = 0. We now use the fact that Z a Aa a Bb dµ Λ = 0 unless A = B and a = − b. It thus follows that ( a m . . . a m l ) F = 0 unless the a n ’s that get integratedare present in pairs with equal and opposite indices. It follows that the sumof the indices of the a n ’s that are left over after integration is done, still isequal to zero. Putting it all together, we have that (cid:16) : a n . . . a n k : Λ (cid:17) F isequal to a sum of terms proportional to a m . . . a m l with m + · · · + m l = 0.It is trivial to see that each such term is quasilocal and we have what wewant. Note that this implies that S F is also quasilocal in the original sense,i.e. considered as a function on the full space. In view of the discussionabove, we see that it is enough to show that I Λ , Λ ′ ( S F ) is quasilocal withrespect to µ Λ ,F .We have thus reduced the original problem to one defined on functions offinitely many variables. Let us show now that we can further assume that S F is smooth. To this end, let O ( τ ) S F [ φ ] = Z S F [ e − τ φ + p − e − τ ψ ] dµ Λ ,F [ ψ ] . It is well-known that O ( τ ) S F is smooth and that O ( τ ) S F → S F in L ( µ Λ ,F )as τ → + . Moreover, since O ( τ ) is diagonal in the basis of Wick products,we have that O ( τ ) S F preserves quasilocality. We have thus reduced theproblem to showing that a C ∞ , cylindrical quasilocal function maps to aquasilocal one under I Λ , Λ ′ .We now need to streamline our notation. First, since F will be held fixed inthe rest of this paper, it shall be omitted (e.g. we’ll just write S instead of S F , dµ Λ instead of dµ Λ ,F and so on). Second, let Λ ′ ( t ) = e − t Λ. We shall This is of course the Ornstein-Uhlenbeck semigroup action, hence the notation. The exact form here is unimportant. We can take Λ ′ to be any smooth function of t which is equal to Λ for t = 0 and which has a strictly negative derivative everywhere. enote by K n,m ( t ) the propagator of the measure µ Λ , Λ ′ ( t ) , i.e. K n,m ( t ) = δ n, − m e − n − e − n ′ ( t ) n + 1) . Also, S ( t ) will stand for I Λ , Λ ′ ( t ) ( S ). Additionally, we will denote the oper-ator I ′ Λ ′ ( t ) , Λ ′ ( t ) (note that this is the linearized map) by U ( t , t ). Finally,the closed subspace of C functions (topologized with the usual ||·|| C norm)whose derivative is uniformly continuous will be denoted by BU C .We now have the following important Lemma. S ( t ) satisfies the following equation S ( t ) = U ( t, S + Z t U ( t, τ ) (cid:18) X n,m ˙ K n,m ( τ ) ∂S ( τ ) ∂a m ∂S ( τ ) ∂a n (cid:19) dτ, where this equation holds as an equation in the Banach space of continuousfunctions t → f ( t ) where t ∈ [0 , ∞ ) and f ( t ) ∈ BU C . The formula above is of course the celebrated Polchinski equation [8] writtenin the “variation of constants” form.
Proof of Lemma.
First, note that since S is C ∞ , then S ( t ) is C ∞ for every t and thus certainly S ( t ) ∈ BU C . Now, suppose f is a smooth function. Weclaim that the map t → U ( t, f is continuous on [0 , ∞ ) and continuouslydifferentiable on (0 , ∞ ) , where U ( t, f = f t is considered as an elementof BU C .To see this, note first that if ∆ t >
0, we have that || f t +∆ t − f t || C = || U ( t + ∆ t, t ) f t − f t || C ≤ Z || f t ( x + y ) − f t ( x ) || C dµ Λ ′ ( t ) , Λ ′ ( t ) ( y ) . Now, since the upper bound and the modulus of continuity of f t are nonin-creasing as functions of t , the latter expression goes to 0 uniformly in t as∆ t → + , as can be shown by the usual elementary argument of breaking theintegral into two parts, one with small y where uniform continuity of f t andits derivative is used, and the rest, which goes to zero since dµ Λ ′ ( t +∆ t ) , Λ ′ ( t ) converges to a delta function. Continuity of t → f t follows at once.To see that t → f t is continuously differentiable on (0 , ∞ ), first note thatas long as t >
0, one can interchange the integral and the derivative with In fact, it is C on [0 , ∞ ) but we are not going to need that. espect to t in dfdt (cid:12)(cid:12) t = t . This is a basic fact which follows from dominatedconvergence. Now, note that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f t +∆ t − f t ∆ t − dfdt (cid:12)(cid:12)(cid:12) t = t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ≤|| f || C Z (cid:12)(cid:12)(cid:12)(cid:12) dµ Λ , Λ ′ ( t +∆ t ) − dµ Λ , Λ ′ ( t ) ∆ t − ddt (cid:0) dµ Λ , Λ ′ ( t ) (cid:1)(cid:12)(cid:12)(cid:12) t = t (cid:12)(cid:12)(cid:12)(cid:12) = || f || C Z (cid:12)(cid:12)(cid:12)(cid:12) ddt (cid:0) dµ Λ , Λ ′ ( t ) (cid:1)(cid:12)(cid:12)(cid:12) t = c − ddt (cid:0) dµ Λ , Λ ′ ( t ) (cid:1)(cid:12)(cid:12)(cid:12) t = t (cid:12)(cid:12)(cid:12)(cid:12) , where c in the last equation is between t and t + ∆ t . It is trivial to see nowthat the last integral goes to zero as ∆ t →
0. We thus have differentiabilityof the map t → f t . The fact that t → dfdt is continuous follows essentially inthe same way as the continuity of t → f t .Now, observe that g → − ln( g ) and h → e − h are local diffeomorphisms whichare inverses of each other from BU C into itself, as long as g is boundedaway from 0. Putting everything together, we have that t → S ( t ) maps into BU C , is continuous on [0 , ∞ ), and is continuously differentiable on (0 , ∞ ).The rest of the proof is a standard argument in the theory of evolution equa-tions in Banach spaces [9, 10]. Consider the function ˜ S ( τ ) = U ( t, τ ) S ( τ ).By similar arguments to the one above, (when we showed that t → S ( t ) iscontinuous and continuously differentiable) we have that ˜ S ( τ ) is a contin-uous function into BU C on [0 , ∞ ), and is continuously differentiable on(0 , ∞ ). Moreover, by a direct calculation, we have that d ˜ Sdτ = U ( t, τ ) (cid:18) X n,m ˙ K n,m ( τ ) ∂S ( τ ) ∂a m ∂S ( τ ) ∂a n (cid:19) . Integrating the above equation between t and t , and then taking the limits t → + , t → t − , we get what we want.The lemma above shows in effect that the function t → S ( t ) is a fixed pointof the following variation of constants mapΦ( f )( t ) = U ( t, S + Z t U ( t, τ ) (cid:18) X n,m ˙ K n,m ( τ ) ∂f ( τ ) ∂a m ∂f ( τ ) ∂a n (cid:19) . Let us make this more precise. Again, we follow the general ideas of thetheory of the evolution equations in Banach spaces. Consider the metricspace X of all continuous functions on [0 , δ ] valued in the closed ball of The reader is encouraged here to go over the introductory discussion in [10] for theabstract setting behind the argument below. enter 0 and radius R in BU C . We shall take R = 2 || S || C and will specify δ below. The metric on X is given by ρ ( f , f ) = sup t ∈ [0 ,δ ] || f ( t ) − f ( t ) || C . Note that Φ( v )( t ) is indeed in BU C . This is due to the regularizing effectof U ( t, τ ) inside the integral. In fact, it is straightforward to show thatthe following inequality || U ( t, τ ) g || C . || g || C √ t − τ , holds.Now, note that for any two BU C functions f and f we trivially have theinequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n,m ˙ K n,m ( τ ) (cid:18) ∂f ( τ ) ∂a m ∂f ( τ ) ∂a n − ∂f ( τ ) ∂a m ∂f ( τ ) ∂a n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C . || f − f || C , where the implicit constant depends on R in general. Using the inequalitiesabove, it is easy to show that ρ (Φ( f ) , Φ( f )) . (cid:16) Z δ √ t − τ dτ (cid:17) ρ ( f , f ) . Since the square root is an integrable function, we can choose δ small enoughsuch that the overall constant in front of ρ ( g , g ) is less than . Using thefact that || U ( t, S || C ≤ || S || C , we have thatsup t ∈ [0 ,δ ] || Φ( g )( t ) || C ≤ sup t ∈ [0 ,δ ] || Φ( g )( t ) − Φ(0) || C + sup t ∈ [0 ,δ ] || Φ(0) || C ≤ ρ ( g,
0) + sup t ∈ [0 ,δ ] || U ( t, S || C < R. Putting everything together, we have that Φ is a contraction which takesthe space X to itself. We thus have that t → S ( t ) is indeed the unique fixedpoint of Φ. Moreover, by starting with any point in the space and repeat-edly applying Φ, we converge to this fixed point. We are going to take thefunction t → U ( t, S as our starting point.Now, let us say that t → f ( t ) in X is quasilocal if for every t , f ( t ) is quasilo-cal with respect to the appropriate measure , i.e. with respect to the measure µ Λ ′ ( t ) . Note that according to this definition, our starting point of the iter-ation t → U ( t, S is quasilocal. We now claim that if t → f ( t ) is quasilocal Recall that U is basically a convolution with a Gaussian. Incidentally, this is wherewe would need the differentiability of α at 0 in case we decide to replace e − x with α ( x )in the ultraviolet cutoff. hen its image by Φ is quasilocal as well. In view of the above, this willimmediately imply that S ( t ) is quasilocal, for t ∈ [0 , δ ], and thus, we willhave that S ( δ ) is quasilocal. Repeating the argument , we can extend thisfact to all of [0 , ∞ ) and obtain that S ( t ) is quasilocal for every t . This wouldconclude the proof of the theorem.It thus remains to demonstrate the claim. Since U ( t , t ) acts diagonally inthe Wick expansion, and thus preserves quasilocality, we will be done if weshow that 2 X n,m ˙ K n,m ( τ ) ∂f ( τ ) ∂a m ∂f ( τ ) ∂a n , is quasilocal if f ( τ ) is quasilocal. Therefore, consider the expression X n,m ˙ K n,m ( τ ) Z : a n . . . a n k : Λ ′ ( τ ) (cid:18) ∂f ( τ ) ∂a m ∂f ( τ ) ∂a n (cid:19) dµ Λ ′ ( τ ) , (8)for some indices n , . . . , n k such that n + · · · + n k = 0. We need to showthat the expression above vanishes. Since τ will be fixed in what remains ofthe proof, we shall drop it, together with the index Λ ′ ( τ ), and will simplywrite f for f ( τ ), dµ instead of dµ Λ ′ ( τ ) , and so on.Let us assume now that g is smooth (and not just in BU C ). It is easy toshow then that its Wick expansion can be differentiated term by term withthe result converging to the appropriate derivative.We now use the fact [6] that for a smooth function g , we have Z : a n . . . a n k : gdµ ≃ Z ∂ k g∂a n . . . ∂a n k dµ, where ≃ means equality up to an irrelevant constant. Therefore, we havethat (8) is equal to a finite sum of terms which are schematically of the form Z (cid:16) f (cid:17) ( A ) (cid:16) f (cid:17) ( B ) dµ, where the bracketed exponents stand for derivatives. It thus follows that if f k denotes the the k -th partial sum of the Wick expansion of f then, since f k → f (and thus ( f k ) ( A ) → ( f ) ( A ) , ( f k ) ( B ) → ( f ) ( B ) in L ( µ )), we havethat Note that || e − S ( t ) || C doesn’t increase as a function of t , and thus || S ( t ) || C is boundedfrom above by some multiple (which is a function of t ) of || S || C . This implies that theestimates that went into showing that Φ is a contraction which preserves the space X , canbe carried through on any compact subinterval of [0 , ∞ ). (cid:16) f k (cid:17) ( A ) (cid:16) f k (cid:17) ( B ) dµ → Z (cid:16) f (cid:17) ( A ) (cid:16) f (cid:17) ( B ) dµ. Putting it all together, we have that X n,m ˙ K n,m Z : a n . . . a n k : (cid:18) ∂f k ∂a m ∂f k ∂a n (cid:19) dµ → X n,m ˙ K n,m Z : a n . . . a n k : (cid:18) ∂f∂a m ∂f∂a n (cid:19) dµ. Rearranging now in the standard way [11, 12], and using the fact that K n,m is proportional to δ n, − m we have that the expression above is proportionalto δ n + ··· + n k , = 0, since we’ve assumed that n + · · · + n k = 0. The proof isthus complete for the case of a smooth f .Now, using the fact that O ( σ ) f → f in L ( µ ) as σ → + , we have that X n,m ˙ K n,m Z : a n . . . a n k : (cid:18) ∂ ( O ( σ ) f ) ∂a m ∂ ( O ( σ ) f ) ∂a n (cid:19) dµ → X n,m ˙ K n,m Z : a n . . . a n k : (cid:18) ∂f∂a m ∂f∂a n (cid:19) dµ. Since the Ornstein-Uhlenbeck semigroup preserves quasilocality, and sincethe limit of a zero sequence is zero, the proof is complete.We finish by noting that the latter part of the proof above can be consideredto be a theorem in viability theory. In effect, we show that a mild solution[9, 10] of the Polchinski’s equation ∂S∂t = 2 X n,m ˙ K n,m ( τ ) ∂ S∂a n ∂a m − ∂S ( τ ) ∂a m ∂S ( τ ) ∂a n , which starts in Q b , remains there. It is easy to see that the evolution gen-erated by the linear term in the PDE above preserves quasilocality (this issimply the map U ( t , t ) above). We do have the knowledge that the nonlin-ear term when acting on a quasilocal S would give a quasilocal expression.However, it doesn’t seem easy to control this term away from the quasilocalsubset which is what is usually required in the viability literature [4, 5]. Theproof above circumvents this by using a variation of constants formula andthe smoothing effect of the U . It is clear that the argument above does notdepend on the precise form of the Polchinski’s equation, but rather on thesmoothing effect of the U , and thus generalizes to any setting where thesame fact holds. ppendix A We gather here a few elementary facts about Wick products. This is stan-dard material [12, 13]. The reason we’re giving it here (apart from thereader’s convenience) is that one needs to be a little careful in using theusual formulas in our setting. This is because, due to the way we have de-fined our measures, the a n ’s are not independent random variables (recallthat a n = a − n ).So, let µ be a Gaussian measure. Define the Wick-ordered exponential via: e i ( t a n + ··· + t k a nk ) : ≡ e i ( t a n + ··· + t k a nk ) (cid:18) Z e i ( t a n + ··· + t k a nk ) (cid:19) − = e i ( t a n + ··· + t k a nk ) e P kα,β =1 t α t β K nα,nβ , where K n α ,n β = R a n α a n β dµ . Now, the Wick monomial is defined by: a n . . . a n k := 1 i k ∂ k ∂t . . . ∂t k (cid:18) : e i ( t a n + ··· + t k a nk ) : (cid:19) t ,...,t k =0 . Using these definitions one immediately gets that Z : a n . . . a n k :: a m . . . a m l : dµ Λ ,L = ( k = l P σ K n n σ (1) K n n σ (2) . . . K n k n σ ( k ) if k = l and that : a n . . . a n k := X P Y { i,j }∈ P (cid:16) − K n i ,n j (cid:17) Y l / ∈ P a n l , (9)where P is a collection of pairs of indices from { , . . . , k } . Appendix B
Recall that a local functional is a linear combination of expressions of theform (3). Clearly, any such expression is of the form Z T d L (cid:16) φ ( x ) , ∂ x φ ( x ) , . . . , ∂ Mx d ,x d ,...,x d φ ( x ) (cid:17) dx, where L ( z , . . . , z N ) is some polynomial in N variables. Let us call thispolynomial the integrand of the local functional. In this appendix, we provethe following N = P Mm =0 (cid:0) m + d − m (cid:1) . roposition. If S is a local functional whose integrand is bounded frombelow by b , then S ∈ Q b (Λ) . Note that this would imply that e.g. R T d ( ∂φ ) + P ( φ ) is in Q b (Λ) as long as P ( z ) is a polynomial in z bounded from below by b . Proof.
Clearly, it is enough to prove this for the case b = 0, which is whatwe are going to assume below. Let ǫ >
0. Let β be a constant (which we’llassume is greater than 1) such that | ∂ mx i ,...,x im φ ( x ) | ≤ β ||| a n ||| for all i , . . . , i m ∈ { , . . . , d } and m ≤ M . Using this estimate, we have that (cid:12)(cid:12)(cid:12)(cid:12) L (cid:16) φ ( x ) , ∂ x φ ( x ) , . . . , ∂ Mx d ,x d ,...,x d φ ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C + C ||| a n ||| A for some constants A, C and C . Choose R such that sZ B c (0 ,R ) (cid:16) C + C ||| a n ||| A (cid:17) dµ Λ < ǫ , and ( C + C ( β ( R + 1)) A ) µ Λ ( B c (0 , R )) < ǫ , where B c (0 , R ) is the complement of the ball of radius R (with respect tothe norm ||| · ||| ). The existence of such an R follows immediately fromFernique’s theorem. Now, let T ( z , . . . , z N ) be a trigonometric polynomialsuch that • T is nonnegative. • | T | ≤ C + C ( β ( R + 1)) A . • sup ( z ,...,z N ) ∈ [ − βR,βR ] N | T ( z , . . . , z N ) − L ( z , . . . , z n ) | < ǫ .To see that such a T exists, first restrict L (cid:16) φ ( x ) , ∂ x φ ( x ) , . . . , ∂ Mx d ,x d ,...,x d φ ( x ) (cid:17) to [ − β ( R + 1) , β ( R + 1)] N , and then extend it periodically. If we now takethe Ces`aro sum of a sufficiently far truncation of the Fourier series of thisperiodization, we get what we want directly from the properties of the Fej´erkernel. Putting everything together, we have that Z (cid:12)(cid:12)(cid:12)(cid:12) L (cid:16) φ ( x ) , . . . , ∂ Mx d ,x d ,...,x d φ ( x ) (cid:17) − T (cid:16) φ ( x ) , . . . , ∂ Mx d ,x d ,...,x d φ ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) dµ Λ ≤ sZ B (0 ,R ) . . . + sZ B c (0 ,R ) . . . < sZ B (0 ,R ) . . . + sZ B c (0 ,R ) (cid:16) C + C ||| a n ||| A (cid:17) dµ Λ + ( C + C ( β ( R + 1)) A ) µ Λ ( B c (0 , R )) <ǫ p µ Λ ( B (0 , r )) + ǫ < ǫ. We thus have that S can be approximated in L ( µ Λ ) by elements of the form R T d T (cid:16) φ ( x ) , ∂ x φ ( x ) , . . . , ∂ Mx d ,x d ,...,x d φ ( x ) (cid:17) dx . Since clearly each such elementis bounded and nonnegative, we will have what we want provided we showthat each such element is in Q . This, in turn, would follow if we show thateach element of the form Z T d e iα φ ( x ) e iα ∂ x φ ( x ) . . . e iα d,...,d ∂ Mxd,...,xd φ ( x ) dx (10)is in Q , where α , α , . . . , α d,d,...,d are constants. To this end, note that Z T d N Y n =0 Y i ,...,i n =1 ,...,d (cid:18) L X l =0 (cid:16) iα i ,...,i n ∂ lx i ,...,x in φ ( x ) (cid:17) l l ! (cid:19)! dx is in Q for every L , converges pointwise to (10), and is bounded (uniformlyin L ) from above by e (cid:0) | α | + | α | + ··· + | α d,...,d | (cid:1) ||| a n ||| . By another application of Fernique’s theorem, we have that they convergeto (10) in L ( µ Λ ), and the proof is complete. Acknowledgments:
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Department of Mathematics, American University of Beirut, Beirut, Lebanon.Email address ::