# Some Comments on Rigorous Finite-Volume Euclidean Quantum Field Path Integrals in the Analytical Regularization Scheme

Authors: Luiz C. L. Botelho
Journal: Advances in Mathematical Physics
 (2011)

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## Abstract

Through the systematic use of the Minlos theorem on the support of cylindrical measures onR∞, we produce several mathematically rigorous finite-volume euclidean path integrals in interacting euclidean quantum fields with Gaussian free measures defined by generalized powers of finite-volume Laplacian operator.


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## Body

## 1. Introduction

Since the result of R.P. Feynman on representing the initial value solution of Schrodinger Equation by means of an analytically time continued integration on a infinite-dimensional space of functions, the subject of Euclidean Functional Integrals representations for Quantum Systems has became the mathematical-operational framework to analyze Quantum Phenomena and stochastic systems as showed in the previous decades of research on Theoretical Physics [1–3].One of the most important open problems in the mathematical theory of Euclidean Functional Integrals is that related to implementation of sound mathematical approximations to these Infinite-Dimensional Integrals by means of Finite-Dimensional approximations outside of the always used [computer oriented] Space-Time Lattice approximations (see [2], [3, chapter 9]). As a first step to tackle upon the above cited problem it will be needed to characterize mathematically the Functional Domain where these Functional Integrals are defined.The purpose of this paper is to present the formulation of Euclidean Quantum Field theories as Functional Fourier Transforms by means of the Bochner-Martin-Kolmogorov theorem for Topological Vector Spaces ([4], [5, Theorem  4.35]) and suitable to define and analyze rigorously Functional Integrals by means of the well-known Minlos theorem ([5, Theorem  4.312] and [6, part 2]) which is presented in full in Appendix A.We thus present studies on the difficult problem of defining rigorously infinite-dimensional quantum field path integrals in general finite volume space timesΩ⊂Rν(ν=2,4,…) by means of the analytical regularization scheme [7].


## 2. Some Rigorous Finite-Volume Quantum Field Path Integral in the Analytical Regularization Scheme

Let us thus start our analysis by considering the Gaussian measure onL2(R2) defined by the finite volume, infrared regularized, and α-power Laplacian acting on L2(RN) as an operatorial quadratic form (j(x)∈L2(RN)) (see Appendix B) (2.1a)Zα,εIR(0)[j]=exp⁡{-12〈j,(χΩ[(-Δ)-α+εIR2]-1χΩ)j〉L2(R2)}≡∫L2(R2)dα,εIR(0)μ[φ]exp⁡(i〈j,φ〉L2(R2)).HereχΩ denotes the multiplication operator defined by the characteristic function χΩ(x) of the compact region Ω⊂R2 and εIR>0 the associated infrared cutoff.It is worth calling the reader attention that due to the infrared regularization introduced on (2.1a), the domain of the Gaussian measure [4, 6] is given by the space of square integrable functions on R2 by the Minlos theorem of Appendix A, since for α>1, the operator defines a trace class operator on L2(R2), namely, (2.1b)Tr⁡∮1(L2(R2)){χΩ[(-Δ)α+εIR2]-1χΩ}=vol(Ω)×[∫d2p|p|2α+εIR2]<∞.This is the only point of our analysis where it is needed to consider the infrared cut off. As a consequence of the above remarks, one can analyze the ultraviolet renormalization program in the following interacting model proposed by us and defined by an interactiongbareV(φ(x)), with V(x) being the Fourier Transformed of an integrable and essentially bounded measurable real function. (It could be as well consider a polynomial interaction of the form Vn,p(x) = minimum of {(φ(x))p,n} with p and n positive integers. Note that Ṽ(k)∈L1(R)∩L∞(R) by hypothesis |∫Ω[∫eikφ(x)·Ṽ(k)dk]d2x|≤vol(Ω)||Ṽ||L1<∞.) Note that V(x) is thus a continuous real function vanishing at the infinite point.Let us show that by defining an ultraviolet renormalized coupling constant (with a finite volumeΩ cutoff built in) (2.2)gbare(α)=gren(1-α)1/2(2-απ-1/4)
one can show that the interaction function (2.3)exp⁡{-gbare(α)∫Ωd2xV(φ(x))}
is an integrable function on L1(L2(R2),dα,εIR(0)μ[φ]) and leads to a well-defined ultraviolet functional integral in the limit of α→1.The proof is based on the following estimates.Since almost everywhere we have the pointwise limit(2.4)exp⁡{-gbare(α)∫d2xV(φ(x))},lim⁡N→∞{∑n=0N(-1)n(gbare(α))nn!∫dk1⋯dkNṼ(k1)⋯Ṽ(kN)×∫Ωdx1⋯dxNeik1φ(x1)⋯eiknφ(xN)∑n=0N(-1)n(gbare(α))nn!}
we have that the upper bound estimate below holds true (2.5a)|ZεIRα[gbare]|≤|∑n=0∞(-1)n(gbare(α))nn!∫Rdk1⋯dknṼ(k1)⋯Ṽ(kn)∫Ωdx1⋯dxn∫dα,εIR(0)μ[φ](ei∑l=1Nklφ(xl))∑n=0∞(-1)n(gbare(α))nn!|
With (2.5b)ZεIRα[gbare]=∫dα,εIR(0)μ[φ]exp⁡{-gbare(α)∫Ωd2xV(φ(x))}we have, thus, the more suitable form  after realizing the d2ki and dα,εIR(0)μ[φ] integrals, respectively (2.6)|ZεIR=0α[gbare]|≤∑n=0∞(gbare(α))nn!(‖Ṽ‖L∞(R))n×|∫Ωdx1⋯dxmdet-1/2[Gα(m)(xi,xj)]1≤i≤m1≤j≤m|.Here[Gα(N)(xi,xj)]1≤i≤N,1≤j≤N denotes the N×N symmetric matrix with the (i,j) entry given by the positive Green-function of the α-Laplacian (without the infrared cut off here!). (2.7)Gα(xi,xj)=|xi-xj|2(α-1)Γ(1-α)Γ(α)(e-iπα4α⋅π1/2).At this point, we call the reader attention that we have the formulae on the asymptotic behavior forα→1 and α<1 (see [7, Appendix  A]) (2.8){lim⁡α→1α>1det-1/2[Gα(N)(xi,xj)]}~e(N/2)πiα×(πN/4⋅2Nα)×(1-α)N/2(|(+1)(-1)N(N-1)|).After substituting (2.8) into (2.6) and taking into account the hypothesis of the compact support of the nonlinearity Ṽ(k), one obtains the finite bound for any value gren>0, with the finite volume cutoff and producing a proof for the convergence of the perturbative expansion in terms of the renormalized coupling constant for the model (2.9)lim⁡α→1|ZεIR=0α[gbare(α)]|≤∑n=0∞(‖Ṽ‖L∞(R))nn!(gren(1-α)1/2)n×(1-α)n/2×(vol(Ω))n≤exp⁡{gren‖Ṽ‖L∞(R)vol(Ω)}<∞.Another important rigorously defined functional integral is to consider the followingα-power Klein Gordon operator on Euclidean space-time (with f∈L2(RN))(2.10)LΩ,f-1=(χΩ(2π)ν/2)[(-Δ)α+fχΩ+m2]-1×(χΩ(2π)ν/2vol(Ω))
with m2 a positive “mass" parameters.Let us note thatℒΩ,f-1 is an operator of trace class on L2(Rν) if and only if the result below holds true (2.11)Tr⁡L2(Rν)(L-1)=∫dνk1k2α+m2=C̅(ν)m((ν/α)-2)×{π2αcosecνπ2α}<∞,
namely, if (2.12)α>ν2.In this case, let us consider the double functional integral with functional domainL2(Rν)(2.13)Z[j,k]=∫dG(0)β[v(x)]×∫dLΩ,v(0)μ[φ]×exp⁡{i∫dνx(j(x)φ(x)+k(x)v(x))},
where the Gaussian functional integral on the fields V(x) has a Gaussian generating functional defined by a ∮1-integral operator with a positive-definite kernel g(|x-y|), namely, (2.14)Z(0)[k]=∫dG(0)β[v(x)]exp⁡{i∫dνxk(x)v(x)}=exp⁡{-12∫Ωdνx∫Ωdνy(k(x)g(|x-y|)k(y))}.A simple direct application of the Fubbini-Tonelli theorem on the exchange of the integration order on (2.13) leads us to the effective λφ4—(Formal) functional integral representation (2.15)Zeff[j]=∫dL(0)μ[φ][φ(x)]×exp⁡{-12∫Ωdνxdνy|φ(x)|2g(|x-y|)|φ(y)|2}×exp⁡{i∫Ωdνxj(x)φ(x)}.Note that if one introduces from the beginning a bare mass parametermbare2 depending on the parameter α, but such that it always satisfies (2.11) one should obtain again (2.15) as a well-defined measure on L2(Rν). Of course that the usual pure Laplacian limit of α→1 on (2.10), there will be needed a renormalization of this mass parameters (limα→1mbare2(α)=+∞!) as much as it has been done in previous studies.Let us continue our examples by showing again the usefulness of the precise determination of the functional-distributional structure of the domain of the functional integrals in order to construct rigorously these path integrals without complicated lattice limit procedures [2].Let us consider a generalRν Gaussian measure defined by the Generating functional on S(Rν) defined by the α-power of the Laplacian operator -Δ acting on S(Rν) with a small infrared regularization mass parameter μ2 as considered in (2.1a) (2.16)Z(0)[j]=exp⁡{-12〈j,((-Δ)α+μ02)-1j〉L2(Rν)}=∫Ealg(S(Rν))dα(0)μ[φ]exp⁡(iφ(j)).An explicit expression in momentum space for the Green function of theα-power of (-Δ)α+μ02 given by
(2.17)((-Δ)+α+μ02)-1(x-y)=∫dνk(2π)νeik(x-y)(1k2α+μ02).HereC̅(ν) is a ν-dependent (finite for ν-values !) normalization factor.Let us suppose that there is a range ofα-power values that can be choosen in such way that one satisfies the constraint below (2.18)∫Ealg(S(Rν))dα(0)μ[φ](‖φ‖L2j(Rν))2j<∞
with j=1,2,…,N and for a given fixed integer N, the highest power of our polynomial field interaction. Or equivalently, after realizing the φ-Gaussian functional integration, with a space-time cuttoff volume Ω on the interaction to be analyzed on (2.16) (2.19)|∫Ωdνx[(-Δ)α+μ02]-j(x,x)|≤|vol(Ω)×(∫dνkk2α+μ02)j|=|[Cν(μ0)((ν/α)-2)×(π2αcosecνπ2α)]j×vol(Ω)|<∞.Forα>ν/2, one can see by the Minlos theorem that the measure support of the Gaussian measure (2.16) will be given by the intersection Banach space of measurable Lebesgue functions on Rν instead of the previous one Ealg(S(Rν)) ([4–6]) (2.20)L2N(Rν)=⋂j=1N(L2j(Rν)).In this case, one obtains that the finite-volumep(φ)2 interactions (2.21)exp⁡{-∑j=1Nλ2j∫Ω(φ2(x))jdx}≤1
are mathematically well defined as the usual pointwise product of measurable functions and for positive coupling constant values λ2j≥0. As a consequence, we have a measurable functional on L1(ℒ2N(Rν);dα(0)μ[φ]) (since it is bounded by the function 1). Thus, it makes sense to consider mathematically the well-defined path integral on the full space Rν with those values of the power α satisfying the constraint (2.17). (2.22)Z[j]=∫L2N(Rν)dα(0)μ[φ]exp⁡{-∑j=1Nλ2j∫Ωφ2j(x)dx}×exp⁡(i∫Rνj(x)φ(x)).Finally, let us consider an interacting field theory in a compact space-timeΩ⊂Rν defined by an iteger even power 2n of the Laplacian operator with Dirichlet Boundary conditions as the free Gaussian kinetic action, namely, (2.23)Z(0)[j]=exp⁡{-12〈j,(-Δ)-2nj〉L2(Ω)}=∫W2n(Ω)d(2n)(0)μ[φ]exp⁡(i〈j,φ〉L2(Ω))
here φ∈W2n(Ω)—the Sobolev space of order n which is the functional domain of the cylindrical Fourier Transform measure of the Generating functional Z(0)[j], a continuous bilinear positive form on W2-n(Ω) (the topological dual of W2n(Ω)) [4–6].By a straightforward application of the well-known Sobolev immersion theorem, we have that for the case of(2.24)n-k>ν2
including k as a real number the functional Sobolev space W2n(Ω) is contained in the continuously fractional differentiable space of functions Ck(Ω). As a consequence, the domain of the Bosonic functional integral can be further reduced to Ck(Ω) in the situation of (2.24) (2.25)Z(0)[j]=∫Ck(Ω)d(2n)(0)μ[φ]exp⁡(i〈j,φ〉L2(Ω)).That is our new result generalizing the Wiener theorem on Brownian paths in the case ofn=1, k=1/2, and ν=1.Since the bosonic functional domain on (2.25) is formed by real functions and not distributions, we can see straightforwardly that any interaction of the form (2.26)exp⁡{-g∫ΩF(φ(x))dνx}
with the nonlinearity F(x) denoting a lower bounded real function (γ>0)(2.27)F(x)≥-γ
is well-defined and is integrable function on the functional space (Ck(Ω),d(2n)(0)μ[φ]) by a direct application of the Lebesgue theorem (2.28)|exp⁡{-g∫ΩF(φ(x))dνx}|≤exp⁡{+gγ}.At this point we make a subtle mathematical remark that the infinite volume limit of (2.25)-(2.26) is very difficult, since one looses the Garding-Poincaré inequality at this limit for those elliptic operators and, thus, the very important Sobolev theorem. The probable correct procedure to consider the thermodynamic limit in our Bosonic path integrals is to consider solely a volume cutoff on the interaction term Gaussian action as in (2.22) and there search for vol(Ω)→∞ [8–11].As a last remark related to (2.23) one can see that a kind of “fishnet" exponential generating functional (2.29)Z(0)[j]=exp⁡{-12〈j,exp⁡{-αΔ}j〉L2(Ω)}
has a Fourier transformed functional integral representation defined on the space of the infinitely differentiable functions C∞(Ω), which physically means that all field configurations making the domain of such path integral has a strong behavior like purely nice smooth classical field configurations.As a last important point of this paper, we present an important result on the geometrical characterization of massive free field on an Euclidean Space-Time [11].Firstly we announce a slightly improved version of the usual Minlos Theorem [4].Theorem 2.1.
LetE be a nuclear space of tests functions and dμ a given σ-measure on its topological dual with the strong topology. Let 〈,〉0 be an inner product in E, inducing a Hilbertian structure on ℋ0=(E,〈,〉0)¯, after its topological completion.We suppose the following.(a)
There is a continuous positive definite functional inℋ0, Z(j), with an associated cylindrical measure dμ.(b)
There is a Hilbert-Schmidt operatorT:ℋ0→ℋ0; invertible, such that E⊂Range(T), T-1(E) is dense in ℋ0 and T-1:ℋ0→ℋ0 is continuous.We have thus, that the support of the measure satisfies the relationship(2.30)supportdμ⊆(T-1)*(H0)⊂E*.At this point we give a nontrivial application of ours of the above cited Theorem2.1.Let us consider a differential inversible operatorℒ:S′(RN)→S(R), together with a positive inversible self-adjoint elliptic operator P:D(P)⊂L2(RN)→L2(RN). Let Hα be the following Hilbert space: (2.31)Hα={S(RN),〈Pαφ,Pαφ〉L2(RN)=〈,〉α¯,forαarealnumber}.We can see that forα>0, the operators below(2.32)P-α:L2(RN)⟶H+αφ⟶(P-αφ),Pα:H+α⟶L2(RN)φ⟶(Pαφ)
are isometries among the following subspaces: (2.33)D(P-α),〈,〉L2)¯,H+α
since (2.34)〈P-αφ,P-αφ〉H+α=〈PαP-αφ,PαP-αφ〉L2(RN)=〈φ,φ〉L2(RN),〈Pαf,Pαf〉L2(RN)=〈f,f〉H+α.If one considersT a given Hilbert-Schmidt operator on Hα, the composite operator T0=PαTP-α is an operator with domain being D(P-α) and its image being the Range (Pα). T0 is clearly an invertible operator and S(RN)⊂Range(T) means that the equation (TP-α)(φ)=f has always a nonzero solution in D(P-α) for any given f∈S(RN). Note that the condition that T-1(f) be a dense subset on Range (P-α) means that (2.35)〈T-1f,P-αφ〉L2(RN)=0
has as unique solution the trivial solution f≡0.Let us suppose too thatT-1:S(RN)→Hα be a continuous application and the bilinear term (ℒ-1(j))(j) be a continuous application in the Hilbert spaces H+α⊃S(RN), namely, if jn→L2j, then ℒ-1:P-αjn→L2ℒ-1P-αj, for {jn}n∈ℤ and (fn,j)∈S(RN).By a direct application of the Minlos Theorem, we have the result(2.36)Z(j)=exp⁡{-12[L-1(j)(j)]}=∫(T-1)*Hαdμ(T)exp⁡(iT(j)).Here the topological space support is given by(2.37)(T-1)*Hα=[(P-αT0Pα)-1]*((Pα(S(RN)))¯)=[(Pα)*(T0-1)*(P-α)*]Pα(S(RN))=PαT0-1(L2(RN)).In the important case ofℒ=(-Δ+m2):S′(RN)→S(RN) and T0T0*=(χΩ(2π)N/2/vol(Ω))(-Δ+m2)-2β×χΩ∈∮1(L2(RN)) since Tr(T0T0*)=(1/2(m2)β)(m2/1)N/2Γ(N/2)Γ(2β-(N/2))/Γ(β)<∞ for β>N/4 with the choice P=(-Δ+m2), we can see that the support of the measure in the path-integral representation of the Euclidean measure field in RN may be taken as the measurable subset below (2.38)supp⁡{d(-Δ+m2)μ(φ)}=(-Δ+m2)+αIΩ(x)(-Δ+m2)+β(L2(RN))
since ℒ-1P-α=(-Δ+m2)-1-α is always a bounded operator in L2(RN) for α>-1.As a consequence each field configuration can be considered as a kind of “fractional distributional" derivative of a square integrable function as written below of the formal infinite volumeΩ→RN. (2.39)φ(x)=[(-Δ+m2)(N/4)+ε-1f](x)
with a function f(x)∈L2(RN) and any given ε>0, even if originally all fields configurations entering into the path integral were elements of the Schwartz Tempered Distribution Spaces S′(RN) certainly very “rough" mathematical objects to characterize from a rigorous geometrical point of view.We have, thus, made a further reduction of the functional domain of the free massive Euclidean scalar field ofS′(RN) to the measurable subset as given by (2.38) denoted by W(RN)(2.40)exp⁡{-12[(-Δ+m2)-1j](j)}=∫S′(RN)d(-Δ+m2)μ(φ)eiφ(j)=∫W(RN)⊂S′(RN)d(-Δ+m2)μ̃(f)ei〈f,(-Δ+m2)(N/4)+ε-1f〉L2(RN).



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