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

Abstract

Through the systematic use of the Minlos theorem on the support of cylindrical measures on 𝑅^∞, 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.

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 times Ω⊂𝑅^𝜈(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 on 𝐿²(𝑅²) defined by the finite volume, infrared regularized, and 𝛼-power Laplacian acting on 𝐿²(𝑅^𝑁) as an operatorial quadratic form (𝑗(𝑥)∈𝐿²(𝑅^𝑁)) (see Appendix B) 𝑍⁽⁰⁾_𝛼,𝜀IR𝑗−1=exp2𝑗,𝜒_Ω(−Δ)^−𝛼+𝜀²_IR⁻¹𝜒_Ω𝑗_𝐿2(𝑅2)≡_𝐿2(𝑅2)𝑑⁽⁰⁾_𝛼,𝜀IR𝜇𝜑exp𝑖𝑗,𝜑_𝐿2(𝑅2). (2.1a)

Here 𝜒_Ω denotes the multiplication operator defined by the characteristic function 𝜒_Ω(𝑥) of the compact region Ω⊂𝑅² and 𝜀_IR>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, Tr_∮1(𝐿²_(𝑅²₎₎𝜒_Ω(−Δ)^𝛼+𝜀²_IR⁻¹𝜒_Ω(Ω)×𝑑=vol²𝑝𝑝^2𝛼+𝜀²_IR<∞. (2.1b)

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 interaction 𝑔_bare𝑉(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 𝑉_𝑛,𝑝(𝑥) = minimum of {(𝜑(𝑥))^𝑝,𝑛} with 𝑝 and 𝑛 positive integers. Note that 𝑉(𝑘)∈𝐿¹(𝑅)∩𝐿^∞(𝑅) by hypothesis |∫_Ω[∫𝑒^{𝑖𝑘𝜑(𝑥)}⋅𝑉(𝑘)𝑑𝑘]𝑑²𝑥|≤vol(Ω)||V||_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 volume Ω cutoff built in) 𝑔_bare(𝛼)=𝑔_ren(1−𝛼)^1/22^−𝛼𝜋^−1/4 (2.2) one can show that the interaction function exp−𝑔_bare(𝛼)_Ω𝑑²𝑥𝑉𝜑(𝑥) (2.3) is an integrable function on 𝐿¹(𝐿²(𝑅²),𝑑⁽⁰⁾_𝛼,𝜀IR𝜇[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 exp−𝑔_bare(𝛼)𝑑²𝑥𝑉𝜑(𝑥),lim_𝑁→∞^𝑁_𝑛=0(−1)^𝑛𝑔_bare(𝛼)^𝑛𝑛!𝑑𝑘₁⋯𝑑𝑘_𝑁𝑉𝑘₁⋯𝑉𝑘_𝑁×_Ω𝑑𝑥₁⋯𝑑𝑥_𝑁𝑒^{𝑖𝑘1𝜑(𝑥1)}⋯𝑒^{𝑖𝑘𝑛𝜑(𝑥𝑁)} (2.4) we have that the upper bound estimate below holds true 𝑍^𝛼_𝜀IR𝑔_bare≤^∞_𝑛=0(−1)^𝑛𝑔_bare(𝛼)^𝑛𝑛!_𝑅𝑑𝑘₁⋯𝑑𝑘_𝑛𝑉𝑘₁⋯𝑉𝑘_𝑛Ω𝑑𝑥₁⋯𝑑𝑥_𝑛𝑑⁽⁰⁾_𝛼,𝜀IR𝜇𝜑𝑒^{𝑖∑𝑁ℓ=1𝑘ℓ𝜑(𝑥ℓ)} (2.5a) With 𝑍^𝛼_𝜀IR𝑔_bare=𝑑⁽⁰⁾_𝛼,𝜀IR𝜇𝜑exp−𝑔_bare(𝛼)_Ω𝑑²𝑥𝑉𝜑(𝑥) (2.5b)we have, thus, the more suitable form after realizing the 𝑑²𝑘_𝑖 and 𝑑⁽⁰⁾_𝛼,𝜀IR𝜇[𝜑] integrals, respectively 𝑍^𝛼_𝜀IR=0𝑔_bare≤^∞_𝑛=0𝑔_{𝑏𝑎𝑟𝑒}(𝛼)^𝑛𝑉𝑛!^{𝐿∞(𝑅)𝑛}×_Ω𝑑𝑥₁⋯𝑑𝑥_𝑚det^−1/2𝐺_𝛼^(𝑚)𝑥_𝑖,𝑥_{𝑗1≤𝑖≤𝑚1≤𝑗≤𝑚}. (2.6)

Here [𝐺_𝛼^(𝑁)(𝑥_𝑖,𝑥_𝑗)]_{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!). 𝐺_𝛼𝑥_𝑖,𝑥_𝑗=|𝑥_𝑖−𝑥_𝑗|^2(𝛼−1)Γ(1−𝛼)Γ(𝛼)𝑒^{−𝑖𝜋𝛼}4^𝛼⋅𝜋^1/2. (2.7)

At this point, we call the reader attention that we have the formulae on the asymptotic behavior for 𝛼→1 and 𝛼<1 (see [7, Appendix  A]) lim_{𝛼→1𝛼>1}det^−1/2𝐺_𝛼^(𝑁)𝑥_𝑖,𝑥_𝑗∼𝑒^{(𝑁/2)𝜋𝑖𝛼}×𝜋^𝑁/4⋅2^𝑁𝛼×(1−𝛼)^𝑁/2(+1)(−1)^𝑁().𝑁−1 (2.8)

After substituting (2.8) into (2.6) and taking into account the hypothesis of the compact support of the nonlinearity 𝑉(𝑘), one obtains the finite bound for any value 𝑔_ren>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 lim_𝛼→1𝑍^𝛼_𝜀IR=0𝑔_bare(𝛼)≤^∞_𝑛=0𝑉_{𝐿∞(𝑅)}^𝑛𝑔𝑛!(1−𝛼^ren)^1/2𝑛×(1−𝛼)^𝑛/2×(vol(Ω))^𝑛≤exp𝑔_ren𝑉_{𝐿∞(𝑅)}vol(Ω)<∞. (2.9)

Another important rigorously defined functional integral is to consider the following 𝛼-power Klein Gordon operator on Euclidean space-time (with 𝑓∈𝐿²(𝑅^𝑁))ℒ⁻¹_Ω,𝑓=𝜒_Ω(2𝜋)^𝜈/2(−Δ)^𝛼+𝑓𝜒_Ω+𝑚²⁻¹×𝜒_Ω(2𝜋)^𝜈/2vol(Ω) (2.10) with 𝑚² a positive “mass" parameters.

Let us note that ℒ⁻¹_Ω,𝑓 is an operator of trace class on 𝐿²(𝑅^𝜈) if and only if the result below holds true Tr_𝐿²_(𝑅^𝜈₎ℒ⁻¹=𝑑^𝜈𝑘1𝑘^2𝛼+𝑚²=𝐶(𝜈)𝑚^{((𝜈/𝛼)−2)}×𝜋2𝛼cosec𝜈𝜋2𝛼<∞, (2.11) namely, if 𝜈𝛼>2. (2.12)

In this case, let us consider the double functional integral with functional domain 𝐿²(𝑅^𝜈)𝑍𝑗,𝑘=𝑑_𝐺⁽⁰⁾𝛽[𝑣(𝑥)]×𝑑_ℒ⁽⁰⁾_Ω,𝑣𝜇𝜑×exp𝑖𝑑^𝜈𝑥𝑗(𝑥)𝜑(𝑥)+𝑘(𝑥)𝑣(𝑥), (2.13) 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, 𝑍⁽⁰⁾[𝑘]=𝑑_𝐺⁽⁰⁾𝛽[𝑣(𝑥)]exp𝑖𝑑^𝜈(𝑥)𝑣(𝑥−1𝑥𝑘)=exp2_Ω𝑑^𝜈𝑥_Ω𝑑^𝜈𝑦𝑘(𝑥)𝑔𝑥−𝑦𝑘𝑦. (2.14)

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 𝑍_eff𝑗=𝑑_ℒ⁽⁰⁾𝜇𝜑𝜑(𝑥−1)×exp2_Ω𝑑^𝜈𝑥𝑑^𝜈𝑦𝜑(𝑥)²𝑔𝑥−𝑦𝜑𝑦²×exp𝑖_Ω𝑑^𝜈𝑥𝑗(𝑥)𝜑(𝑥). (2.15)

Note that if one introduces from the beginning a bare mass parameter 𝑚²_bare 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(𝑅^𝜈). Of course that the usual pure Laplacian limit of 𝛼→1 on (2.10), there will be needed a renormalization of this mass parameters (lim_𝛼→1𝑚²_bare(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 general 𝑅^𝜈 Gaussian measure defined by the Generating functional on 𝑆(𝑅^𝜈) defined by the 𝛼-power of the Laplacian operator −Δ acting on 𝑆(𝑅^𝜈) with a small infrared regularization mass parameter 𝜇² as considered in (2.1a) 𝑍₍₀₎𝑗−1=exp2𝑗,(−Δ)^𝛼+𝜇²₀⁻¹𝑗_{𝐿2(𝑅𝜈)}=_{𝐸alg(𝑆(𝑅𝜈))}𝑑_𝛼⁽⁰⁾𝜇𝜑exp𝑖𝜑𝑗. (2.16)

An explicit expression in momentum space for the Green function of the 𝛼-power of (−Δ)^𝛼+𝜇²₀ given by (−Δ)^+𝛼+𝜇²₀⁻¹=𝑑𝑥−𝑦^𝜈𝑘(2𝜋)^𝜈𝑒^{𝑖𝑘(𝑥−𝑦)}1𝑘^2𝛼+𝜇²₀. (2.17)

Here 𝐶(𝜈) is a 𝜈-dependent (finite for 𝜈-values !) normalization factor.

Let us suppose that there is a range of 𝛼-power values that can be choosen in such way that one satisfies the constraint below _{𝐸alg(𝑆(𝑅𝜈))}𝑑_𝛼⁽⁰⁾𝜇𝜑𝜑_{𝐿2𝑗(𝑅𝜈)}^2𝑗<∞ (2.18) 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) _Ω𝑑^𝜈𝑥(−Δ)^𝛼+𝜇²₀^−𝑗(≤(Ω)×𝑑𝑥,𝑥)vol^𝜈𝑘𝑘^2𝛼+𝜇²₀^𝑗=𝐶_𝜈(𝜇₀)^{((𝜈/𝛼)−2)}×𝜋2𝛼cosec𝜈𝜋2𝛼^𝑗×vol(Ω)<∞. (2.19)

For 𝛼>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 𝑅^𝜈 instead of the previous one 𝐸^alg(𝑆(𝑅^𝜈)) ([4–6]) ℒ_2𝑁𝑅^𝜈=^𝑁_𝑗=1𝐿^2𝑗𝑅^𝜈. (2.20)

In this case, one obtains that the finite-volume 𝑝(𝜑)₂ interactions exp−^𝑁_𝑗=1𝜆_2𝑗Ω𝜑²(𝑥)^𝑗𝑑𝑥≤1 (2.21) are mathematically well defined as the usual pointwise product of measurable functions and for positive coupling constant values 𝜆_2𝑗≥0. As a consequence, we have a measurable functional on 𝐿¹(ℒ_2𝑁(𝑅^𝜈);𝑑_𝛼⁽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 𝑅^𝜈 with those values of the power 𝛼 satisfying the constraint (2.17). 𝑍𝑗=_{ℒ2𝑁(𝑅𝜈)}𝑑_𝛼⁽⁰⁾𝜇𝜑exp−^𝑁_𝑗=1𝜆_2𝑗Ω𝜑^2𝑗(𝑥)𝑑𝑥×exp𝑖_𝑅𝜈𝑗(𝑥)𝜑(𝑥). (2.22)

Finally, let us consider an interacting field theory in a compact space-time Ω⊂𝑅^𝜈 defined by an iteger even power 2𝑛 of the Laplacian operator with Dirichlet Boundary conditions as the free Gaussian kinetic action, namely, 𝑍⁽⁰⁾𝑗−1=exp2𝑗,(−Δ)^−2𝑛𝑗_𝐿2(Ω)=_{𝑊𝑛2(Ω)}𝑑₍₎⁽⁰⁾_2𝑛𝜇𝜑exp𝑖𝑗,𝜑_𝐿2(Ω) (2.23) here 𝜑∈𝑊^𝑛₂(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 𝑊^𝑛₂(Ω)) [4–6].

By a straightforward application of the well-known Sobolev immersion theorem, we have that for the case of 𝜈𝑛−𝑘>2 (2.24) including 𝑘 as a real number the functional Sobolev space 𝑊^𝑛₂(Ω) is contained in the continuously fractional differentiable space of functions 𝐶^𝑘(Ω). As a consequence, the domain of the Bosonic functional integral can be further reduced to 𝐶^𝑘(Ω) in the situation of (2.24) 𝑍⁽⁰⁾𝑗=_𝐶𝑘(Ω)𝑑⁽⁰⁾_(2𝑛)𝜇𝜑exp𝑖𝑗,𝜑_𝐿2(Ω). (2.25)

That is our new result generalizing the Wiener theorem on Brownian paths in the case of 𝑛=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 exp−𝑔_Ω𝐹𝜑(𝑥)𝑑^𝜈𝑥 (2.26) with the nonlinearity 𝐹(𝑥) denoting a lower bounded real function (𝛾>0)𝐹(𝑥)≥−𝛾 (2.27) is well-defined and is integrable function on the functional space (𝐶^𝑘(Ω),𝑑⁽⁰⁾_(2𝑛)𝜇[𝜑]) by a direct application of the Lebesgue theorem exp−𝑔_Ω𝐹𝜑(𝑥)𝑑^𝜈𝑥≤exp+𝑔𝛾. (2.28)

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 𝑍⁽⁰⁾𝑗−1=exp2𝑗,exp{−𝛼Δ}𝑗_𝐿2(Ω) (2.29) 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. Let 𝐸 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 in ℋ₀, 𝑍(𝑗), with an associated cylindrical measure 𝑑𝜇.(b)There is a Hilbert-Schmidt operator 𝑇∶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 support𝑑𝜇⊆𝑇^−1∗ℋ₀⊂𝐸^∗. (2.30)

At this point we give a nontrivial application of ours of the above cited Theorem 2.1.

Let us consider a differential inversible operator ℒ∶𝑆^′(𝑅^𝑁)→𝑆(𝑅), together with a positive inversible self-adjoint elliptic operator 𝑃∶𝐷(𝑃)⊂𝐿²(𝑅^𝑁)→𝐿²(𝑅^𝑁). Let 𝐻_𝛼 be the following Hilbert space: 𝐻_𝛼=𝑆𝑅^𝑁,𝑃^𝛼𝜑,𝑃^𝛼𝜑_{𝐿2(𝑅𝑁)}=⟨,⟩_𝛼,for𝛼arealnumber. (2.31)

We can see that for 𝛼>0, the operators below𝑃^−𝛼∶𝐿²𝑅^𝑁⟶ℋ_+𝛼𝜑⟶𝑃^−𝛼𝜑,𝑃^𝛼∶ℋ_+𝛼⟶𝐿²𝑅^𝑁𝜑⟶𝑃^𝛼𝜑 (2.32) are isometries among the following subspaces: 𝐷(𝑃^−𝛼),⟨,⟩_𝐿²),𝐻_+𝛼 (2.33) since 𝑃^−𝛼𝜑,𝑃^−𝛼𝜑_ℋ+𝛼=𝑃^𝛼𝑃^−𝛼𝜑,𝑃^𝛼𝑃^−𝛼𝜑_{𝐿2(𝑅𝑁)}=𝜑,𝜑_{𝐿2(𝑅𝑁)},⟨𝑃^𝛼𝑓,𝑃^𝛼𝑓⟩_𝐿²_(𝑅^𝑁₎=⟨𝑓,𝑓⟩_𝐻+𝛼. (2.34)

If one considers 𝑇 a given Hilbert-Schmidt operator on 𝐻_𝛼, the composite operator 𝑇₀=𝑃^𝛼𝑇𝑃^−𝛼 is an operator with domain being 𝐷(𝑃^−𝛼) and its image being the Range (𝑃^𝛼). 𝑇₀ is clearly an invertible operator and 𝑆(𝑅^𝑁)⊂Range(𝑇) means that the equation (𝑇𝑃^−𝛼)(𝜑)=𝑓 has always a nonzero solution in 𝐷(𝑃^−𝛼) for any given 𝑓∈𝑆(𝑅^𝑁). Note that the condition that 𝑇⁻¹(𝑓) be a dense subset on Range (𝑃^−𝛼) means that 𝑇⁻¹𝑓,𝑃^−𝛼𝜑_{𝐿2(𝑅𝑁)}=0 (2.35) has as unique solution the trivial solution 𝑓≡0.

Let us suppose too that 𝑇⁻¹∶𝑆(𝑅^𝑁)→𝐻_𝛼 be a continuous application and the bilinear term (ℒ⁻¹(𝑗))(𝑗) be a continuous application in the Hilbert spaces 𝐻_+𝛼⊃𝑆(𝑅^𝑁), namely, if 𝑗−−→_𝑛^𝐿2𝑗, then ℒ⁻¹∶𝑃^−𝛼𝑗−−→_𝑛^𝐿2ℒ⁻¹𝑃^−𝛼𝑗, for {𝑗_𝑛}_𝑛∈ℤ and (𝑓_𝑛,𝑗)∈𝑆(𝑅^𝑁).

By a direct application of the Minlos Theorem, we have the result 𝑍𝑗−1=exp2ℒ⁻¹𝑗𝑗=_{(𝑇−1)∗𝐻𝛼}𝑑𝜇(𝑇)exp𝑖𝑇𝑗. (2.36)

Here the topological space support is given by 𝑇^−1∗ℋ_𝛼=𝑃^−𝛼𝑇₀𝑃^𝛼−1∗(𝑃^𝛼(𝑆(𝑅^𝑁=𝑃)))_𝛼∗𝑇₀^−1∗𝑃^−𝛼∗𝑃^𝛼𝑆𝑅^𝑁=𝑃^𝛼𝑇₀⁻¹𝐿²𝑅^𝑁. (2.37)

In the important case of ℒ=(−Δ+𝑚²)∶𝑆^′(𝑅^𝑁)→𝑆(𝑅^𝑁) and 𝑇₀𝑇^∗₀=(𝜒_Ω(2𝜋)^𝑁/2/vol(Ω))(−Δ+𝑚²)^−2𝛽×𝜒_Ω∈∮₁(𝐿²(𝑅^𝑁)) since Tr(𝑇₀𝑇^∗₀)=(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 𝑅^𝑁 may be taken as the measurable subset below supp𝑑_{(−Δ+𝑚2)}𝜇𝜑=−Δ+𝑚^2+𝛼𝐼_Ω(𝑥)−Δ+𝑚^2+𝛽𝐿²𝑅^𝑁 (2.38) since ℒ⁻¹𝑃^−𝛼=(−Δ+𝑚²)^−1−𝛼 is always a bounded operator in 𝐿²(𝑅^𝑁) 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 volume Ω→𝑅^𝑁. 𝜑(𝑥)=−Δ+𝑚^{2(𝑁/4)+𝜀−1}𝑓(𝑥) (2.39) with a function 𝑓(𝑥)∈𝐿²(𝑅^𝑁) 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^′(𝑅^𝑁) 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 of 𝑆^′(𝑅^𝑁) to the measurable subset as given by (2.38) denoted by 𝑊(𝑅^𝑁)−1exp2−Δ+𝑚²⁻¹𝑗𝑗=_{𝑆′(𝑅𝑁)}𝑑_{(−Δ+𝑚2)}𝜇𝜑𝑒^{𝑖𝜑(𝑗)}=_{𝑊(𝑅𝑁)⊂𝑆′(𝑅𝑁)}𝑑_{(−Δ+𝑚2)}𝜇𝑓𝑒^{𝑖⟨𝑓,(−Δ+𝑚2)(𝑁/4)+𝜀−1𝑓⟩𝐿2𝑁)(𝑅}. (2.40)

Appendices

A. Some Comments on the Support of Functional Measures in Hilbert Space

Let us comment further on the application of the Minlos Theorem in Hilbert Spaces. In this case one has a very simple proof which holds true in general Banach Spaces (𝐸,‖‖).

Let us thus, give a cylindrical measures 𝑑^∞𝜇(𝑥) in the algebraic dual 𝐸^alg of a given Banach Space 𝐸 [4–6].

Let us suppose either that the function ‖𝑥‖ belongs to 𝐿¹(𝐸^alg,𝑑^∞𝜇(𝑥)). Then the support of this cylindrical measures will be the Banach Space 𝐸.

The proof is the following.

Let 𝐴 be a subset of the vectorial space 𝐸^alg (with the topology of pontual convergence), such that 𝐴⊂𝐸^𝑐 (so ‖𝑥‖=+∞) (𝐸 can always be imbeded as a cylindrical measurable subset of 𝐸^alg—just use a Hammel vectorial basis to see that). Let be the sets 𝐴_𝑛={𝑥∈𝐸^alg∣‖𝑥‖≥𝑛}. Then we have the set inclusion 𝐴⊂⋂^∞_𝑛=0𝐴_𝑛, so its measure satisfies the estimates below: 𝜇(𝐴)≤liminf_𝑛𝜇𝐴_𝑛=liminf_𝑛𝜇𝑥∈𝐸^alg∣‖𝑥‖≥𝑛≤liminf_𝑛1𝑛_𝐸^alg‖𝑥‖𝑑^∞𝜇(𝑥)=liminf_𝑛‖𝑥‖_𝐿¹_(𝐸^alg_,𝑑^∞_𝜇)𝑛=0. (A.1)

Leading us to the Minlos theorem that the support of the cylindrical measure in 𝐸^alg is reduced to the own Banach Space 𝐸.

Note that by the Minkowisky inequality for general integrals, we have that ‖𝑥‖²∈𝐿¹(𝐸^alg,𝑑^∞𝜇(𝑥)). Now it is elementary evaluation to see that if 𝐴⁻¹∈∮₁(ℳ), when 𝐸=ℳ, a given Hilbert Space, we have that _ℳalg𝑑^∞_𝐴𝜇(𝑥)⋅‖𝑥‖²=Tr_ℳ𝐴⁻¹<∞. (A.2) This result produces another criterium for supp 𝑑^∞_𝐴𝜇=ℳ (the Minlos Theorem), when 𝐸=ℳ is a Hilbert Space.

It is easy too to see that if _ℳ‖𝑥‖𝑑^∞𝜇(𝑥)<∞ (A.3) then the Fourier-Transformed functional 𝑍𝑗=_ℳ𝑒^{𝑖(𝑗,𝑥)ℳ}𝑑^∞𝜇(𝑥) (A.4) is continuous in the norm topology of ℳ.

Otherwise, if 𝑍(𝑗) is not continuous in the origin 0∈ℳ (without loss of generality), then there is a sequence {𝑗_𝑛}∈ℳ and 𝛿>0, such that ‖𝑗_𝑛‖→0 with 𝛿≤𝑍𝑗_𝑛−1≤_ℳ𝑒^{𝑖(𝑗𝑛,𝑥)ℳ}−1𝑑^∞𝜇(𝑥)≤_ℳ𝑗_𝑛,𝑥𝑑^∞𝜇(𝑥)≤𝑗_𝑛ℳ‖𝑥‖𝑑^∞𝜇(𝑥)⟶0, (A.5) a contradiction with 𝛿>0.

Finally, let us consider an elliptic operator 𝐵 (with inverse) from the Sobelev space 𝑊^−2𝑚(Ω) to 𝑊^2𝑚(Ω). Then by the criterium given by (A.2) if Tr_𝐿²_(Ω)(𝐼+Δ)^+𝑚/2𝐵⁻¹(𝐼+Δ)^+𝑚/2<∞, (A.6) we will have that the path integral written below is well defined for 𝑥∈𝑊^+2𝑚(Ω) and 𝑗∈𝑊^−2𝑚(Ω). Namely, −1exp2𝑗,𝐵⁻¹𝑗_𝐿2(Ω)=_{𝑊2𝑚(Ω)}𝑑_𝐵𝜇(𝑥)exp𝑖𝑗,𝑥_𝐿2(Ω). (A.7)

By the Sobolev theorem which means that the embeded below is continuous (with Ω⊆𝑅^𝜈 denoting a smooth domain), one can further reduce the measure support to the Hölder 𝛼 continuous function in Ω if 2𝑚−(𝜈/2)>𝛼. Namely, we have an easy proof of the famous Wiener Theorem on sample continuity of certain path integrals in Sobolev Spaces 𝑊^2𝑚(Ω)⊂𝐶^𝛼(Ω). (A.8a)

The above Wiener Theorem is fundamental in order to construct nontrivial examples of mathematically rigorous euclidean’s path integrals in spaces 𝑅^𝜈 of higher dimensionality, since it is a trivial consequence of the Lebesgue theorem that positive continuous functions 𝑉(𝑥) generate functionals integrable in {𝑊^2𝑚(Ω),𝑑_𝐵𝜇(𝜑)} of the form below exp−_Ω𝑉𝜑(𝑥)𝑑𝑥∈𝐿¹𝑊^2𝑚(Ω),𝑑_𝐵𝜇𝜑. (A.8b)

As a last important remark on Cylindrical Measures in Separable Hilbert Spaces, let us point to our reader that the support of such above measures is always a 𝜎-compact set in the norm topology of ℳ. In order to see such result let us consider a given dense set of ℳ, namely, {𝑥_𝑘}_𝑘∈𝐼+. Let {𝛿_𝑘}_𝑘∈𝐼+ be a given sequence of positive real numbers with 𝛿_𝑘→0. Let {𝜀_𝑛} be another sequence of positive real numbers such that ∑^∞_𝑛=1𝜀_𝑛<𝜀. Now it is straightforward to see that ℳ⊂⋃^∞_ℎ=1𝐵(𝑥_𝑘,𝛿_𝑘)⊂ℳ and thus limsup𝜇{⋃^𝑛_𝑘=1𝐵(𝑥_𝑘,𝛿_𝑘)}=𝜇(ℳ)=1. As a consequence, for each 𝑛, there is a 𝑘_𝑛, such that 𝜇(⋃^𝑘𝑛_𝑘=1𝐵(𝑥_𝑘,𝛿_𝑘))≥1−𝜀.

Now the sets 𝐾_𝜇=⋂^∞_𝑛=1[⋃^𝑘𝑛_𝑘=1𝐵(𝑥_𝑘,𝛿_𝑘)] are closed and totally bounded, so they are compact sets in ℳ with 𝜇(ℳ)≥1−𝜀. Let us now choose 𝜀=1/𝑛 and the associated compact sets {𝐾_𝑛,𝜇}. Let us further consider the compact sets 𝐾_𝑛,𝜇=⋃^𝑛_ℓ=1𝐾_ℓ,𝜇. We have that 𝐾_𝑛,𝜇⊆𝐾_𝑛+1,𝜇, for any 𝑛 and limsup𝜇(𝐾_𝑛,𝜇)=1. So, Supp𝑑𝜇=⋃^∞_𝑛=1𝐾_𝑛,𝜇, a 𝜎-compact set of ℳ.

We consider now a enumerable family of cylindrical measures {𝑑𝜇_𝑛} in ℳ satisfying the chain inclusion relationship for any 𝑛∈𝐼⁺Supp𝑑𝜇_𝑛⊆Supp𝑑𝜇_𝑛+1. (A.9) Now it is straightforward to see that the compact sets {𝐾_𝑛^(𝑛)}, where Supp𝑑𝜇_𝑚=⋃^∞_𝑛=1𝐾_𝑛^(𝑚), is such that Supp{𝑑𝜇_𝑚}⊆⋃^∞_𝑛=1𝐾_𝑛^(𝑛), for any 𝑚∈𝐼⁺.

Let us consider the family of functionals induced by the restriction of this sequence of measures in any compact 𝐾_𝑛^(𝑛). Namely, 𝜇_𝑛⟶𝐿_𝑛^(𝑛)𝑓=𝐾_𝑛^(𝑛)𝑓(𝑥)⋅𝑑𝜇_𝑝(𝑥). (A.8c) Here 𝑓∈𝐶_𝑏(𝐾_𝑛^(𝑛)). Note that all the above functionals in ⋃^∞_𝑛=1𝐶_𝑏(𝐾_𝑛^(𝑛)) are bounded by 1. By the Alaoglu-Bourbaki theorem they form a compact set in the weak star topology of (⋃^∞_𝑛=1𝐶_𝑏(𝐾_𝑛^(𝑛)))^∗, so there is a subsequence (or better the whole sequence) converging to a unique cylindrical measure 𝜇(𝑥). Namely, lim_{𝑛→∞ℳ}𝑓(𝑥)𝑑𝜇_𝑛(𝑥)=_ℳ𝑓(𝑥)𝑑𝜇(𝑥) (A.8d) for any 𝑓∈⋃^∞_𝑛=1𝐶_𝑏(𝐾_𝑛^(𝑛)).

B. On the Trace Class Property

A straightforward calculation gives the following expression for the trace of the integral operator on (2.1a) Tr_∮1𝜒_Ω(−Δ)^𝛼+𝜀²_IR⁻¹𝜒_Ω=^+∞_−∞^+∞_−∞𝜒_Ω𝑘−𝑝𝜒_Ω𝑝−𝑘𝑝^2𝛼+𝜀²_IR𝑑𝑝𝑑𝑘, (B.1a) where 𝜒_Ω𝑝=12𝜋^+∞_−∞𝜒_Ω(𝑥)𝑒^𝑖𝑝𝑥𝑑²𝑥. (B.1b)

Since by Parserval’s theorem 𝑑𝑘𝜒^+∞_−∞Ω𝑘−𝑝²𝑑²𝑘=^+∞_−∞𝜒_Ω(𝑥)²𝑑²𝑥=vol(Ω) (B.1c) one has the validity of the result written in (2.1b).