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

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.


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 infinitedimensional quantum field path integrals in general finite volume space times Ω ⊂ R ν ν 2, 4, . . .by means of the analytical regularization scheme 7 .

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 L 2 R 2 defined by the finite volume, infrared regularized, and α-power Laplacian acting on L 2 R N as an operatorial quadratic form

2.1a
Here χ Ω denotes the multiplication operator defined by the characteristic function χ Ω x of the compact region Ω ⊂ R 2 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 R 2 by the Minlos theorem of Appendix A, since for α > 1, the operator defines a trace class operator on L 2 R 2 , namely,

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 bare V 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 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 one can show that the interaction function ,ε IR 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 we have that the upper bound estimate below holds true

2.5a
With we have, thus, the more suitable form after realizing the d 2 k i and d 0 α,ε IR μ ϕ integrals, respectively x i , x j 1≤i≤m 1≤j≤m .

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Here G N α 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! .
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

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 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 with m 2 a positive "mass" parameters.
Let us note that L −1 Ω,f is an operator of trace class on L 2 R ν if and only if the result below holds true

2.12
In this case, let us consider the double functional integral with functional domain 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
A simple direct application of the Fubbini-Tonelli theorem on the exchange of the integration order on 2.13 leads us to the effective λϕ 4 -Formal functional integral representation

2.15
Note that if one introduces from the beginning a bare mass parameter m 2 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 L 2 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 → 1 m 2 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 μ 2 as considered in 2.1a

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An explicit expression in momentum space for the Green function of the α-power of −Δ α μ 2 0 given by 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 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
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

2.20
In this case, one obtains that the finite-volume p ϕ 2 interactions exp 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 L 1 L 2N R ν ; d 0 α 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 .

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,

2.23
here ϕ ∈ W n 2 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 W −n 2 Ω the topological dual of W n 2 Ω 4-6 .By a straightforward application of the well-known Sobolev immersion theorem, we have that for the case of including k as a real number the functional Sobolev space W n 2 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

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

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 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 H 0 E, , 0 , after its topological completion.
We suppose the following.
a There is a continuous positive definite functional in H 0 , Z j , with an associated cylindrical measure dμ.
b There is a Hilbert-Schmidt operator T : We have thus, that the support of the measure satisfies the relationship

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 ⊂ L 2 R N → L 2 R N .Let H α be the following Hilbert space: , for α a real number .

2.31
We can see that for α > 0, the operators below are isometries among the following subspaces:

2.34
If one considers T a given Hilbert-Schmidt operator on H α , the composite operator T 0 P α TP −α is an operator with domain being D P −α and its image being the Range P α .T 0 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 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 → L −1 P −α j, for {j n } n∈ and f n , j ∈ S R N .By a direct application of the Minlos Theorem, we have the result

2.36
Here the topological space support is given by

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In the important case of L −Δ m 2 : S R N → S R N and T 0 T * with the choice P −Δ m 2 , 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 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 .
with a function f x ∈ L 2 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

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 L 1 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 0 A n , so its measure satisfies the estimates below: 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 ∈ L 1 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 This result produces another criterium for supp d ∞ A μ M the Minlos Theorem , when E M is a Hilbert Space.
It is easy too to see that if 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 a contradiction with δ > 0.

ΩF
ϕ x d ν x 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 Ω F ϕ x d ν x ≤ exp gγ .
,x M d ∞ μ x A.4is continuous in the norm topology of M.