A Note on Functional Integration, Basis Functions Representation and Strong Coupling Expansion

The nonlocal quantum field theory (QFT) of one-component scalar field $\varphi$ in $D$-dimensional Euclidean spacetime is considered. The generating functional of total Green functions $\mathcal{Z}$ as a functional of external sources $j$, coupling constants $g$, and spatial measures $d\mu$ is studied. An expression for $\mathcal{Z}$ in terms of the abstract integral over the primary field $\varphi$ is given. An expression for $\mathcal{Z}$ in terms of integrals over the primary field and separable HS is obtained by means of a separable expansion of the free theory propagator $G$ over the separable Hilbert space (HS) basis. In terms of the original symbol for the product integral, a novel definition for the functional integration measure $\mathcal{D}\left[\varphi\right]$ over the primary field is given: The argument in favor of such a definition is given in the Appendix. This definition allows to calculate the corresponding functional integral in terms of quadrature. An expression for $\mathcal{Z}$ in terms of an integral over the separable HS with a new integrand is obtained. This is the generating functional $\mathcal{Z}$ in the basis functions representation. For polynomial theories $\varphi^{2n},\, n=2,3,4,\ldots,$ and for a nonpolynomial theory $\mathrm{sinh}^{4}\varphi$, an integral over the separable HS in terms of a power series over the inverse coupling constant $1/\sqrt{g}$ is calculated. Thus, the strong coupling expansion in all theories considered is given.


Introduction
The calculation of functional (path) integrals is now an outstanding problem in quantum field theory [1,2,3] as well as in other branches of theoretical and mathematical physics and infinite dimensional analysis [4,5,6,7,8,9,10,11,12,13,14,15,16]. The expansion over a small coupling constant g, the perturbation theory (PT) over g, does not solve this problem [17,18,19]. Actually, behavior of generating functionals of different Green functions families for large coupling constants g beyond the PT for any interaction Lagrangian is unknown.
In this note, we consider a nonlocal QFT of a single component scalar field in D-dimensional Euclidean spacetime of two types: polynomial and nonpolynomial [20,21,22,23,24,25,26,27,28,29,30,31]. A scalar field theory underlies multiple physical models: Higgs Scalar Particles, the Theory of Critical Phenomena (note an interesting book [32]), the Quantum Theory of Magnetism, Plasma Physics (note an interesting review [33]), the Theory of Developed Turbulence, Nuclear Physics (note an interesting book [34]), etc. And, to the extent that it is the simplest theory, it serves as a building block of more complicated physical models.
We formulate a generating functional of total Green functions Z as a functional of external source j, coupling constant g, and a spatial measure dµ in terms of an abstract functional integral of a primary field ϕ as a formal variable [35,36]. Using a separable expansion of a free theory propagator G over a separable HS basis (similar to [31]), we derive an expression for Z in terms of integrals over the primary field ϕ and over the separable HS (infinite lattice of variables t s ).
In the next step we introduce an original symbol for a product integral and define a novel functional integral measure D [ϕ] over a primary field. This definition allows us to calculate an integral over a primary field by quadrature. The arguments for this definition and comparison of definitions from mathematical analysis and probability theory point of view are given in the Appendix. Thus, we obtain an expression for a generating functional Z in terms of an integral over a Hilbert space -the generating functional Z in a basis functions representation. It is important to note that in this case an integrand is a more complicated expression than initial exponential function.
A generating functional Z in a basis functions representation is open to numerical methods for QFT on lattice with rather efficient computational use of resources, as an alternative to majorize the functional Z within a framework of variational principle: Solving corresponding variational problem to obtain a physically satisfactory estimate for Z [26,31].
A developed theory is applied to generally accepted models of theoretical and mathematical physics: polynomial theories ϕ 2n , n = 2, 3, 4, . . . , and for a nonpolynomial theory sinh 4 ϕ. For these theories we calculate an integral over a separable HS in terms of series over inverse coupling constant 1/ √ g. In other words, we get the strong coupling expansion in all theories considered. Note that such an expansion resembles the hopping parameter expansion in a lattice QFT as well as a high-temperature expansion in statistical physics [36]. However such theories turn out to be non-trivial for any dimension D of spacetime. We verify obtained results by using a nonlocal inverse propagator. Although, if we consider quasilocal ones, we need an additional renormalization scheme but such a scheme does not affect a convergence properties of series in 1/ √ g as a whole.
Thus, we propose a broad mathematical apparatus that allows to go beyond the PT and is able to shed light on a nonperturbative physics, described by various scalar field theories.
Since nonlocal theories study is motivated, in particular, by attempts to construct a quantum theory of gravity (QG) [37,38,39,40] and QFT in a curved spacetime, we would like to point out that the basis functions representation can draw parallels between QG on a continuous spacetime, and QG on a discrete lattice in HS. Such a theory of QG is popular nowadays, in particular, as Loop QG. Also note the paper on the study of nonlocality in string theory [41]. This note has the following structure: in section 2 we propose a general theory, sections 3 and 4 are devoted to polynomial and nonpolynomial QFTs, respectively. In the conclusion section (5), we give a final discussion of all obtained results. The note contains one Appendix.

General Theory
Traditionally we begin with generating functional of total Green functions Z in terms of an abstract integral over primary field ϕ [25,26,31,35,36]: where S 0 is free theory action, S 1 is an interaction action, g and j are functions that we feed to our functional. As for interaction action theories with interaction Lagrangian U (ϕ) = ϕ 2n , sinh 2n ϕ, n = 2, 3, 4, . . . , etc. can be considered. In other words, it is important that U (ϕ → ∞) → ∞ and even function of field ϕ. Let's keep things general for now and later specify a theory. Taking a closer look on free theory action one can rewrite it using basis functions representation (similar to [31]): The operatorL is the inverse Gaussian propagator, the function L(x, y) is the kernel ofL. We will consider this operator nonlocal and expandable in the manner given in the previous expression. An exponent with a free theory action can be rewritten using Gaussian integral trick with auxiliary variables t s : where, for the sake of more convenient notation, vector notation instead of HS index s is used, and a Gaussian measure dσ t over HS is introduced: Everything mentioned gives us Z as (it is important to note that we have rearranged the integration over dσ t and D[ϕ]!) where total source J t = j + i s t s D s = j + i t · D, and its scalar product with a primary field ϕ is: Remaining exponent we represent using continuous product notation (X is the set of all values x): Now it is time to specify functional measure. Its naive definition, leads to a pretty bad story. Instead, consider another version of continuous product (product integral), given not only over the set X, but also over the measure d D x: We come up with a following expressions for generating functional Z: where we introduced the function eter as follows: As already mentioned generating functional Z is a functional of a source j and a coupling constant g. Further in the note we will consider more general construction replacing a measure d D x → dµ(x): Now Z is functional of measure dµ as well. Motivation to do this is to exclude possibilities to face with divergences of a kind d D x = ∞ (if there is one). Our new measure dµ satisfies the normalization condition: For the normalization of measure we use a certain quantity V, so that it is possible to investigate the dependence on the latter, if necessary.

Polynomial theory
In this section we will get to more specific example and choose interaction Lagrangian to be U (ϕ) = ϕ 2n , n = 2, 3, 4, . . . In this case, the function eter has the following form (depending on 2n): Here a change of variables ϕ = ξ 2n √ g was made. Let a source equals zero j = 0. Then we obtain following integral -the Fourier transform of the function e −ξ 2n : which is even and alternating. A Figure 1 shows, how the function +∞ −∞ dξ e −ξ 2n +iwξ evolves with a change of w. For a comparison functions corresponding to ϕ 4 , ϕ 6 and ϕ 8 theories are presented. Next, consider the generating functional Z in case when j = 0: does not depend on t s . Let g(x) be a constant. As far as g = const, the functional C is the function of g: due to normalization condition dµ = V. The expression for Z reads: Function ln (eter 2n (x)) can be expended into series over 1/ √ g (strong coupling expansion): After integration over dµ we obtain: where the matrix in HS D s 1 s 2 = dµ(x)D s 1 (x)D s 2 (x). The expression for Z takes the form of the strong coupling expansion: Here the equality dσ t t s 1 t s 2 = δ s 1 s 2 was used. Since we are primarily interested in the translation-invariant case, the second term in the last line on the expansion can be made even simpler due to Finally, we obtain our generating functional as a series over coupling constant g. It is worth mentioning that first two terms does not depend on measure dµ which is not true for remaining ones: Thus, we have obtained the generating functional Z in terms of series over inverse coupling constant 1/ √ g. Note that we have performed the integration over the field ϕ and the Hilbert space in a certain order.

Nonpolynomial theory
In this section we consider a nonpolynomial theory sinh 4 ϕ. In this case, the interaction Lagrangian U (ϕ) is given by the expression: where ϕ 0 is the dimensional divisor. Thus, the function eter is given by (the source j = 0 as in the polynomial case): and ϕ = ϕ 0 . Expanding the integrand into the Taylor series over 1/ √ g (corresponding to a large coupling constant g) and omitting terms of odd ξ powers (since integration of odd function in symmetric limits yields zero) we obtain: Here Γ(z) is the gamma function. Next, consider the functional integral over measure D [ϕ] with the function eter: .
Expanding the exponent in the first line of the previous expression into a power series up to O 1 g the following expression reads: As soon as the measure dσ t is Gaussian, and the function D s (x) does not depend on t s , the following is true: Assume the following ratio of scales: ϕ 2 0 L (0) = α, where α is a positive constant. In this case, we obtain the following expression for the generating functional Z: For the case g(x) = const the previous expression reads: At the end of the subsection, we note that the critical value of α (where 1/ √ g correction changes sign) is α c = 1.
In conclusion of this section, we note the following: A Figure 2 demonstrates the same behavior in sinh 4 ϕ theory as in the polynomial case. Since the plotted function is not positive-definite, a strong coupling expansion is natural for the primary analysis of such a theory. However, this should not exclude a deeper analysis, which should be the subject of the furthest.

Conclusion
In this note we have studied the generating functional of total Green functions Z as a functional of an external source j, coupling constant g, and spatial measure dµ, both for polynomial and nonpolynomial interaction Lagrangians U (ϕ). An important property of the interaction Lagrangian chosen in this note is that U (ϕ → ∞) → ∞ and even function of field ϕ. If the Lagrangian was to be bounded, e.g. U (ϕ) = sin 4 ϕ, ϕ 2n e −ϕ 2 , n = 2, 3, 4, . . . , et cetera, we could use an expression for Z in terms of the grand canonical partition function. This partition function is a classical series over the interaction constant g in case of proposed nonpolynomial theory, and can be explored with statistical physics methods.
We have introduced an original symbol for a product integral and defined a novel functional integral measure D [ϕ] over a primary field. A discussion of this definition and comparison of definitions are given further in the Appendix. This way, we have calculated the integral over the primary field ϕ in quadrature, remaining with the integral over the HS only and with complicated integrand. As one can see from the plots, the integrand is not a positive definite quantity. For this reason, an interesting question of investigating the phase of such an integrand arises.
To overcome this complexity as well as to develop some mathematical intuition about the object obtained, we have proposed the calculation of Z in terms of the inverse coupling constant 1/ √ g. Alternatively, various inequalities could be used for analysis: Jensen inequality, Hölder inequality, etc. Besides this, as a necessity we have introduced a spatial measure dµ. In a sense, such a measure is equivalent to a smooth switch on-off of interaction with coupling constant g. The physical value of this measure can be an infinitely small D-dimensional hypercube d D x.
We have presented final results for the polynomial theories ϕ 2n , n = 2, 3, 4, . . . , and for the nonpolynomial theory sinh 4 ϕ. In the latter case, an interesting dependence on a dimensionless parameter α = ϕ 2 0 L (0) has arisen. This parameter is the ratio of the ultraviolet length parameter from L (0), and some length parameter from ϕ 0 . The latter in turn determines the scale of the interaction Lagrangian U (ϕ) and must be considered ultraviolet.
An interesting picture shows up: The interaction Lagrangian U (ϕ) is driven by the ultraviolet length scale, but the spatial measure dµ is driven by some infrared length scale. Such a hierarchy of length scales forms a nontrivial nonpolynomial field theory. The critical value of α (where 1/ √ g correction changes sign) is α c = 1.
All results obtained are original in the literature.

Appendix: Probability Theoretic Argument for the New Definition of Functional Integration Measure
In this Appendix we give the argument in favor of the new definition of functional integration measure based on mathematical analysis (brilliant book on mathematical analysis -the book of Vladimir Antonovich Zorich [42]) and probability theory (excellent book on probability theory in QFT -the book of Andreas Wipf [36]). First of all, let us fix a point x in the product integral. In this case, it suffices to investigate an ordinary integral over the field ϕ at a fixed point x. Consider two possibilities. In the first case, we consider the integral over a finite segment [a, b] with splitting ∆ϕ i , the splitting step (maximum splitting) λ (P ), and sampling points ψ i . In the second case, we consider the integral over the whole real axis with the probability density distribution d . The following expressions illustrate this: Type I where the quantity Compare Taylor series expansions over measure dµ in terms of functions f and F : On the other hand, by rewriting the expression above, one can resort to Jensen inequality for the exponent: f ∼ ln e f ⇒ e f ∼ e f , Jensen : e f ≤ e f .
Thus, in a linear and, probably, unique legal order over dµ, the connection of definitions of measures is controlled by Jensen inequality.
If from this moment we assume an additional smallness λ of the function f itself, controlled, by a small coupling constant g and a source j, then we can obtain an order of smallness λ to which both definitions of the measure are equivalent: We see that in the linear order in λ there is an exact equality. For the larger values of the coupling constant g and the source j, a naive definition of a measure leads to a catastrophe, while the definition in this note ensures the existence of a functional integral in a rigorous mathematical sense.