Calculation of Some Integrals over Gaussian Measure with Nuclear Covariance Operator in Separable Hilbert Space

Abstract

The main purpose of this paper is to construct convergent series for the approximate calculation of certain integrals over the Gaussian measure with a nuclear covariance operator, nonlocal propagator, in separable Hilbert space. Such series arise, for example, in the model with the interaction Lagrangian $\sinh^{2(p+1)}\varphi$, where $p \in \mathbb{N}$ and $\varphi$ is the scalar field, although the problem can be solved in general form for a fairly wide class of Lagrangians: an even, strictly convex, continuous, non-negative function with a single zero value for $\varphi=0$ and for $|\varphi|\rightarrow +\infty$ growing faster than $\varphi^{2}$. We strictly define the scattering matrix, $\mathcal{S}$-matrix, at the zero value of the classical field, argument of the $\mathcal{S}$-matrix, of such a theory in terms of the corresponding integral, find the iterated expansion for the integrand (the Gaussian measure doesn't expand) over two orthonormal bases of functions, prove the validity of summation and integration interchange and thus find the expansion of the $\mathcal{S}$-matrix at the zero value of the classical field into the iterated series in powers of the interaction action. The individual terms of the resulting series have the form of a canonical partition function (CPF), and the methods of statistical physics are applicable to them. In particular, we express them in terms of Bell polynomials. It is important to note that such iterated series cannot be reduced to the perturbation theory (PT) series, since in the proposed model the latter diverges as $e^{n^{2}}$, where $n \in \mathbb {N}$ is the PT order. Along the way, we provide detailed mathematical background, including Beppo Levi's monotone convergence theorem (MCT) and Henri Lebesgue's dominated convergence theorem (DCT), without which the presented calculation would be significantly more complex.