Nonlocal Scalar Quantum Field Theory—Functional Integration, Basis Functions Representation and Strong Coupling Expansion

Abstract

Nonlocal quantum field theory (QFT) of one-component scalar field φ in D-dimensional Euclidean spacetime is considered. The generating functional (GF) of complete Green functions Z as a functional of external source j, coupling constant g and spatial measure d μ is studied. An expression for GF Z in terms of the abstract integral over the primary field φ is given. An expression for GF Z in terms of integrals over the primary field and separable Hilbert space (HS) is obtained by means of a separable expansion of the free theory inverse propagator L ^ over the separable HS basis. The classification of functional integration measures D φ is formulated, according to which trivial and two nontrivial versions of GF Z are obtained. Nontrivial versions of GF Z are expressed in terms of 1-norm and 0-norm, respectively. In the 1-norm case in terms of the original symbol for the product integral, the definition for the functional integration measure D φ over the primary field is suggested. In the 0-norm case, the definition and the meaning of 0-norm are given in terms of the replica-functional Taylor series. The definition of the 0-norm generator Ψ is suggested. Simple cases of sharp and smooth generators are considered. An alternative derivation of GF Z in terms of 0-norm is also given. All these definitions allow to calculate corresponding functional integrals over φ in quadratures. Expressions for GF Z in terms of integrals over the separable HS, aka the basis functions representation, with new integrands are obtained. For polynomial theories φ 2 n , n = 2 , 3 , 4 , … , and for the nonpolynomial theory sinh 4 φ , integrals over the separable HS in terms of a power series over the inverse coupling constant 1 / g for both norms (1-norm and 0-norm) are calculated. Thus, the strong coupling expansion in all theories considered is given. “Phase transitions” and critical values of model parameters are found numerically. A generalization of the theory to the case of the uncountable integral over HS is formulated—GF Z for an arbitrary QFT and the strong coupling expansion for the theory φ 4 are derived. Finally a comparison of two GFs Z , one on the continuous lattice of functions and one obtained using the Parseval–Plancherel identity, is given.