The theory of Fourier&ndash';Laplace transformation was developed for the functionals defined on the Gelfand–Shilov spaces of type S and the corresponding generalization of Vladimirov's theorems on functions holomorphic in tubular cones was obtained. The existence of smallest carrier cones was proved for the analytic functionals belonging to the classes $(S^\alpha)'$ and $(S^\alpha_\beta)'$, $\alpha<1$, and analogues of some basic structure theorems (including density and decomposition theorems) of the theory of hyperfunctions were established for these classes. The theory of Lorentz-covariant distributions was extended to ultradistributions, hyperfunctions and analytic functionals. The test function space $S^1_1$ corresponding to Fourier hyperfunctions was proposed as a universal object for formulating local quantum field theory (QFT). An abstract version of Ruelle's theorem on cluster decomposition properties of vacuum expectation values of quantum fields was formulated and proved by using the theory of quasianalytic classes. Namely, if two distributions coincide in an open cone and the supports of their Fourier transforms are separated by a nonzero distance, then both of them have an exponential decrease of order $\geq 1$ inside this cone. An extension of this theorem to analytic functionals was applied to construct the space of scattering states and the scattering matrix for nonlocal interactions of particles. An axiomatic formulation of nonlocal QFT was developed in terms of operator-valued highly singular generalized functions and a new derivation of the spin-statistics relation and CPT symmetry was presented. This derivation covers nonlocal fields and is based on exploiting the notion of analytic wave front set. A simple and general method for the operator realization of Wick-ordered entire functions of the indefinite metric free fields in Fock–Hilbert–Krein space was developed through the use of an appropriate generalization of the Paley–Wiener–Schwartz theorem. A number of papers were devoted to an comparative analysis of the topological obstructions to globally fixing the gauge in non-Abelian gauge theories and in string theory and also to an investigation of geometrical and functional-analytic structure of the infinite-dimensional principle bundle determined by the action of the group of gauge transformations on gauge fields. For non-Abelian gauge theory, the principle bundle was shown to be irreducible to a finite-dimensional subgroup. It was proved that, after an invariant regularization suppressing the ultraviolet divergences, the Gaussian measure of the functional integrals of Yang–Mills theory is supported by those function classes that admit a local gauge choice.
Main publications:
Soloviev M.A. Beyond the theory of hyperfunctions, in: "Developments in Mathematics: The Moscow School", V. Arnold and M. Monastyrsky (eds.). London: Chapman and Hall, 1993, 131-193.
Soloviev M.A. An extension of distribution theory and of the Paley-Wiener-Schwartz theorem related to quantum gauge theory. Commun. Math. Phys., 1997, 184, 579-596.
Soloviev M.A. Wick-ordered entire functions of the indefinite metric free field. Lett. Math. Phys., 1997, 41, 265-277.
