Representation of a quantum field Hamiltonian inp-adic Hilbert space

Abstract

Gaussian measures on infinite-dimensional p-adic spaces are defined and the corresponding L2-spaces of p-adic-valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in such spaces and the formal analogy with the usual Segal representation is discussed. It is found that the parameters of the p-adic infinite-dimensional Weyl group are defined only on some balls. In p-adic Hilbert space, representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. The Hamiltonians with singular potentials are realized as bounded symmetric operators in L2-space with respect to a p-adic Gaussian measure.