p-adic Hilbert space representation of quantum systems with an infinite number of degrees of freedom

Gaussian measures on infinite-dimensional p-adic spaces are introduced and the corresponding L(2)-spaces of p-adic valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in p-adic L(2)-spaces, p-adic Hilbert space representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. Many Hamiltonians with potentials which are too singular to exist as functions over reals are realized as bounded symmetric operators in L(2)-spaces with respect to a p-adic Gaussian measure.