A representation of quantum field Hamiltonian in a $p$-adic Hilbert space

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. There is a formal analogy with the usual Segal representation. But there is also a large topological difference: parameters of the $p$-adic infinite-dimensional Weyl group are defined only on some balls (these balls are additive subgroups). $p$-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.