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Representation of a quantum field Hamiltonian in \(p\)-adic Hilbert space. (English. Russian original) Zbl 0968.46519
Theor. Math. Phys. 112, No. 3, 1081-1096 (1997); translation from Teor. Mat. Fiz. 112, No. 3, 355-374 (1997).
Summary: Gaussian measures on infinite-dimensional \(p\)-adic spaces are defined 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 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 \(L_2\)-space with respect to a \(p\)-adic Gaussian measure.

MSC:
46N50 Applications of functional analysis in quantum physics
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
81T05 Axiomatic quantum field theory; operator algebras
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