Symplectic group and Heisenberg group in p-adic quantum mechanics

TL;DR

This paper explores the mathematical structures of p-adic symplectic and Heisenberg groups in quantum mechanics, providing explicit decompositions, invariants, and representations relevant to p-adic quantum theory.

Contribution

It offers explicit descriptions of p-adic symplectic groups, their subgroups, and Iwasawa decompositions, along with analysis of representations of p-adic Heisenberg groups and associated invariants.

Findings

Explicit formulas for parabolic and compact subgroups of p-adic symplectic groups

Calculation of the Hasse invariant for quadratic forms from Lagrangian triples

Development of Weyl operators and Maslov index in p-adic quantum representations

Abstract

This paper treats mathematically some problems in p-adic quantum mechanics. We first deal with p-adic symplectic group corresponding to the symmetry on the classical phase space. By the filtrations of isotropic subspaces and almost self-dual lattices in the p-adic symplectic vector space, we explicitly give the expressions of parabolic subgroups, maximal compact subgroups and corresponding Iwasawa decompositions of some symplectic groups. For a triple of Lagrangian subspaces, we associated it with a quadratic form whose Hasse invariant is calculated. Next we study the various equivalent realizations of unique irreducible and admissible representation of p-adic Heisenberg group. For the Schrodinger representation, we can define Weyl operator and its kernel function, while for the induced representations from the characters of maximal abelian subgroups of Heisenberg group generated by the isotropic subspaces or self-dual lattice in the p-adic symplectic vector space, we calculate the Maslov index defined via the intertwining operators corresponding to the representation transformation operators in quantum mechanics.