Normal forms, inner products and Maslov indices of general multimode squeezings

TL;DR

This paper develops an algebraic approach to normal forms and inner products of multimode squeezed states, introducing new methods for orthonormalization, normalization, and analyzing degeneracies in squeezing problems.

Contribution

It provides a novel algebraic construction for normal factorization and inner product calculation of multimode squeezed states, including a Maslov index analogue and Jordan decomposition applications.

Findings

Algebraic normal form construction for multimode squeezed states

Explicit formulas for inner products and normalization constants

Application to solvable high-dimensional squeezing problems

Abstract

In this paper we present a pure algebraic construction of the normal factorization of multimode squeezed states and calculate their inner products. This procedure allows one to orthonormalize bases generated by squeezed states. We calculate several correct representations of the normalizing constant for the normal factorization, discuss an analogue of the Maslov index for squeezed states, and show that the Jordan decomposition is a useful mathematical tool for problems with degenerate Hamiltonians. As an application of this theory we consider a non-trivial class of squeezing problems which are solvable in any dimension.