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Geometry Integrability and Quantization

Bloch-Messiah Decompositions of Symmetric Ray Transfer Matrices in \({\mathbb R}^4\)

Clementina D. Mladenova, Ivaïlo M. Mladenov

Published Jan 1, 2025• 0 citations
Publisher

Abstract

Boost matrices, eigenvalues, group factorization, Lorentz group, orthosymplectic matrices, squeezing matrices, Michelson-Morley experiment, FitzGerald-Lorentz contraction, history of special relativity, Lorentz-Poincar ether theory
DOI
10.7546/giq-31-2025-37-58

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The Bloch-Messiah (BM) reduction allows the decomposition of an arbitrarily complicated Gaussian unitary into a very simple scheme in which linear optical components are separated from nonlinear ones. The nonlinear part is due to the squeezing possibly present in the Gaussian unitary. The reduction is usually obtained by exploiting the singular value decomposition (SVD) of the matrices appearing in the Bogoliubov transformation of the given Gaussian unitary. This paper discusses a different approach, where the BM reduction is obtained in a straightforward way. It is based on the Takagi factorization of the (complex and symmetric) squeeze matrix and has the advantage of avoiding several matrix operations of the previous approach (polar decomposition, eigendecomposition, SVD, and Takagi factorization). The theory is illustrated with an application example in which the previous and present approaches are compared.

Physical Review A•Dec 12, 2016•48 citations
Bloch-Messiah Decompositions of Symmetric Ray Transfer Matrices in R4{\mathbb R}^4R4
The Bloch-Messiah (BM) reduction allows the decomposition of an arbitrarily complicated Gaussian unitary into a very simple scheme in which linear optical components are separated from nonlinear ones. The nonlinear part is due to the squeezing possibly present in the Gaussian unitary. The reduction is usually obtained by exploiting the singular value decomposition (SVD) of the matrices appearing in the Bogoliubov transformation of the given Gaussian unitary. This paper discusses a different approach, where the BM reduction is obtained in a straightforward way. It is based on the Takagi factorization of the (complex and symmetric) squeeze matrix and has the advantage of avoiding several matrix operations of the previous approach (polar decomposition, eigendecomposition, SVD, and Takagi factorization). The theory is illustrated with an application example in which the previous and present approaches are compared.