A Note on the Real Fermionic and Bosonic quadratic forms: Their Diagonalization and Topological Interpreation

TL;DR

This paper discusses the effective diagonalization of real fermionic and bosonic quadratic forms, highlighting a gap in the literature for complex hermitian forms and connecting the diagonalization to topological Morse inequalities.

Contribution

It introduces a novel method for diagonalizing real quadratic forms in fermionic and bosonic systems and links this to topological Morse inequalities, a previously overlooked aspect.

Findings

Effective diagonalization method for real quadratic forms

Connection between diagonalization and Morse inequalities

Identification of a gap in the literature for complex hermitian forms

Abstract

We explain in this note how real fermionic and bosonic quadratic forms can be effectively diagonalized. Nothing like that exists for the general complex hermitian forms. Looks like this observation was missed in the Quantum Field theoretical literature. The present author observed it for the case of fermions in 1986 making some topological work dedicated to the problem: how to construct Morse-type inequalities for the generic real vector fields? This idea also is presented in the note.