Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation

Abstract

We prove that, for $X$, $Y$, $A$ and $B$ matrices with entries in a non-commutative ring such that

satisfying suitable commutation relations (in particular, $X$ is a Manin matrix), row-pseudo-commutative matrix (a Manin matrix), the following identity holds:

Furthermore, if also $Y$ is a Manin matrix, $[Y_{ij},Y_{kl}]=0$ for $i\ne k$, $j\ne l$

Here $⟨0\mid$ and $\mid 0⟩$, are respectively the bra and the ket of the ground state, $a^{†}$ and $a$ the creation and annihilation operators of a quantum harmonic oscillator, while ${\bar{\psi }}_i$ and ${\psi }_i$ are Grassmann variables in a Berezin integral. These results should be seen as a generalization of the classical Cauchy–Binet formula, in which $A$ and $B$ are null matrices, and of the non-commutative generalization, the Capelli identity, in which $A$ and $B$ are identity matrices and $[X_{ij},X_{k\ell }]=[Y_{ij},Y_{k\ell }]=0$.