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 $[X_{ij},Y_{k\ell }]=-A_{i\ell }{B}_{kj}$, satisfying suitable commutation relations (in particular, $X$ is a Manin matrix), the following identity holds: $\mathrm{coldet}X\mathrm{coldet}Y=<0|\mathrm{coldet}(aA+X(I-a^{†}B{)}^{-1}Y)|0>$. Furthermore, if also $Y$ is a Manin matrix, $\mathrm{coldet}X\mathrm{coldet}Y=\int \mathcal{D}(\psi ,{\psi }^{†})\exp [{\sum }_{k\ge 0}\frac{1}{k+1}({\psi }^{†}A\psi {)}^k({\psi }^{†}X{B}^{k}Y\psi )]$. Notations: $<0|$, $|0>$, 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 ${\psi }_i^{†}$ 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 $[X_{ij},X_{k\ell }]=[Y_{ij},Y_{k\ell }]=0$.

40 pages

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