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

We prove that, for X, Y, A and B matrices with entries in a non-commutative ring such that [Xij,Ykℓ]=−AiℓBkj, satisfying suitable commutation relations (in particular, X is a Manin matrix), the following identity holds: coldetXcoldetY=<0|coldet(aA+X(I−a†B)−1Y)|0>. Furthermore, if also Y is a Manin matrix, coldetXcoldetY=∫D(ψ,ψ†)exp[∑k≥01k+1(ψ†Aψ)k(ψ†XBkYψ)]. 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 ψ†i and ψi 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 [Xij,Xkℓ]=[Yij,Ykℓ]=0.