We prove that, for , , and matrices with entries in a non-commutative ring such that
satisfying suitable commutation relations (in particular, is a Manin matrix), row-pseudo-commutative matrix (a Manin matrix), the following identity holds:
Furthermore, if also is a Manin matrix, for ,
Here and , are respectively the bra and the ket of the ground state, and the creation and annihilation operators of a quantum harmonic oscillator, while and are Grassmann variables in a Berezin integral. These results should be seen as a generalization of the classical Cauchy–Binet formula, in which and are null matrices, and of the non-commutative generalization, the Capelli identity, in which and are identity matrices and .
Cite this article
Sergio Caracciolo, Andrea Sportiello, Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 1 (2014), no. 1, pp. 1–46DOI 10.4171/AIHPD/1