Abstract
We prove that, for
X
,
Y
,
A
and
B
matrices with entries in a non-commutative ring such that
\hbox{$[X_{ij},Y_{k\ell}]=-A_{i\ell} B_{kj}$},
satisfying suitable commutation relations (in particular,
X
is a Manin matrix), row-pseudo-commutative matrix (a Manin matrix), the following identity holds:
\mathrm {col-det } \ X \ \mathrm { col-det } \ Y \ = \langle 0\mid \mathrm { col-det } \ (aA + X (I-a^{\dagger} B)^{-1} Y)\mid 0\rangle
Furthermore, if also
Y
is a Manin matrix,
[Y_{ij},Y_{kl}]=0
for
i\neq k
,
j\neq l
\mathrm {col-det } \ X \ \mathrm { col-det } \ Y =\int \mathcal{D}(\psi, \bar{\psi}) \exp \big(\sum_{k \geq 0}\frac{(\bar{\psi} A \psi)^{k}}{k+1}(\bar{\psi} X B^k Y \psi)\big)
Here
\langle 0 \mid
and
\mid 0\rangle
, are respectively the bra and the ket of the ground state,
a^{\dagger}
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
.