Berezin-Toeplitz Quantization over Matrix Domains

Abstract

Let Ω be a symplectic manifold, with symplectic form ω, and $\mathcal{H}$ a subspace of L²(Ω, d µ), for some measure µ, admitting a reproducing kernel K. For φ ∈ C^∞(Ω), the Toeplitz operator T _φ with symbol φ is the operator on $\mathcal{H}$ defined by

$$ {{T}_{\phi }}f=P\left( {\phi f} \right),\:f\in \mathcal{H}, $$

where P: L²((Ω), d µ) → $\mathcal{H}$ is the orthogonal projection. Using the reproducing kernel K, this can also be written as

$${T_\phi }f(x) = \int_\Omega {d(y)\phi (y)K(x,y)d\mu (y)} .$$

We explore the possibility of extending the well-known Berezin-Toeplitz quantization to reproducing kernel spaces of vector-valued functions. In physical terms, this can be interpreted as accommodating the internal degrees of freedom of the quantized system. We analyze in particular the vector-valued analogues of the classical Segal-Bargmann space on the domain of all complex matrices and of all normal matrices, respectively, showing that for the former a semi-classical limit, in the traditional sense, does not exist, while for the latter only a certain subset of the quantized observables have a classical limit: in other words, in the semiclassical limit the internal degrees of freedom disappear, as they should. We expect that a similar situation prevails in much more general setups.

Dedicated to Gerard G. Emch on the occasion of his 70th birthday