Berezin quantization and unitary representations of Lie groups

Abstract

In 1974, Berezin proposed a quantum theory for dynamical systems having a Kähler manifold as their phase space. The system states were represented by holomorphic functions on the manifold. For any homogeneous Kähler manifold, the Lie algebra of its group of motions may be represented either by holomorphic differential operators ("quantum theory"), or by functions on the manifold with Poisson brackets, generated by the Kähler structure ("classical theory"). The Kähler potentials and the corresponding Lie algebras are constructed now explicitly for all unitary representations of any compact simple Lie group. The quantum dynamics can be represented in terms of a phase-space path integral, and the action principle appears in the semi-classical approximation.

A new perspective on functional integration

“…Choose a Kähler polarization on a non-trivial symplectic manifold, so that M is a Kähler manifold M of complex dimension n. It is convenient (though not necessary) to take the phase space to be the product manifold M × M with coordinates (ζ, z) and complex dimension 2n. Following the work of Berezin [21] and Bar-Moshe and Marinov [22], we use generalized coherent state wave functions φ ζ ′ (ζ) = exp K(ζ, ζ ′ ) where K(ζ, ζ ′ ) is the Kähler potential. Consider the function φ : M × M → C given by φ(ζ, ζ ′ ) = e K(ζ,ζ a ) and take z M : T → C n such that z M (t b ) = 0 and z M : T → C n such that z M (t a ) = 0.…”

Weyl–Wigner–Moyal formulation of a Dirac quantized constrained system

“…Therefore, the Dirac-Hamilton equations (12,13) can be written in terms of Dirac brackets as follows [11]:ż Φ µ (z) = 0 (16)…”

A Method for Weight Multiplicity Computation Based on Berezin Quantization

“…In the parametrization we use, the principal covariant symbols have the property that their restriction to the largest Schubert cell Σ s ⊂ G/T is polynomial in the affine coordinates of Σ s [5,6]. The reproduction property of is expressed through the relation:…”

“…The Berezin principal symbol of the projector onto the subspace of weight m, restricted to the largest Schubert cell Σ s of G/T is polynomial in the affine coordinates of Σ s in both of its arguments [5,6]. This parametrization is used to compute the multiplicity of the weight m in π λ as follows: Proposition 3.…”