[Paper Review] Coherent States Quantization for Generalized Bargmann Spaces with Formulae for their Attached Berezin Transforms in Terms of the Laplacian on Cn

Zouhaı̈r Mouayn|arXiv (Cornell University)|Jan 2, 2011

Mathematical Analysis and Transform Methods14 references4 citations

TL;DR

This paper presents a coherent states quantization approach for generalized Bargmann spaces in ℂⁿ, deriving their Berezin transforms via an orthonormal basis and an addition formula for Laguerre polynomials. It establishes two new formulae expressing these transforms as functions of the Euclidean Laplacian, offering new mathematical tools for quantum systems with magnetic fields, particularly in modeling diamagnetism in spinless Bose systems.

ABSTRACT

While dealing with a class of generalized Bargmann spaces, we rederive their reproducing kernels from the knowledge of an orthonormal basis by using an addition formula for Laguerre polynomials involving the disk polynomials. We construct for each of these spaces a set of coherent states to apply a coherent states quantization method. This provides us with another way to recover the Berezin transforms attached to these spaces. Finally, two new formulae representing these transforms as functions of the Euclidean Laplacian are established and a possible physics direction for the application of such formulae is discussed.

Motivation & Objective

• To provide a direct derivation of reproducing kernels for generalized Bargmann spaces using an orthonormal basis and an addition formula for Laguerre polynomials.
• To construct coherent states for each generalized Bargmann space using a generalized formalism to enable coherent states quantization.
• To recover Berezin transforms via the coherent states quantization method, offering an alternative to the standard Toeplitz operator approach.
• To derive two new representations of the Berezin transform as functions of the Euclidean Laplacian in ℂⁿ.
• To explore physical applications of these formulae in quantum systems with magnetic fields, particularly in modeling diamagnetism of spinless Bose systems.

Proposed method

• Derive the reproducing kernel of generalized Bargmann spaces using an addition formula involving disk polynomials and Laguerre polynomials.
• Construct coherent states via a generalized formalism by superposing orthonormal basis vectors with phase factors and normalization.
• Apply the Klauder-Berezin coherent state quantization scheme, mapping classical observables to operators via an integral over coherent states.
• Compute the Fourier transform of the Berezin kernel function hₘ(w) to express the Berezin transform as a function of the Laplacian.
• Use known integral identities involving Laguerre polynomials and Bessel functions to evaluate the Fourier transform and derive polynomial expressions in the Laplacian.
• Establish two new formulae for the Berezin transform Bₘ as functions of the Euclidean Laplacian △ℂⁿ, with explicit polynomial coefficients.

Experimental results

Research questions

• RQ1 Can the reproducing kernel of generalized Bargmann spaces be derived directly from an orthonormal basis using an addition formula for Laguerre polynomials?
• RQ2 How can coherent states be systematically constructed for generalized Bargmann spaces to enable quantization?
• RQ3 Can the Berezin transform associated with these spaces be recovered via coherent state quantization, independent of the Toeplitz operator formalism?
• RQ4 What are the explicit representations of the Berezin transform as functions of the Euclidean Laplacian in ℂⁿ?
• RQ5 What physical insights do these Laplacian-based formulae provide for quantum systems with magnetic fields, such as diamagnetic Bose systems?

Key findings

• The reproducing kernel for generalized Bargmann spaces is derived directly from an orthonormal basis using an addition formula involving disk polynomials and Laguerre polynomials.
• Coherent states are successfully constructed for each generalized Bargmann space, enabling a new derivation of the associated Berezin transforms via coherent state quantization.
• The Berezin transform Bₘ is expressed as a convolution with a kernel hₘ(w), whose Fourier transform is computed via Bessel function identities and Laguerre polynomial expansions.
• Two new formulae are established expressing the Berezin transform Bₘ as functions of the Euclidean Laplacian △ℂⁿ, with explicit polynomial coefficients in the Laplacian operator.
• The formulae reveal a precise mathematical link between the magnetic Schrödinger operator (via Bₘ) and the non-magnetic Schrödinger operator (via the Laplacian), mediated by an exponential prefactor and a polynomial in the Laplacian.
• These results provide new analytical tools for studying diamagnetism in spinless Bose systems, where the Laplacian represents the free-particle Hamiltonian and Bₘ encodes Landau level effects.