Representations for creation and annihilation operators

TL;DR

This paper introduces a novel representation of creation and annihilation operators similar to the Bargmann representation, exploring its mathematical structure and potential applications in quantum field theories like Chern-Simons and gravity.

Contribution

A new operator representation is developed, connecting it with existing frameworks and analyzing its implications in gauge-fixed quantum field theories.

Findings

The new representation acts as multiplication and derivation operators.

Relations with the Schrödinger representation are established.

The representation in Chern-Simons theory is not a *-representation, affecting state evolution.

Abstract

A new representation -which is similar to the Bargmann representation- of the creation and annihilation operators is introduced, in which the operators act like "multiplication with" and like "derivation with respect to" a single real variable. The Hilbert space structure of the corresponding states space is produced and the relations with the Schroedinger representation are derived. Possible connections of this new representation with the asymptotic wave functions of the gauge-fixed quantum Chern-Simons field theory and (2+1) gravity are pointed out. It is shown that the representation of the fields operator algebra of the Chern-Simons theory in the Landau gauge is not a *-representation; the consequences on the evolution of the states in the semiclassical approximation are discussed.