On Fock-Bargmann space, Dirac delta function, Feynman propagator, angular momentum and SU(3) multiplicity free

TL;DR

This paper develops new integral representations and generating functions for SU(2) and SU(3) symbols using Fock-Bargmann space, simplifying calculations of angular momentum coupling in quantum mechanics.

Contribution

It introduces novel integral formulas and generating functions for SU(2) and SU(3) symbols based on Fock-Bargmann space, enhancing computational methods in angular momentum theory.

Findings

Derived integral representations of 3j and 6j symbols.

Established generating functions for spherical harmonics and SU(2) characters.

Presented a new expression for SU(3) 3j symbols as a product involving SU(2) symbols.

Abstract

The Dirac delta function and the Feynman propagator of the harmonic oscillator are found by a simple calculation using Fock Bargmann space and the generating function of the harmonic oscillator. With help of the Schwinger generating function of Wigner's D-matrix elements we derive the generating function of spherical harmonics, the quadratic transformations and the generating functions of: the characters of SU (2), Legendre and Gegenbauer polynomials. We also deduce the van der Wearden invariant of 3-j symbols of SU (2). Using the Fock Bargmann space and its complex conjugate we find the integral representations of 3j symbols, function of the series, and from the properties of we deduce a set of generalized hypergeometric functions of SU (2) and from Euler's identity we find Regge symmetry. We find also the integral representation of the 6j symbols. We find the generating function and a new expression of the 3j symbols for SU (3) multiplicity free. Our formula of SU (3) is a product of a constant, 3j symbols of SU (2) by . The calculations in this work require only the Gauss integral, well known to undergraduates.