N = 2 Supersymmetric Harmonic Oscillator: Basic Brackets Without Canonical Conjugate Momenta

TL;DR

This paper derives the fundamental (anti)commutation relations for an N=2 supersymmetric harmonic oscillator using symmetry principles, avoiding the traditional canonical conjugate momentum approach, and confirms their consistency with standard quantization methods.

Contribution

It introduces a symmetry-based method to obtain basic brackets in a supersymmetric quantum system without relying on canonical conjugate momenta.

Findings

Derived (anti)commutators using symmetry principles

Confirmed agreement with standard canonical brackets

Provided a novel approach for quantization in SUSY models

Abstract

We exploit the ideas of spin-statistics theorem, normal-ordering and the key concepts behind the symmetry principles to derive the canonical (anti)commutators for the case of a one (0 + 1)-dimensional (1D) N = 2 supersymmetric (SUSY) harmonic oscillator (HO) without taking the help of the mathematical definition of canonical conjugate momenta with respect to the bosonic and fermionic variables of this toy model for the Hodge theory (where the continuous and discrete symmetries of the theory provide the physical realizations of the de Rham cohomological operators of differential geometry). In our present endeavor, it is the full set of continuous symmetries and their corresponding generators that lead to the derivation of basic (anti)commutators amongst the creation and annihilation operators that appear in the normal mode expansions of the dynamical fermionic and bosonic variables of our present N = 2 SUSY theory of a HO. These basic brackets are in complete agreement with such kind of brackets that are derived from the standard canonical method of quantization scheme.