A Quantum Mechanical Example for Hodge Theory

TL;DR

This paper demonstrates how a specific quantum mechanical model exemplifies Hodge theory through its symmetry structures and conserved charges, linking quantum physics with differential geometry concepts.

Contribution

It provides a concrete physical realization of Hodge theory within the BRST formalism using the Friedberg-Lee-Pang-Ren model, connecting algebraic structures to differential geometry.

Findings

Symmetry operators realize de Rham cohomological operators

Harmonic states identified as physical states

Physicality criteria derived from conserved charges

Abstract

On the basis of (i) the discrete and continuous symmetries (and corresponding conserved charges), (ii) the ensuing algebraic structures of the symmetry operators and conserved charges, and (iii) a few basic concepts behind the subject of differential geometry, we show that the celebrated Friedberg-Lee-Pang-Ren (FLPR) quantum mechanical model (describing the motion of a single non-relativistic particle of unit mass under the influence of the general spatial 2D rotationally invariant potential) provides a tractable physical example for the Hodge theory within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism where the symmetry operators and conserved charges lead to the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. We concisely mention the Hodge decomposition theorem in the quantum Hilbert space of states and choose the harmonic states as the real physical states of our theory. We discuss the physicality criteria w.r.t. the conserved and nilpotent versions of the (anti-)BRST and (anti-)co-BRST charges and the physical consequences that ensue from them.