On relation between geometric momentum and annihilation operators on a two-dimensional sphere

TL;DR

This paper explores the relationship between geometric momentum and annihilation operators on a 2D sphere, revealing how coherent states incorporate extrinsic curvature effects through a new geometric momentum concept.

Contribution

It introduces a geometric momentum dependent on extrinsic curvature and demonstrates how annihilation operators on the sphere relate to this momentum and position operators.

Findings

Coherent states on the sphere are expressed as { extalpha}x+i{eta}p.

Geometric momentum depends on extrinsic curvature and embedding.

Coherent states reflect extrinsic geometric effects beyond intrinsic surface properties.

Abstract

With a recently introduced geometric momentum that depends on the extrinsic curvature and offers a proper description of momentum on two-dimensional sphere, we show that the annihilation operators whose eigenstates are coherent states on the sphere take the expected form {\alpha}x+i{\beta}p, where {\alpha} and {\beta} are two operators that depend on the angular momentum and x and p are the position and the geometric momentum, respectively. Since the geometric momentum is manifestly a consequence of embedding the two-dimensional sphere in the three-dimensional flat space, the coherent states reflects some aspects beyond the intrinsic geometry of the surfaces.