Coherent states and rational surfaces

TL;DR

This paper explores the geometric structure of generalized coherent states, revealing they form rational curves and surfaces via Veronese embeddings, and extends these concepts to indefinite inner product spaces.

Contribution

It introduces a geometric framework for generalized coherent states using algebraic surfaces and extends the theory to indefinite inner product spaces.

Findings

Coherent state spaces form rational curves and surfaces.

Veronese embeddings facilitate parameterization of coherent states.

Extension to indefinite inner product spaces introduces new generalized states.

Abstract

The state spaces of generalised coherent states associated with special unitary groups are shown to form rational curves and surfaces in the space of pure states. These curves and surfaces are generated by the various Veronese embeddings of the underlying state space into higher-dimensional state spaces. This construction is applied to the parameterisation of generalised coherent states, which is useful for practical calculations and provides an elementary combinatorial approach to the geometry of the coherent state space. The results are extended to Hilbert spaces with indefinite inner products, leading to the introduction of a new kind of generalised coherent states.