Representations of fermionic star product algebras and the projectively flat connection

TL;DR

This paper develops a family of fermionic star products extending the fermionic Moyal product, establishing their algebraic properties and compatibility with quantum connections, thus advancing fermionic deformation quantization.

Contribution

It introduces a new class of fermionic star products that generalize existing products and demonstrates their compatibility with quantum geometric structures.

Findings

Star products form an algebra on fermionic functions.

Quantum states are preserved under the star product.

Compatibility with flat and projectively flat connections is established.

Abstract

We construct a family of fermionic star products generalising the fermionic Moyal product. The parameter space contains the polarisations necessary to define a quantum Hilbert space. We find a star product of fermionic functions on sections of the pre-quantum line bundle and show that the star product of any function on a quantum state remain a quantum state. Associativity implies a representation of the fermionic star product algebra on the quantum Hilbert space. The star product is compatible with both the flat connection on the bundle of fermionic functions and the projectively flat connection on the bundle of Hilbert spaces over the space of polarisations.