Representations of fermionic star product algebras and the projectively flat connection

Abstract

We consider a family of fermionic star products generalising the fermionic Moyal product. The parameter space contains polarisations used to define quantum Hilbert spaces in geometric quantisation. For each polarisation, we find a star product of fermionic functions on quantum states and show that the star product of any function on a quantum state remains a quantum state. We establish the associativity of such star products, which yields representations of the fermionic star product algebras on the quantum Hilbert spaces. The family of star products is compatible with both the flat connection on the bundle of fermionic functions and the projectively flat connection on the bundle of Hilbert spaces or the flat connection of the metaplectically corrected bundle over the space of polarisations. Finally, we relate the fermionic star products to the operator formalism.