Star product representation of coherent state path integrals

Abstract

In this paper, we determine the star product representation of coherent path integrals. By employing the properties of generalized delta functions with complex arguments, the Glauber-Sudarshan P-function corresponding to a non-diagonal density operator is obtained. Then, we compute the Husimi-Kano Q-representation of the time evolution operator in terms of the normal star product. Finally, the optical equivalence theorem allows us to express the coherent state path integral as a star exponential of the Hamiltonian function for the normal product.

Integral representation of the star product in Loop Quantum Cosmology

“…It should be noted that the right hand side of the identities (22) agrees with the holonomy-flux Poisson algebra occurring in classical cosmology [2]. Thus, we conclude that the Weyl quantization map Q LQC defined in (10) and the star product ⋆ LQC , provides a one-parameter, associative deformation of the holonomy-flux classical algebra.…”

“…As a consequence, the introduction of the star product induces a deformation of the Poisson algebra in such a way that the information hold in the quantum commutators of any pair of self-adjoint operators is mapped to the deformed algebraic classical structures. Until now, the formalism of deformation quantization has not only provided significant contributions in pure and applied mathematics [15], [16], but it also has proved to be an outstanding tool for the quantum analysis of a broad variety of physical systems [17], [18], [19], recently including the treatment of constrained systems [20], the coherent field quantization [21], [22], and the tomographic representation for fields [23].…”