Star product representation of coherent state path integrals
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Star product representation of coherent state path integrals Jasel Berra-Montiel1,2,a 1 Facultad de Ciencias, Universidad Autónoma de San Luis Potosí Campus Pedregal, Av. Parque Chapultepec
1610, Col. Privadas del Pedregal, San Luis Potosí 78217, SLP, Mexico
2 Dual CP Institute of High Energy Physics, Colima, Mexico
Received: 5 July 2020 / Accepted: 8 November 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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.
1 Introduction The path integral quantization remains up to date one of the main tools for understanding quantum mechanics and quantum field theory; in particular, the formalism has proved to be extremely helpful to study perturbative approximations of a wide diversity of physical phenomena [1]. The notion of path integration was extended to the complex plane with the introduction of coherent states [2–4], motivating a prolific progress on the production of semiclassical methods focused on the analysis of non-integrable systems and quantum chaos [5]. From another perspective, the phase space formulation of quantum mechanics, based on the early works of Wigner [6], Weyl [8] and Moyal [8], enabled to represent the quantum theory as a statistical theory defined on the classical phase space. Within this picture, the employment of coherent states has found a wide-ranging application in quantum optics and quantum information [9,10]. The main reason lies on the pioneering works proposed by Cahill and Glauber [11], where a family of s-parameterized quasi-probability distributions was introduced in order to characterize non-classical effects by means of the overcomplete basis of coherent states, independently of the adopted ordering prescription. Later on, in 1978 P. Sharan showed that the Feynman path integral, in the position and momentum representation, can be expressed as a Fourier transform of the star exponential for the Moyal product. This result was generalized to the case of scalar field theories in [13], by means of Berezin calculus and the c-equivalence of star products. In this paper, we analyze the coherent state path integrals within the context of the star product quantization. By employing the properties of generalized delta functions with com-
a e-mail: [email protected] (corresponding author)
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plex arguments, the Glauber–Sudarshan P-function corresponding to a non-diagonal density operator is obtained. Then, we co
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