Chapter 1 Coherent-State Representations

Monique Combescure, Didier Robert

Lukong Cornelius Fai

This chapter shows how to construct path integral representation using a closure relation based on the eigenstates of the creation or the annihilation operators. It examines how coherent states are used to obtain a path integral representation of the partition function as well as calculating directly the trace defining the partition function. The chapter explores quantum field theory using the Feynman functional and diagrammatic techniques as foundations to apply Quantum Field Theory to a broad range of topics in physics. The minimum uncertainty wave packets for the harmonic oscillator were published as coherent states by Schrodinger. Coherent states have played an important role in many branches of physics and, in particular, quantum field theory and quantum optics. The most important property of coherent states is their overcompleteness in the Fock space, which implies that any vector of the Fock space can be expanded in terms of coherent states.

Marlan O. Scully, Muhammad Suhail Zubairy

As we have seen in the previous chapters, there are quantum fluctuations associated with the states corresponding to classically welldefined electromagnetic fields. The general description of fluctuation phenomena requires the density operator. However, it is possible to give an alternative but equivalent description in terms of distribution functions. In the present chapter, we extend our treatment of quantum statistical phenomena by developing the theory of quasi-classical distributions. This is of interest for several reasons. First of all, the extension of the quantum theory of radiation to involve nonquantum stochastic effects such as thermal fluctuations is needed. This is an important ingredient in the theory of partial coherence. Furthermore, the interface between classical and quantum physics is elucidated by the use of such distributions. The arch type example being the Wigner distribution. In this chapter, we introduce various distribution functions. These include the coherent state representation or the Glauber–Sudarshan Prepresentation. The P -representation is used to evaluate the normally ordered correlation functions of the field operators. As we shall see in the next chapter, the P -representation forms a correspondence between the quantum and the classical coherence theory. This distribution function does not have all of the properties of the classical distribution functions for certain states of the field, e.g., it can be negative. We also discuss the so-called Q -representation associated with the antinormally ordered correlation functions. Other distribution functions and their properties are also presented.

Giampiero V. M. Esposito, Giuseppe Marmo, Gennaro Miele +1 more

Jean-Pierre Antoine, Bagarello, Fabio, Jean‐Pierre Gazeau