The lowering, raising, and number operators are defined and their properties are studied; the eigenfunctions of the harmonic oscillator Hamiltonian are derived; coherent or quasi-classical states are obtained and their properties examined. Several applications are discussed, including, among others, the isotropic harmonic oscillator in N dimensions and a model for a one-dimensional crystal.

This chapter discusses some of the “super” properties of lasers, superfluids, and superconductors in the context of quantum field theory, including their innate property of spontaneous coherence, which can be seen as the opposite of decoherence.

This chapter introduces the formal “second quantization” method for bosons in quantum field theory. It is shown that phonons (sound particles) and photons (light particles) are simple extensions of the physics of a spring-like oscillator. The connection of boson states to classical waves is shown in a discussion of “coherent states.”

The general concept of quantization is discussed, which provides the starting point for the further developments in this book. Starting with the concepts familiar from quantum mechanics, a number of quantum spaces defined via explicit operators on Hilbert space are discussed in detail, including compact and non-compact examples.

In this chapter we introduce the Glauber coherent states of a quantized field as eigenstates of the annihilation operator and as displaced vacuum states. The phase-space picture of coherent states is introduced, along with phase-space probability distributions, namely the Q distribution, the P distribution, and the Wigner function, and their interrelations are discussed.

Appendix E: free, then harmonically bound, massive quantum particle. Lowering and rising operators, displacement operator, number states, coherent states, zero point fluctuations.

We study the canonical commutation relation, and give a complete proof of a fundamental result of Stone and von Neumann: A finite set of operators satisfying (the proper form of) these relations is essentially unique. We also detail why this result miserably fails for infinite sets of operators.

"After reviewing some basic principles of quantum field theory in Chapter 1, we now turn to a series of problems exploring various aspects of functional methods. Although it lacks a robust mathematical foundation in the case of interacting theories (but the situation in this respect is no better within the canonical formalism), the formulation of QFT in terms of path integrals considerably simplifies many manipulations that would otherwise be extremely tedious because of the need to keep track of the ordering of operators.

Besides the conventional representation of expectation values of time-ordered products of field operators and their generating functionals in terms of path integrals, we also briefly discuss the worldline representation for propagators and for one-loop effective actions, which provides an alternative point of view on quantization. Moreover, these ideas are not limited to ordering products of field operators, and can be useful in managing products of other types of non-commuting objects."

Introduces the idea of second quantized operators in the many-particle domain, Fock spaces, field operators, and vacuum states, and outlines how canonical transformations can be applied to solve many-body problems. Coherent states, as eigenstates of the annihilation operator, including the development of Grassmann’s algebra and calculus for fermions, are presented.