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This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.

Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of many particles. It focuses on the foundations of quantum many-particle physics, the many-particle Hilbert spaces to describe large assemblies of interacting systems composed of Bosons or Fermions, which lead to the versatile formalism of second quantization as a convenient and eminently practical language ubiquitous in the mathematical formulation of the theory of many-particle systems of quantum matter. The main objects in which the formalism of second quantization is expressed are the Bosonic or Fermionic creation and annihilation operators that become, in the position basis, the quantum field operators.

This chapter introduces a quantum field theory for interacting boson systems. It develops a mean-field theory to study the superfluid phase. A path integral formulation is then developed to re-derive the superfuid phase, which results in a low energy effective non-linear sigma model. A renormalization group approach is introduced to study the zero temperature quantum phase transition between superfluid and Mott insulator phase, and finite temperature phase transition between superfluid and normal phase. The physics and the importance of symmetry breaking in phase transitions and in protecting gapless excitations are discussed. The phenomenon of superfluidity and superconductivity is also discussed, where the coupling to U(1) gauge field is introduced.

This chapter describes the approximate second quantization and approximate transition to boson picture. Using exact transformation from paulions to bosons, the collective properties of an ideal three-dimensional gas of paulions are considered. The statistics and the collective properties of Frenkel excitons are discussed, taking into account the dynamic exciton-exciton interaction. The fermionic character of Frenkel excitons in one-dimensional molecular crystals is demonstrated.

This chapter aims to place the discussion of holism in quantum theory generally and of the problem of identical particles and quantum statistics specifically in the context of general philosophy. Various debates in metaphysics, both historical and current, broach related subjects and involve analyses of concepts currently used in the philosophy of quantum mechanics. These include concepts of identity, individuation, essence, form, necessity, possibility, modality, contingency, and universality. But this chapter also extends the previous one by introducing Fock space and the transition from elementary quantum mechanics to quantum field theory via 'second quantization’. For this formalism relates clearly to certain ’anti‐essentialist’ approaches in the metaphysics of modality. Finally, in keeping with the empiricism pursued in this book, the general enterprise of metaphysics pertaining to those topics is viewed here within an anti‐metaphysical stance, as having as its value the display of a plurality of interpretations whose very diversity enhances our understanding of the theory.

This chapter introduces the second quantization formalism based on Schrödinger and Heisenberg operators. It defines the temperature and real-time Green functions for Bose and Fermi particles and discusses their analytical properties.

This chapter discusses the case of a many-body system of identical particles. It considers the consequencies of the symmetry of the Hamiltonian under particle permutations in this case. It discusses how to build many-body Boson and Fermion states. It then introduces the formalism of so-called second quantization, which is usefu, in the case of identical particles. It also expresses the many-body Hamiltonian in terms of creation and annihilation operators.

Chapter 5 discussed two possible approaches to constructing a relativistically invariant theory of particle scattering. The first attempt — a frontal assault in which one directly writes down for each scattering sector (i.e., with a specified number of incoming and outgoing particles) a manifestly Lorentz-invariant interaction operator containing momentum-dependent Lorentz scalar amplitudes — led to disaster. The resultant theory led to particle interactions which could not be confined to finite regions of space-time. The second attempt, in which the interaction Hamiltonian is written as a spatial integral of a local, Lorentz (ultra-)scalar field, accomplishes the primary goal of producing a Lorentz-invariant set of scattering amplitudes, but its compliance with the clustering principle remains uncertain. This chapter puts this latter requirement into a precise mathematical framework, called second quantization, so that the process of identifying clustering relativistic scattering theories can be simplified and even to some degree automated.

This chapter analyzes issues that deal with quantum statistics (symmetry of wavefunctions); elements of the second quantization formalism (the occupation-number representation); and the simplest systems with a large number of particles.

The simple harmonic oscillator is well known from basic quantum physics as an elementary model of an oscillating system. The chapter solves this simple model using creation and annihilation operators and shows that the solutions have the characteristics of particles. Linking masses by springs into a chain allows for a generalization of this problem and the solutions are phonons.

Second quantization (SQ) concepts were introduced in chapter 2 as a general tool to treat excitations in molecular collisions for which the dynamics were described in cartesian coordinates. This SQ-formulation, which was derived from the TDGH representation of the wave function, could be introduced if the potential was expanded locally to second order around the position defined by a trajectory. It is, however, possible to use the SQ approach in a number of other dynamical situations, as for instance when dealing with the vibrational excitation of diatomic and polyatomic molecules, or with energy transfer to solids and chemical reactions in the socalled reaction path formulation. Since the formal expressions in the operators are the same, irrespective of the system or dynamical situation, the algebraic manipulations are also identical, and, hence, the formal solution the same. But the dynamical input to the scheme is of course different from case to case. In the second quantization formulation of the dynamical problems, one solves the operator algebraic equations formally. Once the formal solution is obtained, we can compute the dynamical quantities which enter the expressions. The advantage over state or grid expansion methods is significant since (at least for bosons) the number of dynamical operators is much less than the number of states. In order to solve the problem to infinite order, that is, also the TDSE for the system, the operators have to form a closed set with respect to commutations. This makes it necessary to drop some two-quantum operators. Historically, the M = 1 quantum problem, namely that of a linearly forced harmonic oscillator, was solved using the operator algebraic approach by Pechukas and Light in 1966 [131]. In 1972, Kelley [128] solved the two-oscillator (M = 2) problem and the author solved the M = 3 and the general problem in 1978 [129] and 1980 [147], respectively. The general case was solved using graph theory designed for the problem and it will not be repeated here. But the formulas are given in this chapter and in the appendices B and C.

This chapter considers how to build single-particle operators out of creation and annihilation operators, and this already gives enough information to discuss the tight-binding model of solid state physics and the Hubbard model.

Semi-empirical π‐electron theories of conjugated polymers are introduced, starting from the fundamentals of the Born-Oppenheimer approximation and sp-hybridization. The models described include the simplest non-interacting (Hückel) theory, models of electron-nuclear coupling (i.e., the Su-Schrieffer-Heeger and Peierls models), and models of interacting electrons (i.e., the Pariser-Parr-Pople model). All of them are introduced in the second quantized notation.There is also a brief discussion of the role of spatial and electron-hole symmetries, and quantum numbers in classifying the electronic states of π‐ conjugated polymers.

This chapter sets out the second-quantisation treatment for systems of identical bosons or fermions. Key features such as the symmetrisation principle and super-selection rules are presented. Orthogonality and completeness requirements for mode functions are set out, and Fock states are defined in terms of mode occupancy, leading to the definitions of mode annihilation and creation operators and the determination of their commutation (bosons) or anticommutation (fermions) rules. Important two-mode states such as binomial states and relative-phase eigenstates are considered. Field creation and annihilation operators are defined and related to multi-particle position eigenstates. The Hamiltonian is expressed in terms of both mode operators and field operators, state dynamics being treated via Liouville–von Neumann, master or Matsubara equations for the density operator. Normal ordering is introduced and applied to expressions for the vacuum state projector. Multi-particle position probabilities are considered and related to normally ordered quantum correlation functions.

Focusing on systems of many identical particles, Chapter 2 introduces appropriate operators to describe their properties in terms of Schwinger’s “measurement symbols.” The latter are then factorized into “creation” and “annihilation” operators, whose fundamental properties and commutation/anticommutation relations are derived in conjunction with the Pauli exclusion principle. This leads to “second quantization” with the Hamiltonian, number, linear and angular momentum operators expressed in terms of the annihilation and creation operators, as well as the occupation number representation. Finally, the concept of coherent states, as eigenstates of the annihilation operator, having minimum uncertainty, is introduced and discussed in detail.

In this chapter we discuss theories which are rigorous in their formulation but which in order to be useful need to be modified by introducing approximations of some kind. The approximations we are interested in are those which involve introduction of classical mechanical concepts, that is, the classical picture and/or classical mechanical equations of motion in part of the system. At this point, we wish to distinguish between “the classical picture,” which is obtained by classical the limit ħ → 0 and the appearance of “classical equations of motion.” The latter may be extracted from the quantum mechanical formulation without taking the classical limit—but, as we shall see later by introducing a certain parametrization of quantum mechanics. Thus there are two ways of introducing classical mechanical concepts in quantum mechanics. In the first method, the classical limit is defined by taking the limit ħ → 0 either in all degrees of freedom (complete classical limit) or in some degrees of freedom (semi-classical theories). We note in passing that the word semi-classical has been used to cover a wide variety of approaches which have also been referred to as classical S-matrix theories, quantum-classical theories, classical path theory, hemi-quantal theory, Wentzel Kramer-Brillouin (WKB) theories, and so on. It is not the purpose of this book to define precisely what is behind these various acronyms. We shall rather focus on methods which we think have been successful as far as practical applications are concerned and discuss the approximations and philosophy behind these. In the other approach, the ħ-limit is not taken—at least not explicitl—but here one introduces “classical” quantities, such as, trajectories and momenta as parameters, and derives equations of motion for these parameters. The latter method is therefore one particular way of parameterizing quantum mechanics. We discuss both of these approaches in this chapter. The Feynman path-integral formulation is one way of formulating quantum mechanics such that the classical limit is immediately visible [3]. Formally, the approach involves the introduction of a quantity S, which has a definition resembling that of an action integral [101].

This chapter changes our viewpoint and get rid of wave functions entirely and develops the occupation number representation. The chapter shows that bosons are described by commuting operators and fermions are described by anticommuting operators.