Special relativity implies that mass can be created from energy — it implies particle production. A relativistic quantum theory is of necessity a theory of many particles. We shall therefore first describe systems of many identical particles but in the simpler context of non-relativistic theory. As we shall soon see, many-particle systems can be described by field operators that create and destroy particles, with these operators acting in a vastly enlarged vector space that spans states with an arbitrary number of particles, including a state of no particles — the vacuum state. Both Bose and Fermi statistics are handled in this way, and the resulting formalism is compact, convenient, and powerful. For the Bose case, the functional integral representation of the quantum many-particle system is direct generalization of the coherent-state functional integral developed in the previous chapter. To describe the analogous formulation for the Fermi case, we must first develop the theory of anticommuting variables. The power and utility of field operator and functional integral methods will be illustrated by applying it to systems in thermal equilibrium, first to free Fermi and Bose gases, and then to a model of liquid helium which exhibits the phenomenon of spontaneous symmetry breaking, a phenomenon of great importance in condensed matter physics as well as in elementary particle theory.
Special relativity implies that mass can be created from energy — it implies particle production. A relativistic quantum theory is of necessity a theory of many particles. We shall therefore first describe systems of many identical particles but in the simpler context of non-relativistic theory. As we shall soon see, many-particle systems can be described by field operators that create and destroy particles, with these operators acting in a vastly enlarged vector space that spans states with an arbitrary number of particles, including a state of no particles — the vacuum state. Both Bose and Fermi statistics are handled in this way, and the resulting formalism is compact, convenient, and powerful. For the Bose case, the functional integral representation of the quantum many-particle system is direct generalization of the coherent-state functional integral developed in the previous chapter. To describe the analogous formulation for the Fermi case, we must first develop the theory of anticommuting variables. The power and utility of field operator and functional integral methods will be illustrated by applying it to systems in thermal equilibrium, first to free Fermi and Bose gases, and then to a model of liquid helium which exhibits the phenomenon of spontaneous symmetry breaking, a phenomenon of great importance in condensed matter physics as well as in elementary particle theory.
Up to a point, the quantum theory developed in chapter 5 was quite general. However, the systems we had in mind were non-relativistic ones consisting of a fixed number of particles. In this chapter, we extend the theory to deal with systems in which the number of particles can change. There are several reasons for wanting to do this. The most obvious is that we need a method of describing high-energy scattering and decay processes in which particles can be created and destroyed. A second is that, when we try to make quantum theory consistent with special relativity, we encounter difficulties (discussed in chapter 7) that can be resolved only in this more general setting. The final reason is that, even for systems of non-relativistic particles, the mathematics rapidly becomes intractable as the number of particles increases. A useful device for dealing with large systems is, roughly speaking, to imagine adding an extra particle, which serves as a theoretical probe of the state of the system. To put the matter another way, the method of second quantization developed in this chapter provides a means of dealing with the entire system by considering only a few particles at a time.
Up to a point, the quantum theory developed in Chapter 5 was quite general. However, the systems we had in mind were non-relativistic ones consisting of a xed number of particles. In this chapter, we extend the theory to deal with systems in which the number of particles can change. There are several reasons for wanting to do this. The most obvious is that we need a method of describing high-energy scattering and decay processes in which particles can be created and destroyed. A second is that, when we try to make quantum theory consistent with special relativity, we encounter diculties (discussed in Chapter 7) that can be resolved only in this more general setting. The nal reason is that, even for systems of non-relativistic particles, the mathematics rapidly becomes intractable as the number of particles increases. A useful device for dealing with large systems is, roughly speaking, to imagine adding an extra particle, which serves as a theoretical probe of the state of the system. To put the matter another way, the method of second quantization developed in this chapter provides a means of dealing with the entire system by considering only a few particles at a time.
This chapter introduces and applies the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. Second quantization formalism treats systems of bosons or fermions with a fixed or variable number of particles in a unified way. The chapter reviews the key aspects of this formalism. The formalism of second quantization treats systems of bosons with a fixed or variable number of particles in a unified manner. In many physical processes, the particle number does change. Different quantum operators are expressed most naturally in different representations, which make basis changes an essential issue in quantum physics. Quantum field theory is a universal tool that has applications ranging from atomic, molecular, and particle physics to condensed matter and statistical physics as well as modern quantum chemistry. The fundamental difference between classical and quantum mechanics relates to the concept of indistinguishability of identical particles.
Up to this point we have not had much to say about the detailed structure of the Hamiltonian operator H . This operator can be defined by giving all its matrix elements between states with arbitrary numbers of particles. Equivalently, as we shall show here, any such operator may be expressed as a function of certain operators that create and destroy single particles. We saw in Chapter 1 that such creation and annihilation operators were first encountered in the canonical quantization of the electromagnetic field and other fields in the early days of quantum mechanics. They provided a natural formalism for theories in which massive particles as well as photons can be produced and destroyed, beginning in the early 1930s with Fermi's theory of beta decay. However, there is a deeper reason for constructing the Hamiltonian out of creation and annihilation operators, which goes beyond the need to quantize any pre-existing field theory like electrodynamics, and has nothing to do with whether particles can actually be produced or destroyed. The great advantage of this formalism is that if we express the Hamiltonian as a sum of products of creation and annihilation operators, with suitable non-singular coefficients, then the S -matrix will automatically satisfy a crucial physical requirement, the cluster decomposition principle, which says in effect that distant experiments yield uncorrelated results. Indeed, it is for this reason that the formalism of creation and annihilation operators is widely used in non-relativistic quantum statistical mechanics, where the number of particles is typically fixed.
The monograph is devoted to the main aspects of theoretical description of quantum many-body systems with Bose-Einstein or Fermi-Dirac statistics. We provide a review of modern methods of theoretical research, both analytical and numerical, as well as experimental realizations of unique effects and phenomena in quantum systems. The main attention is given to the theoretical study of exotic states in degenerate gases of atoms and quasiparticles. We develop a methology to accurately determine chemical potentials and other thermodynamic characteristics of ideal Bose and Fermi gases. These allow us to describe the ultraslow-light phenomenon in ultracold atomic gases in the framework of the microscopic approach that accounts for the internal degrees of freedom of bound states. The formalism relies on the linear response theory applied for the case of weak perturbation by the external elecromagnetic field. We also study Bose-Einstein condensation of photons in ideal atomic gases in different regimes of degeneracy of atomic and photonic components. A special attention is given to the Bogoliubov method of quasiaverages and related theory of a weakly interacting Bose gas with condensate. We extend the Bogoliubov theory to examine the spinor condensate and its magnetic states, as well as the condensate of heteronuclear Fermi-Fermi molecules. We also study a spatially periodic state of Bose-Einstein condensate, which can serve as a model for supersolid state of matter and develop the hydrodynamic theory of the latter. We outline the main statements of the Fermi liquid theory including the Pomeranchuk stability conditions of the normal state. It is shown that the violation of stability conditions results in certain phase transitions associated with the breaking of translational symmetry and rotational symmetry in the momentum space. We review a progress in the theoretical description of ultracold Fermi gases consisting of neutral atoms in the presence of the optical lattice spatial modulation. In case of multicomponent systems, a number of unique magnetically-ordered states are theoretically studied and predicted. We also develop a theory of highly dispersive spinful excitons in cobaltite compounds that explains certain unusual magnetic properties and effects in the resonant inelastic X-ray scattering on these materials.
Formal functional integrals are commonly used as theoretical tools and as sources of intuition for predicting phase transitions of many-body systems in Condensed Matter Physics. In this thesis, we derive rigorous versions of these functional integrals for two types of quantum many-particle systems. We begin with a brief review of quantum statistical mechanics in Chapter 2 and the formalism of coherent states in Chapter 3, which form the basis for our analysis in Chapters 4 and 5. In Chapter 4, we study a mixed gas of bosons and/or fermions interacting on a finite lattice, with a general Hamiltonian that preserves the total number of particles in each species. We rigorously derive a functional integral representation for the partition function, which employs a large-field cutoff for the boson fields. We then expand the resulting “action” in powers of the fields and find a recursion relation for the coefficients. In the case of a two-body interaction (such as the Coulomb interaction), we also find bounds on the coefficients, which give a domain of analyticity for the action. This domain is large enough for use of the action in the functional integral, provided that the large-field cutoffs are taken to grow not too quickly. In Chapter 5, we study a system of electrons and phonons interacting in a finite lattice, using the Holstein Hamiltonian. Again, we rigorously derive a coherent-state functional integral representation for the partition function of this system and then prove that the “action” in the functional integral is an entire-analytic function of the fields. However, since the Holstein Hamiltonian does not preserve the total number of bosons, the approach from Chapter 4 requires some modification. In particular, we repeatedly use Duhamel expansions in powers of the interaction, rather than sums over particle numbers.