Abstract 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.
Abstract 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.
In this chapter we describe a technique to deal with identical particles that is called second quantization . Despite being a technique, second quantization helps a lot in understanding physics. One can learn and endlessly repeat newspaper-style statements particles are fields , fields are particles without grasping their meaning. Second quantization brings these concepts to an operational level. Since the procedure of second quantization is slightly different for fermions and bosons, we have to treat the two cases separately. We start with considering a system of identical bosons, and introduce creation and annihilation operators (CAPs) that change the number of particles in a given many-particle state, thereby connecting different states in Fock space. We show how to express occupation numbers in terms of CAPs, and then get to the heart of the second quantization: the commutation relations for the bosonic CAPs. As a next step, we construct field operators out of the CAPs and spend quite some time (and energy) presenting the physics of the particles in terms of these field operators. Only afterward, do we explain why all this activity has been called second quantization. In Section 3.4 we then present the same procedure for fermions, and we mainly focus on what is different as compared to bosons. The formalism of second quantization is summarized in Table 3.1.
Abstract 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.
Abstract This chapter provides an overview of the phase space method for treating the physics of systems of bosons and fermions in regimes where quantum degeneracy occurs. Key concepts such as modes, creation and annihilation operators, distribution functions in phase space depending on c-number (bosons) or Grassmann (fermions) phase space variables, quantum correlation functions, and phase space integrals are introduced, together with how the dynamics of the system density operator as described via Liouville–von Neumann or master equations is translated into Fokker–Planck equations for the distribution function, and then how phase space integrals for quantum correlation functions can be determined via Langevin stochastic equations that are equivalent to the Fokker–Planck equations, and which involve stochastic phase space variables. The extension of the phase space approach to systems with large mode numbers, involving field annihilation and creation operators, distribution functionals, and functional Fokker–Planck and Langevin stochastic field equations is also reviewed.
Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functions of phase space variables, leading to expressions for quantum correlation functions in terms of phase space integrals in which phase space variables for each mode—c-numbers for bosons, Grassmann numbers for fermions—replace mode annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functions, with distinct phase space variables for annihilation and creation operators; however, symmetrically ordered (Wigner type) distribution functions are also considered, along with unnormalised B distribution functions, which lead to phase space integrals for Fock state populations and coherences. Characteristic functions are first defined and then shown to be related to distribution functions via phase space integrals. The existence and symmetry properties of distribution functions—which are non-unique and non-analytic for bosons—is demonstrated using Bargmann coherent-state projectors.
Abstract This chapter defines and determines the properties of the important (though non-physical) boson and fermion coherent states, which are specified in terms of c-number (bosons) or Grassmann (fermions) variables and their complex conjugates, and defined via unitary displacement operators acting on the vacuum state. Important non-orthogonality, overcompleteness and coherence properties of coherent states are obtained. Coherent states are shown to be eigenstates of mode annihilation operators, with eigenvalues given by the c-number or Grassmann variable specifying the state. Related (unnormalised) Bargmann states are defined, depending only on the variable and not its complex conjugate, and are used to obtain useful representations of physical states and canonical forms for operators in terms of Bargmann state projectors, as well as obtaining important results such as for traces of operator products in terms of phase space integrals. Mode creation operators acting on Bargmann states are shown as equivalent to differentiation.
Second quantization in nonrelativistic quantum mechanics is given in a simplified way using eigenstates of the number operator. The field creation operator is shown to be the operator which adds a particle with a delta-function wave function to the system, while the field annihilation operator removes a particle with a delta-function wave function from the system. The field creation operator is proved to be the Hermitian conjugate of the annihilation operator. The commutation (anticommutation) relations for boson (fermion) field operators are proved from the symmetry (antisymmetry) of the many-particle wave function. The second quantized form of one- and two-particle operators is obtained from their form in wave mechanics using the definition of the field creation and annihilation operators.