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Journal of Mathematical Physics

Number operators for composite particles in nonrelativistic many-body theory

Wesley E. Brittin, A.Y. Sakakura

Published Aug 1, 1980• 5 citations
Publisher

Abstract

Commuting physical occupation number operators for composite particles are constructed using projection operator techniques. The composite particle occupation number operators are constructed from creation and annihilation operators of the elementary particles which make up the many-body system. They appear as positive operators in any given second quantized theory and represent observables within the framework of that theory. Bose-type composites have number operators with eigenvalues 0,1,2,..., and Fermi-type composites have number operators with eigenvalues 0,1. There does not arise here any problem having to do with exchange symmetry—exchange symmetry is exact, since the number operators act in the Fock space of the elementary particles. The composite particle number operators may be used in the construction of theories of composite particle reactions or equilibrium from a first principles standpoint. The construction used here not only establishes the existence of composite particle number operators but also provides some computational machinery which hopefully will aid in more practical applications.

DOI
10.1063/1.524698

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Commuting physical occupation number operators for composite particles are constructed using projection operator techniques. The composite particle occupation number operators are constructed from creation and annihilation operators of the elementary particles which make up the many-body system. They appear as positive operators in any given second quantized theory and represent observables within the framework of that theory. Bose-type composites have number operators with eigenvalues 0,1,2,..., and Fermi-type composites have number operators with eigenvalues 0,1. There does not arise here any problem having to do with exchange symmetry—exchange symmetry is exact, since the number operators act in the Fock space of the elementary particles. The composite particle number operators may be used in the construction of theories of composite particle reactions or equilibrium from a first principles standpoint. The construction used here not only establishes the existence of composite particle number operators but also provides some computational machinery which hopefully will aid in more practical applications.
A new fundamental theory of composite particles in nonrelativistic many-body systems is developed. The theory is constructed making use solely of the physical Fock space for the basic elementary particles which comprise the many-body system. In this manner exchange symmetry in the elementary particles is exact. Physical composite particle creation and annihilation operators are introduced and these operators satisfy exact Bose (Fermi) commutation (anticommutati$\stackrel{\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}{\mathrm{o}}$n) relations depending on whether the composites correspond to bosons or to fermions. Commuting physical occupation number operators for composite particles are then constructed in the usual manner from the composite creation and annihilation operators. These number operators are highly dressed in terms of the basic elementary particle operators. Finally, the foundations of statistical mechanics for bound composite particles are formulated in terms of the appropriate occupation number operators.
The many-body problem for a system of composite particles is formulated in a way which takes explicit account of the composite nature of these particles and allows a clear separation between interatomic and intraatomic interactions. A second-quantization representation, fully equivalent to the conventional representation in which nuclei and electrons appear explicitly, is developed in terms of atomic annihilation and creation operators satisfying elementary Bose or Fermi commutation relations. All effects of the composite nature of the atoms are exactly contained in the interatomic and intraatomic matrix elements and in certain exchange integrals. An application is made to the problem of Bose condensation of fermion pairs.
Presented herein is a second-quantized formulation for a system of interacting nuclei and electrons expressed in terms of the creation and annihilation operators of various species of atoms and ions which can be formed by combining various numbers of the "elementary" particles. Processes such as ionization and recombination appear naturally. Explicit expressions for interatomic forces including exchange effects are generated from a "cluster expansion" reminiscent of the Ursell expansion.
We present a formalism for the description of many-particle systems when the particles are themselves composites of several elementary constituents. We investigate specifically the case of composites made up of two fermions, the method being easily extended to more general composites. The formalism, although its development is guided by the knowledge that there are fermion constituents in the composites, is given entirely in the composite language and requires no reference to these underlying fermions. Thus, the procedure maintains all required symmetries at every step. Composite field operators satisfying all symmetries are defined, these fields obeying neither Bose nor Fermi commutation rules, but more complicated ones. The many-composite dynamics is studied via the Green's functions formed from composite operator products. A Hartree-Fock-like approximation is introduced and leads to the generation of composite wave functions describing the distortion of the composite internal states by the presence of other composites in the system.
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 We propose a new generalization of the standard (anti-)commutation relations for creation and annihilation operators of bosons and fermions. These relations preserve the usual symmetry properties of bosons and fermions. Only the standard (anti-)commutator relation involving one creation and one annihilation operator is deformed by introducing fractional coefficients, containing a positive integer parameterp. The Fock space is determined by the classical definition. The new relations are chosen in such a way that the total occupation number in the system has the maximum valuep. From the actions of creation and annihilation operators in the Fock space, a group theoretical framework is determined, and from here the correspondence with known particle statistics is established.