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Journal of Mathematical Physics

Fock–Tani representation for composite particles in a soluble model

Marvin D. Girardeau

Published Sep 1, 1980• 9 citations
Publisher

Abstract

The transformation from the usual Fock representation to the Fock–Tani representation for composite particles is carried out for a simplified model of the composite which allows closed-form expressions for the transformed quantities. A ’’statistical renormalization’’, in which single particle energies and interactions become dependent on the composite particle occupation number, plays an essential rôle in the solution, allowing absorption of unlinked terms into the definition of the renormalized energies.

DOI
10.1063/1.524693

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The transformation from the usual Fock representation to the Fock–Tani representation for composite particles is carried out for a simplified model of the composite which allows closed-form expressions for the transformed quantities. A ’’statistical renormalization’’, in which single particle energies and interactions become dependent on the composite particle occupation number, plays an essential rôle in the solution, allowing absorption of unlinked terms into the definition of the renormalized energies.
A method of constructing ’’atomic second quantization’’ representations is described, based on the introduction of redundant modes (’’ideal atom variables’’) which are then given physical content by carrying out a suitable unitary transformation, the ’’generalized Tani transformation.’’ In such a representation bound atoms or molecules are described by elementary Bose or Fermi operators, the field operators for nuclei and electrons referring only to unbound particles. The Hamiltonian thus obtained contains not only nucleus–nucleus, electron–electron, and nucleus–electron Coulomb interactions, but also atom–atom, atom–nucleus, and atom–electron Coulomb and exchange interactions, including breakup and recombination terms. All possible scattering and reaction channels are exhibited simultaneously in this transformed Hamiltonian. The method is applicable to any species or mixture of species of composite particles, but for simplicity the derivation is restricted here to the case of atomic hydrogen.
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.
In previous publications a first-principles method was developed for constructing second-quantization representations for systems of composite bound states and their constituents. It is based on the introduction of redundant modes (’’ideal atom variables’’) which are given physical content by carrying out a suitable unitary transformation. A single transformed Hamiltonian explicitly and simultaneously exhibits all kinematically possible scattering and reaction channels. Most of the previous results were limited to the case of two fermion composites. This restriction is removed herein, in order to allow a first-principles approach to the many-particle quantum dynamics and statistics of chemical and nuclear reactions. The kinematical aspects of the relevant transformation are examined.
A second-quantized representation for nonrelativistic systems with composite particles is given in which composite particles of two-fermion constituents are described as elementary bosons. The Tani transformation of Girardeau is used to introduce elementary composite fields. An equivalent linear equation of motion method—introducing a ’’d matrix’’—is used in place of the usual nonlinear equation of motion method. This and a diagram formalism are developed to calculate a Hamiltonian and subsidiary condition for this model system and to derive some properties of the Tani transformation.
Abstract A generalization of the quantum mechanical representation transformation is presented, in this paper. It is shown that, as an important example, the Boson-expansion method, commonly employed in nuclear physics, corresponds to such a generalized transformation. Using this generalization, we were able to construct a special representation called the “Composite Particle Representation”. In the composite, particle representation, the composite particle degrees of freedom are included, as well as the original particle degrees of freedom. The former is introduced in order that the motion of certain particle clusters can be described as separate entities in a many-body system. This representation is shown to be exactly equivalent to the usual quantum mechanical representation which includes only the original particle degrees of freedom. Many applications of this theory are expected, in particular in the study of hadrons from the quark point of view and the Interacting Boson Model in nuclei.