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Journal of Mathematical Physics

Second-quantization representation for a nonrelativistic system of composite particles. II. Kinematical properties of the multispecies Tani transformation

Marvin D. Girardeau

Published Dec 1, 1978• 16 citations
Publisher

Abstract

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.

DOI
10.1063/1.523615

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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 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.
Abstract The salient features of generalized second‐quantization representations for nonrelativistic systems of composite particles are reviewed and their application to a reformulation of the quantum theory of reactive collisions is discussed. Such representations allow the properties of the bound composite states to be built explicitly into the algebra of states and observables. A single unperturbed Hamiltonian simultaneously describes the free propagation of the various species of bound composites as well as of their unbound constituents, while the interaction Hamiltonian describes only true scattering and reaction processes. The inclusion of the binding of all composites in the unperturbed Hamiltonian cures the divergences in the Born series arising from bound‐state poles. Unstable composites can be included in a natural way, leading to an explicit representation for the kinematics and dynamics of their decay and their contribution to collision phenomena.
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.
A practical method is developed to deal with the second quantization of the many-body system containing the composite particles.In our treatment,the modes associated with composite particles are regarded approximately as independent ones compared with those of unbound particles.The field operators of the composite particles thus arise naturally in the second quantization Hamiltonian.To be emphasized,the second quantization Hamiltonian has the regular structures which correspond clearly to different physical processes.
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.
We develop recursion relations, in particle number, for all (unprojected) Jain composite fermion (CF) wave functions. These recursions generalize a similar recursion originally written down by Read for Laughlin states, in mixed first-second quantized notation. In contrast, our approach is purely second-quantized, giving rise to an algebraic, ``pure guiding center'' definition of CF states that de-emphasizes first quantized many-body wave functions. Key to the construction is a second-quantized representation of the flux attachment operator that maps any given fermion state to its CF counterpart. An algebra of generators of edge excitations is identified. In particular, in those cases where a well-studied parent Hamiltonian exists, its properties can be entirely understood in the present framework, and the identification of edge state generators can be understood as an instance of ``microscopic bosonization.'' The intimate connection of Read's original recursion with ``nonlocal order parameters'' generalizes to the present situation, and we are able to give explicit second-quantized formulas for nonlocal order parameters associated with CF states.