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Physical Review Letters

General Many-Body Formalism for Composite Quantum Particles

Monique Combescot, Odile Betbeder-Matibet

Published May 20, 2010• 27 citations
Publisher

Abstract

This Letter provides a formalism capable of exactly treating Pauli blocking between n-fermion particles. This formalism is based on an operator algebra made of commutators and anticommutators which contrasts with the usual scalar formalism of Green functions developed half a century ago for elementary quantum particles. We also provide the diagrams which visualize the very specific many-body physics induced by fermion exchanges between composite quantum particles.

DOI
10.1103/physrevlett.104.206404

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This Letter provides a formalism capable of exactly treating Pauli blocking between n-fermion particles. This formalism is based on an operator algebra made of commutators and anticommutators which contrasts with the usual scalar formalism of Green functions developed half a century ago for elementary quantum particles. We also provide the diagrams which visualize the very specific many-body physics induced by fermion exchanges between composite quantum particles.
For half a century, Feynman diagrams have provided an enlightening way of representing many-body effects between elementary fermions and bosons. They however are quite inappropriate to visualize fermion exchanges taking place between a large number of composite quantum particles. We propose to replace them by "Shiva diagrams" for cobosons made of two fermions and by Shiva-like and "Kali diagrams" for cofermions made of three fermions. We also show how these fermion exchanges formally appear in a many-body theory appropriate to composite quantum particles. This theory relies on an operator algebra based on commutators and anticommutators, the usual scalar algebra based on Green functions being valid for elementary bosons or fermions having strict commutation relations, only.
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.
A formalism for treating the many-body problem of composite particles is presented for the case of composite particles consisting of two fermions. Commutation relations for composite-particle operators are derived, as well as a sum rule satisfied by the composite-particle Green's function. In an approximation that shuts off the interactions between composite particles in a consistent manner, the dynamical equation for the one-composite-particle Green's function is solved and the distribution function for the composite particles obtained. Possible applications to real systems are discussed.
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.
In this research, we develop a many-body Green's function approach for interacting composite bosons in the Bose–Einstein condensed phase. The resummation scheme dealing with the carrier–carrier exchange between many-body composite bosons in cold condensation is manifested as a diagrammatic technique through the introduction of a set of Dyson equations for composite bosons. We diagrammatically calculate the specific heat capacity of neutral composite bosons, made of mass-imbalanced fermions with opposite charges, up to the first orders of the Pauli exchange and Coulomb interaction. Condensate phenomena due to the effect of the carrier–carrier exchange are discussed. In comparison with the approximation that composite bosons behave like interacting fundamental bosons, our result shows that the energy reduction from the carrier–carrier exchange stabilizes composite-boson condensates.