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Chinese Physics C

A BOSON-FERMION DESCIPTION OF THE NUCLEAR MANY-BODY PROBLEM

Z. Yang

Published Jan 1, 1978• 0 citations
OpenAlex

Abstract

A new formalism of the nuclear many-body problem is established in which one ean work with various physical state vector spaces consisting of both Fermions and Bosons, all being equivalent to the original one consisting only of Fermions. With the help of the usual commutation relations and anticommutation relations between the annihilation and creation operators of Boson and Fermion, a generalized state vector space is established, which contains all the physical spaces each being equivalent to the original physical space in terms of pure Fermion operators. Transformation between the state vectors in various equivalent physical spaces are constructed. Basic operators in the original state space are transformed into effective operators in the new physical spaces.

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Second quantization

Yuli V. Nazarov, Jeroen Danon

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.

Cambridge University Press eBooks•Jan 3, 2013•0 citations
A new formalism of the nuclear many-body problem is established in which one ean work with various physical state vector spaces consisting of both Fermions and Bosons, all being equivalent to the original one consisting only of Fermions.
With the help of the usual commutation relations and anticommutation relations between the annihilation and creation operators of Boson and Fermion, a generalized state vector space is established, which contains all the physical spaces each being equivalent to the original physical space in terms of pure Fermion operators. Transformation between the state vectors in various equivalent physical spaces are constructed. Basic operators in the original state space are transformed into effective operators in the new physical spaces.
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.
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.
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 the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrödinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate r1,r2,... and momentum p1, p2, . . . , boson creation and annihilation operators, b†iα, biα. The index i runs over the number of relevant degrees of freedom, while the index α runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i ≠ j, . . . [biα, b†jα´] = 0, [biα, bjα´] = 0,. . . . . .[bjα, b†iα´] = 0, [b†jα, b†iα´] = 0,. . . . . . [biα, b†iα´] = ẟαα´, [biα, b†iα´] = 0, [b†iα, b†iα´] = 0. . . .
A new many-body theory for fermions is proposed wihich is based on the SO(2N+1) Lie algebra of the fermion operators consisted of the annihilation-creation operators and the pair operators. A New cannonical transformation, which is the extension of the Bogoliubov transformation to the SO(2N+1) group, is introduced. A new bose representation for the fermion Lie operators is obtained by mapping the fermion Lie operators into the regular representation of the SO(2N+1) group. The annihilation-creation operators and the pair operators of fermions are represented by the closed first order differential operators on the SO(2N+1) group. An exact representation of fermion wavefunctions in a form similar to the wavefunction of the generator coordinate method is obtained making use of the SO(2N+1) canonical transformation. The physical fermion space is shown to be the irreducible spinor representation of the SO(2N+1) group. The dynamics of fermions in the bose representation space is shown to represent rotations of a 2N+1 dimensional rotator.
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.