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arXiv:2409.14789 · Journal of Physics A Mathematical and Theoretical

Generalized boson and fermion operators with a maximal total occupation property

Nedialka I. Stoilova, Joris Van Der Jeugt

Published Sep 6, 2024• 0 citations
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Abstract

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.

DOI
10.1088/1751-8121/ad77fd

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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.
A basis of products of one-particle vectors (symmetric or antisymmetric for bosons or fermions) is defined, and related to the occupation-number basis in an abstract Fock space, a generalization of the wave-mechanical configuration representation of the space of second quantization. Creation and destruction operators are simply defined in the product-vector basis and shown to have their usual properties in occupation-number representation. Algebraic properties of product vectors are developed and illustrated. The representation of operators in the product-vector basis is described, together with a brief discussion of the effects on these operators of transformations from one set of one-particle vectors to another. A simple treatment of density and reduced density operators for systems of bosons or fermions is given in the product-vector basis. The relationship between degeneracy in the Fock-space spectrum of an additive operator NA1 and degeneracy in the spectrum of one-particle operator A1 is obtained.
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.
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.
We consider Fock representations of the Q-deformed commutation relations ∂s∂t†=Q(s,t)∂t†∂s+δ(s,t) for s,t∈T. Here T:=Rd (or more generally T is a locally compact Polish space), the function Q:T2→C satisfies |Q(s,t)|≤1 and Q(s,t)=Q(t,s)¯, and ∫T2h(s)g(t)δ(s,t) σ(ds)σ(dt):=∫Th(t)g(t) σ(dt), σ being a fixed reference measure on T. In the case, where |Q(s,t)|≡1, the Q-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev. These generalized statistics contain anyon statistics as a special case (with T=R2 and a special choice of the function Q). The related Q-deformed Fock space F(H) over H:=L2(T→C,σ) is constructed. An explicit form of the orthogonal projection of H⊗n onto the n-particle space Fn(H) is derived. A scalar product in Fn(H) is given by an operator Pn≥0 in H⊗n which is strictly positive on Fn(H). We realize the smeared operators ∂t† and ∂t as creation and annihilation operators in F(H), respectively. Additional Q-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ∂s†∂t†=Q(t,s)∂t†∂s†, ∂s∂t=Q(t,s)∂t∂s, valid for those s,t∈T for which |Q(s, t)| = 1.
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
The expansion of single- particle transition operator( defined as the product of creation and annihilation operator,namely a+iajin single- particle state is obtained through the form of single- operator in occupation particle number representation,and the commutation relations for transition operator and creation( annihilation) operator are also given,then based on these conclusions,the form of general two- and n- particle operators in occupation particle number representation can be derived succinctly without distinguishing the statistical property of identical particle system.