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arXiv:hep-th/9405139 · Journal of Physics A Mathematical and General

Identical particles and permutation group

E Celeghini, M Rasetti, G Vitiello

Published Apr 21, 1995• 8 citations
arXivHTMLPDF

Abstract

Second quantization is studied. Creation and annihilation operators are shown to be related on the same basis to both the algebra h(1) and to the superalgebra osp(1/2) which are shown to be compatible with both Bose and Fermi statistics. The two algebras are completely equivalent in the one-mode sector but, because of the grading of osp(1/2), differ in the many-particle case. The possibility of an unorthodox quantum field theory is suggested.

DOI
10.1088/0305-4470/28/8/003

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arXiv:1812.10732•Dec 27, 2018•1 citation
Second quantization is studied. Creation and annihilation operators are shown to be related on the same basis to both the algebra h(1) and to the superalgebra osp(1/2) which are shown to be compatible with both Bose and Fermi statistics. The two algebras are completely equivalent in the one-mode sector but, because of the grading of osp(1/2), differ in the many-particle case. The possibility of an unorthodox quantum field theory is suggested.
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.
Summary of the lectures: 1. Creation and annihilation operators for the system of indistinguishable particles 1.1 The permutation group and the states of a system of indistinguishable particles 1.2 Dimension of the Hilbert space of a system of indistinguishable particles 1.3 Definition and properties of the creation and annihilation operators 1.4 TheFockspace 1.5 The representations of state vectors and operators 1.5.1 N-particlewave-functions 1.5.2 The second-quantization representation of operators 1.5.3 Examples 1.6 Evolution of operators in the Heisenberg picture 1.7 Statistical operators of indistinguishable particles 1.7.1 The averages of the s-particle operators 1.7.2 The general structure of the one-particle statistical operator 2 Quadratic Hamiltonian and the diagonalization 2.1 Diagionalization of the Hamiltonian quadratic in the fermion operators 2.2 Diagionalization of the Hamiltonian quadratic in the boson operators 2.2.1 Diagonalization of the quadratic bosonic Hamiltonian possessing a zero mode 2.3 Long range order, condensation and the Bogoliubov spectrum of weakly interacting Bose gas 2.3.1 The excitation spectrum of weakly interacting Bose gas inabox 2.3.2 The excitation spectrum of an interacting Bose gas in an externalpotential 2.4 The Jordan-Wigner transformation: fermionization of interacting 1D spin chains
This chapter describes a very elegant way of accounting for the symmetry of the states and the operators of systems of many identical particles and illustrates its use in a few simple calculations. It presents operators in identical particle systems that remove particles from and add particles to the system. This formal similarity is the reason the creation and annihilation operator formalism is called second quantization; one-particle wave functions appear to have become operators which create and annihilate particles, while single particle expectation values appear to have become operators for physical quantities. The Hanbury-Brown and Twiss experiment provides a simple way of observing this tendency of bosons to clump together. Basically, the experiment measures the probability of observing two photons simultaneously at different points in a beam of incoherent light. The actual measuring apparatus uses a half silvered mirror, to split the beam into some identical beams; this avoids the problem of one detector casting a shadow on the other.
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.)