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Bogolubov transformation for distinguishable particles

Markus Holzmann, Franck Laloë

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

The Bogoliubov transformation is generally derived in the context of identical bosons with the use of "second quantized'' $a$ and $a^{\dagger}$ operators (or, equivalently, in field theory). Here, we show that the transformation, together with its characteristic energy spectrum, can also be derived within the Hilbert space of distinguishable particles, obeying Boltzmann statistics; in this derivation, ordinary dyadic operators play the role usually played by the $a$ and $a^{\dagger}$ operators; therefore, breaking the symmetry of particle conservation is not necessary.

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Bogoliubov transformation for distinguishable particles

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The Bogoliubov transformation is generally derived in the context of identical bosons with the use of "second quantized" a and a† operators (or, equivalently, in field theory). Here, we show that the transformation, together with its characteristic energy spectrum, can also be derived within the Hilbert space of distinguishable particles, obeying Boltzmann statistics; in this derivation, ordinary dyadic operators play the rote usually played by the a and a† operators; therefore, breaking the symmetry of particles conservation is not necessary.

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The thermal Bogoliubov transformation in thermo field dynamics is generalized in two respects. First, a generalization of the $\\alpha$--degree of freedom to tilde non--conserving representations is considered. Secondly, the usual $2\\times2$ Bogoliubov matrix is extended to a $4\\times4$ matrix including mixing of modes with non--trivial multiparticle correlations. The analysis is carried out for both bosons and fermions.

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We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.

Tsinghua report and review in physics•Feb 1, 2014•0 citations
The Bogoliubov transformation is generally derived in the context of identical bosons with the use of "second quantized'' aaa and a†a^{\dagger}a† operators (or, equivalently, in field theory). Here, we show that the transformation, together with its characteristic energy spectrum, can also be derived within the Hilbert space of distinguishable particles, obeying Boltzmann statistics; in this derivation, ordinary dyadic operators play the role usually played by the aaa and a†a^{\dagger}a† operators; therefore, breaking the symmetry of particle conservation is not necessary.
The Bogoliubov transformation is generally derived in the context of identical bosons with the use of "second quantized" a and a† operators (or, equivalently, in field theory). Here, we show that the transformation, together with its characteristic energy spectrum, can also be derived within the Hilbert space of distinguishable particles, obeying Boltzmann statistics; in this derivation, ordinary dyadic operators play the rote usually played by the a and a† operators; therefore, breaking the symmetry of particles conservation is not necessary.
Second-quantized operators are derived algebraically from first-quantized counterparts, thus providing an alternative to all existing methods, most of which rely upon demonstrating the equivalence of matrix elements of the two operators. This derivation establishes an operator identity between a second-quantized operator and a first-quantized operator whose domain of definition is extended explicitly from an N-particle Hilbert space to Fock space. The existence of such an identity eliminates the need to address the issue of the representations used for the two operators, which often obscures the underlying simple relationship between them.
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
The thermal Bogoliubov transformation in thermo field dynamics is generalized in two respects. First, a generalization of the alpha\\alphaalpha--degree of freedom to tilde non--conserving representations is considered. Secondly, the usual 2times22\\times22times2 Bogoliubov matrix is extended to a 4times44\\times44times4 matrix including mixing of modes with non--trivial multiparticle correlations. The analysis is carried out for both bosons and fermions.
For strongly pair-correlated systems, a new transformation is derived similar in form to the usual Bogoliubov-Valatin transformation, but with the transformation coefficients replaced by operators. The original fermion operators are now replaced by two sets of independent operators, the pair operators and the quasi-fermions . The transformation conserves the particle number. Among other applications, it is particularly suited for the construction of a microscopic theory of the type of the interacting boson model. The new transformation given here also serves to provide a connection between the usual ph picture of a phonon and the pp picture-of bosons in IBM. However, the boson picture that evolves from the present formulation differs from the ideal boson picture in IBM..
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.