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Physical Review Letters

Generalized Bosons and Coherent States

E. Celeghini, Mario Rasetti

Published Apr 20, 1998• 9 citations
Publisher

Abstract

The characteristic features of Bose-Einstein quantum statistics are derived resorting to the notion of coproduct associated with any Lie-Hopf structure and to generalized coherent states. It is shown that, while Boltzmann particles are well described by coherent states of $h\left(1\right)$, orthodox bosons require the ${D}_{1/2}^{\left(+\right)}$ representation of SU(1,1). A whole class of inequivalent generalized bosons is then defined by multimode coherent states of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SU}}\left(1,1\right)$. Known experimental results are in agreement with our generalizations---as well as with the Bose hypothesis---but experiments with discrete spectra, such as the recent ones on Bose condensation, could discriminate among all these possibilities.

DOI
10.1103/physrevlett.80.3424

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The characteristic features of Bose-Einstein quantum statistics are derived resorting to the notion of coproduct associated with any Lie-Hopf structure and to generalized coherent states. It is shown that, while Boltzmann particles are well described by coherent states of h(1)h\left(1\right)h(1), orthodox bosons require the D1/2(+){D}_{1/2}^{\left(+\right)}D1/2(+)​ representation of SU(1,1). A whole class of inequivalent generalized bosons is then defined by multimode coherent states of SU\ifmmode~\else˜\fi(1,1)\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{SU}}\left(1,1\right)SU\ifmmode~\else˜\fi(1,1). Known experimental results are in agreement with our generalizations---as well as with the Bose hypothesis---but experiments with discrete spectra, such as the recent ones on Bose condensation, could discriminate among all these possibilities.
A recipe is given to reproduce algebraically the Bose prescription in Quantum Statistics. Bosons are thus defined as coherent states of the [Formula: see text] representation of the Lie-Hopf algebra su(1, 1), while h(1) is shown to be related to Boltzmann statistics. The technical power of group theory is used to show that black-body radiation formula as well as all other experimental results obtained in continuum limit do not require Bose distribution but are compatible with many other less symmetric ones. This may have dramatical effects on the predictions on Bose condensation.
Usual quantum statistics is written in Fock space but it is not an algebraic theory. We show that at a deeper level it can be algebraically formalized defining the different statistics as (multi-mode) coherent states of the appropriate (but different from the usual ones) Lie-Hopf groups. The traditional connection between groups and statistics, established in vacuum, is indeed subverted by the interaction with the thermal bath. We show indeed that h(1), related in quantum field theory to bosons, must be used to define in presence of a bath the Boltzmann statistics while, to build the Bose statistics, we have to take into account su(1,1). Astonishing to describe fermions we are forced to use not the superalgebra h(1|1) but su(2) in the fundamental representation. Higher representations of su(2) allow also to give a possible definition of anyon statistics with generalized Pauli principle. Physical implications are discussed; the results is more general then the usual on the discrete spectrum, but everything collapses to standard theory when the continuum limit is performed.
The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.
Abstract. The role of the background in bosonic quantum statistics is discussed in the frame of a new approach in terms of coherent states. Bosons are indeed detected in different physical situations where they exhibit different and apparently unconnected properties. Besides Bose gas we consider bosons in particle physics and bosons in harmonic traps. In particle physics bosons are dealt with in a context where the number of observed particles is finite: here the relevant feature are the canonical commutation relations, which we shall show to be related to a Boltzmann-like distribution. A further case is Bose condensate in harmonic traps, where discrete spectrum leads us to predict, below Tc, a new critical temperature. The unified approach proposed shows that all differences can be ascribed to the environment.
The role of the background in bosonic quantum statistics is discussed in the frame of a new approach in terms of coherent states. Bose gas is not the only physical situation where bosons are detected. In particle physics bosons are dealt with in a context where the number of observed particles is finite: here the relevant feature are the canonical commutation relations, which we shall show to be related to a Boltzmann-like distribution. A further case is Bose condensate in harmonic traps, where discrete spectrum leads us to predict a new critical temperature. All these cases are analyzed in the unified approach proposed, showing that all differences can be ascribed to the role of the background.