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arXiv:2005.04223 · Journal of Mathematical Physics

Algebra, coherent states, generalized Hermite polynomials, and path integrals for fractional statistics—Interpolating from fermions to bosons

Satish Ramakrishna

Published Oct 1, 2020• 0 citations
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Abstract

This article constructs the Hilbert space for the algebra αβ − eiθβα = 1 that provides a continuous interpolation between the Clifford and Heisenberg algebras—this particular form is inspired by the properties of anyons. We study the eigenvalues of a generalized number operator (N=βα) and construct the Hilbert space, classified by values of a complex coordinate (λ0): the eigenvalues lie on a circle. For θ being an irrational multiple of 2π, we get an infinite-dimensional representation; however, for a rational multiple (MN) of 2π, it is finite-dimensional, parameterized by the complex coordinate λ0. The case for N = 2, θ = π is the usual Clifford algebra for fermions, while the case for N = ∞, θ = 0 is the Heisenberg algebra of bosons, albeit with two copies for positive and negative eigenvalues. We find a smooth transition from the fermion to the boson situation as N → ∞ from N = 2. After constructing the Hilbert space from the algebra, the cases for N = 2, 3 can be mapped to the “fuzzy sphere” of SU(2), while for general N, the “fuzzy pancake” is found to be the correct representation. Then, we motivate the study of coherent states, which are the eigenstates of α (the lowering operator), labeled by complex numbers for non-zero λ0. We specialize the study of coherent states to the very interesting case of λ0 = 0 and construct a calculus of the generalized Grassmann variables that result, applying it to compute a partition function for these particles. We then make some remarks about extending this study to that of anyons.

DOI
10.1063/5.0022407

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This article constructs the Hilbert space for the algebra αβ − eiθβα = 1 that provides a continuous interpolation between the Clifford and Heisenberg algebras—this particular form is inspired by the properties of anyons. We study the eigenvalues of a generalized number operator (N=βα) and construct the Hilbert space, classified by values of a complex coordinate (λ0): the eigenvalues lie on a circle. For θ being an irrational multiple of 2π, we get an infinite-dimensional representation; however, for a rational multiple (MN) of 2π, it is finite-dimensional, parameterized by the complex coordinate λ0. The case for N = 2, θ = π is the usual Clifford algebra for fermions, while the case for N = ∞, θ = 0 is the Heisenberg algebra of bosons, albeit with two copies for positive and negative eigenvalues. We find a smooth transition from the fermion to the boson situation as N → ∞ from N = 2. After constructing the Hilbert space from the algebra, the cases for N = 2, 3 can be mapped to the “fuzzy sphere” of SU(2), while for general N, the “fuzzy pancake” is found to be the correct representation. Then, we motivate the study of coherent states, which are the eigenstates of α (the lowering operator), labeled by complex numbers for non-zero λ0. We specialize the study of coherent states to the very interesting case of λ0 = 0 and construct a calculus of the generalized Grassmann variables that result, applying it to compute a partition function for these particles. We then make some remarks about extending this study to that of anyons.
This article constructs the Hilbert space for the algebra $αβ- e^{i θ} βα= 1 $ that provides a continuous interpolation between the Clifford and Heisenberg algebras. This particular form is inspired by the properties of anyons. We study the eigenvalues of a generalized number operator (N=βα{\cal N} = βαN=βα) and construct the Hilbert space, classified by values of a complex coordinate (λ0λ_0λ0​): the eigenvalues lie on a circle. For θθθ being an irrational multiple of 2π2 π2π, we get an infinite-dimensional representation, however for a rational multiple (MN\frac{M}{N}NM​) of 2π2 π2π, it is finite-dimensional, parametrized by the complex coordinate λ0λ_0λ0​. The case for N=2 ; θ=πN=2 \: ; \: θ=πN=2;θ=π is the usual Clifford algebra for fermions, while the case for N=∞ ; θ=0N=\infty \: ; \: θ=0N=∞;θ=0 is the Heisenberg algebra of bosons, albeit with two copies for positive and negative eigenvalues. We find a smooth transition from the fermion to the boson situation as N→∞N \rightarrow \inftyN→∞ from N=2N=2N=2. After constructing the Hilbert space from the algebra, the cases for N=2,3N=2,3N=2,3 can be mapped to SU(2)SU(2)SU(2). Then, we motivate the study of coherent states, rather generally. The coherent states are eigenstates of ααα, the annihilation operator and are labeled by complex numbers for non-zero λ0λ_0λ0​.
The aim of this paper is to construct coherent states à la Perelomov and à la Barut–Girardello for a polynomial Weyl–Heisenberg algebra. This generalized Weyl–Heisenberg algebra, denoted by , depends on r real parameters and is an extension of the one-parameter algebra (Daoud and Kibler 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su (1, 1) algebra (for κ > 0), the su (2) algebra (for κ < 0) and the h 4 ordinary Weyl–Heisenberg algebra (for κ = 0). For finite-dimensional representations of and , where is a truncation of order s of in the sense of Pegg–Barnett, a connection is established with k -fermionic algebras (or quon algebras). This connection makes it possible to use generalized Grassmann variables for constructing certain coherent states. Coherent states of the Perelomov type are derived for infinite-dimensional representations of and for finite-dimensional representations of and through a Fock–Bargmann analytical approach based on the use of complex (or bosonic) variables. The same approach is applied for deriving coherent states of the Barut–Girardello type in the case of infinite-dimensional representations of . In contrast, the construction of coherent states à la Barut–Girardello for finite-dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized Grassmann (or k -fermionic) variables. Some of the results are applied to su (2), su (1, 1) and the harmonic oscillator (in a truncated or not truncated form). This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
It is the aim of this paper to show how to construct á la Perelomov and á la Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction of á la Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.
It is the aim of this paper to show how to construct Perelomov and Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction of Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.
We construct ladder operators, C̃ and C̃†, for a multistep rational extension of the harmonic oscillator on the half plane, x ≥ 0. These ladder operators connect all states of the spectrum in only infinite-dimensional representations of their polynomial Heisenberg algebra. For comparison, we also construct two different classes of ladder operator acting on this system that form finite-dimensional as well as infinite-dimensional representations of their respective polynomial Heisenberg algebras. For the rational extension, we construct the position wavefunctions in terms of exceptional orthogonal polynomials. For a particular choice of parameters and for the three lowest weights μ = −5, −3, and 5, we construct the coherent states, eigenvectors of C̃ with generally complex eigenvalues, z, as superposition of subsets of the energy eigenvectors. Then, we calculate the properties of these coherent states, looking for classical or nonclassical behavior. We calculate the energy expectations as functions of |z|. We plot position probability densities for the coherent states and for the even and odd cat states formed from these coherent states. We plot the Wigner functions for a particular choice of z. For these coherent states on one arm of a beamsplitter, we calculate the two excitation number distributions and the linear entropies of the output states. We plot the standard deviations in x and find squeezing in the regime considered in one of the cases. By plotting the Mandel Q parameters for the coherent states as functions of |z|, we find that the number statistics is sub-Poissonian in all cases.
The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter θ. θ = 0, π correspond to bosons and fermions, respectively; at intermediate values, we say that we have fractional statistics. In two dimensions, θ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve nonabelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of nonabelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern–Simons terms. The crossings of one-dimensional anyons on a ring are unidirectional, such that a fractional phase θ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently, the anyon behavior predicted for quasiparticles in the ν = 1/3 fractional quantum Hall state has been observed in both scattering and interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.