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 (${\cal N} = βα$) and construct the Hilbert space, classified by values of a complex coordinate ($λ_0$): the eigenvalues lie on a circle. For $θ$ being an irrational multiple of $2 π$, we get an infinite-dimensional representation, however for a rational multiple ($\frac{M}{N}$) of $2 π$, it is finite-dimensional, parametrized by the complex coordinate $λ_0$. The case for $N=2 \: ; \: θ=π$ is the usual Clifford algebra for fermions, while the case for $N=\infty \: ; \: θ=0$ 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 $N \rightarrow \infty$ from $N=2$. After constructing the Hilbert space from the algebra, the cases for $N=2,3$ can be mapped to $SU(2)$. Then, we motivate the study of coherent states, rather generally. The coherent states are eigenstates of $α$, the annihilation operator and are labeled by complex numbers for non-zero $λ_0$.
Operators, refered to as k-fermion operators, that interpolate between boson and fermion operators are introduced through the consideration of two noncommuting quon algebras. The deformation parameters for these quon algebras are roots of unity connected to an integer k. The case k=2 corresponds to fermions and the limiting case k going to infinity to bosons. Generalized coherent states and supercoherent states are investigated. The Dirac quantum phase operator and the Fairlie-Fletcher-Zachos algebra are also considered.
Interpolating functional method is a powerful tool for studying the behavior of a quantity in the intermediate region of the parameter space of interest by using its perturbative expansions at both ends. Recently several interpolating functional methods have been proposed, in addition to the well-known Pade approximant, namely the "Fractional Power of Polynomial" (FPP) and the "Fractional Power of Rational functions" (FPR) methods. Since combinations of these methods also give interpolating functions, we may end up with multitudes of the possible approaches. So a criterion for choosing an appr…
Starting from determinants at finite temperature obeying an intermediate boundary condition between the periodic (bosonic) and antiperiodic (fermionic) cases, we find results which can be mapped onto the ones obtained from anyons for the second virial coefficient. Using this approach, we calculate the corresponding higher virial coefficients and compare them with the results known in the literature.
A light boson $U$ from an extra $U(1)$ interpolates between a generalized dark photon coupled to $Q,\ B$ and $L_i$ (or $B-L$), plus possibly dark matter, a dark $Z$ coupled to the $Z$ current, and one axially coupled to quarks and leptons. We identify the corresponding $U(1)_F$ symmetries, with $F= γ_Y Y+γ_B B+γ_{L_i}L_i+γ_A F_A+γ_{F'}F'\!+γ_d F_d$, $F_d$ acting in a dark sector and $F'$ on possible semi-inert BEH doublets uncoupled to quarks and leptons. The $U$ current is obtained from the $U(1)_F $ and $Z$ currents, with a mixing determined by the spin-0 BEH fields. The charge $Q_U$ of chir…
We show that the kinetic approach to statistical mechanics permits an elegant and efficient treatment of fractional exclusion statistics. By using the exclusion-inclusion principle recently proposed [Phys. Rev. E49, 5103 (1994)] as a generalization of the Pauli exclusion principle, which is based on a proper definition of the transition probability between two states, we derive a variety of different statistical distributions interpolating between bosons and fermions. The Haldane exclusion principle and the Haldane-Wu fractional exclusion statistics are obtained in a natural way as particular …
Motivated by Haldane's exclusion statistics, we construct creation and annihilation operators for $g$-ons using a bosonic algebra. We find that $g$-ons appear due to the breaking of a descrete symmetry of the original bosonic system. This symmetry is intimately related to the braid group and we demonstrate a link between exclusion statistics and fractional statistics.
We review some basic elements on k-fermions, which are objects interpolating between bosons and fermions. In particular, we define k-fermionic coherent states and study some of their properties. The decomposition of a Q-uon into a boson and a k-fermion leads to a definition of fractional supercoherent states. Such states involve bosonic coherent states and k-fermionic coherent states. We construct an Hamiltonian which generalizes the ordinary (or Z_2-graded) supersymmetric oscillator Hamiltonian. Our Hamiltonian describes a fractional (or Z_k-graded) supersymmetric oscillator for which the fra…
The Hamiltonian for a fractional supersymmetric oscillator is derived from three approaches. The first one is based on a decomposition in which a Q-uon gives rise to an ordinary boson and a k-fermion (a k-fermion being an object interpolating between boson and fermion). The second one starts from a generalized Weyl-Heisenberg algebra. Finally, the third one relies on the quantum algebra Uq(sl(2)) where q is a root of unity.
In noncommutative space to maintain Bose-Einstein statistics for identical particles at the non-perturbation level described by deformed annihilation-creation operators when the state vector space of identical bosons is constructed by generalizing one-particle quantum mechanics it is explored that the consistent ansatz of commutation relations of phase space variables should simultaneously include space-space noncommutativity and momentum-momentum noncommutativity, and a new type of boson commutation relations at the deformed level is obtained. Consistent perturbation expansions of deformed an…
The aim of this work is to further study the fractional bosonic string theory. In particular, we wrote the energy-momentum tensor in the fractional conformal gauge and study their symmetries. We introduced the Virasoro operators of all orders. In fact, we found the same $L_0 (\widetilde{L}_0)$ operator originally defined in the work of fractional bosonic string up to a shift transformation. Also, we compute the algebra of our Fractional Virasoro Operators, finding that the satifies the $Witt$ algebra. Lastly, we showed that in the boundary of our theory we recover the lost conservation law ass…
An outstanding idea originally introduced by Greenberg is to investigate whether there is equivalence between intermediate statistics, which may be different from anyonic statistics, and q-deformed particle algebra. Also, a model to be studied for addressing such an idea could possibly provide us some new consequences about the interactions of particles as well as their internal structures. Motivated mainly by this idea, in this work, we consider a q-deformed Fermi gas model whose statistical properties enable effectively us to study interpolating statistics. Starting with a generalized Fermi-…
The $K=4$ fractional superstring Fock space is constructed in terms of $\bZ_4$ parafermions and free bosons. The bosonization of the $\bZ_4$ parafermion theory and the generalized commutation relations satisfied by the modes of various parafermion fields are reviewed. In this preliminary analysis, we describe a Fock space which is simply a tensor product of $\bZ_4$ parafermion and free boson Fock spaces. It is larger than the Lorentz-covariant Fock space indicated by the fractional superstring partition function. We derive the form of the fractional superconformal algebra that may be used as t…
We pin down the origin of a peculiar dispersion relation of domain wall fluctuation, the so-called ripplon, in a superfluid-superfluid interface. A ripplon has a dispersion relation $ω\propto k^{3/2}$ due to the nonlocality of the effective Lagrangian, which is mediated by gapless superfluid phonons in the bulk. We point out the analogy to the longitudinal phonon in the two-dimensional Wigner crystal.
We develop the basis of the two dimensional generalized quantum statistical systems by using results on $r$-generalized Fibonacci sequences. According to the spin value $s$ of the 2d-quasiparticles, we distinguish four classes of quantum statistical systems indexed by $ s=0,1/2:mod(1)$, $s=1/M:mod(1)$, $s=n/M:mod(1)$ and $0\leq s\leq 1:mod(1)$. For quantum gases of quasiparticles with $s=1/M:mod(1)$, $M\geq 2,$, we show that the statistical weights densities $ρ_M$ are given by the integer hierarchies of Fibonacci sequences. This is a remarkable result which envelopes naturally the Fermi and Bo…
Employing a thermodynamic interpretation of gravity based on the holographic principle and assuming underlying particle statistics, fermionic or bosonic, for the excitations of the holographic screen leads to Modified Newtonian Dynamics (MOND). A connection between the acceleration scale $a_0$ appearing in MOND and the Fermi energy of the holographic fermionic degrees of freedom is obtained. In this formulation the physics of MOND results from the quantum-classical crossover in the fermionic specific heat. However, due to the dimensionality of the screen, the formalism is general and applies t…
For each lattice one can define a free boson theory propagating on the corresponding torus. We give an alternative definition where one employs any automorphism of the group $M^*/M$. This gives a wealth of conformal data, which we realize as some bosonic theory, in all the `regular' cases. We discuss the generalization to affine theories. As a byproduct, we compute the gauss sum for any lattice and any diagonal automorphism.
Two complementary approaches of N = 2 fractional supersymmetric quantum mechanics of order k are studied in this article. The first one, based on a generalized Weyl-Heisenberg algebra W(k) (that comprizes the affine quantum algebra Uq(sl(2)) with q to k = 1 as a special case), apparently contains solely one bosonic degree of freedom. The second one uses generalized bosonic and k-fermionic degrees of freedom. As an illustration, a particular emphasis is put on the fractional supersymmetric oscillator of order k.
Scattering amplitudes of the spin-4/3 fractional superstring are shown to satisfy spurious state decoupling and cyclic symmetry (duality) at tree-level in the string perturbation expansion. This fractional superstring is characterized by the spin-4/3 fractional superconformal algebra---a parafermionic algebra studied by Zamolodchikov and Fateev involving chiral spin-4/3 currents on the world-sheet in addition to the stress-energy tensor. Examples of tree scattering amplitudes are calculated in an explicit c=5 representation of this fractional superconformal algebra realized in terms of free bo…
The chiral algebra of the symmetric product orbifold of a single-boson CFT corresponds to a "higher spin square" algebra in the large $N$ limit. In this note, we show that a symmetrized collection of $N$ bosons defines a similar structure that we refer to as the higher spin rectangle algebra. We explore the relation of this algebra to the higher spin square algebra. The existence of such a truncated algebra hints at bulk theories interpolating between Vasiliev higher spin theory and string theory.
Topological insulators are characterized by the presence of gapless surface modes protected by time-reversal symmetry. In three space dimensions the magnetoelectric response is described in terms of a bulk theta term for the electromagnetic field. Here we construct theoretical examples of such phases that cannot be smoothly connected to any band insulator. Such correlated topological insulators admit the possibility of fractional magnetoelectric response described by fractional theta/pi. We show that fractional theta/pi is only possible in a gapped time reversal invariant system of bosons or f…
The topological $p$-wave pairing of composite fermions, believed to be responsible for the 5/2 fractional quantum Hall effect (FQHE), has generated much exciting physics. Motivated by the parton theory of the FQHE, we consider the possibility of a new kind of emergent "superconductivity" in the 1/3 FQHE, which involves condensation of clusters of $n$ composite bosons. From a microscopic perspective, the state is described by the $n\bar{n}111$ parton wave function ${\cal P}_{\rm LLL} Φ_nΦ_n^*Φ_1^3$, where $Φ_n$ is the wave function of the integer quantum Hall state with $n$ filled Landau levels…
