We introduce a generalized Fock space for a recently proposed operatorial deformation of the Heisenberg-Weyl (HW) algebra, aimed at describing statistics different from the Bose or Fermi ones. The new Fock space is obtained by the tensor product of the usual Fock space and the space spanned by the eigenstates of the deformation operator ĝ. We prove a “statistical Ehrenfest-like theorem”, stating that the expectation values of the ladder operators of the generalized HW algebra — taken in the ĝ-subspace — are creation and annihilation operators defined in the usual Fock space and obeying the ordinary statistics, according to the ĝ-eigenvalues. Moreover, such a “statistics” operator ĝ can be regarded as the generator of a boson-fermion superselection rule. As a consequence, the generalized Fock space decomposes into incoherent sectors, and therefore one gets a density matrix diagonal in the ĝ eigenstates. This leads, under suitable conditions, to the possibility of continuously interpolating between different statistics. In particular, it is necessary to assume a nonstandard Liouville-Von Neumann equation for the density matrix, of the type already considered e.g. in the framework of quantum gravity. It is also preliminarily shown that our formalism leads in a natural way — due to the very properties of the operator ĝ — to a grading of the HW algebra, and therefore to a supersymmetrical scheme.
Abstract We propose a new generalization of the standard (anti-)commutation relations for creation and annihilation operators of bosons and fermions. These relations preserve the usual symmetry properties of bosons and fermions. Only the standard (anti-)commutator relation involving one creation and one annihilation operator is deformed by introducing fractional coefficients, containing a positive integer parameterp. The Fock space is determined by the classical definition. The new relations are chosen in such a way that the total occupation number in the system has the maximum valuep. From the actions of creation and annihilation operators in the Fock space, a group theoretical framework is determined, and from here the correspondence with known particle statistics is established.
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) 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 (
NM) of
2π, it is finite-dimensional, parametrized 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
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
λ0.
We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as "infinite", Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are : new algebras for infinite statistics, q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, "doubly-infinite" statistics, many representations of orthostatistics, Hubbard statistics and its variations. Key Words : Fock space -- multioscillator algebras -- q-deformation -- quantum groups -- general...
Recently an f -deformed Fock space which is spanned by | n 〉 λ was introduced. These bases are the eigenstates of a deformed non-Hermitian Hamiltonian. In this contribution, we will use rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by | n 〉 λ bases, to a new one, spanned by extended two-parameters bases | n 〉 λ 1 , λ 2 . These bases are now the eigenstates of a non-Hermitian Hamiltonian H λ 1 ,λ 2 = a † λ 1 ,λ 2 a +½, where a † λ 1 , λ 2 and a are, respectively, the deformed creation and ordinary bosonic annihilation operators. The bases | n 〉 λ 1 , λ 2 are non-orthogonal (squeezed states), but normalizable. Then, we deduce the new representations of coherent and squeezed states in our two-parameter Fock space. Finally, we discuss the quantum statistical properties, as well as the non-classical properties of the obtained states numerically.
The recent discoveries of new forms of quantum statistics require a close look at the under-lying Fock space structure. This exercise becomes all the more important in order to provide a general classification scheme for various forms of statistics, and establish interconnections among them whenever it is possible. We formulate a theory of generalized Fock spaces, which has a three tired structure consisting of Fock space, statistics and algebra. This general formalism unifies various forms of statistics and algebras, which were earlier considered to describe different systems. Besides, the formalism allows us to construct many new kinds of quantum statistics and the associated algebras of creation and destruction operators. Some of these are: orthostatistics, null statistics or statistics of frozen order, quantum group based statistics and its many avatars, and ‘doubly-infinite’ statistics. The emergence of new forms of quantum statistics for particles interacting with singular potential is also highlighted.
We consider the two-fermion system whose states are subjected to the superselection rule forbidding the superposition of states with fermionic and bosonic statistics. This implies that separable states are described only by diagonal density matrices. Moreover, we find the explicit formula for the entanglement of formation, which in this case cannot be calculated properly using Wootters's concurrence. We also discuss the problem of the choice of tensor product decomposition in a system of two fermions with the help of Bogoliubov transformations of creation and annihilation operators. Finally, we show that there exist states which are separable with respect to all tensor product decompositions of the underlying Hilbert space.