Q-deformed Commutation Research Articles

One-parameter generalization of the Böttcher-Wenzel inequality and its application to open quantum dynamics

In this paper, we introduce a one-parameter generalization of the famous Böttcher-Wenzel (BW) inequality in terms of a q-deformed commutator. For n×n matrices A and B, we consider the inequalityRe〈[B,A],[B,A]q〉≤c(q)‖A‖2‖B‖2, where 〈A,B〉=tr(A⁎B) is the Hilbert-Schmidt inner product, ‖A‖ is the Frobenius norm, [A,B]=AB−BA is the commutator, and [A,B]q=AB−qBA is the q-deformed commutator. We prove that when n=2, or when A is normal with any size n, the optimal bound is given byc(q)=(1+q)+2(1+q2)2. We conjecture that this is also true for any matrices, and this conjecture is perfectly supported for n up to 15 by numerical optimization. When q=1, this inequality is exactly the BW inequality. When q=0, this inequality leads the sharp bound for the r-function which is recently derived for the application to universal constraints of relaxation rates in open quantum dynamics.

Bounding the Frobenius norm of a q-deformed commutator

For two n×n complex matrices A and B, we define the q-deformed commutator as [A,B]q:=AB−qBA for a real parameter q. In this paper, we investigate a generalization of the Böttcher-Wenzel inequality which gives the sharp upper bound of the (Frobenius) norm of the commutator. In our generalisation, we investigate sharp upper bounds on the q-deformed commutator. This generalization can be studied in two different scenarios: firstly bounds for general matrices, and secondly for traceless matrices. For both scenarios, partial answers and conjectures are given for positive and negative q. In particular, denoting the Frobenius norm by ||.||F, when A or B is normal, we prove the following inequality to be true and sharp: ||[A,B]q||F2≤(1+q2)||A||F2||B||F2 for positive q. Also, we conjecture that the same bound is true for positive q when A or B is traceless. For negative q, we conjecture other sharp upper bounds to be true for the generic scenarios and the scenario when A or B is traceless. All conjectures are supported with numerics and proved for n=2.

Fock representations of Q-deformed commutation relations

We consider Fock representations of the Q-deformed commutation relations ∂s∂t†=Q(s,t)∂t†∂s+δ(s,t) for s,t∈T. Here T:=Rd (or more generally T is a locally compact Polish space), the function Q:T2→C satisfies |Q(s,t)|≤1 and Q(s,t)=Q(t,s)¯, and ∫T2h(s)g(t)δ(s,t) σ(ds)σ(dt):=∫Th(t)g(t) σ(dt), σ being a fixed reference measure on T. In the case, where |Q(s,t)|≡1, the Q-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev. These generalized statistics contain anyon statistics as a special case (with T=R2 and a special choice of the function Q). The related Q-deformed Fock space F(H) over H:=L2(T→C,σ) is constructed. An explicit form of the orthogonal projection of H⊗n onto the n-particle space Fn(H) is derived. A scalar product in Fn(H) is given by an operator Pn≥0 in H⊗n which is strictly positive on Fn(H). We realize the smeared operators ∂t† and ∂t as creation and annihilation operators in F(H), respectively. Additional Q-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ∂s†∂t†=Q(t,s)∂t†∂s†, ∂s∂t=Q(t,s)∂t∂s, valid for those s,t∈T for which |Q(s, t)| = 1.

Wick Calculus for Noncommutative White Noise Corresponding to q-Deformed Commutation Relations

We derive the Wick calculus for test and generalized functionals of noncommutative white noise corresponding to q-deformed commutation relations with \(q\in (-1,1)\). We construct a Gel’fand triple centered at the q-deformed Fock space in which both the test, nuclear space and its dual space are algebras with respect to the addition and the Wick multiplication. Furthermore, we prove a Vage-type inequality for the Wick product on the dual space.

Hermite type ladders in q-Weyl algebras

In this paper we show that the q-deformed commutator for a ladder is related to the usual commutator of the modified ladder. In addition, we consider a family of ladders in the (q, h)-Weyl algebra, which gives a factorization of certain second order difference equations.

THREE-PARAMETER (TWO-SIDED) DEFORMATION OF HEISENBERG ALGEBRA

A three-parametric two-sided deformation of Heisenberg algebra (HA), with p, q-deformed commutator in the L.H.S. of basic defining relation and certain deformation of its R.H.S., is introduced and studied. The third deformation parameter μ appears in an extra term in the R.H.S. as pre-factor of Hamiltonian. For this deformation of HA we find novel properties. Namely, we prove it is possible to realize this (p, q, μ)-deformed HA by means of some deformed oscillator algebra. Also, we find the unusual property that the deforming factor μ in the considered deformed HA inevitably depends explicitly on particle number operator N. Such a novel N-dependence is special for the two-sided deformation of HA treated jointly with its deformed oscillator realizations.

Q-Deformed Bosons at Quasi-classical Limit of SO(3)

When a boson interacts with another to form a composite system with SO(3) dynamic symmetry, it is shown that there exists the q-deformed bosonic excitation satisfying the q-deformed Heisenberg commutation relation in the quasi-classical limit that the angular momentum j for SO(3) is large, but not infinite. In second quantization this quasi-excitation is associated with the boson realization of SO(3) Lie algebra. Physically, the phenomena of q-deformed excitation can happen in many models of quantum dynamics, such as super emission from a system of many identical two-level atoms, the spin wave in Heisenberg chain, the high spin precession and the coherent output of Bose–Einstein atoms in a trap. Especially, in these models, the deformation parameter q is well defined intrinsically by a conservative quantity, such as the total atomic number and the angular momentum.

A NOTE ON GAUSSIAN INTEGRALS OVER PARA-GRASSMANN VARIABLES

We discuss the generalization of the connection between the determinant of an operator entering a quadratic form and the associated Gaussian path-integral valid for Grassmann variables to the para-Grassmann case [θp+1=0 with p=1(p>1) for Grassmann (para-Grassmann) variables]. We show that the q-deformed commutation relations of the para-Grassmann variables lead naturally to consider q-deformed quadratic forms related to multiparametric deformations of GL (n) and their corresponding q-determinants. We suggest a possible application to the study of disordered systems.

An algebraicq-deformed form for shape-invariant systems

A quantum deformed theory applicable to all shape-invariant bound-state systems is introduced by defining q-deformed ladder operators. We show these new ladder operators satisfy new q-deformed commutation relations. In this context we construct an alternative q-deformed model that preserve the shape-invariance property presented by primary system. q-deformed generalizations of Morse, Scarf, and Coulomb potentials are given as examples.

Analysis of an atomicJ=0toJ=1two-photon transition as a test of the spin-statistics connection for photons

Using the q-deformed commutator formalism (q mutators), we have calculated the two-photon transition amplitude connecting a J = 0 atomic ground state to a J= I atomic excited state of the same parity. We find, in agreement with a semiclassical calculation, that this transition amplitude vanishes for two equal-frequency photons if the photons are traditional bosons with q = 1. If q < 1 (i.e., if the spin-statistics connection is violated for photons), then the amplitude is nonzero and is proportional to (1 - q) ). Thus such an experiment, originally proposed by Budker and DeMille, provides a sensitive test of the spin-statistics connection for photons within the q-mutator formalism.

New Application of Noncanonical Maps in Quantum Mechanics

One invertible and one unitary operator can be used to reproduce the effect of a q-deformed commutator of annihilation and creation operators. The original annihilation and creation operators are mapped into new operators, not conjugate to each other, whose standard commutator equals the identity plus a correction proportional to the original number operator. The consistency condition for the existence of this new set of operators is derived, by exploiting the Stone theorem on 1-parameter unitary groups. The above scheme leads to modified “equations of motion” which do not preserve the properties of the original first-order set for annihilation and creation operators. Their relation with commutation relations is also studied.

Critical Phenomenon of a Consistent q-Deformed Squeezed State

Within a self-consistent framework of the q-deformed Heisenberg algebra and its equivalent framework of q-deformed boson commutation relations, which relate to the under-cutting phenomenon of Heisenberg's minimal uncertainty relation, special q-deformed squeezed states are constructed. Besides the similar local maximum squeezing as the one in the undeformed case, new strong squeezing appears when the amplitude of the related coherent state increases to large values. A critical phenomenon appears at a large value of the amplitude: the variance of one component of the quadrature of the light field approaches zero, but the variance of the corresponding conjugate quantity remains finite, which is a surprising deviation from Heisenberg's uncertainty relation. The qualitative character exposed by this q-squeezed state may provide some evidence about q-deformed effects in current experiments.

The U(1)-invariant field theories with normal field operators

In this paper we consider a very general U(1)-invariant field theory such that a field operator commutes with its adjoint, that corresponds to a theory of a charged bosonic particle. We show that from such an invariance follows the existence of particles and antiparticles associated with the same physical state. The field operator always turns out to be the sum of a particle creation and an antiparticle annihilation operator. We study in detail the case when creation and annihilation operators factorize and show that such operators are closely related to q-deformed commutation relations.

Phase transition in a q-deformed Lipkin model

Assuming q-deformed commutation relations for the fermions, an extension of the standard Lipkin Hamiltonian is presented. The usual quasi-spin representation of the standard Lipkin model is also obtained in this q-deformed framework. A variationally obtained energy functional is used to analyse the phase transition associated with the spherical symmetry breaking. The only phase transitions in this q-deformed model are of second order. As an outcome of this analysis a critical parameter is obtained which is dependent on the deformation of the algebra and on the number of particles.

Q-OSCILLATORS, NON-KÄHLER MANIFOLDS AND CONSTRAINED DYNAMICS

It is shown that q-deformed quantum mechanics (systems with q-deformed Heisenberg commutation relations) can be interpreted as an ordinary quantum mechanics on Kähler manifolds, or as a quantum theory with second- (or first-) class constraints.

Q-Canonical transformations for many q-bosons and new q–k symmetry

A new symmetry is identified for the many particle q-Bose system, which permits carrying out a linear q-canonical transformation, i.e., one which preserves the q-deformed commutation rules. The symmetry is exhibited when the set of q-modes is partitioned into "positive" and "negative," each subset having its own type of deformation. For plane-wave modes, labelled by wave vector k, this means separate positive and negative wave vectors. Then, identifying the new symmetry: (q → q−1, k → −k) the transformation can be implemented. The new aspect is the simultaneous transformation of wave vector k and parameter q.

Quantum mechanics with q-deformed commutators and periodic variables

A q-deformed commutator for arbitrary q is derived from a variable with a periodic boundary condition such as an azimuthal angle phi (0<or= phi <2 pi ). A Hamiltonian can be written down in an Hermitian form for q=ealpha or q=ei alpha with alpha in R, and its eigenfunctions and eigenvalues are obtained. Algebraic structures, W1+ infinity and Uq(sl2), of this model and introductions of gauge interactions are discussed. Extensions to many variables and some elementary examples are presented.

Quantum mechanics for q-deformed rotators

We study a system for particles rotating on rings, where only discrete changes of an angle variable ϕ are allowed. The momentum operator for such discrete changes p(−)ϕ obeys a q-deformed commutator [eiϕ, p(−)ϕ]q = ih̷. A hamiltonian Hq which coincides with that for a rotator in the limit q → 1 is introduced. Finiteness of the energy spectrum and an infinite-fold degeneracy for every state with a fixed energy appear.

Statistical distribution of gases which obey q-deformed commutation relations

The statistical distribution of a system of q-gases which obey the q-deformed commutation relations, is derived by using the principle of detailed balance. We also study the Bose-Einstein condensation of the q-bosons in a box and find that the condensation temperature will rise as the deformation parameter moves away from q = 1.

GL q( N)-covariant quantum algebras and covariant differential calculus

We consider GL q ( N)-covariant quantum algebras with generators satisfying quadratic polynomial relations. We show that, up to some inessential arbitrariness, there are only two kinds of such quantum algebras, namely, the algebras with q-deformed commutation and q-deformed anticommutation relations. The connection with the bicovariant differential calculus on the linear quantum groups is discussed.