Commutative Research Articles

The q-analogue of the Higgs algebra

In this paper, the q-generalization of the Higgs algebra is considered. The realization of this algebra is shown in an explicit form using a nonlinear transformation of the creation-annihilation operators of the q-harmonic oscillator. This transformation is the performance of two operations, namely, a “correction” using a function of the original Hamiltonian, and raising to the fourth power the creation and annihilation operators of a q-harmonic oscillator. The choice of the “correcting” function is justified by the standard form of commutation relations for the operators of the metaplectic realization Uq(SU(1,1)). Further possible directions of research are briefly discussed to summarize the results obtained. The first direction is quite obvious. It is the consideration of the problem when the dimension of the operator space increases or for any value N. The second direction can be associated with the analysis of the relationship between q-generalizations of the Higgs and Hahn algebras.

The Weyl–Mellin quantization map

In this paper, we present a quantization of the functions of spacetime, i.e. a map, analog to Weyl map, which reproduces the [Formula: see text]-Minkowski commutation relations, and it has the desirable properties of mapping square integrable functions into Hilbert–Schmidt operators, as well as real functions into symmetric operators. The map is based on Mellin transform on radial and time coordinates. The map also defines a deformed ∗ product which we discuss with examples.

Statistical model describing Bose-Einstein and Fermi-Dirac statistics.

Beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles, a unified quantum statistics, which may interpolate smoothly between Bose-Einstein and Fermi-Dirac statistics, is proposed. The commutation relations for such particle creation and annihilation operators are derived, and statistical partition function and thermodynamical properties of an ideal gas of the particles are investigated. Besides the Bose-Einstein condensation, another kind of particle condensation may appear.

Quantisation of Lorentz invariant scalar field theory in non-commutative space-time and its consequence

Quantisation of Lorentz invariant scalar field theory in Doplicher-Fredenhagen-Roberts (DFR) space-time, a Lorentz invariant, non-commutative space-time is studied. We use an approach to quantisation that is based on the equations of motion alone and derive the equal time commutation relation between Doplicher-Fredenhagen-Roberts-Amorim (DFRA) scalar field and its conjugate, which has non-commutative dependent modifications, but the corresponding creation and annihilation operators obey usual algebra. We show that imposing the condition that the commutation relation between the field and its conjugate is same as that in the commutative space-time leads to a deformation of the algebra of quantised oscillators. Both these deformed commutation relations derived are valid to all orders in the non-commutative parameter. By analysing the first non-vanishing terms which are θ3 order, we show that the deformed commutation relations scale as 1/λ4, where λ is the length scale set by the non-commutativity of the space-time. We also derive the conserved currents for DFRA scalar field. Further, we analyse the effects of non-commutativity on Unruh effect by analysing a detector coupled to the DFRA scalar field, showing that the Unruh temperature is not modified but the thermal radiation seen by the accelerated observer gets correction due to the non-commutativity of space-time.

On the Bogolyubov endomorphisms of the renormalized square of white noise algebra

We introduce the quadratic analogue of the Bogolyubov endomorphisms of the canonical commutation relations (CCR) associated with the re-normalized square of white noise algebra (RSWN-algebra). We focus on the structure of a subclass of these endomorphisms: each of them is uniquely determined by a quadruple [Formula: see text], where [Formula: see text] are linear transformations from a test-function space [Formula: see text] into itself, while [Formula: see text] is anti-linear on [Formula: see text] and [Formula: see text] is real. Precisely, we prove that [Formula: see text] and [Formula: see text] are uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text], into itself. Under some additional analytic conditions on [Formula: see text] and [Formula: see text], we discover that we have only two equivalent classes of Bogolyubov endomorphisms, one of them corresponds to the case [Formula: see text] and the other corresponds to the case [Formula: see text]. Finally, we close the paper by building some examples in one and multi-dimensional cases.

Quasi-Hopf twist and elliptic Nekrasov factor

We investigate the quasi-Hopf twist of the quantum toroidal algebra of {mathfrak{gl}}_1 as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter p is identified as the elliptic modulus. Computing the quasi-Hopf twist of the R matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on ℝ4× T2. The same R matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum KZ equation for the trace of intertwiners. We also show that it allows a solution which is factorized into the elliptic Nekrasov factors and the triple elliptic gamma function.

Detecting deformed commutators with exceptional points in optomechanical sensors

Models of quantum gravity imply a modification of the canonical position-momentum commutation relations. In this paper, working with a binary mechanical system, we examine the effect of quantum gravity on the exceptional points of the system. On the one side, we find that the exceedingly weak effect of quantum gravity can be sensed via pushing the system towards a second-order exceptional point, where the spectra of the non-Hermitian system exhibits non-analytic and even discontinuous behavior. On the other side, the gravity perturbation will affect the sensitivity of the system to deposition mass. In order to further enhance the sensitivity of the system to quantum gravity, we extend the system to the other one which has a third-order exceptional point. Our work provides a feasible way to use exceptional points as a new tool to explore the effect of quantum gravity.

Quadratic-exponential functionals of Gaussian quantum processes

This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust performance criteria in control problems for open quantum harmonic oscillators (OQHOs) driven by bosonic fields. We develop a randomised representation for the QEF using a Karhunen–Loeve expansion of the quantum process on a bounded time interval over the eigenbasis of its two-point commutator kernel, with noncommuting position-momentum pairs as coefficients. This representation holds regardless of a particular quantum state and employs averaging over an auxiliary classical Gaussian random process whose covariance operator is specified by the commutator kernel. This allows the QEF to be related to the moment-generating functional of the quantum process and computed for multipoint Gaussian states. For stationary Gaussian quantum processes, we establish a frequency-domain formula for the QEF rate in terms of the Fourier transform of the quantum covariance kernel in composition with trigonometric functions. A differential equation is obtained for the QEF rate with respect to the risk sensitivity parameter for its approximation and numerical computation. The QEF is also applied to large deviations and worst-case mean square cost bounds for OQHOs in the presence of statistical uncertainty with a quantum relative entropy description.

General Wick's theorem for bosonic and fermionic operators

Wick's theorem provides a connection between time ordered products of bosonic or fermionic fields, and their normal ordered counterparts. We consider a generic pair of operator orderings and we prove, by induction, the theorem that relates them. We name this the General Wick's Theorem (GWT) because it carries Wick's theorem as special instance, when one applies the GWT to time and normal orderings. We establish the GWT both for bosonic and fermionic operators, i.e. operators that satisfy c-number commutation and anticommutation relations respectively. We remarkably show that the GWT is the same, independently from the type of operator involved. By means of a few examples, we show how the GWT helps treating demanding problems by reducing the amount of calculations required.

Infrared-safe scattering without photon vacuum transitions and time-dependent decoherence

Scattering in 3 + 1-dimensional QED is believed to give rise to transitions between different photon vacua. We show that these transitions can be removed by taking into account off-shell modes which correspond to Liénard-Wiechert fields of asymptotic states. This makes it possible to formulate scattering in 3 + 1-dimensional QED on a Hilbert space which furnishes a single representation of the canonical commutation relations (CCR). Different QED selection sectors correspond to inequivalent representations of the photon CCR and are stable under the action of an IR finite, unitary S-matrix. Infrared divergences are cancelled by IR radiation. Using this formalism, we discuss the time-dependence of decoherence and phases of out-going density matrix elements in the presence of classical currents. The results demonstrate that although no information about a scattering process is stored in strictly zero-energy modes of the photon field, entanglement between charged matter and low energy modes increases over time.

On the group-theoretical approach to relativistic wave equations for arbitrary spin

Formulating a relativistic equation for particles with arbitrary spin remains an open challenge in theoretical physics. In this study, the main algebraic approaches used to generalize the Dirac and Kemmer–Duffin equations for arbitrary-spin particles are investigated. It is proved that an irreducible relativistic equation formulated using spin matrices satisfying the commutation relations of the anti-de Sitter group leads to inconsistent results, mainly as a consequence of the violation of unitarity and the appearance of a mass spectrum that does not reflect the physical reality of elementary particles. However, the introduction of subsidiary conditions resolves the problem of unitarity and restores the physical meaning of the mass spectrum. The equations obtained by these approaches are solved and the physical nature of the solutions is discussed.

Nonlinear corrections in the quantization of a weakly nonideal Bose gas at zero temperature

In the present paper, quantization of a weakly nonideal Bose gas at zero temperature along the lines of the well-known Bogolyubov approach is performed. The analysis presented in this paper is based, in addition to the steps of the original Bogolyubov approach, on the use of nonoscillation modes (which are also solutions of the linearized Heisenberg equation) for recovering the canonical commutation relations in the linear approximation, as well as on the calculation of the first nonlinear correction to the solution of the linearized Heisenberg equation which satisfies the canonical commutation relations at the next order. It is shown that, at least in the case of free quasi-particles, consideration of the nonlinear correction automatically solves the problem of nonconserved particle number, which is inherent to the original approach.

An extended (3+1)-dimensional Jimbo–Miwa equation: Symmetry reductions, invariant solutions and dynamics of different solitary waves

The Lie symmetry method is used to obtain a variety of closed-form wave solutions for the extended (3[Formula: see text]+[Formula: see text]1)-dimensional Jimbo–Miwa (JM) Equation, which describes certain interesting higher-dimensional waves in ocean studies, marine engineering, and other fields. By applying the Lie symmetry technique, we explicitly investigate all the possible vector fields, commutation relations of the considered vectors, and various symmetry reductions of the equation. Based on three stages of Lie symmetry reductions, the JM equation is reduced to several nonlinear ordinary differential equations (NLODEs). Consequently, abundant closed-form wave solutions are achieved, including arbitrary functional parameters. Evolutionary dynamics of some analytic wave solutions are demonstrated through three-dimensional plots based on numerical simulation. Consequently, singular soliton, kink waves, periodic oscillating wave profiles, combined singular soliton profiles, curved-shaped multiple solitons, and periodic multiple solitons with parabolic wave profiles are demonstrated by taking advantage of symbolic computation work. The obtained analytical wave solutions, which include arbitrary independent functions and other constants of the governing equation, could be used to enrich the advanced dynamical behaviors of solitary wave solutions. Furthermore, the study of conservation laws is investigated via the Ibragimov technique for Lie point symmetries.

The metric nature of matter

We construct a metric structure on a configuration space of gauge connections and show that it naturally produces a candidate for a non-perturbative, 3+1 dimensional Yang-Mills-Dirac quantum field theory on a curved background. The metric structure is an infinite-dimensional Bott-Dirac operator and the fermionic sector of the emerging quantum field theory is generated by the infinite-dimensional Clifford algebra required to construct this operator. The Bott-Dirac operator interacts with the HD(M) algebra, which is a non-commutative algebra generated by holonomy-diffeomorphisms on the underlying manifold, i.e. parallel-transforms along flows of vector fields. This algebra combined with the Bott-Dirac operator encode the canonical commutation and anti-commutation relations of the quantised bosonic and fermionic fields. The square of the Bott-Dirac operator produces both the Yang-Mills Hamilton operator and the Dirac Hamilton operator as well as a topological Yang-Mills term alongside higher-derivative terms and a metric invariant.

Universality of Weyl unitaries

Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p×p unitary matrices given by the diagonal matrix whose entries are the p-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by u and v, satisfy up=vp=1p (the p×p identity matrix) and the commutation relation uv=ζvu, where ζ is a primitive p-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if u and v are any d×d unitary matrices such that up=vp=1d and uv=ζvu, then there exists a unital completely positive linear map ϕ:Mp(C)→Md(C) such that ϕ(u)=u and ϕ(v)=v. We also show, moreover, that any two pairs of p-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent, but that the assertion for three such unitaries fails.There is a standard tensor-product construction involving the Pauli matrices that produces irreducible sequences of anticommuting selfadjoint unitary matrices of arbitrary length. The matrices in this sequence are called Weyl-Brauer unitary matrices [11, Definition 6.63]. This standard construction is generalised herein to the case p≥3, producing a sequence of matrices that we also call Weyl-Brauer unitary matrices. We show that the Weyl-Brauer unitary matrices, as a g-tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory.

Drinfeld Hecke algebras for symmetric groups in positive characteristic

We investigate deformations of skew group algebras arising from the action of the symmetric group on polynomial rings over fields of arbitrary characteristic. Over the real or complex numbers, Lusztig’s graded affine Hecke algebra and analogs are all isomorphic to Drinfeld Hecke algebras, which include the symplectic reflection algebras and rational Cherednik algebras. Over fields of prime characteristic, new deformations arise that capture both a disruption of the group action and also a disruption of the commutativity relations defining the polynomial ring. We classify deformations for the symmetric group acting via its natural (reducible) reflection representation.

Stirling operators in spatial combinatorics

We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol (m)n can be extended from a natural number m∈N to the falling factorials (z)n=z(z−1)⋯(z−n+1) of an argument z from F=R or C, and Stirling numbers of the first and second kinds are the coefficients of the expansions of (z)n through zk, k≤n and vice versa. When taking into account spatial positions of elements in a locally compact Polish space X, we replace N by the space of configurations—discrete Radon measures γ=∑iδxi on X, where δxi is the Dirac measure with mass at xi. The spatial falling factorials (γ)n:=∑i1∑i2≠i1⋯∑in≠i1,…,in≠in−1δ(xi1,xi2,…,xin) can be naturally extended to mappings M(1)(X)∋ω↦(ω)n∈M(n)(X), where M(n)(X) denotes the space of F-valued, symmetric (for n≥2) Radon measures on Xn. There is a natural duality between M(n)(X) and the space CF(n)(X) of F-valued, symmetric continuous functions on Xn with compact support. The Stirling operators of the first and second kind, s(n,k) and S(n,k), are linear operators, acting between spaces CF(n)(X) and CF(k)(X) such that their dual operators, acting from M(k)(X) into M(n)(X), satisfy (ω)n=∑k=1ns(n,k)⁎ω⊗k and ω⊗n=∑k=1nS(n,k)⁎(ω)k, respectively. In the case where X has only a single point, the Stirling operators can be identified with Stirling numbers. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations.

Abstract ladder operators and their applications

We consider a rather general version of ladder operator Z used by some authors in few recent papers, [H 0, Z] = λZ for some , , and we show that several interesting results can be deduced from this formula. Then we extend it in two ways: first we replace the original equality with formula [H 0, Z] = λZ[Z †, Z], and secondly we consider [H, Z] = λZ for some , H ≠ H †. In both cases many applications are discussed. In particular we consider factorizable Hamiltonians and Hamiltonians written in terms of operators satisfying the generalized Heisenberg algebra or the pseudo-bosonic commutation relations.

A Variety of New Traveling Wave Packets and Conservation Laws to the Nonlinear Low-Pass Electrical Transmission Lines via Lie Analysis

This research is based on computing the new wave packets and conserved quantities to the nonlinear low-pass electrical transmission lines (NLETLs) via the group-theoretic method. By using the group-theoretic technique, we analyse the NLETLs and compute infinitesimal generators. The resulting equations concede two-dimensional Lie algebra. Then, we have to find the commutation relation of the entire vector field and observe that the obtained generators make an abelian algebra. The optimal system is computed by using the entire vector field and using the concept of abelian algebra. With the help of an optimal system, NLETLs convert into nonlinear ODE. The modified Khater method (MKM) is used to find the wave packets by using the resulting ODEs for a supposed model. To represent the physical importance of the considered model, some 3D, 2D, and density diagrams of acquired results are plotted by using Mathematica under the suitable choice of involving parameter values. Furthermore, all derived results were verified by putting them back into the assumed equation with the aid of Maple software. Further, the conservation laws of NLETLs are computed by the multiplier method.

Quantum mechanical four-dimensional non-polarizing beamsplitter

Some quantum optics researchers might not realize that classical electromagnetism predicts a mathbf {pi } phase shift between S- and P-polarized reflection and might think the reflection coefficients of the transverse modes are independent, or that such a phase shift has no measurable consequences. In this paper, we discuss theoretical grounds to define elements of a 4x4 matrix to represent the beamsplitter, accurately accounting for transverse polarization modes and phase relations between them. We also provide experimental evidence confirming this matrix representation. From a scientific point of view, the paper addresses a non-trivial equivalence between the classical fields Fresnel formalism and the canonical commutation relations of the quantized photonic fields. That the formalism can be readily verified with a simple experiment provides further benefit. The beamsplitter expression derived can be applied in the field of quantum computing.