Commutative Research Articles

SU (1,1) the hidden dynamical symmetry group for an exact bound state of the Hulthen potential

In this research work, an exact normalised bound eigenstate and energy eigenvalue of the Hulthen potential, for the zero angular momentum within the framework of the non-relativistic time independent Schrödinger wave equation were obtained using the Nikiforov–Uvarov method. Also, their self-adjoint differential creation and annihilation (ladder) operators were derived, andfrom the commutation relations their Casimir operator was defined. These results showed that the non-compact group is the dynamical group for this exact bound state, described by the representation with a spectrum bounded below.

Differential equation for partition functions and a duality pseudo-forest

We consider finite quantum systems defined by a mixed set of commutation and anti-commutation relations between components of the Hamiltonian operator. These relations are represented by an anti-commutativity graph that contains necessary and sufficient information for computing the full quantum partition function. We derive a second-order differential equation for an extended partition function z[β, β′, J] that describes a transformation from a “parent” partition function z[0, β′, J] (or anti-commutativity graph) to a “child” partition function z[β, 0, J] (or anti-commutativity graph). The procedure can be iterated, and then one forms a pseudo-forest of duality transformations between quantum systems, i.e., a directed graph in which every vertex (or quantum system) has at most one incoming edge (from its parent system). The pseudo-forest has a single tree connected to a constant partition function, many pseudo-trees connected to self-dual systems, and all other pseudo-trees connected to closed cycles of transformations between mutually dual systems. We also show how the differential equation for the extended partition function can be used to study disordered systems.

On families of twisted power partial isometries

We consider families of power partial isometries satisfying twisted commutation relations with deformation parameters $\lambda_{ij}\in\mathbb C$, $|\lambda_{ij}|=1$. Irreducible representations of such a families are described up to the unitary equivalence. Namely any such representation corresponds, up to the unitary equivalence, to irreducible representation of certain higher-dimensional non-commutative torus.

Lie symmetries, invariant solutions and phenomena dynamics of Boiti–Leon–Pempinelli system

The present work deals with highly nonlinear dispersive water equation named Boiti–Leon–Pempinelli (BLP) system, which is a well known two dimensional generalization of sinh-Gordon equation. The BLP system describes interaction of horizontal wave component propagating in plane in infinite narrow channel with constant depth. The Lie symmetry method is employed to derive infinitesimal generators under invariance conditions of one-parameter transformations. The invariance conditions ensure the reducibility of independent variables and preserve the characteristics of system. The commutative relations of infinitesimal vectors show infinite dimensional algebra of symmetries. Thereafter, similarity variables are derived using infinitesimal generators, which lead to first symmetry reductions. Thus, a repeated process of symmetry reductions provide equivalent system of ODEs. These ODEs are integrated and thus some invariant solutions of BLP system are constructed under parametric constraints. The derived solutions are general than previous researches [J Q Yu et al 2010 Exact solutions and conservation laws of (2 + 1)-dimensional Boiti–Leon–Pempinelli equation, Appl. Math. Comput. 216 2293–2300; M Kumar, R Kumar, 2014 On new similarity solutions of the boiti–leon–pempinelli system, Commun. Theor. Phys. 61 121–126; M Kumar et al 2015 Some more similarity solutions of the (2 + 1)-dimensional BLP system, Comput. Math. Appl. 70 212–221; J Fei et al 2015 Symmetry reduction and explicit solutions of the (2 + 1)-dimensional boiti–leon–pempinelli system, Appl. Math. Comput. 268 432–438; M Kumar et al 2021 Some more invariant solutions of (2 + 1)-water waves, Int. J. Appl. Comput. Math. 7 18] as they contain all four arbitrary functions involved in infinitesimals and several free parameters as well. The deduction of previous results obtained by taking particular choices of arbitrary functions and constants, ensure novelty and importance of these solutions as well as the authenticity and effectiveness of Lie symmetry method. To demonstrate the distinct physical structures of phenomena, these solutions are expanded via numerical simulation. Thus, doubly soliton, line soliton, multi soliton, soliton fusion and fission nature have been analyzed. The graphical representations are seemed useful to understand the dynamics of BLP system and propagation of dispersive water waves in infinite narrow channel.

Noncommutative quantum cosmology for the FLRW model in Unimodular Gravity

It is well known that classical Unimodular Gravity (UG) is equivalent to General Relativity, but the equivalence at the quantum level is not clear. Here we exhibit, in the UG framework, the Quantum Cosmology (QC) for a Friedmann–Lemaitre–Robertson–Walker (FLRW) universe, considering an inflationary scenario and also introducing a deformation on the commutation relations for the minisuperspace variables (NCC).

BMS-invariant free scalar model

The BMS (Bondi--van der Burg--Metzner--Sachs) symmetry arises as the asymptotic symmetry of flat spacetime at null infinity. In particular, the BMS algebra for three-dimensional flat spacetime (${\mathrm{BMS}}_{3}$) is generated by the super-rotation generators that form a Virasoro subalgebra with central charge ${c}_{L}$, together with mutually commuting super-translation generators. The super-rotation and supertranslation generators have nontrivial commutation relations with another central charge ${c}_{M}$. In this paper, we study a free scalar theory in two dimensions exhibiting ${\mathrm{BMS}}_{3}$ symmetry, which can also be understood as the ultrarelativistic limit of a free scalar ${\mathrm{CFT}}_{2}$ in the flipped representation. Upon canonical quantization on the highest weight vacuum, the central charges are found to be ${c}_{L}=2$ and ${c}_{M}=0$. Because of the vanishing central charge ${c}_{M}=0$, the theory features novel properties: there exist primary states which form a multiplet, and the Hilbert space can be organized by an enlarged version of BMS modules dubbed the staggered modules. We further calculate correlation functions and the torus partition function, the latter of which is also shown explicitly to be modular invariant. Is it interesting to note that our model has vanishing ${c}_{M}$, a feature also shared by the so-called flat space chiral gravity in Bagchi et al. [Flat-Space Chiral Gravity, Phys. Rev. Lett. 109, 151301 (2012)].

Lax Operator Algebras and Applications to τ-Symmetries for Multilayer Integrable Couplings

The algebraic structures of zero curvature representations are furnished for multilayer integrable couplings associated with matrix spectral problems, both discrete and continuous. The key elements are a class of matrix loop algebras consisting of block matrices with blocks of the same size. As illustrative examples, isospectral and non-isospectral integrable couplings and the corresponding commutator relations of their Lax operators are computed explicitly in the cases of the Volterra lattice hierarchy and the AKNS hierarchy, along with their τ-symmetry algebras.

Anyonic-parity-time symmetry in complex-coupled lasers.

Non-Hermitian Hamiltonians, and particularly parity-time (PT) and anti-PT symmetric Hamiltonians, play an important role in many branches of physics, from quantum mechanics to optical systems and acoustics. Both the PT and anti-PT symmetries are specific instances of a broader class known as anyonic-PT symmetry, where the Hamiltonian and the PT operator satisfy a generalized commutation relation. Here, we study theoretically these novel symmetries and demonstrate them experimentally in coupled lasers systems. We resort to complex coupling of mixed dispersive and dissipative nature, which allows unprecedented control on the location in parameter space where the symmetry and symmetry breaking occur. Moreover, tuning the coupling in the same physical system allows us to realize the special cases of PT and anti-PT symmetries. In a more general perspective, we present and experimentally validate a new relation between laser synchronization and the symmetry of the underlying non-Hermitian Hamiltonian.

Weak solutions to the Cauchy problem of the time-dependent Thomas–Fermi equations

In this paper, we are concerned with the existence of weak solutions of the time-dependent Thomas–Fermi equations. We derive approximate solutions by the fractional step Lax–Friedrichs scheme and establish uniform boundedness of approximate solutions. Based on the uniform energy-type estimates, we establish that the entropy dissipation measures of the weak solution of the one-dimensional time-dependent Thomas–Fermi equations for weak entropy–entropy flux pairs, generated by compactly supported C0∞ test functions, are confined in a compact set in Hloc−1. We prove that the Young measure must be a Dirac measure by the Tartar–Murat commutator relation. The convergence of approximate solutions is established by using the compensated compactness method.

Quantum field theory for spin operator of the photon

All elementary particles in nature can be classified as fermions with half-integer spin and bosons with integer spin. Within quantum electrodynamics (QED), even though the spin of the Dirac particle is well defined, there exist open questions on the quantized description of spin of the gauge field particle -- the photon. Using quantum field theory, we discover the quantum operators for the spin angular momentum (SAM) $\boldsymbol{S}_{M}=(1/c)\int d^{3}x\boldsymbol{\pi}\times\boldsymbol{A}$ and orbital angular momentum (OAM) $\boldsymbol{L}_{M}=-(1/c)\int d^{3}x\pi^{\mu}\boldsymbol{x}\times\boldsymbol{\nabla}A_{\mu}$ of the photon, where $\pi^{\mu}$ is the conjugate canonical momentum of the gauge field $A^{\mu}$. We also reveal a perfect symmetry between the angular momentum commutation relations for Dirac fields and Maxwell fields. We derive the well-known OAM and SAM of classical electromagnetic fields from the above defined quantum operators. Our work shows that the spin and OAM operators commute which is important for simultaneously observing and separating the SAM and OAM. The correct commutation relations of orbital and spin angular momentum of the photon have applications in quantum optics, topological photonics as well as nanophotonics and can be extended in the future for the spin structure of nucleons.

Quantized nilradicals of parabolic subalgebras of and algebras of coinvariants

Let PJ be the standard parabolic subgroup of SLn associated to a subset J of simple roots, and let be the standard Levi decomposition. We let and denote the quantized algebras of regular functions on PJ and LJ , respectively. Following work of the first author, we study the quantum analogue of an induced coaction and the corresponding subalgebra of coinvariants. It was previously shown that the smash product algebra is isomorphic to In view of this, – while it is not a Hopf algebra – can be viewed as a quantum analogue of the coordinate ring In this article we prove that when is nonzero and not a root of unity, is isomorphic to a quantum Schubert cell algebra associated to a certain parabolic element w in the Weyl group of In this setting, the quantum Schubert cell algebra is a q-deformation of the universal enveloping algebra of the nilradical of a parabolic subalgebra of We also compute explicit commutation relations among the Lusztig root vectors for these particular quantized nilradicals and we give explicit algebra isomorphisms from to

Time and frequency as quantum continuous variables

We present a second quantization description of frequency-based continuous variables quantum computation in the subspace of single photons. For this, we define frequency and time operators using the free-field Hamiltonian and its Fourier transform, and we show that these observables, when restricted to the one photon per mode subspace, reproduce the canonical position-momentum commutation relations. As a consequence, frequency and time operators can be used to define a universal set of gates in this particular subspace. We discuss the physical implementation of these gates as well as their effect on single photon states, and show that frequency and time variables can also be used to implement continuous variables quantum information protocols, in the same way than polarization is currently used as a two-dimensional quantum variable.

Lie symmetries and exact solutions for a fourth‐order nonlinear diffusion equation

In this paper, we consider a fourth‐order nonlinear diffusion partial differential equation, depending on two arbitrary functions. First, we perform an analysis of the symmetry reductions for this parabolic partial differential equation by applying the Lie symmetry method. The invariance property of a partial differential equation under a Lie group of transformations yields the infinitesimal generators. By using this invariance condition, we present a complete classification of the Lie point symmetries for the different forms of the functions that the partial differential equation involves. Afterwards, the optimal systems of one‐dimensional subalgebras for each maximal Lie algebra are determined, by computing previously the commutation relations, with the Lie bracket operator, and the adjoint representation. Next, the reductions to ordinary differential equations are derived from the optimal systems of one‐dimensional subalgebras. Furthermore, we study travelling wave reductions depending on the form of the two arbitrary functions of the original equation. Some travelling wave solutions are obtained, such as solitons, kinks and periodic waves.

Quantum Euclidean spaces with noncommutative derivatives

Quantum Euclidean spaces, as Moyal deformations of Euclidean spaces, are the model examples of noncompact noncommutative manifold. In this paper, we study the quantum Euclidean space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example of semifinite spectral triple with nonflat geometric structure. We develop an abstract symbol calculus for the pseudo-differential operators with noncommuting derivatives. We also obtain a local index formula in our setting via the computation of the Connes–Chern character of the corresponding spectral triple.

Hecke correspondences for smooth moduli spaces of sheaves

We define functors on the derived category of the moduli space ℳ of stable sheaves on a smooth projective surface (under Assumptions A and S below), and prove that these functors satisfy certain commutation relations. These relations allow us to prove that the given functors induce an action of the elliptic Hall algebra on the K-theory of the moduli space ℳ, thus generalizing the action studied by Nakajima, Grojnowski and Baranovsky in cohomology.

On the R-matrix realization of quantum loop algebras

We consider rr-matrix realization of the quantum deformations of the loop algebras \tilde{\mathfrak{g}}𝔤̃ corresponding to non-exceptional affine Lie algebras of type \widehat{\mathfrak{g}}=A^{(1)}_{N-1}𝔤̂=AN−1(1), B^{(1)}_nBn(1), C^{(1)}_nCn(1), D^{(1)}_nDn(1), A^{(2)}_{N-1}AN−1(2). For each U_q(\tilde{\mathfrak{g}})Uq(𝔤̃) we investigate the commutation relations between Gauss coordinates of the fundamental \mathbb{L}𝕃-operators using embedding of the smaller algebra into bigger one. The new realization of these algebras in terms of the currents is given. The relations between all off-diagonal Gauss coordinates and certain projections from the ordered products of the currents are presented. These relations are important in applications to the quantum integrable models.

Towards a supersymmetric w1+∞ symmetry in the celestial conformal field theory

The ${w}_{1+\ensuremath{\infty}}$ symmetry algebra appears in the Einstein--Yang--Mills theory, proposed recently by Strominger. In this paper, we derive the supersymmetric ${w}_{1+\ensuremath{\infty}}$ symmetry by using the known results on the operator product expansions (OPEs) between the graviton, gravitino, gluon, and gluino in the supersymmetric version of the above theory. We calculate the four additional commutator relations between the soft currents explicitly. In addition, we analyze the works of Odake et al. and Pope et al. and introduce the additional symmetry current that corresponds to the celestial gluino operator. Through this procedure, all seven commutator relations can be connected to the ones associated with the supersymmetric ${w}_{1+\ensuremath{\infty}}$ algebra with $SU(N)$ symmetry under the restrictions of wedge modes.

Investigation of wave solutions and conservation laws of generalized Calogero–Bogoyavlenskii–Schiff equation by group theoretic method

This work is focused to analyze the generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) by the Lie symmetry method. GCBS equation has been utilized to explain the wave profiles in soliton theory. GCBSE was constructed by Bogoyavlenskii and Schiff in different ways (explained in the introduction section). With the aid of Lie symmetry analysis, we have computed the symmetry generators of the GCBSE and commutation relation. We observed from the commutator table, translational symmetries make an Abelian algebra. Then by using the theory of Lie, we have discovered the similarity variables, which are used to convert the supposed nonlinear partial differential equation (NLPDE) into a nonlinear ordinary differential equation (NLODE). Using the new auxiliary method (NAM), we have to discover some new wave profiles of GCBSE in the type of few trigonometric functions. These exits some parameters which we give to some suitable values to attain the different diagrams of some obtained solutions. Further, the GCBSE is presented by non-linear self-adjointness, and conserved vectors are discovered corresponding to each generator.

Path-integral derivation of the equations of the anomalous Hall effect

A path integral (Lagrangian formalism) is used to derive the effective equations of motion of the anomalous Hall effect with Berry's phase on the basis of the adiabatic condition $|E_{n\pm1}-E_{n}|\gg 2\pi\hbar/T$, where $T$ is the typical time scale of the slower system and $E_{n}$ is the energy level of the fast system. In the conventional definition of the adiabatic condition with $T\rightarrow {\rm large}$ and fixed energy eigenvalues, no commutation relations are defined for slower variables by the Bjorken-Johnson-Low prescription except for the starting canonical commutators. On the other hand, in a singular limit $|E_{n\pm1}-E_{n}|\rightarrow \infty$ with specific $E_{n}$ kept fixed for which any motions of the slower variables $X_{k}$ can be treated to be adiabatic, the non-canonical dynamical system with deformed commutators and the Nernst effect appear. In the Born-Oppenheimer approximation based on the canonical commutation relations, the equations of motion of the anomalous Hall effect is obtained if one uses an auxiliary variable $X_{k}^{(n)}=X_{k}+{\cal A}^{(n)}_{k}$ with Berry's connection ${\cal A}^{(n)}_{k}$ in the absence of the electromagnetic vector potential $eA_{k}(X)$ and thus without the Nernst effect. It is shown that the gauge symmetries associated with Berry's connection and the electromagnetic vector potential $eA_{k}(X)$ are incompatible in the canonical Hamiltonian formalism. The appearance of the non-canonical dynamical system with the Nernst effect is a consequence of the deformation of the quantum principle to incorporate the two incompatible gauge symmetries.

Many-body probes for quantum features of spacetime

Many theories of quantum gravity can be understood as imposing a minimum length scale the signatures of which can potentially be seen in precise table top experiments. In this work, we inspect the capacity for correlated many-body systems to probe non-classicalities of spacetime through modifications of the commutation relations. We find an analytic derivation of the dynamics for a single mode light field interacting with a single mechanical oscillator and with coupled oscillators to first order corrections to the commutation relations. Our solution is valid for any coupling function as we work out the full Magnus expansion. We numerically show that it is possible to have superquadratic scaling of a nonstandard phase term, arising from the modification to the commutation relations, with coupled mechanical oscillators.