Commutation Relations Of Operators Research Articles

Commutation relations of vertex operators for Uq(sl^(M|N))

We consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra Uq(sl^(M|N)) by using bosonization. We show that vertex operators give a representation of the graded Zamolodchikov-Faddeev algebra by direct computation. Invertibility relations of type-II vertex operators for N > M are very similar to those of type-I for M > N.

Real Gentile Statistical Systems: N-Annulenes.

Gentile statistics, which is famous for its advantages in dealing with composite particle systems, is a kind of fractional statistics. Lately, researchers are concentrating on finding real systems that obey Gentile statistics. Five years ago, we discovered that the cyclic hydrocarbon polyenes called N-annulenes in the Hückel model had certain correspondence with Gentile oscillators. According to their rotation and dihedral symmetry, these two systems had the same energy levels and partition functions. In this paper, we give further discussions. We discuss the transformations of wave functions between N-annulenes and Gentile oscillators, the creation and annihilation operators in site pictures of N-annulenes, the coherent state, and the mathematical proofs of the intermediate commutation relations of operators. All of our works prove that N-annulenes in the Hückel model are real Gentile statistical systems and offer a new algebraic method to deal with the problems of electron-electron and electron-phonon interactions.

On the initial conditions of scalar and tensor fluctuations in f(R,phi ) gravity

We have considered the perturbation equations governing the growth of fluctuations during inflation in generalized scalar tensor theory f(R,phi ). We have found that the scalar metric perturbations at very early times are negligible compared to the scalar field perturbation, just like general relativity. At sufficiently early times, when the physical momentum of perturbation mode, q / a is much larger than the Hubble parameter H, i.e. q/agg H, we have obtained the metric and scalar field perturbation in the form of WKB solutions up to an undetermined coefficient. Then we have quantized the scalar fluctuations and expanded the metric and the scalar field perturbations with the help of annihilation and creation operators of the scalar field perturbation. The standard commutation relations of annihilation and creation operators fix the unknown coefficient. Going over to the gauge invariant quantities which are conserved beyond the horizon, we have obtained the initial condition of the generalized Mukhanov–Sasaki equation. Then a similar procedure is performed for the case of tensor metric perturbation. As an example of the generalized Mukhanov–Sasaki equation and its initial condition, we have proposed a power-law functional form as f(R,phi )=f_0 R^m phi ^n and obtained an exact inflationary solution. In this background, then we have discussed how the scalar and tensor fluctuations grow.

Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations**Supported by the National Science Foundation of China under Grant No. 11371244 and the Applied Mathematical Subject of SSPU under Grant No. XXKPY1604

In this paper, based on a discrete spectral problem and the corresponding zero curvature representation, the isospectral and nonisospectral lattice hierarchies are proposed. An algebraic structure of discrete zero curvature equations is then established for such integrable systems. the commutation relations of Lax operators corresponding to the isospectral and non-isospectral lattice flows are worked out, the master symmetries of each lattice equation in the isospectral hierarchyand are generated, thus a τ-symmetry algebra for the lattice integrable systems is engendered from this theory.

Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus

We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations and stochastic calculus. We study a family of representations of the canonical commutation relations (CCR)-algebra (an infinite number of degrees of freedom), which we call admissible. The family of admissible representations includes the Fock-vacuum representation. We show that, to every admissible representation, there is an associated Gaussian stochastic calculus, and we point out that the case of the Fock-vacuum CCR-representation in a natural way yields the operators of Malliavin calculus. We thus get the operators of Malliavin’s calculus of variation from a more algebraic approach than is common. We further obtain explicit and natural formulas, and rules, for the operators of stochastic calculus. Our approach makes use of a notion of symmetric (closable) pairs of operators. The Fock-vacuum representation yields a maximal symmetric pair. This duality viewpoint has the further advantage that issues with unbounded operators and dense domains can be resolved much easier than what is possible with alternative tools. With the use of CCR representation theory, we also obtain, as a byproduct, a number of new results in multi-variable operator theory which we feel are of independent interest.

Dissipative and nonunitary solutions of operator commutation relations

We study the (generalized) semi-Weyl commutation relations $$ U_gAU_g^*=g(A) \quad \text{ on }\quad \Dom(A), $$ where $A$ is a densely defined operator and $G\ni g\mapsto U_g$ is a unitary representation of the subgroup $G$ of the affine group $\cG$, the group of affine transformations of the real axis preserving the orientation. If $A$ is a symmetric operator, the group $G$ induces an action/flow on the operator unit ball of contractive transformations from $\Ker (A^*-iI)$ to $\Ker (A^*+iI)$. We establish several fixed point theorems for this flow. In the case of one-parameter continuous subgroups of linear transformations, self-adjoint (maximal dissipative) operators associated with the fixed points of the flow give rise to solutions of the (restricted) generalized Weyl commutation relations. We show that in the dissipative setting, the restricted Weyl relations admit a variety of non-unitarily equivalent representations. In the case of deficiency indices $(1,1)$, our general results can be strengthened to the level of an alternative.

Commutation relations for unitary operators II

Let f be a regular non-constant symbol defined on the d-dimensional torus Td with values on the unit circle. Denote respectively by κ and L, its set of critical points and the associated Laurent operator on l2(Zd). Let U be a suitable unitary local perturbation of L. We show that the operator U has finite point spectrum and no singular continuous component away from the set f(κ). We apply these results and provide a new approach to analyze the spectral properties of GGT matrices with asymptotically constant Verblunsky coefficients. The proofs are based on positive commutator techniques. We also obtain some propagation estimates.

Theory of radiation pressure on magneto–dielectric materials

We present a classical linear response theory for a magneto–dielectric material and determine the polariton dispersion relations. The electromagnetic field fluctuation spectra are obtained and polariton sum rules for their optical parameters are presented. The electromagnetic field for systems with multiple polariton branches is quantized in three dimensions and field operators are converted to 1–dimensional forms appropriate for parallel light beams. We show that the field–operator commutation relations agree with previous calculations that ignored polariton effects. The Abraham (kinetic) and Minkowski (canonical) momentum operators are introduced and their corresponding single–photon momenta are identified. The commutation relations of these and of their angular analogues support the identification, in particular, of the Minkowski momentum with the canonical momentum of the light. We exploit the Heaviside–Larmor symmetry of Maxwell’s equations to obtain, very directly, the Einsetin–Laub force density for action on a magneto–dielectric. The surface and bulk contributions to the radiation pressure are calculated for the passage of an optical pulse into a semi–infinite sample.

Commutation relations for unitary operators I

Let U be a unitary operator defined on some infinite-dimensional complex Hilbert space H. Under some suitable regularity assumptions, it is known that a positive commutation relation between U and an auxiliary self-adjoint operator A defined on H allows to prove that the spectrum of U has locally no singular continuous spectrum and a finite point spectrum. We show that these conclusions still hold under weak regularity hypotheses and without any gap condition. As an application, we study the spectral properties of the Floquet operator associated to some perturbations of the quantum harmonic oscillator under resonant AC-Stark potential.

A deformation of affine Hecke algebra and integrable stochastic particle system

We introduce a deformation of the affine Hecke algebra of type which describes the commutation relations of the divided difference operators found by Lascoux and Schützenberger and the multiplication operators. Making use of its representation we construct an integrable stochastic particle system. It is a generalization of the q-Boson system due to Sasamoto and Wadati. We also construct eigenfunctions of its generator using the propagation operator. As a result we get the same eigenfunctions for the -Boson process obtained by Povolotsky.

Weak commutation relations of unbounded operators: Nonlinear extensions

We continue our analysis of the consequences of the commutation relation \documentclass[12pt]{minimal}\begin{document}$[S, T]\break = {\bb 1}$\end{document}[S,T]=1, where S and T are two closable unbounded operators. The weak sense of this commutator is given in terms of the inner product of the Hilbert space \documentclass[12pt]{minimal}\begin{document}${\mathcal {H}},$\end{document}H, where the operators act. We also consider what we call, adopting a physical terminology, a nonlinear extension of the above commutation relations.

Weak commutation relations of unbounded operators and applications

Four possible definitions of the commutation relation [S, T] = 11 of two closable unbounded operators S, T are compared. The weak sense of this commutator is given in terms of the inner product of the Hilbert space \documentclass[12pt]{minimal}\begin{document}${\mathcal H}$\end{document}H where the operators act. Some consequences on the existence of eigenvectors of two numberlike operators are derived and the partial O*-algebra generated by S, T is studied. Some applications are also considered.

On the axiomatic approach to quasi-classical field theory

We study the connection between quasi-classical field theory and axiomatic statements of the quantum field theory, Schwinger source theory, and the Lehmann-Symanzik-Zimmermann (LSZ) formalism. The classical Schwinger source is connected with the classical field; the LSZ R-function is connected with the quantum field operator. The axioms of the quantum field theory are written in the context of the quasi-classical expansion. In the considered approach, the stationary action principle and canonical commutation relations for field operators are obtained as corollaries and are not postulated as initial statements of the theory.

Experimental Realization of Programmable Quantum Gate Array for Directly Probing Commutation Relations of Pauli Operators

We experimentally demonstrate an advanced linear-optical programmable quantum processor that combines two elementary single-qubit programmable quantum gates. We show that this scheme enables direct experimental probing of quantum commutation relations for Pauli operators acting on polarization states of single photons. Depending on a state of two-qubit program register, we can probe either commutation or anticommutation relations. Very good agreement between theory and experiment is observed, indicating high-quality performance of the implemented quantum processor.

Experimental verification of the commutation relation for Pauli spin operators using single-photon quantum interference

We report experimental verification of the commutation relation for Pauli spin operators using the single-photon polarization state. The experimental quantum operation corresponding to the commutator, [ σ z , σ x ] = k σ y , showed process fidelity of 0.94 compared to the ideal σ y operation and | k | is determined to be 2.12 ± 0.18 .

Second quantized representation of observables for orthofermions

Starting from a state vector, which is antisymmetric with exchange of orbital indices only, we provide an alternative approach for the commutation relations for the creation and annihilation operators of orthofermions, particles obeying a new exclusion principle that is more exclusive than the Pauli‘s exclusion principle, forbidding the occupation of both up and down spin fermions in the same orbital state. It is found that the form of the second quantized representation of the physical variables, expressed by one or two body operators, is the same as that of fermions, although the commutation relations of orthofermion creation and annihilation operators are quite different from that of fermions. This finding is useful in the study of strongly correlated electron systems, described by Hubbard or Hubbard like Hamiltonians where the double occupancy of any lattice site by the electrons is avoided.

Intermediate-statistics spin waves

In this paper, we show that spin waves, the elementary excitation of the Heisenberg magneticsystem, obey a kind of intermediate statistics with a finite maximum occupation numbern. We construct an operator realization for the intermediate statistics obeyed by magnons,the quantized spin waves, and then construct a corresponding intermediate-statisticsrealization for the angular momentum algebra in terms of the creation and annihilationoperators of the magnons. In other words, instead of the Holstein–Primakoff representation,a bosonic representation subject to a constraint on the occupation number, we present anintermediate-statistics representation with no constraints. In this realization, the maximumoccupation number is naturally embodied in the commutation relation of creation andannihilation operators, while the Holstein–Primakoff representation is a bosonic operatorrelation with an additional putting-in-by-hand restriction on the occupation number. Wededuce the intermediate-statistics distribution function for magnons from theintermediate-statistics commutation relation of the creation and annihilation operatorsdirectly, which is a modified Bose–Einstein distribution. On the basis of these results, wecalculate the dispersion relations for ferromagnetic and antiferromagnetic spin waves. Therelations between the intermediate statistics that magnons obey and the othertwo important kinds of intermediate statistics, Haldane–Wu statistics and thefractional statistics of anyons, are discussed. We also compare the spectrum of theintermediate-statistics spin wave with the exact solution of the one-dimensionals = 1/2 Heisenberg model, which is obtained by the Bethe ansatz method. For ferromagnets, wetake the contributions from the interaction between magnons (the quartic contribution),the next-to-nearest-neighbor interaction, and the dipolar interaction into account forcomparison with the experiment.

THE ALGEBRAIC STRUCTURE OF ZERO CURVATURE REPRESENTATIONS ASSOCIATED WITH INTEGRABLE COUPLINGS

The commutator of enlarged vector fields was explicitly computed for integrable coupling systems associated with semidirect sums of Lie algebras. An algebraic structure of zero curvature representations is then established for such integrable coupling systems. As an application example of this algebraic structure, the commutation relations of Lax operators corresponding to the enlarged isospectral and nonisospectral AKNS flows are worked out, and thus a τ-symmetry algebra for the AKNS integrable couplings is engendered from this theory.

COMMUTATION RELATIONS AND SELF-CONSISTENCY IN ELECTRODYNAMICS FOR THE FIELDS IN TIME-VARYING MEDIA

In linear media with time-dependent parameters, various commutation relations for the field operators obtained from the Lewis–Riesenfeld invariant operator method are calculated. We investigated whether our development is self-consistent or not by evaluating the Heisenberg equation of motion for field operators using the associated commutation relation.

Commutation relations of Hecke operators for Arakawa lifting

T. Arakawa, in his unpublished note, constructed and studied a theta lifting from elliptic cusp forms to automorphic forms on the quaternion unitary group of signature $(1, q)$. The second named author proved that such a lifting provides bounded (or cuspidal) automorphic forms generating quaternionic discrete series. In this paper, restricting ourselves to the case of $q=1$, we reformulate Arakawa's theta lifting as a theta correspondence in the adelic setting and determine a commutation relation of Hecke operators satisfied by the lifting. As an application, we show that the theta lift of an elliptic Hecke eigenform is also a Hecke eigenform on the quaternion unitary group. We furthermore study the spinor $L$-function attached to the theta lift.