On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.     

General chiral gauge theories on the lattice

We still extend the large class of Dirac operators decribing massless fermions on the lattice found recently, only requiring that such operators decompose into Weyl operators. After deriving general relations and constructions of operators, we study the basis representations of the chiral projections. We then investigate correlation functions of Weyl fermions for any value of the index, stressing the related conditions for basis transformations and their consequences, and getting the precise behaviors under gauge transformations and CP transformations. Various further developments include considerations of the explicit form of the effective action and of a representation of the general correlation functions in terms of alternating multilinear forms. For comparison we also consider gauge-field variations and their respective applications. Finally we compare with continuum perturbation theory.     

On the <Emphasis Type="Italic">s</Emphasis>-parameterized Quantization Scheme for Dirac’s Representation theory

Dirac is the founder of quantum mechanical representation theory. By virtue of the technique of integration within an ordered product (IWOP) of operators we introduce s-parameterized form of quantum mechanical coordinate and momentum representations, which are complete. We then point out that s-parameterized representation’s completeness relation is accompanied with operators’ s-ordering, the special cases s=1,0,−1 correspond to normal-ordering, Weyl ordering and antinormal-ordering, respectively. The s-parameterized form of the coherent state representation and the entangled state representation are also derived. In our view, the operators’ s-ordering should be traced back to s-parameterized form of the completeness relation of quantum mechanical coordinate and momentum representations, which is more fundamental. Many operator identities can be derived by virtue of the above mentioned s-parameterized representation’s completeness relations.     

Orthogonality between scales in a renormalization group for fermions

Having in mind the development of a technical tool to treat fermionic systems, we propose a Kadanoff-Wilson block renormalization transformation employing unusual averages (an inevitable artifact due to the specificity of lattice fermions and to the desired transformation properties). The free propagator is decomposed into operators associated to different momentum scales and with orthogonal relations, and the effective actions generated from the Dirac operator by the transformations present uniform exponential decay. We argue to show the usefulness of the formalism to study correlation functions of interacting fermions.     

ON THE IRREDUCIBLE REPRESENTATIONS OF THE SIMPLE LIE GROUPS I——THE TENSOR BASIS FOR THE INFINITESIMAL GENERATORS OF THE CLASSICAL SIMPLE LIE GROUPS

In this paper, we analyse the commutation relations of the infinitesimal generatorsof all simple classical Lie groups and establish a new basis for these generators, calledthe tensor basis. In tensor basis, the infinitesimal, generators can be written as somescalar operators, some sets of angular momentum operators and some sets of irreducibletensor operators. The commutation relations, of these operators are very simple andhave many regularities. By means of the method that has been used in the earlier papers, "On the irre-ducible representations of the compact simple Lie groups of rank 2, I,II,III" and thetensor basis, all the irreducible representations of the classical simple Lie groups canbe calculated systematically.     

Entirety of Quantum Uncertainty and Its Experimental Verification

As a foundation of quantum physics, uncertainty relations describe ultimate limit for the measurement uncertainty of incompatible observables. Traditionally, uncertainty relations are formulated by mathematical bounds for a specific state. Here we present a method for geometrically characterizing uncertainty relations as an entire area of variances of the observables, ranging over all possible input states. We find that for the pair of position and momentum operators, Heisenberg's uncertainty principle points exactly to the attainable area of the variances of position and momentum. Moreover, for finite-dimensional systems, we prove that the corresponding area is necessarily semialgebraic; in other words, this set can be represented via finite polynomial equations and inequalities, or any finite union of such sets. In particular, we give the analytical characterization of the areas of variances of(a) a pair of one-qubit observables and(b) a pair of projective observables for arbitrary dimension,and give the first experimental observation of such areas in a photonic system.     

Dirac Induction for Loop Groups

Using a coset version of the cubic Dirac operators for affine Lie algebras, we give an algebraic construction of the Dirac induction homomorphism for loop group representations. With this, we prove a homogeneous generalization of the Weyl–Kac character formula and show compatibility with Dirac induction for compact Lie groups.     

Exact Solution of D-Dimensional Klein-Gordon Oscillator with Minimal Length

Specific modifications of the usual canonical commutation relations between position and momentum operators have been proposed in the literature in order to implement the idea of the existence of a minimal observable length. Here, we study a consequence of this proposal in relativistic quantum mechanics by solving in the momentum space representation the Klein-Gordon oscillator in arbitrary dimensions. The exact bound states spectrum and the corresponding momentum space wave function are obtained. Following Chang et al, （Phys. Rev. D 65 （2002） 125027）, we discuss constraint that can be placed on the minimal length by measuring the energy levels of an electron in a penning trap.     

Mean motion of Dirac electrons in a gravitational field

A sequence of Foldy-Wouthuysen transformations applied to the Dirac equation coupled to a background gravitational field is used to obtain evolution equations for the mean position, mean momentum, and mean spin operators. These equations are compared with the analogous Papapetrou equations for a classical gravitationally coupled pole-dipole particle.     

Equivalent Forms of Dirac Equations in Curved Space-times and Generalized de Broglie Relations

One may ask whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We first transform any Dirac equation into an equivalent canonical form, sometimes used in particular cases to solve Dirac equations in a curved space-time. This canonical form is needed to apply Whitham’s Lagrangian method. The latter method, unlike the Wentzel–Kramers–Brillouin method, places no restriction on the magnitude of Planck’s constant to obtain wave packets and furthermore preserves the symmetries of the Dirac Lagrangian. We show by using canonical Dirac fields in a curved space-time that the probability current has a Gordon decomposition into a convection current and a spin current and that the spin current vanishes in the Whitham approximation, which explains the negligible effect of spin on wave packet solutions, independent of the size of Planck’s constant. We further discuss the classical-quantum correspondence in a curved space-time based on both Lagrangian and Hamiltonian formulations of the Whitham equations. We show that the generalized de Broglie relations in a curved space-time are a direct consequence of Whitham’s Lagrangian method and not just a physical hypothesis as introduced by Einstein and de Broglie and by many quantum mechanics textbooks.     

Baxter operators for arbitrary spin

We construct Baxter operators for the homogeneous closed XXX spin chain with the quantum space carrying infinite- or finite-dimensional s?₂ representations. All algebraic relations of Baxter operators and transfer matrices are deduced uniformly from Yang-Baxter relations of the local building blocks of these operators. This results in a systematic and very transparent approach where the cases of finite- and infinite-dimensional representations are treated in analogy. Simple relations between the Baxter operators of both cases are obtained. We represent the quantum spaces by polynomials and build the operators from elementary differentiation and multiplication operators. We present compact explicit formulae for the action of Baxter operators on polynomials.     

Generally covariant quantization and the Dirac field

Canonical Hamiltonian field theory in curved spacetime is formulated in a manifestly covariant way. Second quantization is achieved invoking a correspondence principle between the Poisson bracket of classical fields and the commutator of the corresponding quantum operators. The Dirac theory is investigated and it is shown that, in contrast to the case of bosonic fields, in curved spacetime, the field momentum does not coincide with the generators of spacetime translations. The reason is traced back to the presence of second class constraints occurring in Dirac theory. Further, it is shown that the modification of the Dirac Lagrangian by a surface term leads to a momentum transfer between the Dirac field and the gravitational background field, resulting in a theory that is free of constraints, but not manifestly hermitian.     

The physical meaning of the de Sitter invariants

We study the Lie algebras of the covariant representations transforming the matter fields under the de Sitter isometries. We point out that the Casimir operators of these representations can be written in closed forms and we deduce how their eigenvalues depend on the field’s rest energy and spin. For the scalar, vector and Dirac fields, which have well-defined field equations, we express these eigenvalues in terms of mass and spin obtaining thus the principal invariants of the theory of free fields on the de Sitter spacetime. We show that in the flat limit we recover the corresponding invariants of the Wigner irreducible representations of the Poincaré group.     

General Formalism for Mapping of Two-Mode Classical Canonical Transformations to Quantum Unitary operators

Two types of canonical transformations in two-mode classical phase space are mapped into the quantum mechanical Hilbert space to produce some new normally ordered unitary operators. These operators are evaluated in the coordinate (momentum) representations using the "integration within ordered product technique, and the mapping is maniferrtly apparent in the derivation. New generalixed coherent states are constructed in terms of these operators, and the uncertainty relations for these states are analysed.     

无穷深势阱中相对论粒子的矩阵元及其经典近似

对于无穷深势阱中自旋为０(满足Klein-Gordon方程)和自旋为1/2(满足Dirac方程)的相对论粒子, 分别计算了坐标、动量以及速度算符的矩阵元. 在大量子数极限下, 这些矩阵元给出相应的经典物理量(这里是狭义相对论中的有关量), 并且满足正确的经典关系. 从而表明, Heisenberg对应原理对这样的相对论体系也适用. 关键词： 无穷深势阱 Klein-Gordon方程 Dirac方程 Heisenberg对应原理     

Dynamical and structural symmetries for the highest Landau levels on the <Emphasis Type="Italic">AdS</Emphasis><Subscript>2</Subscript>

The aim of the paper is to use the recurrence relations with respect to both indices of the associated Legendre functions for the extraction of the Dirac quantization condition and dynamical symmetry group U(1, 1) corresponding to the highest Landau levels on the hyperbolic plane with uniform magnetic field B. Irreducible representations of the su(2) algebra are obtained by the ladder differential operators which change B by 1/2 unit and mode number by one unit. Two different classes of the irreducible representations of SU(1, 1) with the even and odd boson numbers 2B − 1/2 are extracted for the Bargmann indices 1/4 and 3/4, respectively. Finally, we show that shape invariance symmetry is realized by the ladder operators which shift only the magnetic field B by 1/2 unit.     

QCD Dirac spectrum and components of the gauge field

We analyze the relation between the Dirac spectrum and the gauge field in SU(3) lattice QCD. We focus on how a certain component of the gauge field is related to the Dirac spectrum. First, we consider momentum components of the gauge field. It turns out that the broad momentum region is relevant for the low-lying Dirac spectrum and topological charges. The connection with chiral random matrix theory is also discussed. Second, we consider an SU(2) subgroup component of the SU(3) gauge field. The SU(2) subgroup component behaves like the SU(2) gauge field in the low-lying Dirac spectrum.     

Should There Be a Spin-Rotation Coupling for a Dirac Particle?

It was argued by Mashhoon that a spin-rotation coupling term should add to the Hamiltonian operator in a rotating frame, as compared with the one in an inertial frame. For a Dirac particle, the Hamiltonian and energy operators H and E in a given reference frame were recently proved to depend on the tetrad field. We argue that this non-uniqueness of H and E really is a physical problem. We show that a tetrad field contains two informations about local rotation, which usually do not coincide. We compute the energy operator in the inertial and the rotating frame, using three different tetrad fields. We find that Mashhoon’s term is there if the spatial triad rotates as does the reference frame—but then it is also there in the energy operator for the inertial frame. In fact, if one uses the same given tetrad field, the Dirac Hamiltonian operators in two reference frames in relative rotation differ only by the angular momentum term. If the Mashhoon effect is to exist for a Dirac particle, the tetrad field must be selected in a specific way for each reference frame.     

Pryce’s mass-center operators and the anomalous velocity of a spinning electron

In the present work, we develop a method to derive the anomalous velocity of a spinning electron. From Dirac equation, the relationships among the expectation values of the Pryce’s mass-center operator, the position operator, the spin operator and the canonical momentum operator are investigated. By requiring that the center of mass for a classical spinning electron is related to the expectation value of Pryce’s mass-center operator, one can obtain a classical expression for the position of the electron. With the classical equations of motion, the anomalous velocity of a spinning electron can be easily obtained. It is shown that two factors contribute to the anomalous velocity: one is dependent on the selection of Pryce’s mass-center operators and the other is a type-independent velocity expressed by the rotational velocity and the Lorentz force.