Coupling Between the Group—Related Coherent States

When two representations of the Lie algebra are coupled,the coupling integral kernels are presented to relate the coupled to uncoupled group-related coherent states,These kernels have a connection with usual coupling coefficients.The explicit expressions of these kernels for SU(2),SO(4) and SUq(2) are given.When the direct product of three representations is formed in two ways,the recoupling integral kernels relating to the coupled group-related coherent states corresponding to two different schemes are introduced,and the relations between these kernels and the general recoupling coefficients are obtained.The properties of these kerels are discussed.     

On the algebra of field operators. The weak commutant and integral decompositions of states

It is shown that every continuous representation of a nuclear and separable *-algebra on a separable Hilbert space has a certain integral decomposition into representations with a trivial weak commutant. This result is used to obtain a decomposition of Wightman functionals into extremal states. An example is given of an extremal Wightman functional which does not have the cluster property.     

Mean entropy of states in classical statistical mechanics

The equilibrium states for an infinite system of classical mechanics may be represented by states over AbelianC* algebras. We consider here continuous and lattice systems and define a mean entropy for their states. The properties of this mean entropy are investigated: linearity, upper semi-continuity, integral representations. In the lattice case, it is found that our mean entropy coincides with theKolmogorov-Sinai invariant of ergodic theory.     

Coherent States for Hopf Algebras

Families of Perelomov coherent states are defined axiomatically in the context of unitary representations of Hopf algebras. A global geometric picture involving locally trivial noncommutative fibre bundles is involved in the construction. If, in addition, the Hopf algebra has a left Haar integral, then a formula for noncommutative resolution of identity in terms of the family of coherent states holds. Examples come from quantum groups.      

Relation of the tomographic representation of states to the bargmann representation of states

We introduce some equivalent forms of a map realizing the connection between the Bargmann and tomographic representations of states and observables. We perform the same task for a dual tomographic map. In spite of the fact that, due to the analyticity of the Bargmann representation, there exist many forms for such a map, we restrict ourselves to integral transforms. To perform our calculations, we also introduce a new technique for disentangling the SU(1, 1)-group operators.     

Focusing of electromagnetic waves into a uniaxial crystal

We derive integral representations suitable for studying the focusing of electromagnetic waves through a plane interface into a uniaxial crystal. To that end we start from existing exact solutions for the transmitted fields due to an arbitrary three-dimensional (3D) wave that is incident upon a plane interface separating two uniaxial crystals with arbitrary orientation of the optical axis in each medium. Then we specialize to the case in which the medium of the incident wave is isotropic and derive explicit expressions for the dyadic Green's functions associated with the transmitted fields as well as integral representations suitable for asymptotic analysis and efficient numerical evaluation. Relevant integral representations for focused 3D electromagnetic waves are also given. Next we consider the special case in which (i) the incident field is a two-dimensional (2D) TM wave and (ii) the optical axis in the crystal lies in the plane of incidence, implying that we have a 2D vectorial problem, and derive dyadic Green's functions, integral representations suitable for asymptotic and numerical treatment, and integral representations for focused TM fields. Numerical results for focused 2D TM fields based on these integral representations as well as corresponding experimental results will be presented in forthcoming papers.     

Path integrals for the quantum microcanonical ensemble

Path integral representations for the quantum microcanonical ensemble are presented. In the quantum microcanonical ensemble, two operators are of primary interest. First, rhoinsertion mark=delta(E-Hinsertion mark) corresponds to the microcanonical density matrix and can be used to calculate expectation values. Second, Ninsertion mark=Theta(E-Hinsertion mark) can give the number of states with energy E(n) and Theta(x,x('),E)=. A path integral formalism leads to exact integral representations for Omega(x,x('),E) and Theta(x,x('),E). We present both phase space and configuration space forms. For simple systems, such as the free particle and harmonic oscillator, exact solutions are possible. For more complicated systems, expansion schemes or numerical evaluations are required. A perturbative calculation and numerical integration results are presented for the quantum anharmonic oscillator.     

Hilbert space representations of Lie algebras

We describe a new approach to the general theory of unitary representations of Lie groups which makes use of the Gelfand-Segal construction directly on the universal enveloping algebra of any Lie algebra. The crucial observation is that Nelson's theory of analytic vectors allows the characterisation of certain states on the universal enveloping algebra such that the corresponding representations of the universal enveloping algebra are the infinitesimal part of unitary representations of the associated simply connected Lie group. In the first section of the paper we show that with the aid of Choquet's theory of representing measures one can derive a simple new approach to integral decomposition theory along these lines.In the second section of the paper we use these methods to study the irreducible unitary representations of general semi-simple Lie groups. We give a simple proof that theK-finite vectors studied by Harish-Chandra [5] are all analytic vectors. We also give new proofs of some of Godement's results [2] characterising spherical functions of height one, at least for unitary representations. Compared with [2] our method has the possible advantage of obtaining the characterisations by infinitesimal methods instead of using an indirect argument involving functions on the group. We point out that while being purely algebraic in nature, this approach makes almost no use of the deep and difficult theorems of Harish-Chandra concerning the universal enveloping algebra [5].Our work is done in very much the same spirit as that of Power's recent paper [8]. The main difference is that by concentrating on a more special class of positive states we are able to carry the analysis very much further without difficulty.     

Integral representations for the Lagrange polynomials,Shively’s pseudo-Laguerre polynomials,and the generalized Bessel polynomials

Motivated essentially by their potential for applications in the mathematical, physical, and statistical sciences, the object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing the main results presented here, the corresponding integral representations are derived for familiar simpler classes of hypergeometric polynomials such as (for example) the Lagrange polynomials, Shively’s pseudo-Laguerre polynomials, and generalized Bessel polynomials. Each of the integral representations derived in this paper may be also viewed as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

Landau-Lifschitz铁磁方程的Hamilton理论和规范变换

对完全各向同性Heisenberg铁磁链的LandauLifschitz方程的Hamilton理论建立中，Hamilton量的坐标积分和谱参数积分两种表示式不能协调地从单一守恒量导出的问题，利用规范变换完善地解决了.并可推广后处理非各向同性铁磁链的LandauLifschitz方程的Hamilton理论. 关键词： 规范变换 LandauLifschitz方程 守恒量 Hamilton理论     

Integral representations of the Macdonald symmetric polynomials

Multiple-integral representations of the (skew-)Macdonald symmetric polynomials are obtained. Some bosonization schemes for the integral representations are also constructed.JSPS fellow.     

Establishing path integral in the entangled state representation for Hamiltonians in quantum optics

Based on two mutually conjugate entangled state representations, we establish the path integral formalism for some Hamiltonians of quantum optics in entangled state representations. The Wigner operator in the entangled state representation is presented. Its advantages are explained.     

Projective representations of the inhomogeneous Hamilton group: Noninertial symmetry in quantum mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries include the Weyl–Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl–Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves the Heisenberg commutation relations invariant is essentially a projective representation of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of the Hamilton equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.     

Lorentz covariance and kinetic charge

There is a one-to-one correspondence between inequivalent covariant displaced Fock representations of the free relativistic field and the 1-cohomology of the Poincaré group with coefficients in the 1-particle space.Representations with positive energy are obtained from cocycles with finite energy which have particle-like properties and are interpreted as condensed states of matter without a sharply defined mass.The 1-cohomology groups ofP ₊ are calculated. These are trivial in 3- or 4-dimensional space-time, or if the mass is non-zero. Non-trivial cocycles for subgroups lead to representations in whichP-invariance is spontaneously broken. We recoverP-invariance in a direct integral representation possessing a gauge group, and a superselection structure labelled by the velocities of the condensed states of matter which are the cocycles determining each irreducible component of the representation. A model in 4-dimensional space-time is constructed.     

On the relation between the fractional Brownian motion and the fractional derivatives

The definition and simulation of fractional Brownian motion are considered from the point of view of a set of coherent fractional derivative definitions. To do it, two sets of fractional derivatives are considered: (a) the forward and backward and (b) the central derivatives, together with two representations: generalised difference and integral. It is shown that for these derivatives the corresponding autocorrelation functions have the same representations. The obtained results are used to define a fractional noise and, from it, the fractional Brownian motion. This is studied. The simulation problem is also considered.     

Proof of the Wehrl-type Entropy Conjecture for Symmetric {SU(N)} Coherent States

The Wehrl entropy conjecture for coherent (highest weight) states in representations of the Heisenberg group, which was proved in 1978 and recently extended by us to the group \({SU(2)}\), is further extended here to symmetric representations of the groups \({SU(N)}\) for all N. This result gives further evidence for our conjecture that highest weight states minimize group integrals of certain concave functions for a large class of Lie groups and their representations.     

The matrix elements of irreducible representations of the group SO(n,1) in integral form

Using the method of induced representations, the matrix elements of unitary irreducible representations of the group SO(n,1) are found in integral form.     

Coherent states and induced representations

A generalization of the notion of coherent states is given. The following one-to-one correspondences are pointed out: (1) between covariant overcomplete systems of coherent states and a class of covariant semi-spectral measures; (2) between covariant semispectral measures and unitary irreducible subrepresentations of induced representations of Lie groups; (3) between unitary irreducible representations of Lie groups with covariant overcomplete systems of coherent states and unitary irreducible subrepresentations of induced representations, whose representation spaces are reproducing kernel Hilbert spaces.