共查询到20条相似文献,搜索用时 31 毫秒
1.
Leonid I. Korogodsky 《Communications in Mathematical Physics》1994,163(3):433-460
A quantum analogue of the groupSU(1,1)Z
2—the normalizer ofSU(1, 1) inSL
2(C)—is introduced and studied. Although there isno correctly defined tensor product in the category of *-representations of the quantum algebraC[SU(1, 1)]
q
of regular functions, some categories of *-representations ofC[SU(1, 1)Z
2]
q
turn out to be endowed with a certainZ
2-graded structure which can be considered as a super-generalization of the monoidal category structure. This quantum effect may be considered as a step to understanding the concept of quantum topological locally compact group.In fact, there seems to be afamily of quantum groupsSU(1, 1)Z
2 parameterized by unitary characters T
1 of the fundamental group of the two-dimensional symplectic leaf ofSU(1, 1)/T, whereT is the subgroup of diagonal matrices.It is shown that thequasi-classical analogues of the results of the paper are connected with the decomposition of Schubert cells of the flag manifoldSL
2(C)R/B (whereB is the Borel subgroup of upper-triangular matrices) into symplectic leaves.Supported by the Rosenbaum Fellowship. 相似文献
2.
We introduce o(p+1q+1)-invariant bilinear differential operators on the space of tensor densities on Rn generalizing the well-known bilinear sl2-invariant differential operators in the one-dimensional case, called Transvectants or Rankin–Cohen brackets. We also consider already known linear o(p+1q+1)-invariant differential operators given by powers of the Laplacian. 相似文献
3.
We give the algebra
q
/*
dual to the matrix Lorentz quantum group
q
of Podles-Woronowicz, and Watamuraet al. As a commutation algebra, it has the classical form
q
/*
U
q
(sl(2, )) U
q
(sl(2, )). However, this splitting is not preserved by the coalgebra structure which we also give. For the derivation, we use a generalization of the approach of Sudbery, viz. tangent vectors at the identity. 相似文献
4.
Kaoru Ikeda 《Letters in Mathematical Physics》1991,23(2):121-126
We consider the Hamiltonian systems on the Poisson structure of GL() which is introduced from the quantum group GL
q
() by the so-called quasi-classical limit of GL
q
(). Furthermore, we show that the Toda lattice hierarchy is a Hamiltonian system of this structure. 相似文献
5.
We establish a previously conjectured connection betweenp-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which interpolate between the zonal spherical functions of related real andp-adic symmetric spaces. The elliptic quantum algebras underlie theZ
n
-Baxter models. We show that in then limit, the Jost function for the scattering offirst level excitations in the 1+1 dimensional field theory model associated to theZ
n
-Baxter model coincides with the Harish-Chandra-likec-function constructed from the Macdonald polynomials associated to the root systemA
1. The partition function of theZ
2-Baxter model itself is also expressed in terms of this Macdonald-Harish-Chandrac-function, albeit in a less simple way. We relate the two parametersq andt of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular thep-adic regimes in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of q-deforming Euler products.Work supported in part by the NSF: PHY-9000386 相似文献
6.
A relation between the magnetic susceptibility(k, ) of an interacting 1-D Fermi system and the dielectric function(q) of a 2-D Coulomb gas is established. By applying a cluster-expansion technique and by using known results for the pair-correlation function of the Coulomb gas we obtain a number of expressions for(q) which apply in different regions of theq-plane and in different temperature intervals. These results supplement the existing picture of the transition from non-metallic to metallic behaviour occurring in the 2-D Coulomb gas as the temperature increases. The relation between(q) and(k, ) is then used to derive explicit expressions for(k, ) from these results for(q). The change in the dielectric response of the 2-D Coulomb gas is reflected by a change in the magnetic response of the 1-D Fermi system: as a function of the spin non-flip coupling constant
the susceptibility of the Fermi system changes from normal paramagnetic behaviour to non-magnetic behaviour characteristic of a bound singlet-spin ground state, as
decreases. Our result for the gap in the spin excitation spectrum of the Fermi system is in agreement with the results of other authors.On sabbatical leave from Institut für Theoretische Physik, TU Hannover, GermanyWork at U.C.S.B. supported in part by the National Science Foundation 相似文献
7.
It is shown by numerical simulations for a random, one-dimensional surface defined by the equationx
3=(x
1), where the surface profile function (x
1) is a stationary, stochastic, Gaussian process, that the transverse correlation lengtha of the surface roughness is a good measure of the mean distance d between consecutive peaks and valleys on the surface. In the case that the surface height correlation function (x
1)(x
1)/2(x
1)=W (|x
1–x
1|) has the Lorentzian formW(|x
1|)=a
2/(x
1
2
+a
2) we find that d=0.9080a; when it has the Gaussian formW(|x
1|)=exp(–x
1
2
/a
2), we find that d=1.2837a; and when it has the nonmonotonic formW(|x
1|)=sin(x
1/a)/(x
1/a), we find that d=1.2883a. These results suggest that d is larger, the faster the surface structure factorg(|Q|) [the Fourier transform ofW(|x
1|)] decays to zero with increasing |Q|. We also obtain the functionP(itx
1), which is defined in such a way that, ifx
1=0 is a zero of (x
1),P(x
1)dx
1 is the probability that the nearest zero of (x
1) for positivex
1 lies betweenx
1 andx
1+dx
1. 相似文献
8.
Daowen Qiu 《International Journal of Theoretical Physics》2002,41(9):1627-1639
Quantum automata are mathematical models for quantum computing. We analyze the existing quantum pushdown automata, propose a q quantum pushdown automata (qQPDA), and partially clarify their connections. We emphasize some advantages of our qQPDA over others. We demonstrate the equivalence between qQPDA and another QPDA. We indicate that qQPDA are at least as powerful as the QPDA of Moore and Crutchfield with accepting words by empty stack. We introduce the quantum languages accepted by qQPDA and prove that every -q quantum context-free language is also an -q quantum context-free language for any (0, 1) and (0, 1). 相似文献
9.
We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group C
q
[ SL2], using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for q an odd rth root of unity that its eigenvalues are given by q-integers [m]
q
for m=0,1...,r–1 offset by the constant background curvature. We fully solve the Dirac equation for r=3. 相似文献
10.
Derek W. Robinson 《Communications in Mathematical Physics》1990,132(1):217-243
We define and analyze Lipschitz spaces ,q associated with a representationxgV(x) of the Lie algebrag by closed operatorsV(x) on the Banach space together with a heat semigroupS. If the action ofS satisfies certain minimal smoothness hypotheses with respect to the differential structure of (,g,V) then the Lipschitz spaces support representations ofg for which productsV(x)V(y) are relatively bounded by the Laplacian generatingS. These regularity properties of the ,q can then be exploited to obtain improved smoothness properties ofS on . In particularC
4-estimates on the action ofS automatically implyC
-estimates. Finally we use these results to discuss integrability criteria for (,g,V).Dedicated to Res Jost and Arthur Wightman 相似文献
11.
A certain class of unitary representations of Uq((2,)) has the property of being simultanenously a representation of for a particular choice of (q). Faddeev has proposed to unify the quantum groups Uq((2,)) and into some enlarged object for which he has coined the name ``modular double'. We study the R-operator, the co-product and the Haar-measure for the modular double of Uq((2,)) and establish their main properties. In particular it is shown that the Clebsch-Gordan maps constructed in [PT2] diagonalize this R-operator. 相似文献
12.
Consider the perturbed harmonic oscillator Ty=-y+x2y+q(x)y in L2(), where the real potential q belongs to the Hilbert space H={q, xq L2()}. The spectrum of T is an increasing sequence of simple eigenvalues n(q)=1+2n+n, n 0, such that n 0 as n. Let n(x,q) be the corresponding eigenfunctions. Define the norming constants n(q)=limxlog |n (x,q)/n (-x,q)|. We show that for some real Hilbert space and some subspace Furthermore, the mapping :q(q)=({n(q)}0, {n(q)}0) is a real analytic isomorphism between H and is the set of all strictly increasing sequences s={sn}0 such that The proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials. We use ideas from the analysis of the inverse problem for the operator -ypy, p L2(0,1), with Dirichlet boundary conditions on the unit interval. There is no literature about the spaces We obtain their basic properties, using their representation as spaces of analytic functions in the disk. 相似文献
13.
U. Meyer 《Letters in Mathematical Physics》1995,35(2):91-97
Associated to the standard SU
q
(n) R-matrices, we introduce quantum spheresS
q
2n-1
, projective quantum spaces
q
n-1
, and quantum Grassmann manifoldsG
k(
q
n
). These algebras are shown to be homogeneous spaces of standard quantum groups and are also quantum principle bundles in the sense of T. Brzeziski and S. Majid. 相似文献
14.
S. Majid 《Communications in Mathematical Physics》1993,156(3):607-638
Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groupsU
q
(g). They have the same FRT generatorsl
± but a matrix braided-coproductL=LL, whereL=l
+
Sl
–, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matricesBM
q(2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum doubleD(U
q
(sl
2)) (also known as the quantum Lorentz group) is the semidirect product as an algebra of two copies ofU
q
(sl
2), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups. 相似文献
15.
S. Fujita 《Foundations of Physics》1991,21(4):439-457
If no property of a system of many particles discriminates among the particles, they are said to be indistinguishable. This indistinguishability is equivalent to the requirement that the many-particle distribution function and all of the dynamic functions for the system be symmetric. The indistinguishability defined in terms of the discrete symmetry of many-particle functions cannot change in the continuous classical statistical limit in which the number density n and the reciprocal temperature become small. Thus, microscopic particles like electrons must remain indistinguishable in the classical statistical limit although their behavior can be calculated as if they move following the classical laws of motion. In the classical mechanical limit in which quantum cells of volume (2)3 are reduced to points in the phase space, the partition functionTr{exp(–) for N identical bosons (fermions) approaches (2)–3N(N!) ... d3r1 d3p1 ... d3rN d3pN
exp(–H). The two factors, (2)–3N and (N!)–1, which are often added in anad hoc manner in many books on statistical mechanics, are thus derived from the first principles. The criterion of the classical statistical approximation is that the thermal de Broglie wavelength be much shorter than the interparticle distance irrespective of any translation-invariant interparticle interaction. A new derivation of the Maxwell velocity distribution from Boltzmann's principle is given with the assumption of indistinguishable classical particles. 相似文献
16.
Michael B. Mensky 《Foundations of Physics》1992,22(9):1173-1193
The path-integral approach to quantum theory of continuous measurements has been developed in preceding works of the author. According to this approach the measurement amplitude determining probabilities of different outputs of the measurement can be evaluated in the form of a restricted path integral (a path integral in finite limits). With the help of the measurement amplitude, maximum deviation of measurement outputs from the classical one can be easily determined. The aim of the present paper is to express this variance in a simpler and transparent form of a specific uncertainty principle (called the action uncertainty principle, AUP). The most simple (but weak) form of AUP is S, whereS is the action functional. It can be applied for simple derivation of the Bohr-Rosenfeld inequality for measurability of gravitational field. A stronger (and having wider application) form of AUP (for ideal measurements performed in the quantum regime) is |
t
(S[q]/q(t))q(t)dt|, where the paths [q] and [q] stand correspondingly for the measurement output and for the measurement error. It can also be presented in symbolic form as (Equation) (Path) . This means that deviation of the observed (measured) motion from that obeying the classical equation of motion is reciprocally proportional to the uncertainty in a path (the latter uncertainty resulting from the measurement error). The consequence of AUP is that improving the measurement precision beyond the threshold of the quantum regime leads to decreasing information resulting from the measurement. 相似文献
17.
A kinetic theory for the constitutive Theological relations of rapid granular shear flow of hard circular disks, characterized by a coefficient of restitutione and a surface roughness coefficient, is formulated. From a set of general constitutive equations for single-particle dynamical variables, the approximate expressions for the limit of small and large dimensionless dissipative parameterR
t
are obtained. HereR
t
is defined as the ratio /, where is the fluctuation of translational velocity from the mean flow velocity, is the diameter of a disk, and is the shear rate. At smallR
t
the theoretical predictions can be compared with exact computer simulation results of granular dynamics that are also reported. The agreement between theory and simulation is better than expected; the present theory is accurate up to high packing density in this region ofR
t
. 相似文献
18.
Don Zagier 《Communications in Mathematical Physics》1992,147(1):199-210
Following Greenberg and others, we study a space with a collection of operatorsa(k) satisfying the q-mutator relationsa(l)a
(k)a(l)=
k,l
(corresponding forq=±1 to classical Bose and Fermi statistics). We show that then!×n! matrixA
n
(q) representing the scalar products ofn-particle states is positive definite for alln ifq lies between –1 and +1, so that the commutator relations have a Hilbert space representation in this case (this has also been proved by Fivel and by Bozejko and Speicher). We also give an explicit factorization ofA
n
(q) as a product of matrices of the form(1–q
jT)±1 with 1jn andT a permutation matrix. In particular,A
n
(q) is singular if and only ifq
M=1 for some integerM of the formk
2–k, 2kn. 相似文献
19.
Simba A. Mutangadura 《Communications in Mathematical Physics》1990,132(3):581-592
Let be an infinite dimensional Hilbert space and () the set of all (orthogonal) projections on . A comparative probability on () is a linear preorder on () such thatOP1,1O and such that ifP┴R,Q┴R, thenPQP+RQ+R for allP, Q, R in (). We give a sufficient and necessary condition for to be implemented in a canonical way by a normal state onB(), the bounded linear operators on . 相似文献
20.
We consider some models of classical statistical mechanics which admit an investigation by means of the theory of dominant ground states. Our models are related to the Gibbs ensemble for the multidimensional SOS model with symmetric constraints
x
m/2. The main result is that for 0, where 0 does not depend onm, the structure of thermodynamic phases in the model is determined by dominant ground states: for an evenm a Gibbs state is unique and for an oddm the number of space-periodic pure Gibbs states is two. 相似文献