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1.
Let (, d) be a first-order differential *-calculus on a *-algebra
. We say that a pair (, F) of a *-representation of
on a dense domain
of a Hilbert space and a symmetric operator F on
gives a commutator representation of if there exists a linear mapping : L(
) such that (adb) = (a)i[F, (b) ], a, b
. Among others, it is shown that each left-covariant *-calculus of a compact quantum group Hopf *-algebra
has a faithful commutator representation. For a class of bicovariant *-calculi on
, there is a commutator representation such that F is the image of a central element of the quantum tangent space. If
is the Hopf *-algebra of the compact form of one of the quantum groups SL
q
(n+1), O
q
(n), Sp
q
(2n) with real trancendental q, then this commutator representation is faithful. 相似文献
2.
Stan Gudder 《Foundations of Physics》1999,29(6):877-897
This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the -effect algebra of effects (fuzzy events)
and the set of probability measures
on a measurable space
. An observable
is defined, where
is the value space of X. It is noted that there exists a one-to-one correspondence between states on
and elements of
and between observables
and -morphisms from
to
. Various combinations of observables are discussed. These include compositions, products, direct products, and mixtures. Fuzzy stochastic processes are introduced and an application to quantum dynamics is considered. Quantum effects are characterized from among a more general class of effects. An alternative definition of a statistical map
is given and it is shown that any statistical map has a unique extension to a statistical operator. Finally, various combinations of statistical maps are discussed and their relationships to the corresponding combinations of observables are derived. 相似文献
3.
Mauro Napsuciale 《Foundations of Physics》2003,33(5):741-768
In this work we review the derivation of Dirac and Weinberg equations based on a principle of indistinguishability for the (j,0) and (0,j) irreducible representations (irreps) of the homogeneous Lorentz group (HLG). We generalize this principle and explore its consequences for other irreps containing j1. We rederive Ahluwalia–Kirchbach equation using this principle and conclude that it yields
equations of motion for any representation containing spin j and lower spins. We also use the obtained generators of the HLG for a given representation to explore the possibility of the existence of first order equations for that representation. We show that, except for j=
, there exists no Dirac-like equation for the (j,0)(0,j) representation nor for the (
,
) representation. We rederive Kemmer–Duffin–Petieau (KDP) equation for the (1,0)(
,
)(0,1) representation by this method and show that the (1,
)(
,1) representation satisfies a Dirac-like equation which describes a multiplet of
with masses m and m/2, respectively. 相似文献
4.
Recently, a class of
-invariant scalar quantum field theories described by the non-Hermitian Lagrangian
=
()
2
+g
2
(i) was studied. It was found that there are two regions of . For <0 the
-invariance of the Lagrangian is spontaneously broken, and as a consequence, all but the lowest-lying energy levels are complex. For 0 the
-invariance of the Lagrangian is unbroken, and the entire energy spectrum is real and positive. The subtle transition at =0 is not well understood. In this paper we initiate an investigation of this transition by carrying out a detailed numerical study of the effective potential V
eff
(c) in zero-dimensional spacetime. Although this numerical work reveals some differences between the <0 and the >0 regimes, we cannot yet see convincing evidence of the transition at =0 in the structure of the effective potential for
-symmetric quantum field theories. 相似文献
5.
Let
be the Haag--Kastler net generated by the
(2) chiral current algebra at level 1. We classify the SL(2,
)-covariant subsystems
by showing that they are all fixed points nets
H
for some subgroup H of the gauge automorphisms group SO(3) of
. Then, using the fact that the net
1 generated by the
(1) chiral current can be regarded as a subsystem of
, we classify the subsystems of
1. In this case, there are two distinct proper subsystems: the one generated by the energy-momentum tensor and the gauge invariant subsystem
. 相似文献
6.
The purpose of this Letter is to develop further the Gauss diagram approach to finite-type link invariants. In this context we introduce Gauss diagrams counterparts to chord diagrams, 4T relation, weight systems arising from Lie algebras, and an algebra of unitrivalent graphs modulo the STU relation. The counterparts, respectively, are arrow diagrams, 6T relation, weights arising from the solutions of the classical Yang–Baxter equation, and an algebra
of acyclic arrow graphs (modulo an oriented version
of the STU relation). The algebra
encodes, in a graphical form, the main properties of Lie bialgebras, similarly to the well-known relation of the algebra of unitrivalent graphs to Lie algebras. We show that the oriented
and
relations hold, and that
is isomorphic to the algebra
of arrow diagrams. As an application, we consider an appropriate link-homotopy version
of the algebra
. Using this algebra, we construct a Gauss diagram invariants of string links up to link-homotopy, with values both in the algebra
and in . We observe that this construction gives the universal Milnor's link-homotopy -invariants. 相似文献
7.
We consider the crystal structure of the level zero extremal weight modules V() using the crystal base of the quantum affine algebra constructed in Duke Math. J.
99 (1999), 455–487. This approach yields an explicit form for extremal weight vectors in the U
– part of each connected component of the crystal, which are given as Schur functions in the imaginary root vectors. We show the map
induces a correspondence between the global crystal base of V() and elements
. 相似文献
8.
9.
Marcus Pivato 《Journal of statistical physics》2003,110(1-2):247-267
If
, and
is a finite (nonabelian) group, then
is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation
which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of
. We characterize when MCA are group endomorphisms of
, and show that MCA on
inherit a natural structure theory from the structure of
. We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure. 相似文献
10.
Let
be von Neumann algebras acting on a Hilbert space
and let
be a common cyclic and separating vector. We say that
have the modular intersection property with respect to
if(1)
-half-sided modular inclusions,(2)
(If (1) holds the strong limit exists.) We show that under these conditions the modular groups of
and
generate a 2-dim. Lie group.This observation is the basis for obtaining group representations of Sl(2,
)/Z
2 generated by modular groups. 相似文献
11.
Wolfgang Drechsler 《Foundations of Physics》1999,29(9):1327-1369
A massless electroweak theory for leptons is formulated in a Weyl space, W4, yielding a Weyl invariant dynamics of a scalar field , chiral Dirac fermion fields L and R, and the gauge fields , A, Z, W, and W
, allowing for conformal rescalings of the metric g and all fields with nonvanishing Weyl weight together with the corresponding transformations of the Weyl vector fields, , representing the D(1) or dilatation gauge fields. The local group structure of this Weyl electroweak (WEW) theory is given by
—or its universal coverging group
for the fermions—with
denoting the electroweak gauge group SU(2)W × U(1)Y. In order to investigate the appearance of nonzero masses in the theory the Weyl symmetry is explicitly broken by a term in the Lagrangean constructed with the curvature scalar R of the W4 and a mass term for the scalar field. Thereby also the Z and W gauge fields as well as the charged fermion field (electron) acquire a mass as in the standard electroweak theory. The symmetry breaking is governed by the relation D
2 = 0, where is the modulus of the scalar field and D denotes the Weyl-covariant derivative. This true symmetry reduction, establishing a scale of length in the theory by breaking the D(1) gauge symmetry, is compared to the so-called spontaneous symmetry breaking in the standard electroweak theory, which is, actually, the choice of a particular (nonlinear ) gauge obtained by adopting an origin,
, in the coset space representing , with
being invariant under the electromagnetic, gauge group U(1)e.m.. Particular attention is devoted to the appearance of Einstein's equations for the metric after the Weyl symmetry breaking, yielding a pseudo-Riemannian space, V4, from a W4 and a scalar field with a constant modulus
. The quantity
affects Einstein's gravitational constant in a manner comparable to the Brans-Dicke theory. The consequences of the broken WEW theory are worked out and the determination of the parameters of the theory is discussed. 相似文献
12.
Michael Keyl 《International Journal of Theoretical Physics》1998,37(1):375-385
The major subject of algebraic quantum fieldtheory is the study of nets of local C*-algebras, i.e.,maps
(
) assigning to each open,relatively compact region of space-time (M, g) aC*-algebra
(
), whose self-adjoint elements describe localobservables measurable in the region
. A question discussed recently in a number ofpapers is how much information about the geometricstructure of the underlying space-time (M, g) is encoded in the algebraicstructure of the net
(
). Followingthese ideas, it is demonstrated in this paper howspace-time-related concepts like causality and observerscan be described in a purely algebraic way, i.e., using only thelocal algebras
(
).These results are then used to show how the space-time(M, g) can be reconstructed from the set
loc := {
(
)|
M open,
compact} of local algebras. 相似文献
13.
Haluk Beker 《Foundations of Physics》1998,28(6):999-1004
A purely algebraic perturbation theory based on deforming the generators of the dynamical group SU(1, 1) is applied to the l = 0 Morse potential problem with
. In particular, perturbations of the form
and
are treated explicitly. 相似文献
14.
The product of two real spectral triples
and
, the first of which is necessarily even, was defined by A.Connes as
given by
and, in the even-even case, by
. Generically it is assumed that the real structure
obeys the relations
,
,
, where the
-sign table depends on the dimension n modulo 8 of the spectral triple. If both spectral triples obey Connes'
>-sign table, it is seen that their product, defined in the straightforward way above, does not necessarily obey this
-sign table. In this Letter, we propose an alternative definition of the product real structure such that the
-sign table is also satisfied by the product. 相似文献
15.
We consider the asymmetric exclusion process (ASEP) in one dimension on sites i=1,...,N, in contact at sites i=1 and i=N with infinite particle reservoirs at densities
a
and
b
. As
a
and
b
are varied, the typical macroscopic steady state density profile ¯(x), x[a,b], obtained in the limit N=L(b–a), exhibits shocks and phase transitions. Here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile
, so that
is the large deviation functional, a quantity similar to the free energy of equilibrium systems. We find, as in the symmetric, purely diffusive case q=1 (treated in an earlier work), that
is in general a non-local functional of (x). Unlike the symmetric case, however, the asymmetric case exhibits ranges of the parameters for which
is not convex and others for which
has discontinuities in its second derivatives at (x)=¯(x). In the latter ranges the fluctuations of order
in the density profile near ¯(x) are then non-Gaussian and cannot be calculated from the large deviation function. 相似文献
16.
C denotes either the conformal group in 3+1 dimensions, PSO(4, 2), or in one chiral dimension, PSL(2,
). Let U be a unitary, strongly continuous representation of C satisfying the spectrum condition and inducing, by its adjoint action, automorphisms of a von Neumann algebra
. We construct the unique inner representation
of the universal covering group of C implementing these automorphisms.
satisfies the spectrum condition and acts trivially on any U-invariant vector. This means in particular: Conformal transformations of a field theory having positive energy are weak limit points of local observables. Some immediate implications for chiral subnets are given. We propose the name Borchers–Sugawara construction. 相似文献
17.
We develop a technique for the construction of integrable models with a 2 grading of both the auxiliary (chain) and quantum (time) spaces. These models have a staggered disposition of the anisotropy parameter. The corresponding Yang–Baxter equations are written down and their solution for the gl(N) case is found. We analyze in details the N = 2 case and find the corresponding quantum group behind this solution. It can be regarded as the quantum group
, with a matrix deformation parameter q
such that (q
)2 = q
2. The symmetry behind these models can also be interpreted as the tensor product of the (–1)-Weyl algebra by an extension of
q
(gl(N)) with a Cartan generator related to deformation parameter –1. 相似文献
18.
The fusion rules for the (p,q)-minimal model representations of the Virasoro algebra are shown to come from the group
in the following manner. There is a partition
into disjoint subsets and a bijection between
and the sectors
of the (p,q)-minimal model such that the fusion rules
correspond to
where
. 相似文献
19.
We consider a discrete Schrödinger operator H=–+V acting in l
2(
d
), with periodic potential V supported by the subspace surface {0}×
d
2. We prove that the spectrum of H is purely absolutely continuous, and that surface waves oscillate in the longitudinal directions to the surface. We also find an explicit formula for the generalized spectral shift function introduced by the author in Helv. Phys. Acta.
72 (1999), 93–122. 相似文献
20.
L. Burakovsky 《Foundations of Physics》1998,28(10):1595-1605
We show that linear Regge trajectories for mesons and glueballs, and the cubic mass spectrum associated with them, determine a relation between the masses of the meson and the scalar glueball,
, which implies
MeV. We also discuss relations between the masses of the scalar, tensor and 3-- glueballs,
, which imply
MeV. 相似文献