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1.
S. Doplicher R. V. Kadison D. Kastler Derek W. Robinson 《Communications in Mathematical Physics》1967,6(2):101-120
We study pairs {
, } for which
is aC*-algebra and is a homomorphism of a locally compact, non-compact groupG into the group of *-automorphisms of
. We examine, especially, those systems {
, } which are (weakly) asymptotically abelian with respect to their invariant states (i.e. |A
g
(B) —
g
(B)A 0 asg for those states such that (
g
(A)) = (A) for allg inG andA in
). For concrete systems (those with
-acting on a Hilbert space andg
g
implemented by a unitary representationg U
g
on this space) we prove, among other results, that the operators commuting with
and {U
g
} form a commuting family when there is a vector cyclic under
and invariant under {U
g
}. We characterize the extremal invariant states, in this case, in terms of weak clustering properties and also in terms of factor and irreducibility properties of {
,U
g
}. Specializing to amenable groups, we describe operator means arising from invariant group means; and we study systems which are asymptotically abelian in mean. Our interest in these structures resides in their appearance in the infinite system approach to quantum statistical mechanics. 相似文献
2.
The theorem that each derivation of aC*-algebra
extends to an inner derivation of the weak-operator closure (
)– of
in each faithful representation of
is proved in sketch and used to study the automorphism group of
in its norm topology. It is proved that the connected component of the identity in this group contains the open ball of radius 2 with centerl and that each automorphism in extends to an inner automorphism of (
)–.Research conducted with the partial support of the NSF and ONR. 相似文献
3.
Let be a closed * derivation in aC* algebra
which commutes with an ergodic action of a compact group on
. Then generates aC* dynamics of
. Similar results are obtained for non-ergodic actions on abelianC* algebras and on the algebra of compact operators.Research supported by N.S.F. 相似文献
4.
Within the general framework ofC*-algebra approach to mathematical foundation of statistical mechanics, we prove a theorem which gives a natural explanation for the appearance of the chemical potential (as a thermodynamical parameter labelling equilibrium states) in the presence of a symmetry (under gauge transformations of the first kind). As a symmetry, we consider a compact abelian groupG acting as *-automorphisms of aC*-algebra
(quasi-local field algebra) and commuting (elementwise) with the time translation automorphisms
t
of
. Under a technical assumption which is satisfied by examples of physical interest, we prove that the set of all extremal
t
-KMS states (pure phases) ofG-fixed-point subalgebra
(quasi-local observable algebra) of
satisfying a certain faithfulness condition is in one-to-one correspondence with the set of all extremalG-invariant
t
·
t
-KMS states – of
with varying over one-parameter subgroups ofG (the specification of being the specification of the chemical potential), where the correspondence is that the restriction of – to
is . 相似文献
5.
Using the Godement mean of positive-type functions over a groupG, we study -abelian systems {
, } of aC*-algebra
and a homomorphic mapping of a groupG into the homomorphism group of
. Consideration of the Godement mean off(g)U
g
withf a positive-type function overG andU a unitary representation ofG first yields a generalized mean-ergodic theorem. We then define the Godement mean off(g) (
g
(A)) withA
and a covariant representation of the system {
, } for which theG-invariant Hilbert space vectors are cyclic and study its properties, notably in relation with ergodic and weakly mixing states over
. Finally we investigate the discrete spectrum of covariant representations of {
, } (i.e. the direct sum of the finite-dimensional subrepresentations of the associated representations ofG).On leave of absence from Istituto di Fisica G. Marconi Piazzale delle Scienze 5 — Roma. 相似文献
6.
It has been suggested that a possible classical remnant of the phenomenon of target-space duality (T-duality) would be the equivalence of the classical string Hamiltonian systems. Given a simple compact Lie groupG with a bi-invariant metric and a generating function suggested in the physics literature, we follow the above line of thought and work out the canonical transformation generated by together with an Ad-invariant metric and a B-field on the associated Lie algebra
ofG so thatG and
form a string target-space dual pair at the classical level under the Hamiltonian formalism. In this article, some general features of this Hamiltonian setting are discussed. We study properties of the canonical transformation including a careful analysis of its domain and image. The geometry of the T-dual structure on
is lightly touched. We leave the task of tracing back the Hamiltonian formalism at the quantum level to the sequel of this paper. 相似文献
7.
S. L. Woronowicz 《Communications in Mathematical Physics》1972,28(3):221-235
Let
be aC*-algebra and
be an opposite algebra. Notions of exact andj-positive states of
are introduced. It is shown, that any factor state of
can be extended to a pure exactj-positive state
of
. The correspondence
generalizes the notion of the purifications map introduced by Powers and Størmer. The factor states 1 and 2 are quasi-equivalent if and only if their purifications
and
are equivalent. 相似文献
8.
Konrad Osterwalder 《Communications in Mathematical Physics》1973,29(1):1-14
An elementary proof of Araki's duality theorem for free fields is presented. The theorem says that for a certain class of regionsO in Minkowski space, the commutant of
(O), the von Neumann algebra generated by all observables belonging to measurements withinO, is exactly
(O), whereO is the spacelike separated complement ofO.Supported by the National Science Foundation under grants GP 24003, GP 30819X and GP 31239X. 相似文献
9.
Karl Kraus 《Communications in Mathematical Physics》1967,7(2):99-111
Concrete C*-algebras, interpreted physically as algebras of observables, are defined for quantum mechanics and local quantum field theory.Aquantum mechanical system is characterized formally by a continuous unitary representation up to a factorU
g
of a symmetry group
in Hilbert space and a von Neumann algebra on invariant with respect toU
g
. The set
of all operatorsX such thatU
g
X U
g
–1
, as a function ofg
, is continuous with respect to the uniform operator topology, is aC*-algebra called thealgebra of observables. The algebra is shown to be the weak (or strong) closure of
.Infield theory, a unitary representation up to a factorU(a, ) of the proper inhomogeneous Lorentz group
and local von Neumann algebras C for finite open space-time regionsC are assumed, with the usual transformation properties of
underU(a, ). The collection of allXC giving uniformly continuous functionsU (a, )X U
–1 (a, ) on
is then a localC*-algebra
, called thealgebra of local observables. The algebra
is again weakly (or strongly) dense in
c
. The norm-closed union
of the
for allC is calledalgebra of quasilocal observables (or quasilocal algebra).In either case, the group
is represented by automorphisms V
g
resp. V(a, ) — with V
g
X=U
g
X U
g
–1
— of theC*-algebra
, and this is astrongly continuous representation of
on the Banach space
. Conditions for V (a, ) can then be formulated which correspond to the usualspectrum condition forU (a, ) in field theory.Work supported in part by the Deutsche Forschungsgemeinschaft. 相似文献
10.
Couch and Torrence suggest that the vacuum Einstein equations admit a larger class of asymptotically flat solutions than those exhibiting the peeling property. Starting with the assumption that
, (d/dr)
and (/x
A
)
, wherex
A
(A = 2, 3) are angular coordinates, they show that
, where 1 2 and 1<0;
, where 2 1 and 1< 1; and 4 and 3 peel as they would under the stronger peeling conditions. The Winicour-Tamburino energy-momentun and angular momentum integrals for these solutions, in general, diverge. In fact, since Couch and Torrence determine only the radial dependence of the solution, it is not clear that the solutions are well defined. We find that the stronger assumption
, (d/dr)
, and (/x
A
)
does result in well-defined solutions for which both the energy-momentum and angular momentum intergrals are not only finite but result in the same expressions as are obtained for peeling space-times. This assumption appears to be the minimal assumption that is necessary for investigating outgoing radiation at null infinity.In part based on a dissertation by Stephanie Novak and submitted to Syracuse University in partial fulfillment of the requirement for the Ph.D. degree. 相似文献
11.
J. De Cannière 《Communications in Mathematical Physics》1982,84(2):187-205
Let be a state on aC*-dynamical system
. For each of the following properties of : (1) is -K MS with respect to for some given , 0<+, (2) is either a KMS state or a ground state, necessary and sufficient conditions are given involving only the spectral subspaces of
associated with . The results provide a new insight in the concept of passivity, introduced by W. Pusz and S. L. Woronowicz.Aangesteld navorser N.F.W.O., Belgium, on leave from Katholieke Universiteit Leuven. Research partially supported by N.A.T.O. 相似文献
12.
Let be an invariant state of theC*-system {
,G, } on a locally compact noncommutative groupG. Assume further that is extremal -invariant for an action of an amenable groupH which is -asymptotically abelian and commutes with . Denoting byF
AB,G
AB the corresponding two point functions, we give criteria for the fulfillment of the KMS condition with respect to some one parameter subgroup of the center ofG based on the existence of a closable mapT such thatTF
AB=G
AB for allA,B
. Closability is either inL
(G),B(G) orC
(G), according to clustering properties for . The basic mathematical technique is the duality theory for noncompact, noncommutative locally compact groups.This work is supported in part by the National Science Foundation, Grant MCS 79-03041 相似文献
13.
Let
be a finite-dimensional complex simple Lie algebra and Uq(
) the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of Uq(
), an affinization ofV is an irreducible representationVV of the quantum affine algebra Uq(
) which containsV with multiplicity one and is such that all other irreducible Uq(
)-components ofV have highest weight strictly smaller than the highest weight ofV. There is a natural partial order on the set of Uq(
) classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if
is of typeA, B, C, F orG, the minimal affinization is unique up to Uq(
)-isomorphism; (ii) if
is of typeD orE and is not orthogonal to the triple node of the Dynkin diagram of
, there are either one or three minimal affinizations (depending on ). In this paper, we show, in contrast to the regular case, that if Uq(
) is of typeD
4 and is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of .As a by-product of our methods, we disprove a conjecture according to which, if
is of typeA
n,every affinization is isomorphic to a tensor product of representations of Uq(
) which are irreducible under Uq(
) (in an earlier paper, we proved this conjecture whenn=1).Both authors were partially supported by the NSF, DMS-9207701. 相似文献
14.
R. F. Streater 《Communications in Mathematical Physics》1966,2(1):354-374
The dynamical variables of a classical system form a Lie algebra
, where the Lie multiplication is given by the Poisson bracket. Following the ideas ofSouriau andSegal, but with some modifications, we show that it is possible to realize
as a concrete algebra of smooth transformations of the functionals on the manifold
of smooth solutions to the classical equations of motion. It is even possible to do this in such a way that the action of a chosen dynamical variable, say the Hamiltonian, is given by the classical motion on the manifold, so that the quantum and classical motions coincide. In this realization, constant functionals are realized by multiples of the identity operator. For a finite number of degrees of freedom,n, the space of functionals can be made into a Hilbert space using the invariant Liouville volume element; the dynamical variablesF become operators
in this space. We prove that for any hamiltonianH quadratic in the canonical variablesq
1...q
n
,p
1...p
n
there exists a subspace
1 which is invariant under the action of
and
, and such that the restriction of
to
1 form an irreducible set of operators. Therefore,Souriau's quantization rule agrees with the usual one for quadratic hamiltonians. In fact, it gives the Bargmann-Segal holomorphic function realization. For non-linear problems in general, however, the operators
form a reducible set on any subspace of invariant under the action of the Hamiltonian. In particular this happens for
. Therefore,Souriau's rule cannot agree with the usual quantization procedure for general non-linear systems.The method can be applied to the quantization of a non-linear wave equation and differs from the usual attempts in that (1) at any fixed time the field and its conjugate momentum may form a reducible set (2) the theory is less singular than usual.For a particular wave equation
, we show heuristically that the interacting field may be defined as a first order differential operator acting onc
-functions on the manifold of solutions. In order to make this space into a Hilbert space, one must define a suitable method of functional integration on the manifold; this problem is discussed, without, however, arriving at a satisfactory conclusion.On leave from Physics Department, Imperial College, London SW7.Work partly supported by the Office of Scientific Research, U.S. Air Force. 相似文献
15.
LetS() be the group of finite permutations on countably many symbols. We exhibit an embedding ofS() into a UHF-algebra
of Glimm typen
such that, if is a *-derivation vanishing onS() and satisfying °=0, where is the unique trace on
, then admits an extension which is the generator of aC*-dynamics.Work supported in part by NSF 相似文献
16.
Sergio Doplicher Daniel Kastler Derek W. Robinson 《Communications in Mathematical Physics》1966,3(1):1-28
Starting from aC*-algebra
and a locally compact groupT of automorphisms of
we construct a covariance algebra
with the property that the corresponding *-representations are in one-to-one correspondence with covariant representations of
i.e. *-representations of
in which the automorphisms are continuously unitarily implemented. We further construct for relativistic field theory an algebra
yielding the *-representations of
in which the space time translations have their spectrum contained inV. The problem of denumerable occurence of superselection sectors is formulated as a condition on the spectrum of
. Finally we consider the covariance algebra
built with space translations alone and show its relevance for the discussion of equilibrium states in statistical mechanics, namely we restore in this framework the equivalence of uniqueness of the vacuum, irreducibility and a weak clustering property.On leave of absence from Istituto di Fisica G. Marconi — Roma. 相似文献
17.
R. D. S. Yadava 《Zeitschrift für Physik B Condensed Matter》1992,86(1):93-99
An analysis of the ac conductivity ac(), and the ac dielectric constant, (), of the metal-insulator percolation systems is presented in the critical regime near the transition threshold. It is argued that the polarization and relaxation of the finite fractal metallic clusters play dominant roles in controlling the dynamic response of the system on both sides of the threshold. The relaxation time constant of a fractal cluster is shown to scale with its size as
withd
t
= 4 – 2d +d
c
+ /, whered is tge Euclidean dimension, andd
c
, , and are the scaling indices for the charging, the dc conductivity, and the correlation length respectively. The average time dependent response of the system is shown to scale with a new time scale
, where is the correlation length and 0 is a microscopic time constant. It is shown that at frequencies
and
with /dt 1, in close agreement with experiments. The effects of the anomalous transport along the infinite cluster and the medium polarizability are also discussed. 相似文献
18.
We prove that Gibbs states for the Hamiltonian
, with thes
x varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if
withJ
xy i.i.d. bounded random variables with zero average, 1 in one dimension, and 2 in two dimensions. 相似文献
19.
Given a net
of finite-dimensional real Lie algebras contracting into a Lie algebra, a representation
of is constructed explicitly as limit of a net (
l
) of representations, each
l
being a representation of
on a complex Hilbert space
l
. Conditions are imposed on the net (
l
) implying that the carrier space of
contain a-stable set of vectors which are analytic for all, where is a basis of. As a corollary, the corresponding result for contractions of representations of simply connected finite-dimensional real Lie groups is derived.Supported by the Swiss National Science Foundation 相似文献
20.
Suppose that there is given a Wightman quantum field theory (QFT) whose Euclidean Green functions are invariant under the Euclidean conformal groupSO
e
(5,1). We show that its Hilbert space of physical states carries then a unitary representation of the universal (-sheeted) covering group* of the Minkowskian conformal group SO
e
(4, 2)2. The Wightman functions can be analytically continued to a domain of holomorphy which has as a real boundary an -sheeted covering
of Minkowski-spaceM
4. It is known that* can act on this space
and that
admits a globally*-invariant causal ordering;
is thus the natural space on which a globally*-invariant local QFT could live. We discuss some of the properties of such a theory, in particular the spectrum of the conformal HamiltonianH=1/2(P
0+K
0).As a tool we use a generalized Hille-Yosida theorem for Lie semigroups. Such a theorem is stated and proven in Appendix C. It enables us to analytically continue contractive representations of a certain maximal subsemigroup
of to unitary representations of*. 相似文献