The Algebra of Observables

2002 ◽

Vol 17 (17) ◽

pp. 2331-2349 ◽

Cited By ~ 3

Author(s):

GERRIT HANDRICH

Keyword(s):

Rest Frame ◽

Poisson Algebra ◽

Quantum Algebra ◽

Substantial Part ◽

Quantum Case ◽

Generators And Relations ◽

Defining Relations ◽

Algebra Of Observables ◽

The One ◽

The Relationship

To postulate correspondence for the observables only is a promising approach to a fully satisfying quantization of the Nambu–Goto string. The relationship between the Poisson algebra of observables and the corresponding quantum algebra is established in the language of generators and relations. A very valuable tool is the transformation to the string's rest frame, since a substantial part of the relations are solved. It is the aim of this paper to clarify the relationship between the fully covariant and the rest frame description. Both in the classical and in the quantum case, an efficient method for recovering the covariant algebra from the one in the rest frame is described. Restrictions on the quantum defining relations are obtained, which are not taken into account when one postulates correspondence for the rest frame algebra. For the part of the algebra studied up to now in explicit computations, these further restrictions alone determine the quantum algebra uniquely — in full consistency with the further restrictions found in the rest frame.

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COVARIANT DESCRIPTION OF PARAMETRIZED NONRELATIVISTIC HAMILTONIAN SYSTEMS

International Journal of Modern Physics A ◽

10.1142/s0217751x04018063 ◽

2004 ◽

Vol 19 (15) ◽

pp. 2473-2493 ◽

Cited By ~ 7

Author(s):

MAURICIO MONDRAGÓN ◽

MERCED MONTESINOS

Keyword(s):

Phase Space ◽

Gauge Transformation ◽

Hamiltonian Systems ◽

Symplectic Structure ◽

Canonical Formalism ◽

Extended Phase Space ◽

Extended Phase ◽

Algebra Of Observables ◽

Covariant Description ◽

Constraint Surface

The various phase spaces involved in the dynamics of parametrized nonrelativistic Hamiltonian systems are displayed by using Crnkovic and Witten's covariant canonical formalism. It is also pointed out that in Dirac's canonical formalism there exists a freedom in the choice of the symplectic structure on the extended phase space and in the choice of the equations that define the constraint surface with the only restriction that these two choices combine in such a way that any pair (of these two choices) generates the same gauge transformation. The consequence of this freedom on the algebra of observables is also discussed.

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Algebra of observables

Lectures on Quantum Mechanics ◽

10.1007/978-0-387-37748-3_8 ◽

2007 ◽

pp. 143-160

Keyword(s):

Algebra Of Observables

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Dynamics of Interacting Classical and Quantum Systems

Open Systems & Information Dynamics ◽

10.1142/s1230161211000236 ◽

2011 ◽

Vol 18 (04) ◽

pp. 339-351 ◽

Cited By ~ 7

Author(s):

Dariusz Chruściński ◽

Andrzej Kossakowski ◽

Giuseppe Marmo ◽

E. C. G. Sudarshan

Keyword(s):

Characteristic Feature ◽

Quantum Dynamics ◽

Quantum Discord ◽

Main Idea ◽

Quantum Correlations ◽

Quantum Systems ◽

Quantum States ◽

Algebra Of Observables ◽

Classical Quantum ◽

True Quantum

We analyze the dynamics of coupled classical and quantum systems. The main idea is to treat both systems as true quantum ones and impose a family of superselection rules which imply that the corresponding algebra of observables of one subsystem is commutative and hence may be treated as a classical one. Equivalently, one may impose a special symmetry which restricts the algebra of observables to the 'classical' subalgebra. The characteristic feature of classical-quantum dynamics is that it leaves invariant a subspace of classical-quantum states, that is, it does not create quantum correlations as measured by the quantum discord.

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QUANTUM BACKREACTION (CASIMIR) EFFECT WITHOUT INFINITIES: ALGEBRAIC ANALYSIS

International Journal of Modern Physics Conference Series ◽

10.1142/s2010194512007490 ◽

2012 ◽

Vol 14 ◽

pp. 376-382

Author(s):

ANDRZEJ HERDEGEN

Keyword(s):

Hilbert Space ◽

Quantum Theory ◽

Quantum System ◽

Casimir Effect ◽

Quantum Field ◽

Expectation Value ◽

Physical Systems ◽

Algebra Of Observables ◽

General Terms ◽

External Conditions

Casimir effect, in most general terms, is the backreaction of a quantum system responding to an adiabatic change of external conditions. This backreaction is expected to be quantitatively measured by a change in the expectation value of a certain energy observable of the system. However, for this concept to be applicable, the system has to retain its identity in the process. Most prevailing tendencies in the analysis of the effect seem to ignore this question. In general, a quantum theory is defined by an algebra of observables, whose representations by operators in a Hilbert space define concrete physical systems described by the theory. A quantum system retains its identity if both the algebra as well as its representation do not change. We discuss the resulting restrictions for admissible models of changing external conditions. These ideas are applied to quantum field models. No infinities arise, if the algebraic demands are respected.

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SUPERSELECTION STRUCTURE AND LOCALIZED CONNES’ COCYCLES

Reviews in Mathematical Physics ◽

10.1142/s0129055x95000086 ◽

1995 ◽

Vol 07 (01) ◽

pp. 133-160 ◽

Cited By ~ 1

Author(s):

HANS-WERNER WIESBROCK

Keyword(s):

Field Theories ◽

Local Theory ◽

Quantum Field Theories ◽

Quantum Field ◽

Localization Property ◽

Algebra Of Observables ◽

Local Algebras ◽

Möbius Group ◽

Modular Groups

Let ρ be a localized endomorphism of the universal algebra of observables of a chiral conformal quantum field theory on a circle, see [16, 17, 23] or Chapter 1. Then ρ transforms covariant under the Möbius group. As was pointed out by D. Guido and R. Longo, [23], the covariance transformations are implemented by [Formula: see text] where Ad ∆it are modular groups to local algebras w.r.t. the vacuum vector, ut is a Connes-Radon-Nikodym-Cocycle. Using the localization property of ρ, one gets, at least for regular nets, localization properties of the cocycles. In this work we will do some steps into the opposite direction. Given a localized Connes’ cocycle of a local algebra. We will construct a localized endomorphism on the whole net. The features of this approach are twofold. Firstly sectors of finite and infinite statistical dimensions are handled on the same footing. Secondly it is a local theory right from the beginning. Moreover, soliton-like sectors can easily be incorporated. We will sketch on the last part. The program is carried through for a special class of conformal quantum field theories, the strongly additive ones.

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THE MASTER WARD IDENTITY

Reviews in Mathematical Physics ◽

10.1142/s0129055x02001454 ◽

2002 ◽

Vol 14 (09) ◽

pp. 977-1049 ◽

Cited By ~ 28

Author(s):

M. DÜTSCH ◽

F.-M. BOAS

Keyword(s):

Ward Identity ◽

Brst Symmetry ◽

Quantum Field ◽

Field Equations ◽

Normalization Condition ◽

Abelian Gauge ◽

Algebra Of Observables ◽

Local Construction ◽

Perturbative Quantum ◽

Equal Time Commutation

In the framework of perturbative quantum field theory (QFT) we propose a new, universal (re)normalization condition (called 'master Ward identity') which expresses the symmetries of the underlying classical theory. It implies for example the field equations, energy-momentum, charge- and ghost-number conservation, renormalized equal-time commutation relations and BRST-symmetry. It seems that the master Ward identity can nearly always be satisfied, the only exceptions we know are the usual anomalies. We prove the compatibility of the master Ward identity with the other (re)normalization conditions of causal perturbation theory, and for pure massive theories we show that the 'central solution' of Epstein and Glaser fulfills the master Ward identity, if the UV-scaling behavior of its individual terms is not relatively lowered. Application of the master Ward identity to the BRST-current of non-Abelian gauge theories generates an identity (called 'master BRST-identity') which contains the information which is needed for a local construction of the algebra of observables, i.e. the elimination of the unphysical fields and the construction of physical states in the presence of an adiabatically switched off interaction.

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