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• 1
Electronic Resource
Springer
Communications in mathematical physics 28 (1972), S. 331-348
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We examine field theories with a compact groupG of exact internal gauge symmetries so that the superselection sectors are labelled by the inequivalent irreducible representations ofG. A particle in one of these sectors obeys a parastatistics of orderd if and only if the corresponding representation ofG isd-dimensional. The correspondence between representations of the observable algebra and representations ofG extends to a mapping of the intertwining operators for these representations preserving linearity, tensor products and conjugation. Although we assume no explicit commutation property between fields, the commutation relations of fields of the same irreducible tensor character underG at spacelike separations are largely determined by the statistics parameter of the corresponding sector. For fields of conjugate irreducible tensor character the observable part of the commutator (anticommutator) vanishes at spacelike separations if the corresponding sector has para-Bose (para-Fermi) statistics.
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• 2
Electronic Resource
Springer
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract Starting from the principles of local relativistic Quantum Theory without long range forces, we study the structure of the set of superselection sectors (charge quantum numbers) and its implications for the particle aspects of the theory. Without assuming the commutation properties (or even the existence) of unobservable fields connecting different sectors (charge-carrying fields), one has a particle-antiparticle symmetry, an intrinsic notion of statistics for identical particles, and a spin-statistics theorem. Particles in “pseudoreal sectors” cannot be their own antiparticles (a variant of Carruthers' theorem). We also show how scattering states and transition probabilities are obtained in this frame.
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• 3
Electronic Resource
Springer
Communications in mathematical physics 37 (1974), S. 273-286
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract In a dilatation-invariant theory it is shown that there is a unique locally normal dilatation-invariant state. Furthermore a gauge transformation of a local algebra cannot be implemented by a unitary operator from the local algebra. If the local field algebras are factors then so are the local observable algebras. The superselection structure of the theory can be determined locally.
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• 4
Electronic Resource
Springer
Communications in mathematical physics 15 (1969), S. 173-200
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We wish to study the construction of charge-carrying fields given the representation of the observable algebra in the sector of states of zero charge. It is shown that the set of those covariant sectors which can be obtained from the vacuum sector by acting with “localized automorphisms” has the structure of a discrete Abelian group . An algebra of fields $$\mathfrak{F}$$ can be defined on the Hilbert space of a representation π of the observable algebra $$\mathfrak{A}$$ which contains each of the above sectors exactly once. The dual group of acts as a gauge group on $$\mathfrak{F}$$ in such a way that $$\pi (\mathfrak{A})$$ is the gauge invariant part of $$\mathfrak{F}.\mathfrak{F}$$ is made up of Bose and Fermi fields and is determined uniquely by the commutation relations between spacelike separated fields.
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• 5
Electronic Resource
Springer
Communications in mathematical physics 172 (1995), S. 187-220
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations. We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper. In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundleTS 2 of the 2-sphere. The relations with Conne's theory of the standard model will be studied elsewhere.
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• 6
Electronic Resource
Springer
Communications in mathematical physics 51 (1976), S. 107-119
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract IfA μ is a vector field satisfying ∂μ A v −∂ v A μ=0 can one find a scalar field φ such thatA μ=∂μφ? A novel quantum analogue of this classical problem incorporating locality is introduced and is shown to generate those super-selection sectors whose charge can be strictly localized. In a 2-dimensional space-time there are further possibilities; in particular, soliton sectors can be generated by this procedure.
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• 7
Electronic Resource
Springer
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract The algebra of observables has two distinct local structures. The first, derived from the localization of measurements, gives rise to an additive net structure. The second, derived from the support properties of infinitestimal operations, gives rise to a sheaf structure. It is also shown how an additive net of field algebras acted on by a compact gauge group of the first kind generates an additive net of observable algebras.
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• 8
Electronic Resource
Springer
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract The problem of determining the physically relevant states acquires a new dimension in curved spacetime where there is, in general, no natural definition of a vacuum state. It is argued that there is a unique local quasiequivalence class of physically relevant states and it is shown how this class can be specified for the free Klein-Gordon field on a Robertson-Walker spacetime by using the concept of an adiabatic vacuum state. Any two adiabatic vacuum states of order two are locally quasiequivalent.
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• 9
Electronic Resource
Springer
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract Starting from an algebra of fields $$\mathfrak{F}$$ and a compact gauge group of the first kind ℊ, the observable algebra $$\mathfrak{A}$$ is defined as the gauge invariant part of $$\mathfrak{F}$$ . A gauge group of the first kind is shown to be automatically compact if the scattering states are complete and the mass and spin multiplets have finite multiplicity. Under reasonable assumptions about the structure of $$\mathfrak{F}$$ it is shown that the inequivalent irreducible representations of $$\mathfrak{A}$$ (“sectors”) which occur are in one-to-one correspondence with the inequivalent irreducible representations of ℊ and that all of them are “strongly locally equivalent”. An irreducible representation of $$\mathfrak{A}$$ satisfies the duality property only if the sector corresponds to a 1-dimensional representation of ℊ. If ℊ is Abelian the sectors are connected to each other by localized automorphisms.
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• 10
Electronic Resource
Springer
Communications in mathematical physics 155 (1993), S. 123-134
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We characterize the automorphisms of aC *-algebra which extend to automorphisms of the crossed product by a compact group dual. The case where the inclusion is equipped with a group of automorphisms commuting with the dual action is also treated. These results are applied to the analysis of broken gauge symmetries in Quantum Field Theory to draw conclusions on the structure of the degenerate vacua on the field algebra.
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