<i>Gauge Theory and Variational Principles</i>

Bleecker, David; Orzalesi, C. A.

doi:10.1063/1.2915268

Cited by 184 publications

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“…However, at least for the examples considered in these notes, the topology will be always given explicitly. 10 Still another equivalent definition consists in saying that x y if and only if the constant sequence (x, x, x, · · ·) converges to y. It is worth noticing that in a T 0 -space the limit of a sequence needs not be unique so that the constant sequence (x, x, x, · · ·) may converge to more than one point.…”

Section: Order and Topologymentioning

confidence: 99%

“…with {x} the closure of the one point set {x} 10 . From (3.6) it is clear that the relation is reflexive and transitive, x x, x y , y z ⇒ x z .…”

Section: Order and Topologymentioning

confidence: 99%

“…In the usual approach to gauge theories, one constructs connections on a principal bundle P → M with structure group a finite dimensional Lie group G. Associated with this bundle there is a sequence of infinite dimensional (Hilbert-Lie) groups which looks remarkably similar to the sequence (8.50) [10,100],…”

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confidence: 99%

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An Introduction to Noncommutative Spaces and their Geometries

Landi¹

1997

Lecture Notes in Physics

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Section: Order and Topologymentioning

confidence: 99%

“…with {x} the closure of the one point set {x} 10 . From (3.6) it is clear that the relation is reflexive and transitive, x x, x y , y z ⇒ x z .…”

Section: Order and Topologymentioning

confidence: 99%

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An Introduction to Noncommutative Spaces and their Geometries

Landi¹

1997

Lecture Notes in Physics

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“…8 We recall that there are some other (equivalent) definitions of spinor bundles that we are not going to introduce in this paper as, e.g., the one given in [6] in terms of mappings from P Spin e 1,3 to some appropriate vector space.…”

Section: Definition 11 a Real Spinor Bundle For M Is A Vector Bundlementioning

confidence: 99%

The bundles of algebraic and Dirac–Hestenes spinor fields

Mosna

Rodrigues

2004

Journal of Mathematical Physics

204

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The main objective of this paper is to clarify the ontology of Dirac-Hestenes spinor fields (DHSF ) and its relationship with even multivector fields, on a Riemann-Cartan spacetime (RCST) M =(M, g, ∇, τg, ↑) admitting a spin structure, and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE ) in the case where M is a Lorentzian spacetime (the general case when M is a RCST will be discussed in another publication). To this aim we introduce the Clifford bundle of multivector fields (Cℓ(M, g)) and the left (Cℓ l We also obtain a representation of the DE Cℓ l in the Clifford bundle Cℓ(M, g). It is such equation that we call the DHE and it is satisfied by Clifford fields ψΞ ∈ sec Cℓ(M, g). This means that to each DHSF Ψ ∈ sec Cℓ l Spin e 1,3 (M ) and to each spin frame Ξ ∈ sec P Spin e 1,3 (M ), there is a well-defined sum of even multivector fields ψΞ ∈ sec Cℓ(M, g) (EMFS ) associated with Ψ. Such an EMFS is called a representative of the DHSF on the given spin frame. And, of course, such a EMFS (the representative of the DHSF ) is not a spinor field. With this crucial distinction between a DHSF and its representatives on the Clifford bundle, we provide a consistent theory for the covariant derivatives of Clifford and spinor fields of all kinds. We emphasize that the DE Cℓ l and the DHE, although related, are equations of different mathematical natures. We study also the local Lorentz invariance and the electromagnetic gauge invariance and show that only for the DHE such transformations are of the same mathematical nature, thus suggesting a possible link between them. *

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“…The notion of gauge symmetry, in general terms, expresses certain redundancies in the mathematical description of the interactions considered. In mathematics, by gauge theory one usually refers to gauge theories of the Yang-Mills type with the underlying geometry given by a principal G-bundle over a smooth orientable (compact) manifold endowed, in addition, with a (semi-)Riemannian structure (see, for instance, in [Ble '81], [MM '92], [MMF '95], [Nab '00] and [Trau '80]). This notion of gauge theory, however, is clearly far too restrictive when considered from a physical point of view.…”

Section: Introductionmentioning

confidence: 99%

Gauge theories of Dirac type

Tolksdorf

Thumstädter

2006

Journal of Mathematical Physics

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A specific class of gauge theories is geometrically described in terms of fermions. In particular, it is shown how the geometrical frame presented naturally includes spontaneous symmetry breaking of Yang-Mills gauge theories without making use of a Higgs potential. In more physical terms, it is shown that the Yukawa coupling of fermions, together with gravity, necessarily yields a symmetry reduction provided the fermionic mass is considered as a globally well-defined concept. The structure of this symmetry breaking is shown to be compatible with the symmetry breaking that is induced by the Higgs potential of the minimal Standard Model. As a consequence, it is shown that the fermionic mass has a simple geometrical interpretation in terms of curvature and that the (semi-classical) "fermionic vacuum" determines the intrinsic geometry of space-time. We also discuss the issue of "fermion doubling" in some detail and introduce a specific projection onto the "physical sub-space" that is motivated from the Standard Model.

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Gauge Theory and Variational Principles

Cited by 184 publications

References 0 publications

An Introduction to Noncommutative Spaces and their Geometries

An Introduction to Noncommutative Spaces and their Geometries

The bundles of algebraic and Dirac–Hestenes spinor fields

Gauge theories of Dirac type

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