Metric-affine gravity and the Nester–Witten 2-form

Int. J. Geom. Methods Mod. Phys.

2005

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Starting from the general concept of a Lie derivative of an arbitrary differentiable map, we develop a systematic theory of Lie differentiation in the framework of reductive G-structures P on a principal bundle Q. It is shown that these structures admit a canonical decomposition of the pull-back vector bundle i * P (T Q) ≡ P ×Q T Q over P . For classical G-structures, i.e. reductive G-subbundles of the linear frame bundle, such a decomposition defines an infinitesimal canonical lift. This lift extends to a prolongation Γ-structure on P . In this general geometric framework the concept of a Lie derivative of spinor fields is reviewed. On specializing to the case of the Kosmann lift, we recover Kosmann's original definition. We also show that in the case of a reductive G-structure one can introduce a "reductive Lie derivative" with respect to a certain class of generalized infinitesimal automorphisms, and, as an interesting by-product, prove a result due to Bourguignon and Gauduchon in a more general manner. Next, we give a new characterization as well as a generalization of the Killing equation, and propose a geometric reinterpretation of Penrose's Lie derivative of "spinor fields". Finally, we present an important application of the theory of the Lie derivative of spinor fields to the calculus of variations.

Section: Gauge-natural Bundlesmentioning

confidence: 99%

Section: An Application To the Calculus Of Variationsmentioning

confidence: 99%

Section: An Application To the Calculus Of Variationsmentioning

confidence: 99%

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The Lie Derivative of Spinor Fields: Theory and Applications

Godina

Int. J. Geom. Methods Mod. Phys.

2005

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“…From the physical point of view, this framework enables one to treat at the same time, under a unifying formalism, natural field theories such as general relativity, gauge theories, as well as bosonic and fermionic matter field theories (cf. [8,9,12,25]). For k = 1 we have, of course, the identification L 1 M ∼ = LM , where LM is the usual (principal) bundle of linear frames over M (cf., e.g., [18]).…”

Section: Gauge-natural Bundlesmentioning

confidence: 99%

“…This is the most general notion of a (gauge-natural) Lie derivative of spinor fields and the appropriate one for most situations of physical interest (cf. [12,25]): the generality of Ξ might be disturbing, but is the unavoidable indication that S(M ) is not a natural bundle. If we wish nonetheless to remove such a generality, we must choose some canonical (not natural) lift of ξ onto SO(M, g).…”

Section: Lie Derivatives On Reductive G-structuresmentioning

confidence: 99%

Reductive G-structures and Lie derivatives

Godina

Journal of Geometry and Physics

2003

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Reductive G-structures on a principal bundle Q are considered. It is shown that these structures, i.e. reductive G-subbundles P of Q, admit a canonical decomposition of the pull-back vector bundle i * P (T Q) ≡ P ×Q T Q over P . For classical G-structures, i.e. reductive G-subbundles of the linear frame bundle, such a decomposition defines an infinitesimal canonical lift. This lift extends to a prolongation Γ-structure on P . In this general geometric framework the theory of Lie derivatives is considered. Particular emphasis is given to the morphisms which must be taken in order to state what kind of Lie derivative has to be chosen. On specializing the general theory of gauge-natural Lie derivatives of spinor fields to the case of the Kosmann lift, we recover the result originally found by Kosmann. We also show that in the case of a reductive G-structure one can introduce a "reductive Lie derivative" with respect to a certain class of generalized infinitesimal automorphisms. This differs, in general, from the gauge-natural one, and we conclude by showing that the "metric Lie derivative" introduced by Bourguignon and Gauduchon is in fact a particular kind of reductive rather than gauge-natural Lie derivative.

Einstein-Dirac theory on gauge-natural bundles

Reports on Mathematical Physics

2003

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We present a clear-cut example of the importance of the functorial approach of gauge-natural bundles and the general theory of Lie derivatives for classical field theory, where the sole correct geometrical formulation of Einstein (-Cartan) gravity coupled with Dirac fields gives rise to an unexpected indeterminacy in the concept of conserved quantities.Comment: LaTeX (20 pages, 2 runs). Enlarged introduction, corrected typos, updated reference