Pin groups in general relativity

Janssens, Bas

doi:10.1103/physrevd.101.021702

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…The space A of (smooth) connections in P admits n D ≈ 64 local degrees of freedom. For a given reduction σ, the space Aσ of preserving connections (also in P) has only n d local degrees of freedom 7 .…”

Section: Preserving Connectionsmentioning

confidence: 99%

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Connections and Frame Bundle Reductions

Lachieze-Rey

2020

Preprint

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In general relativity, the gravitational potential is represented by the Levi-Civita connection, the only symmetric connection preserving the metric. On a differentiable manifold, a metric identifies with an orthogonal structure, defined as a Lorentz reduction of the frame bundle. The Levi-Civita connection appears as the only symmetric connection preserving the reduction. This paper presents generalization of this process to other aproaches of gravitation: Weyl structure with Weyl connections, teleparallel structures with Weitzenböck connections, unimodular structure, similarly appear as frame bundle reductions, with preserving connections.To each subgroup H of the linear group GL correspond reduced structures, or H-structures. They are subbundles of the frame bundle (with GL as principal group), with H as principal group. A linear connection in a manifold M is a principal connection on the frame bundle. Given a reduction, the corresponding preserving connections on M are the linear connections which preserve it. I also show that the time gauge used in the 3+1 formalism for general relativity similarly appears as the result of a bundle reduction.

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Section: Preserving Connectionsmentioning

confidence: 99%

“…Each section is a tetrad. " The configuration space1 of general relativity can thus be seen as the space of sections of the bundle Fr/SO " [7].…”

Section: Lorentz Structure and Time Gaugementioning

confidence: 99%

Connections and Frame Bundle Reductions

Lachieze-Rey

2020

Preprint

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Complete Barrett-Crane model and its causal structure

2022

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Spinor Representation of $O(3)$ for $S_4$

Kitabayashi¹,

Yasuè²

2018

Preprint

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All possible permutations in the discrete S 4 group are classified by three rotation angles associated with the orthogonal group O(3). We construct a spinor representation 2 D of O(3), which is transformed by three 4×4 matrices corresponding to three Pauli matrices in SO( 3). An irreducible decomposition of 2 D ⊗ 2 D supplies a vector representation of 3 of O(3), thereby, of S 4 . Our construction is consistent with the mathematical fact that O(3) = SO(3) × Z 2 . The Z 2 parity in the spinorial space is described by a block off-diagonal matrix as the spinorial parity operator, whose eigenvalues are ±1 consistent with Z 2 .

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Pin groups in general relativity

Abstract: Out of the eight possible Pin groups that can be used to describe the transformation behaviour of fermions under parity and time reversal, we show that only two are compatible with general relativity.

Cited by 3 publications

References 21 publications

Connections and Frame Bundle Reductions

Connections and Frame Bundle Reductions

Complete Barrett-Crane model and its causal structure

Spinor Representation of $O(3)$ for $S_4$

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