Dirac Equation in Non-Riemannian Geometries

The geometrical formulation of gravity is not unique and can be set up in a variety of spacetimes. Even though the gravitational sector enjoys this freedom of different geometrical interpretations, consistent matter couplings have to be assured for a steady foundation of gravity. In generalised geometries, further ambiguities arise in the matter couplings unless the minimal coupling principle (MCP) is adopted that is compatible with the principles of relativity, universality and inertia. In this work, MCP is applied to all standard model gauge fields and matter fields in a completely general (linear) affine geometry. This is also discussed from an effective field theory perspective. It is found that the presence of torsion generically leads to theoretical problems. However, symmetric teleparallelism, wherein the affine geometry is integrable and torsion-free, is consistent with MCP. The generalised Bianchi identity is derived and shown to determine the dynamics of the connection in a unified fashion. Also, the parallel transport with respect to a teleparallel connection is shown to be free of second clock effects.

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“…Using the formulae (38) and the constancy dγ a = 0 of the Dirac matrices, we can alternatively write…”

Section: Lemma 4 the Generalised Noether Identity Resulting From The Diffeomorphism Invariance Of I Ismentioning

confidence: 99%

“…where we have used that Dη ab = −Q ab (see (29) below), and the solution reads [36,37] (see also [38])…”

Section: Lemma 3 Consider Mcp In the Hermitian Theory Of Dirac The Action Is Unaffected By A Real Affine Generalisation Of The Metric Conmentioning

confidence: 99%

The coupling of matter and spacetime geometry

Jiménez

Heisenberg

Koivisto

2020

Class. Quantum Grav.

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“…Note that the torsion and non-metricity terms Q [λµ] λ + T µλ λ which appear in [14,15] cancel in our case, as they are both proportional to A µ .…”

Section: Induced Electromagnetism From Non-metricitymentioning

confidence: 75%

Projective symmetries and induced electromagnetism in metric-affine gravity

Janssen

Jiménez-Cano

2018

Physics Letters B

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We present a framework in which the projective symmetry of the Einstein-Hilbert action in metric-affine gravity is used to induce an effective coupling between the Dirac lagrangian and the Maxwell field. The effective U (1) gauge potential arises as the trace of the non-metricity tensor Q µa a and couples in the appropriate way to the Dirac fields to in order to allow for local phase shifts. On shell, the obtained theory is equivalent to Einstein-Cartan-Maxwell theory in presence of Dirac spinors.

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“…This result is not so clear for spin 1/2 fields. Indeed it is not trivial how to generalize the Dirac equation to non-Riemannian spacetimes, as the minimal coupling prescription applied to the Minkowskian spin 1/2 field equation (MCP) gives a different result than when applied in the Minkowskian spin 1/2 Lagrangian (MCPL) [79]. Also recently, it has been claimed that the MCP and MCPL give different dynamics for matter fields in general in Riemann-Cartan space-times [80].…”

Section: Introductionmentioning

confidence: 99%

Minimal coupling in presence of non-metricity and torsion

Delhom

2020

Preprint

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We deal with the question of what are and what are not minimal coupling prescriptions in presence of torsion and/or non-metricity, carefully explaining while the naive substitution ∂ → ∇ is not actually a minimal coupling prescritpion. We will also investigate whether minimal coupling prescriptions at the level of the action (MCPL) or at the level of field equations (MCP) lead to different dynamics. To that end, we will first write the Euler-Lagrange equations for matter fields in terms of the covariant derivatives of a general non-Riemannian space, and derivate the form of the associated Noether currents and charges. Then we will see that, if the correct minimal coupling prescriptions is applied, for spin 0 and 1 fields the results of MCPL and MCP are equivalent, while for spin 1/2 fields there is a difference if one applies the MCP or the MCPL, since the former leads to charge violation. We will also explicitly show how the usual naive substitution ∂ → ∇ is not a minimal coupling prescription for spin 0, 1/2 and 1 fields in presence of torsion and/or non-metricity.1 Here we use Riemannian to imply that the connection is metric-compatible, and non-Riemannian for otherwise. Notice that Mathematicians have another meaning for Riemannian, which is related to the signature of the metric. Our metrics will be assumed to have Lorentzian signature.

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Dirac Equation in Non-Riemannian Geometries

Cited by 8 publications

References 15 publications

The coupling of matter and spacetime geometry

The coupling of matter and spacetime geometry

Projective symmetries and induced electromagnetism in metric-affine gravity

Minimal coupling in presence of non-metricity and torsion

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