Minimal coupling in presence of non-metricity and torsion
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Regular Article - Theoretical Physics
Minimal coupling in presence of non-metricity and torsion Adrià Delhoma Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Universidad de Valencia, Burjassot, 46100 Valencia, Spain
Received: 6 May 2020 / Accepted: 5 August 2020 © The Author(s) 2020
Abstract We deal with the question of what it means to define a minimal coupling prescription in presence of torsion and/or non-metricity, carefully explaining while the naive substitution ∂ → ∇ introduces extra couplings between the matter fields and the connection that can be regarded as non-minimal in presence of torsion and/or non-metricity. We will also investigate whether minimal coupling prescriptions at the level of the action (MCPL) or at the level of field equations (MCPF) 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 nonRiemannian space, and derivate the form of the associated Noether currents and charges. Then we will see that if the minimal coupling prescriptions is applied as we discuss, for spin 0 and 1 fields the results of MCPL and MCPF are equivalent, while for spin 1/2 fields there is a difference if one applies the MCPF or the MCPL, since the former leads to charge violation.
δ
List of symbols
∇g
M X g g dVg
d
n-dimensional space-time manifold. Interior product on M. Set of vector fields on M. A metric structure of M. Determinant of the metric g. Generic volume form on M. Volume form associated to the metric g. In a coordinate frame it reads dVg = √ −gd x μ1 ∧ ... ∧ d x μn . Hodge dual operator associated to the volume form ε. It acts on differential forms on M. Exterior derivative of differential forms on M.
ε Div ∂ ∂μ ∇ μν α ∇M
C(g)
i ψ Aμ ϒμi (ϒ Ni R )μ ψ
ϒμ
ψ
(ϒ g )μ a e-mail:
[email protected] (corresponding author)
0123456789().: V,-vol
Co-differential operator associated to the volume form ε, defined by δ ≡ d . D’Alambertian operator associated to the volume form , defined by ≡ dδ + δ d. Divergence operator on M. It can be defined as in (1) or as Div = d . Formal symbol meaning a general partial derivative of a tensor or spinor field without the need for specifying a frame. Partial derivative associated to the coordinate frame {x μ } on M. Affine connection. Covariant derivative associated to an affine connection . Connection symbols associated to the affine connection (typically associated to the tensorial representations). Covariant derivative of Minkowski spacetime. In a cartesian inertial frame it coincides with ∂. Covariant derivative associated to the LeviCivita connection of g. Levi-Civita connection of g. Its connection coefficients are given by Cμν α . Collection of matter fields labelled by i, each of them belonging to an arbitrary representation of the Lorentz group Scalar field. Spin 1/2 field. Spin 1 field. Connection coefficients of in the representation corresponding to the matter field
i . N
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