Abstract
As a modified gravity theory that introduces new gravitational degrees of freedom, the generalized SU(2) Proca theory (GSU2P for short) is the non-Abelian version of the well-known generalized Proca theory where the action is invariant under global transformations of the SU(2) group. This theory was formulated for the first time in Phys. Rev. D 94 (2016) 084041, having implemented the required primary constraint-enforcing relation to make the Lagrangian degenerate and remove one degree of freedom from the vector field in accordance with the irreducible representations of the Poincar\'e group. It was later shown in Phys. Rev. D 101 (2020) 045008, ibid 045009, that a secondary constraint-enforcing relation, which trivializes for the generalized Proca theory but not for the SU(2) version, was needed to close the constraint algebra. It is the purpose of this paper to implement this secondary constraint-enforcing relation in GSU2P and to make the construction of the theory more transparent. Since several terms in the Lagrangian were dismissed in Phys. Rev. D 94 (2016) 084041 via their equivalence to other terms through total derivatives, not all of the latter satisfying the secondary constraint-enforcing relation, the work was not so simple as directly applying this relation to the resultant Lagrangian pieces of the old theory. Thus, we were motivated to reconstruct the theory from scratch. In the process, we found the beyond GSU2P.
Highlights
Whether a classical description of the gravitational interaction is fundamental or effective remains a mystery
Since the Lagrangian given in the previous expression vanishes in the decoupling limit Aaμ → ∇μπa, because of the antisymmetry of Aaμν, it is free of the Ostrogradski instability
Why is it that in the old GSU2P there exists only one? Leaving aside the fact that L 14;2 might be unhealthy in its decoupling limit, the reason lies in a small mistake in the conditions of Eq (37) in Ref. [103] to make the primary constraint-enforcing relation vanish that prevented the authors of that work from finding a second parity-violating Lagrangian piece
Summary
Whether a classical description of the gravitational interaction is fundamental or effective remains a mystery. The temporal gauge setup and the cosmic triad are two of the four possible setups that are compatible with a spatial spherical symmetry and that are realized under an internal SU(2) symmetry [100,101,102] This was the main motivation behind the formulation of what was baptized as the generalized SU(2) Proca theory (GSU2P for short) [103] The purpose of this paper is to build from scratch the GSU2P, paying attention to the two caveats already mentioned and following a style of construction based on the decomposition of a first-order derivative ∂μAaν of the vector field Aaμ into its symmetric, Saμν ≡ ∂μAaν þ ∂νAaμ; ð1Þ and antisymmetric part, Aaμν ≡ ∂μAaν − ∂νAaμ: ð2Þ Employing this decomposition will simplify things and allow us to deal with a lower number of Lagrangian building blocks as compared with Ref. The sign convention is the (þ þ þ) according to Misner, Thorne, and Wheeler [113]
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