Conservation of Energy-Momentum of Matter as the Basis for the Gauge Theory of Gravitation

Gauging the higher-spin-like symmetries by the Moyal product

“…The well-known but still poorly understood property of all free relativistic field theories on Minkowski spacetime, is that they possess an infinite tower of abelian rigid symmetries with a corresponding tower of conserved currents represented by Lorentz tensor fields of all ranks s ≥ 1 (to which one usually refers to as spin-s currents). 1 As the rigid symmetry variations and corresponding conserved currents contain a number of derivatives growing with s, one can speak of higher derivative or higher spin (HS) symmetries. 2 The low spin sector (s ≤ 2) of these symmetries plays an immensely important role, as spin-1 describes internal symmetries and gives conservation of electric or non-Abelian charges while spin-2 describes translations in spacetime and leads to conservation of energy and momentum.…”

“…It is not inconceivable that the degrees of freedom associated with these symmetries may account for dark matter, a missing puzzle in the standard cosmological model. 1 Likewise, in supersymmetric free field theories there is also an infinite tower of half-integer spin symmetries and corresponding currents. We shall not discuss half-integer spin symmetries in this paper.…”

“…2 We shall use both terms interchangeably, depending on the context. 3 See [1] for recent review and developments.…”

“…1 and usingK s (û) = η abû aûb )S (0) [φ] = d d x ∂ µ φ ∂ µ φ † = α φ |K s (û)|α φ = Tr K s (û)|α φ α φ | . (C.1)…”

Generalized Birkhoff theorem in the Poincaré gauge gravity theory

“…Here we discuss the Poincaré gauge (PG) gravity theory which offers a physically meaningful extension of Einstein's GR to the case when the spin of matter is included as a source of the gravitational field along with the mass of matter [24][25][26][27][28][29]. The canonical spin and the energy-momentum currents [30] underlie the corresponding gauge scheme as the Noether currents for the Poincaré group G = T 4 ⋊ SO(1, 3) which is the semidirect product of the 4-parameter group T 4 of spacetime translations and the 6-parameter local Lorentz group SO(1, 3). In the framework of the consistent Yang-Mills-Kibble-Utiyama field-theoretic approach, the Poincaré gauge potentials are identified with the coframe 1-form ϑ α = e α i dx i ("translational potential" corresponding to the T 4 subgroup) and the local Lorentz connection 1form Γ αβ = − Γ βα = Γ i αβ dx i ("rotational potential" corresponding to the SO(1, 3) subgroup) which give rise to the Riemann-Cartan geometry on the spacetime manifold [31][32][33].…”