Abstract
It has long been known that photons and gravitons may appear as vector and tensor Goldstone modes caused by spontaneous Lorentz invariance violation (SLIV). Usually this approach is considered for photons and gravitons separately. We develop the emergent electrogravity theory consisting of the ordinary QED and the tensor-field gravity model which mimics the linearized general relativity in Minkowski spacetime. In this theory, Lorentz symmetry appears incorporated into higher global symmetries of the length-fixing constraints put on the vector and tensor fields involved, A_{mu }^{2}=pm M_{A}^{2} and H_{mu nu }^{2}=pm M_{H}^{2} (M_{A} and M_{H} are the proposed symmetry breaking scales). We show that such a SLIV pattern being related to breaking of global symmetries underlying these constraints induces the massless Goldstone and pseudo-Goldstone modes shared by photon and graviton. While for a vector field case the symmetry of the constraint coincides with Lorentz symmetry SO(1, 3) of the electrogravity Lagrangian, the tensor-field constraint itself possesses much higher global symmetry SO(7, 3), whose spontaneous violation provides a sufficient number of zero modes collected in a graviton. Accordingly, while the photon may only contain true Goldstone modes, the graviton appears at least partially to be composed of pseudo-Goldstone modes rather than of pure Goldstone ones. When expressed in terms of these modes, the theory looks essentially nonlinear and contains a variety of Lorentz and CPT violating couplings. However, all SLIV effects turn out to be strictly cancelled in the lowest order processes considered in some detail. How this emergent electrogravity theory could be observationally different from conventional QED and GR theories is also briefly discussed.
Highlights
This approach itself has had a long history, dating back to the model of Nambu [17] for QED with a nonlinearly realized Lorentz symmetry for the underlying vector field
While for a vector field case the symmetry of the constraint coincides with Lorentz symmetry S O(1, 3) of the electrogravity Lagrangian, the tensor-field constraint itself possesses much higher global symmetry S O(7, 3), whose spontaneous violation provides a sufficient number of zero modes collected in a graviton
As our calculations show, all spontaneous Lorentz invariance violation (SLIV) effects turn out to be strictly cancelled in the low order physical processes involved once the tensor-field gravity part of the electrogravity theory is properly extended to general relativity (GR). This can be taken as an indication that in the electrogravity theory physical Lorentz invariance is preserved in this approximation
Summary
This approach itself has had a long history, dating back to the model of Nambu [17] for QED with a nonlinearly realized Lorentz symmetry for the underlying vector field. We consider it regarding them both in the so-called electrogravity theory where together with the Nambu QED model [17] with its gauge invariant Lagrangian (3) we propose the linearized Einstein– Hilbert kinetic term for the tensor field preserving a diffeomorphism (diff) invariance We show that such a combined SLIV pattern, conditioned by the constraints (1) and (7), induces the massless Goldstone modes which appear shared among photon and graviton. Thereby, the length-fixing constraints (1) and (7) put on the vector and tensor fields appear as the gauge fixing conditions rather than sources of the actual Lorentz violation just as it was in the pure nonlinear QED framework [17] From this viewpoint, if this cancellation were to work in all orders, one could propose that emergent theories, like as the electrogravity theory, are not different from conventional gauge theories.
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