Abstract
The background field method is used to linearize the Weyl invariant scalar-tensor gravity, coupled with a Stueckelberg field. For a generic background metric, this action is found to be not invariant, under both diffeomorphism and generalized Weyl symmetry, the latter being a combination of gauge and Weyl transformations. Interestingly, the quadratic Lagrangian, emerging from a background of Minkowski metric, respects both the transformations, independently. Becchi-Rouet-Stora-Tyutin (BRST) symmetry of scalar-tensor gravity coupled with a Stueckelberg-like massive gauge particle, possessing diffeomorphism and generalized Weyl symmetry, reveals that in both the cases, negative norm states with unphysical degrees of freedom do exist. We then show that, by combining diffeomorphism and generalized Weyl symmetries, all the ghost states decouple, thereby removing the unphysical redundancies of the theory. During this process, the scalar field does not represent any dynamic mode, yet modifies the usual harmonic gauge condition through non-minimal coupling with gravity.
Highlights
Quantization of gravity has been a challenge due to its highly non-linear nature
For the cases of pure gravity [12], as well as gravity minimally coupled to a dilaton scalar field variable [13], such equations were obtained by identifying proper gauge-fixing against the inherent diffeomorphism redundancy of such systems under generic infinitesimal coordinate transformation
The background field approach [13] is applied to expand the complete metric in powers of quantum fluctuations hμν [12], with the corresponding dynamics governed by their second order contributions, as the first order terms vanish since the classical background fields are always taken to be on-shell [13]
Summary
Quantization of gravity has been a challenge due to its highly non-linear nature. For this purpose, several important methods e.g., tetrad formalism and background field method, have been developed [11,12]. Using the same background field expansion, a generic Weyl-invariant gravitational model with dilatations, in presence of a gauge field, has been quantized through suitable definition of renormalization group flows [15] The latter case, including the STG suitably coupled with Abelian gauge field, is of importance in understanding light-matter interaction during conformal phases of the universe (e.g., inflation) and in higher dimensional gravity models with compactification (e.g., Kaluza–Klein gravity and several string theory models). Presently we leave the proof of the same for later work, and we limit ourselves to the treelevel dynamics, with physical states identified To this end, the corresponding Faddeev–Popov ghosts [20] are identified, with invariance under two independent supersymmetric transformations, BRST and anti-BRST [21,22,23,24], which are both nilpotent and anticommuting. We conclude after summarizing the obtained result and point out future directions of work
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