Abstract
We perform the manifestly covariant quantization of a scale invariant gravity with a scalar field, which is equivalent to the well-known Brans-Dicke gravity via a field redefinition of the scalar field, in the de Donder gauge condition (or harmonic gauge condition) for general coordinate invariance. First, without specifying the expression of a gravitational theory, we write down various equal-time (anti-)commutation relations (ETCRs), in particular, those involving the Nakanishi-Lautrup field, the FP ghost, and the FP antighost only on the basis of the de Donder gauge condition. It is shown that choral symmetry, which is a Poincar${\rm{\acute{e}}}$-like $IOSp(8|8)$ supersymmetry, can be derived from such a general action with the de Donder gauge. Next, taking the scale invariant gravity with a scalar field as a classical theory, we derive the ETCRs for the gravitational sector involving the metric tensor and scalar fields. Moreover, we account for how scale symmetry is spontaneously broken in quantum gravity, thereby showing that the dilaton is a massless Nambu-Goldstone particle.
Highlights
A residual symmetry which is left behind after taking a certain gauge-fixing condition for the gauge invariance, has far played an important role in quantum field theory
The conformal symmetry is a typical residual symmetry, which is left in a theory after taking the conformal gauge for the world-sheet diffeomorphism [or general coordinate transformation (GCT)] and the Weyl symmetry [2]
In the most recent study, we have shown that using the simplest scalar-tensor gravity [3] the restricted Weyl symmetry (RWS) and general coordinate invariance generate conformal symmetry in four dimensions in a flat Minkowski background [4–6]
Summary
A residual symmetry which is left behind after taking a certain gauge-fixing condition for the gauge invariance, has far played an important role in quantum field theory. One of our motivations is to relax this situation and show that the choral symmetry does not depend on the expression of the classical gravity but completely comes from the de Donder gauge condition for GCT in the BRST formalism [10] For this purpose, without the knowledge of the classical Lagrangian we derive various equal-time (anti-)commutation relations (ETCRs) for the Nakanishi-Lautrup field, the FaddeevPopov (FP) ghost, and the FP antighost only on the basis of the de Donder gauge condition.
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