Quantized Brans-Dicke theory: Phase transition, strong coupling limit, and general relativity

We show that Friedmann-Robertson-Walker geometry with a flat spatial section in quantized (Wheeler deWitt quantization) Brans-Dicke (BD) theory reveals a rich phase structure owing to anomalous breaking of a classical symmetry, which maps the scale factor $a\mapsto \lambda a$ for some constant $\lambda$. In the weak coupling ($\omega$) limit, the theory goes from a symmetry preserving phase to a broken phase. The existence of a phase boundary is an obstruction to another classical symmetry [see V. Faraoni, Phys. Rev. D 59, 084021 (1999).] (which relates two BD theories with different couplings) admitted by BD theory with scale invariant matter content, i.e., ${T^{\mu }}_{\mu }=0$. Classically, this prohibits the BD theory from reducing to general relativity (GR) for scale invariant matter content. We show that a strong coupling limit of both BD and GR preserves the symmetry involving the scale factor. We also show that with scale invariant matter content (radiation, i.e., $P=\frac{1}{3}\rho$), the quantized BD theory does reduce to GR as $\omega \to \infty$, which is in sharp contrast to classical behavior. This is a first known illustration of a scenario where quantized BD theory provides an example of anomalous symmetry breaking and resulting binary phase structure. We make a conjecture regarding the strong coupling limit of the BD theory in a generic scenario.

Figure 1

Phase structure in $(k,\omega )$ plane. The red (dark shaded) region is where symmetry is broken due to quantum effects, while in the yellow (lightly shaded) region, the symmetry is preserved. The thick blue line represents the phase wall. The dotted red line is supposed to be at $\omega =\infty$. The dotted green line below which we have the yellow (lightly shaded) region is at $\omega =-1.5$.