Quantized Brans Dicke Theory: Phase Transition and Strong Coupling (Large $ω$) Limit & General Relativity

We show that Friedmann-Robertson-Walker (FRW) geometry with 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 $a\mapstoλa$ for some constant $λ$. In the weak coupling ($ω$) limit, the theory goes from a symmetry preserving phase to a broken phase. The existence of phase boundary is an obstruction to another classical symmetry [arXiv:gr-qc/9902083] (which relates two BD theory with different coupling) admitted by BD theory with scale invariant matter content i.e $T^μ{}_μ=0$. Classically, this prohibits the BD theory to reduce to General Relativity (GR) for scale invariant matter content. We show that strong coupling limit of BD and GR both preserves the symmetry involving scale factor. We also show that with a scale invariant matter content (radiation i.e $P=\frac{1}{3}ρ$), the quantized BD theory does reduce to GR as $ω\rightarrow\infty$, which is in sharp contrast to classical behavior. This is a first known illustration of a scenario, where quantized BD theory provides example of anomalous symmetry breaking and resulting binary phase structure. We make a conjecture regarding strong coupling limit of BD theory in generic scenario.