Is it true that in canonically quantised gravity "non-traceless" and "non-transverse" gravitons are allowed to exist as virtual particles, whereas any graviton that corresponds to a "time-warping" plane wave is not allowed to exist, even as virtual particle? And if so, could this be why gravity is non-renormalisable for high energies?
Rephrasing my 2nd question:
Is perturbative quantum-gravity non-renormalisable because time-warping doesn't occur at all in the low energy limit and so cannot be accounted for through the interaction-picture?
I will explain why I would think to ask this:
I have tried to canonically quantise linearised gravity in exact parallel with the canonical quantisation of spin-1 vector fields by looking at a gauge in which the EOM (in this case the linearised EFE) become a simple wave-equation, and then using the Gupta-Bleuler technique to enforce the gauge conditions.
Using the Gupta-Bleuler approach on the gauge conditions that turn the linearised EFE into a simple-wave equation we get the requirements:
$$h^+_{0\mu} |\psi\rangle = 0 \tag{1}$$ $$h_{i}^{+i} |\psi\rangle = 0 \tag{2}$$ $$\partial^{\mu}h^{+}_{\mu\nu} |\psi\rangle = 0 \tag{3}$$
Where $$h^+_{\mu\nu} = \int\frac{d^{3}k}{(2\pi)^{3}2\omega_{\vec{k}}} \epsilon_{\mu}^r\epsilon_{\nu}^s a_{rs}(\vec{k})e^{-ikx} \tag{4}$$
Just as the gauge condition $\partial_{\mu}A^{\mu}=0$ ensures that no scalar- or longitudinally-polarised photons appear in any observables in QED, although these polarisation CAN contribute to scattering as virtual particles,
Equations (2) & (3) result in the disappearing of the earlier mentioned "non-traceless" and "non-transverse" gravitons from the observables, although it doesn't rule out their contributions to scattering amplitudes.
However equations (1) seem not to be so forgiving towards any "time warping gravitons". The conditions simply leads to the result that the creation operators associated with these "time-warping" gravitons destroy any state, preventing them from contributing to any process, even as virtual particles.