Why should the canonical quantisations of general relativity (e.g. ADM, LQG, etc) be, in principle, expected to give a correct quantum theory?

Question

In the canonical quantisation approach to quantum gravity, one intends to formulate GR in the Hamiltonian formulation and quantise it the usual way. This can be the quantisation of the ADM formalism or some modification of it (like in LQG).

However, there seems to be a question mark on this procedure, which I haven't seen addressed. In the quantisation of ordinary field theories (non dynamical spacetime), one chooses a Cauchy hypersurface and imposes the commutation relations:

$$[\phi (x), n_{\mu} (y) \frac {\partial L}{\partial (\partial _{\mu} \phi ) }(y)]=i\hbar \delta ^3 (x-y)\tag{1}$$

where $n_{\mu}$ is future directed normal to the hypersurface with unit length. This specific quantisation recipe is known to be experimentally correct. Also, the theories quantised using this recipe are formally independent of the choice of Cauchy hypersurface (c.f. this and this)

In principle, one could choose to impose the CCR without the requirement that $n_{\mu}$ is of unit length. The length of $n^{\mu} (x)$ could be arbitrary and $x$ dependent. This would correspond to choosing an arbitrary Cauchy foliation of spacetime, writing the Hamiltonian formulation and imposing the CCR. The time-parameter $t$ of the arbitrary foliation $\Sigma _t$ is not required to be physical time given by the metric, but it can be any time-like parameter. This quantisation would mathematically be fine, but experimentally incorrect, because it would, in general, lead to commutation relations that disagree with $(1)$.

But e.g. when we talk about quantising the ADM formulation, the above is exactly what we are doing. Since the metric is the dynamical field $\phi$, we don't have the choice to impose the CCR according to $(1)$, because we can't define $n^{\mu}$ without a background metric. Instead, what we intend to in the canonical quantisation of GR (e.g. ADM, LQG) is to choose an arbitrary Cauchy foliation $\Sigma _t$, switch to the Hamiltonian formulation, and impose the CCR $$[q_{ab} (x), \pi ^{ab}(y)] = i\hbar \delta (x-y)$$ and the constraints.

Why is this recipe expected to give correct results for quantisation of GR, while it is expected to give incorrect quantisations for ordinary field theories in non dynamical spacetime?

Answer 1

Canonical quantization gives different results not only under different time parametrizations, it gives different results even under non-trivial phase-space transformations. In particular, if your phase-space transformation mixes momenta and coordinates, you get a different quantization because of the ambiguity of the $p,x$ ordering in the transformation. For normal matter fields, a switch in the foliation induces a linear transform that mixes fields and field momenta, so this is ok. However, for the GR field variables these transformations are nonlinear from the outset, which also adds operator-ordering ambiguities to the mix.

Additionally, even in the case of matter fields one does not need to go so far as to use a foliation that has a non-unit $n^\mu$ to get inequivalent experimental predictions. For example, the foliation could correspond to the proper time of oscillating observers (that is, experiencing periodic acceleration). In that case $n^\mu = u^\mu_{\rm osc}$ and is normalized. Yet I can guarantee you that the quantization would not "naturally" lead to the same vacua as you obtain from the canonical quantization in the frames of inertial observers. In fact, the Unruh effect is a prediction that comes from transforming the inertial-observer vacuum into the "quantum frame" of uniformly accelerated observers.

So, quantization can be a finicky procedure. However, the hope (that has never materialized) was that the story with ADM quantization would be similar to quantization of non-gravitational fields in flat space-time. There the quantization superficially depends on the choice of the family of inertial observers, yet one obtains a well-behaved representation of the full Poincaré group. In other words, you can do the following:

A) apply a classical Poincaré transform to the observers and fields and canonically quantize, or

B) canonically quantize and then apply the quantum representation of the same Poincaré transform as in A)

It turns out that the procedures A) and B) always give the same result, so canonical quantization is Poincaré-covariant. The hope of ADM quantization was that it would do the same, just with the group of symmetries of General relativity - the full, infinite-dimensional diffeomorphism group instead of the Poincaré group. In other words, the quantization and diffeomorphisms would in some sense commute or, the canonical quantization would turn out to be diffeomorphism-covariant. However, this was never demonstrated to be the case, at least in a sufficiently general non-perturbative sense.