Quantum gravity as a metaphysical problem

Given that we have an ether theory of gravity, that means, a theory of gravity based on condensed matter theory in a classical Newtonian framework, with the ether fulfilling classical equations of condensed matter theory like the continuity and Euler equations, the problem of quantization of gravity looks quite differently: We know how to quantize condensed matter theories. All we have to do is to follow this scheme.

In detail, once we have a continuity equation, we can construct an "atomic ether" model with the number of ether atoms being conserved. The details of such a construction are, of course, a separate issue, given that all the matter fields, that means, all the fields of the Standard Model of particle physics and possibly some extension of it have to be described by this "atomic ether". This is what is considered in my cell lattice model.

But, of course, one can consider the question what happens with all the many known problems of quantization of GR which make "quantum gravity" so hard that it remained essentially unsolved up to now. It appears that all these problems are, essentially, metaphysical problems. It is easy to construct a quantum theory of gravity using only well-known standard techniques of standard quantum field theory. Namely:

Instead of trying, then, to remove the regularization to get a well-defined field theory, we simply stop here. The regularized lattice theory is already a well-defined theory of quantum gravity. It does not look beautiful, is in conflict with some beautiful principles of relativistic symmetry? Indeed. But, sorry, this is a metaphysical problem.

In a little bit more detail, this has been described in this paper.

The particular quantum gravity problems and how they disappear

Topological foam

GR allows solutions with non-trivial topology. In the classical domain, this is not a big problem, because the equations restrict the possibility of a change of topology in a quite strong way. But in the quantum domain, in particular if we consider vacuum fluctuations, the classical equations no longer restrict modifications of the topology. Thus, at small distances below Planck length. one would have to consider also all possible fluctuations of the topology. How to do such things is completely unclear.

This problem disappears in the Lorentz ether in a quite trivial way - the topology is fixed from the start and trivial, the vacuum fluctuations cannot change the topology too.

The problem with the stress-energy-momentum density of the gravitational field

That there is no well-defined physical definition of a stress-energy-momentum density of the gravitational field is already a problem of classical GR. There are only a lot of different proposals for stress-energy-momentum pseudo-tensors. While they allow to define conservation laws, they are not covariant, but depend on the choice of coordinates. Thus, following the standard interpretation, cannot be used to define physical objects, except one violates one of the central principles of the spacetime interpretation, the equivalence principle.

In classical GR, this problem is not really important, last but not least, there is no way to measure the energy of the gravitational field, thus, no problem appears if there is no preferred stress-energy-momentum tensor.

The problem becomes already more serious in semiclassical theory. Here, the matter fields are handled as quantum fields. But even in this case, we really need only the stress-energy-momentum tensor of the matter fields, thus, the failure to define one for the gravitational field is not a decisive problem too.

But in quantum GR, this problem becomes a serious one.

In the Lorentz ether, we have a well-defined stress-energy-momentum density of the gravitational field. Thus, the problem does not even exist.

The key which makes the difference here is general covariance. The energy and momentum conservation laws are a consequence of Noether's theorem applied to translational symmetry. The Lagrangian does not change if one adds constants to the preferred coordinates. In a covariant formalism, the Lagrangian has a symmetry \(\mathfrak{x}^\mu(x) \to \mathfrak{x}^\mu(x) + c^\mu\). As a consequence, the Euler-Lagrange equations for the preferred coordinates \(\mathfrak{x}^\mu(x)\) become conservation laws.

But if we have not only translational symmetry but general covariance, thus, no dependence on the preferred coordinates at all, we have also no Euler-Langrange equation for the preferred coordinates, and the Noether conservation laws simply disappear.

Thus, the problem is a consequence of the background independence of GR, which does not exist in GLET.

No non-degenerated Hamilton formalism

There is no Hamilton formalism for GR. There are constructions which have something in common with a Hamilton formalism, namely the ADM formalism. The ADM formalism is not a Hamilton formalism in the usual sense, but a generalization of the Hamilton formalism which contains constraints. Such constraints are necessary because of the symmetries of the Lagrange formalism.

The equations contain, together with equations which have the usual form known from the Hamilton formalism, also constraint equations, in particular the diffeomorphism contraint (which follows from the diffeomorphism symmetry for spatial diffeomorphisms which leave the foliation (the time coordinate) unchanged, and the Hamilton constraint, which corresponds to diffeomorphism symmetries which change the time coordinate.

In GLET, the problem does not appear, simply because its origin - the diffeomorphism symmetry of GR - does not exist. The Hamilton formalism of GLET can be constructed from the Lagrange formalism in the same straightforward way as in every field theory. The problems with the Hamilton formalism in GR are side effects of relativistic symmetry, which is fundamental in GR but explicitly broken in GLET.

The problem of time

These constraints lead to serious problems for quantization. While the problems with the diffeomorphism constraint have been claimed to be solved (I doubt they have been really solved, see below), the problems with the Hamilton constraint seem to be much more serious, if not unsolvable.

This problem is named "problem of time".

We can omit here the technical details of this problem, and simply believe those who have tried to solve it that it is a really difficult problem. That this problem simply does not exist in GLET follows directly from the fact that its origin - the Hamilton constraint - does not exist in the Hamilton formalism of GLET, which does not have such constraints because it does not have the symmetry which created the necessity to introduce them into the Hamilton formalism.

No regularizations with relativistic symmetry leads to necessity of renormalization

We simply use an arbitrary lattice regularization of the field theory. But note that such a lattice regularization replaces the space \(\mathbb{R}^3\) by a lattice \(\mathbb{Z}^3\) (or, to get a finite number of degrees of freedom, \(\mathbb{Z}_N^3\) of a big cube with periodic boundary conditions). Time is left continuous. So, this construction from the start destroys any relativistic symmetry. This makes the regularization unacceptable for those who want to have a quantum gravity theory with relativistic symmetry.

So, either one can find a regularization which has, somehow, relativistic symmetry, or one can use regularizations only as an intermediate tool, useful to construct in some way a field theory which, in the limit, has again relativistic symmetry.

This program has in some way or another to solve the problem that GR is non-renormalizable. So, the limit fails to define a reasonable theory.

As an approach which uses a lattice regularization which has, in some way, a relativistic symmetry, is Loop Quantum Gravity.

The quantum hole problem

A covariant equation defines the solution only modulo an arbitrary coordinate transformation. As a consequence, the solution of a covariant equation is not defined completely by the initial and boundary conditions, because these conditions do not fix the coordinates inside, in a hole in spacetime which does not contain the boundary or the hypersurface where the initial values are defined.

This observation was known to Einstein already before he finished GR, and motivated him to look for equations which are not covariant (Einstein named this "Lochbetrachtung"). But further considerations showed him that this is not dangerous - all what physically matters, say, intersections of light rays with trajectories, does not depend on the coordinates used inside the hole. Thus, all what can be empirically measured can be predicted by covariant equations too.

But while everything is fine in classical GR, the situation looks different in quantum GR. In classical GR, we always have only one classical metric, and all we can measure is defined by this metric. But in quantum gravity, we have to consider superpositional states of different metrics.

In such a situation, we can generalize classical covariance in two different ways: Quantum gravity may have a covariance if the same coordinate transformation is applied to all metrics in the superposition. Or it may be covariant in a much stronger sense, so that we can apply different coordinate transformations to different metrics in the same superpositional state. Which is the correct generalization?

In my paper "The background as a quantum observable: Einstein's hole argument in a quasiclassical context" I argue that the result of a quite simple quasiclassical thought experiment can be defined only if the relative position of events defined on different metrics is well-defined. That means, covariance is possible only in the weak sense, when only the same coordinate transformation can be applied to all metrics in the superpositional state. Then, for every event on one metric we have a well-define "same event" on every other metric in that superpositional state.

The information about which is the same event for another metric becomes, therefore, observable in principle by such experiments.

But this information is not provided by GR. If it would exist, it would allow us to define the preferred inertial coordinates of the vacuum solution (the Minkowski space) on all solutions of GR.

How a theory of quantum gravity can work without such preferred coordinates but nonetheless predict the results of the thought experiment remains unclear.