Why is there a “problem of time” in Quantum Gravity?

The problem arises due to a property of general relativity called reparameterisation invariance. In fact it is not unique to GR but arises in any system that has this property, but in most systems we can use a process called deparameterisation to recover the time dependence. What is different about quantum gravity is that for technical reasons deparameterisation does not work, and it is still an unresolved question how the time dependence can be recovered.

For the details see the Wikipedia article on the Hamiltonian constraint. This is somewhat mathematical and I'm not going to reproduce all the details here. Instead I'll try to explain in simple terms why it happens.

In GR we use a quantity called the proper time to describe the evolution of the system. For example the four-velocity and four-acceleration are the first and second derivatives of the position four-vector $(t,x,y,z)$ with respect to the proper time, and the geodesic equation is written using the proper time. The proper time has a physical meaning, but it turns out that we can write the geodesic equation replacing the proper time by any affine parameter and it still gives the correct results. Indeed this has to be done for light rays because their proper time is always zero. This works because what we are observing are the coordinates $(t,x,y,z)$ and not the parameter (proper time or otherwise) being used to calculate them.

The fact we can replace the proper time by any affine parameter means the theory is reparameterisation invariant, and as discussed in the Wikipedia article I linked above, when we use Hamiltonian mechanics to describe such a system we find that the value of the Hamiltonian is always zero. This is a problem because the Hamiltonian describes the time evolution of the system and if it is zero then the system does not evolve with time.

Now this happens in purely classical mechanics, and an example is discussed in the Wikipedia article. Such systems obviously do evolve with time, and the reason we run into the problem is down to a choice of the time variable that is unphysical. Recovering the actual physically meaningful time is done using a straightforward process called deparameterisation, and this gives us the equations of motion of the system.

But here we run into the problem with quantum gravity. In this case the deparameterisation process doesn't work (for technical reasons that are involved), so we cannot recover a time variable and we cannot describe the time evolution of the system. We are left with equations describing a system that cannot change with time. This is the notorious problem of time. Much effort has been expended to try and deparameterise and find a suitable time variable, but so far with no success.

The problem of time arises in the attempt to unifiy quantum mechanics and generally relativity.

In quantum mechanics, time is universal or absolute. However in general (and special) relativity, time is relative and considered to be a dimension that’s interwoven with our three spatial dimensions, into a four-dimensional spacetime.

Unifying quantum mechanics and general relativity will require reconciling theses absolute and relative notions of time. Hence the phrase “the problem of time”.

It is true that special relativity and quantum mechanics are unified within the constructs of quantum field theory, but quantum field theory still does not encapsulate the gravitational interaction. The electromagnetic force, the strong nuclear force and the weak nuclear force are unified in the standard model of particle physics which is explained by the mathematical formulation of quantum field theory. The gravitational force is not included in this unification.

A theory for gravity and quantum mechanics needs to encompass a consistent description of time.

In quantum mechanics and in special relativity we deal with a reference frame in which time is determined from a single master clock. In general relativity we make comparison between local reference frames in which time is determined from different clocks. In quantum mechanics we need to compare an initial (prepared) quantum state with a final (measured) quantum state. In general relativity these states may be defined in different local reference frames, meaning that they must be described in different Hilbert spaces - just as vectors are defined locally in a vector space defined at each point.

A connection is used to compare vectors defined at different points by parallel transport. But parallel transport does not apply to states in Hilbert space. We need a corresponding idea to a connection, but applied to Hilbert space rather than vector space.

Eduard Prugovecki (1994, Class. Quantum Grav. 11 1981; 1995, Principles of Quantum General Relativity, Singapore: World Scientific; 1996, Class.Quant.Grav. 13 1007-1022) considered quantum propagation in curved spacetime represented by a globally hyperbolic Lorentzian manifold $(M, g)$, using parallel transport governed by a quantum connection obtained by extending the Levi-Civita connection from the Lorentz frame bundle to the Poincaré frame bundle over $(M, g)$. He found that this quantum connection gives rise to infinitesimal parallel transport coinciding with special relativistic quantum evolution.

I redeveloped essentially the same idea in a much simpler form in Mathematical Implications of Relationism, and called it a teleconnection between the Hilbert spaces for the initial and final states. It gives a completely consistent formulation of quantum mechanics in a general relativistic context.