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

It’s often said that the problem of time exists because time is treated as absolute in quantum mechanics but not so in General Relativity, see e.g. A list of inconveniences between quantum mechanics and (general) relativity?. However Quantum Field Theory merges QM with Special Relativity, which treats time as relative (time dilation, etc). What actually leads to the problem of time in Quantum Gravity?

• One answer to the question you link explicitly mentions a paper giving a survey of the actual problems one might see. Can you be more explicit what you want as an answer here that is not covered by that? Commented Jan 17, 2021 at 11:56
• @ACuriousMind I didn't link that paper, it was edited in by a moderator. I'll go read it though. Commented Jan 17, 2021 at 12:00
• It's not quantum theory that treats time specially, but the Schrödinger picture (SP) that does. The Heisenberg picture (HP) doesn't treat time specially. When applied to field theory, the Heisenberg equations are partial differential equations with all four coordinates treated the same and the HP state is timeless (in contrast to the SP state). So, the conflict is not Quantum Theory versus General Relativity, but Quantum Theory (SP) versus Quantum Theory (HP). The Problem of Time, arising from quantizing the Hamiltonian formulation for General Relativity, is that even the SP state is timeless. Commented Jan 1 at 0:35

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.

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.

• Why in special relativity the time should be determined from a single master clock ? Actually, there is no reference system in SRT which is preferential with respect to the other possible reference systems (and if I make a choice on a system, I could do the same in GRT, couldn't I)? Commented Jan 17, 2021 at 15:57
• @FredericThomas, that is how we define time in special relativity, using a master clock and a synchronisation procedure. We cannot do the same in general relativity as we can only define local inertial reference frames. Commented Jan 17, 2021 at 16:04
• you mean, in GRT there is no synchronisation procedure ? Commented Jan 17, 2021 at 16:25
• @FredericThomas, it is perhaps more accurate to say that there are many, and arbitrary, "synchronisation" procedures, but in general clocks do not remain synchronised even when stationary with respect to each other. Commented Jan 17, 2021 at 17:13
• It gives a completely consistent formulation of quantum mechanics in a general relativistic context - do you claim to have unified QM and GR? Commented Jan 18, 2021 at 3:29

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 QFT doesn't time just refer to the time of the observer? The invariant quantity is the Hamiltonian (which is invariant after a Lorentz boost) this notion of time exists in GR as well for inertial frames. I do not understand " time is universal or absolute"? Commented Jan 17, 2021 at 10:17
• Classical quantum mechanics (without relativity as in qft) treats time as not being relative. Ie, it treats time as being absolute and not frame dependent. Commented Jan 17, 2021 at 10:26
• Agreed but the OP has asked: "However Quantum Field Theory merges QM with Special Relativity, which treats time as relative (time dilation, etc)" Commented Jan 17, 2021 at 10:29
• To which you say "does not encapsulate the gravitational interaction" but the problem of time is not that Standard Model does not include gravitational potential (that's the problem of merging some version of Quantum Gravity with the standard model). Commented Jan 17, 2021 at 10:38
• That’s correct. But the problem arises when you try to combine QM with general (not special) relativity. The OP thought the problem of time arises when you combine QM with special relativity, which is clearly not the case. Commented Jan 17, 2021 at 10:53