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I was reading a Wikipedia article about the Problem of time, which states:
quantum mechanics regards the flow of time as universal and absolute, whereas general relativity regards the flow of time as malleable and relative.
My question is, do we have any evidence that flow of time is not relative in quantum scale? and if so, at what scale flow of time "changes" from absolute to relativistic?
"Do we have any evidence that flow of time is not relative in quantum scale?" No, we do not. In fact, quite the reverse; Wikipedia is not always reliable!
Firstly, the basic quantum equations simply assume that time is absolute (Newtonian) and not relative, they make no claim that this accurately represents reality. This is what the Wikipedia statement means.
But it is not really true anyway. Relativistic quantum theory was originated by Paul Dirac in the 1930s and has come a long way since. And we have plenty of evidence that relativistic speeds do indeed affect the flow of time at the quantum scale. For example unstable (radioactive) particles can be made to live longer by accelerating them to near-lightspeed.
The more you study Quantum Mechanics, Quantum Field Theory and the mathematical procedure of quantization, the more you understand that "quantumness" is orthogonal to relativity.
A simple way of seeing this intuition is as follows. Canonical Quantization is a procedure that applies to the symplectic geometry of the phase space. But what is phase space, really? The usual definition is, it's the space parametrized by the values of coordinates and momenta chosen at an arbitrary instance of time $t$. It looks like this makes a reference to "time" and so the phase space and quantization should ultimately depend on this single instance of time, which naively looks to be contradictory to the spirit of relativity.
However, there's a different, much more profound way of defining the phase space that happens to coincide with the definition above for almost all physical systems. The phase space is, no more and no less than, the space of solutions to the equations of motion. There's no reference to an instant of time in this definition, or in fact even a reference to time itself.
For a system described by an equation of motion that's second order in some parameter $t$ that we can call time, these two definitions coincide: you have two degrees of freedom per point for a second-order equation, these are your coordinate and momentum.
However you can see where I'm going with this. Quantization is a procedure that deforms geometry on the phase space, which can be defined in a time-independent way. You could conjecture that quantization itself can be carried out in a time-independent way and that relativistic systems can be quantized. That conjecture would be absolutely correct.
There are a plethora of examples. Quantum Field Theories are fully compatible with Special Relativity. Topological Quantum Field Theories have the same extended group of symmetries (diffeomorphisms) as General Relativity does, serving as toy examples proving that conceptually quantization can be applied to General Relativistic theories, if it weren't for technical, mathematical problems.
I think the view that time is a dimension stretching from zero to infinity with an interconnected network of static worldlines stretched out in it is quite problematic. There indeed is no motion involved but we all clearly see motion without the past or future being present (the past is present in our minds but the actual past is gone). The view poses the problem then how we can move in time if everything is laid down in advance.
You can solve this by claiming that particles follow the rigid worldlines deterministically or following the deterministic chance laws of quantum mechanics. This makes the worldlines rather fuzzy. How is time defined here? Just as everywhere. Like the ideal clock ticking on the time coordinate of spacetime. I'm not sure why time should be considered differently from time in classical theories except from the fact that it enters in the energy-time uncertainty relation. Which means the better you can determine the clock time, the less you can determine energy since you need time to do that and the more time you have, the better you can determine a particle's velocity, hence it's energy.
So where does that leave us? Is future time already present on the evolving spacetime manifold of the universe? Are imaginary clocks ticking at all points in space, be it in the past, present, or future? You can assume this obviously, leading to the (partial) block universe. But what does this tell us about the nature of time? Very little.
You can envision time as the unfolding irreversible process constituting the universe. With imaginary ideal (perfectly periodic) clocks placed in it to compare the process and sub-processes with. You can even imagine (and I take a rather big leap here) macroscopic unidirectional irreversible thermodynamic time emerging (together with space) from a virtual circular perfectly periodic time in the beginning of the universe. The real mystery remains why it's going from the past to the future. In principle it could have started at the end but in reversed motion. An irreversible thermodynamic process can't be reversed, obviously (to reverse the whole universe we have to stand outside of it because it's impossible to reverse oursélves). But there is no law forbidding it to take place.
There is still a lot to learn.
