Imagine, a deep empty universe consisting of only one particle muon. Will it decay?
As there isn't any change in its surroundings and thus time will lose its meaning. But if the muon will decay in that situation, doesn't it prove that time is something deeper than our experience or what a clock shows!
This question seems to be based on a perception of a muon as a sort of little entity sitting in otherwise empty space. But that is not the universe as it is described by physics. The universe is full everywhere of all the various fields of the Standard Model (or of some more sophisticated model which subsumes the Standard Model). Once you have that universe then you have decay of muons. But if you are thinking of some other set of physical ideas then you are not talking about muons. You are using the same word to refer to some other thing. In that case what it will do will depend on what it is and what behaviours are open to it. But since this has not been described, no answer is possible.
If the question is merely pointing out that the concept of development over time requires that there be the possibility of change, then of course that is true. It has been pointed out since antiquity. Perhaps it was first noticed and pondered in the paleolithic period; I don't know.
Physics is fundamentally an experimental science. The laws of physics are based on experiments, all of which are conducted in this universe. We have no access to sterile universes like the one described in the question in order to test theories about such universes, and we have no indication that the results of experiments in such universes would be informative regarding the physics of our universe.
All we can say is that according to the laws of physics in our universe isolated muons decay at an intrinsic proper rate that appears to be unaffected by any external consideration.
We happen to live in a universe where today we know space is throughout permeated by quantum fields, which includes the respective field of the muon, and its decay products, the electron and neutrinos. You can think of the muon as the excitation of its respective field, a manifestation of energy stored in the quantum field. This form of energy can transform into other forms of energy if the laws of physics allows. Since to our knowledge the laws allow for the muon to decay, it will do so at a certain probability, and will transform into its decay products.
Muon decay almost always produces at least three particles, which must include an electron of the same charge as the muon and two types of neutrinos.
https://en.wikipedia.org/wiki/Muon
Since vacuum is not empty, and quantum fields can couple to each other, the muon can transform into other forms of excitations in other fields.
We think about particle decay in terms of couplings of quantum fields to each other: an excitation in one field can decay into excitations in others.
Why are muons considered to be elementary particles in the Standard Model?
You say time will lose its meaning in an otherwise empty universe, but I do not think time will lose its meaning, since the muon itself is a quantum mechanical clock if you will.
Since the quantum fields permeate all of space, including the deepest intergalactic voids of space, this decay will happen everywhere in the universe, even in an empty region (and I assume your question is about this), where no other forms of clusters of matter exist, even as your question states, not even a single other particle (excitation). Bottom line, the void of empty space is not empty, it is permeated by all the quantum fields, and since to our knowledge the laws of physics are the same everywhere in the known universe, this together with the probabilistic nature of QM is enough for the decay to happen.
As already said if there is no other quantum field in your universe then the muon will not decay. You have to keep in mind that a quantum field fills the entire universe and that the muon is nothing but an excitation of the muon/anti-muon field. Now if there are the other known quantum fields in your universe, then the muon will decay, even if the other fields don't have any excitation. Now theoretically time is two things:
If there is one field with one excitation in the universe then we have two answers. First, we can say that time flows on it but the muon will not decay because there is no coupling with any other field, this is the relativistic point of view. Secondly one can reasonably say that since there is only one thing in the universe, and because it does not seem to depend on time, then time is a useless parameter and then drop it from the equations.
Now you can describe a decay without "time". In fact, you can organize all the possible initial and final states of the fields in whatever order you like and with any type of labels, you will be able to have the same result as with "time": saying that a state is an initial or final state is a statement for the physicist because mathematically all that matters is a $|\Psi\rangle$ and a $\langle \Psi'|$. We organize them from past to future because this is what we experience every day, but in the end, all that matters is transitions from one state to another.
Why wouldn't it decay normally? If you assert that an external clock is needed, you can imagine two muons. One can be the clock for the other. Will they decay?
If so, when? What is the clock for the first of them to decay?
If not, then what about a lot of muons? Is our universe as a whole any different? What is the external clock for our universe?
The natural clock for muon decay is the timescale determined by our fundamental constants.
There isn't such a thing as "otherwise empty space".
If the quantum fields that permeate spacetime, permit only muons, and no decay to anything else, then only muons can exist.
If they permit anything else, eventually some muons will decay (tunnel, transform...) Into some of those other things.
Or at least, that's our best current understanding........
Does this other universe consist of other quantum fields (e.g. electrons, neutrinos, W bosons)? I don’t think this is really the intent of your question, but if those fields do exist and there is something for the muon to decay to (and something to decay through), then there isn’t a reason to think it wouldn’t decay.
This is similar to the “If a tree falls in a forest ...” question, and, just like that question, it is a philosophical question and not a scientific one. That is not a value judgement - there are many interesting and important questions that simply are not amenable to scientific analysis.
If the universe is truly empty apart from one muon then we can never know whether it will decay or not because there can be nothing else in the universe to observe the muon. Not only is the question beyond our current state of knowledge, it is fundamentally unanswerable in a scientific sense.
The muon may decay, or it may not decay, or it may exist in a superposition of undecayed and decayed states, or it may turn into a small pink unicorn. There is absolutely no way of knowing.
This question can be generalized to whether or not anything can "happen" in empty Minkowski space where there are no external indications of time progressing. If you were to look at the muon's world line in an empty universe or a non-empty universe, the result at a certain point along that line will always be the same (assuming it doesn't interact with anything before that point). It doesn't matter if you're looking at the time dimension or one of the three spacial dimensions. This is true for anything, not just muons.
A muon is a kind of clock in relativity theory. But in an empty universe there would be no other clock to compare it to and no relativity. . Without relativity the universe as we know it is not defined and cannot exist.The answer to your question is therefore no
