Proton Decay: The Mystery of Ergodicity

[Carlos L. Janer]

Why does everyone assume that particles decay is an ergodic random process? After all T is not a symmetry of the Standard Model and I don't see any reason why ensamble averages should be equal to time averages.

 

[Carlos L. Janer]

I'm going to elaborate on my question. If protons are indeed unstable their decay rate is extremely small. Experiments on proton decay are designed to find out if this decay rate is either exactly 0 or an extremely small number. For these experiments to make any sense, ergodicity should be hold exactly and I think it doesn't since T is nota an exact symmetry of the Standard Model.

 

[Orodruin]

T is time reversal. For ergodicity to hold you need invariance under time translations. This is indeed a symmetry of the SM and any QFT where the Lagrangian is not explicitly time-dependent.

 

[Carlos L. Janer]

So ergodicity just means energy conservation?

 

[mfb]

Energy conservation is a consequence of ergodicity.

Both have nothing to do with (broken) T reversal symmetry.

 

[Carlos L. Janer]

Even though I feel intimidated by your status, it's hard for me to understand why energy conservation means that a random process is ergodic. I mean what I say and I'm sorry if I'm wasting your time.

 

[Carlos L. Janer]

My previous comment was referred to Orodruin's answer.

 

[Carlos L. Janer]

Has anybody seen a mathematical proof of ergodicity in any QFT book or is it just and ad hoc hypothesis? I would appreciate it if anyone could give me an answer.

 

[mfb]

You cannot prove laws of physics. It is not completely impossible to imagine a world where physical constants change over time (although one has to be very careful concerning the question which time).

It does not really matter for the proton lifetime, because both experiments and theories make statements about the current decay rate, and convert this to lifetimes for convenience. If protons are unstable and their decay rate will look different in 10³⁰ years: very interesting, but not that relevant for current measurements.

 

[Carlos L. Janer]

Correct me, please, if I'm wrong: You're actually saying that ergodicity IS a law of fundamental physics.

 

[mfb]

All our observations so far are in agreement with it (that's the best you can get in physics). We don't know if it is exact, but it has to be very close. Even if it is not exact, it does not matter for proton decay searches.

 

[Carlos L. Janer]

OK I give up trying to understand it. Just because you can't wait 10^30 years to see if a proton is stable or not, you assume that this stochastic process is ergodic. You assume that looking at large ammounts of protons for a short period of time is exactely the same thing as waiting for a single proton to decay for extremely long times. You assume this because that's what you measure in very unstable particles: both results approximately agree. (By the way, ergodicity is a theorem that you can prove in statistical mechanics under rather strong assumptions that are not true in high energy physics). If you tell me that this is an ad hoc fundamental law of the Standard Model I can live with that. But I'd like to know what the ad hoc hypotheses of the Standard Model are. Ergodicity seems to be one of them and, although it's not self evident to me and I don't like tacit hypothese, I guess I'll have to live with it.

 

[mfb]

you physicists ASSUME that this stochastic process is ergodic

Not in places where it would matter.

The measurement result is something along the line of "we set an upper limit of 1 decay in 10³⁰ protons in one year (in 2015)". This can be compared to various supersymmetry, GUT, and whatever approaches that predict 1 out of 10²⁹ (ruled out), 10³¹ (open) or any other number per year.

For convenience, those numbers are quoted as lifetimes, assuming strictly exponential decay. But that is just a fixed conversion that does not impact the actual physics impact behind it - the comparison between theory and experiment.

All our physical laws are assumptions. The good ones are backed by many precise experiments. The existence of something called electrons is just an assumption. But it is in excellent agreement with experiments, and there is no alternative theory without electrons that would agree with experiments.

The neutrino masses are small compared to other particle masses, but large enough to have their mass differences measured - if we can measure something it is clearly not negligible. If the laws of physics change over time then this change has to be so small that we cannot measure it today - which means for the current experimental precision it is negligible.

 

[Carlos L. Janer]

OK, now I get it. You define lifetime on an ensable average because that's all you can do and measure (and all you can hope for in QMs). Checking ergodicity is out of the question in this case. Thank you very much indeed for your patience!