Relativity, time, and quantum mechanics

[anuttarasammyak]

Cosmic-ray muons correspond to wave packets in free space, and their lifetime extension can be explained by the Lorentz factor. Muons in a circular accelerator are wave packets governed by a Hamiltonian that includes the electromagnetic potential, and strictly speaking they should not have a definite velocity; however, in practice, a Lorentz factor based on velocity is often applied successfully.

 

[PeterDonis]

Different momentum components have slightly different velocities

On what basis are you making this claim? The "different momentum components" are not observables; they are just a convenient basis in the Hilbert space. So I don't see how they correspond to velocities; there are no velocity operators corresponding to them.

 

[anuttarasammyak]

So even in free space QM tells us nothing about velocity ?

 

[Nugatory]

So even in free space QM tells us nothing about velocity ?

If we take the math strictly at face value, which means ruthlessly resisting the temptation to indulge even very plausible interpretational claims: $p$ is an observable so $p/m$ is also an observable that we might reasonably call velocity. And if we're considering non-relativistic QM (not applicable to the questions that started this thread) $x$ is an observable with expectation value $<x>$ so we might reasonably call $\frac{d<x>(t)}{dt}$ the expectation value of the velocity.

In practice it's easier to treat a free particle classically unless and until there is some interaction with some non-zero potential in which quantum effects are relevant.

 

[anuttarasammyak]

x is an observable with expectation value <x> so we might reasonably call d<x>(t)dt the expectation value of the velocity.

Thank you. Re:#30 Can we call $\frac{<p>}{m}$, relativistically $<\frac{pc^2}{E}>,$ as well as $\frac{d}{dt}<x>(t)$ the expectation value of the velocity which can be used for getting Lorentz factor ?

 

[anuttarasammyak]

I would greatly appreciate it if you experts could provide a rough sketch of how quantum field theory explains this phenomenon. As a layperson, I find it difficult to grasp the physical meaning behind terms such as annihilation and creation operators, probability amplitudes, and so on, rather than just the mathematical formalism.

I have read that in relativistic quantum theory the decay rate is computed from the weak-interaction matrix element using the bound-state wavefunction:

$$\Gamma_{bound}=<\psi_{1s}|\hat{\Gamma}|\psi_{1s}>$$
The momentum distribution of the bound muon plays an important role.
The decay width is a Lorentz-invariant quantity derived within quantum field theory.

I understand only part of this, but I have the impression that an explanation based on the Lorentz factor is not necessary in this context. Looking back at #28, it seems that p plays a role in Γ in QFT, while v plays a role in τ in classical relativity; they appear to offer complementary perspectives though the latter is not correct now I know.

 

[Nugatory]

Thank you. Re:#30 Can we call $\frac{<p>}{m}$, relativistically $<\frac{pc^2}{E}>,$ as well as $\frac{d}{dt}<x>(t)$ the expectation value of the velocity which can be used for getting Lorentz factor ?

Of course not. You’re trying to apply non-relativistic quantum mechanics to a problem in which relativistic effects matter.

 

[anuttarasammyak]

Am I correct in understanding that both ⟨v⟩ = ⟨p⟩/m and ⟨v⟩ = d⟨x⟩/dt are valid in the non-relativistic regime for a free particle?　
If yes, I thought $<v>=<\frac{pc}{\sqrt{p^2c^2+m^2c^4}}>$ is an improved rlativistic version of $<v>=\frac{p}{m}$.

 

[renormalize]

This is a follow-up to my post #27 (which so far has not received any specific comments), in which I drew from the 1982 Silverman reference on muonic atoms to promote the idea that bound quantum particles can be usefully characterized by both velocity and time dilation. First, I quote some thread-responses that push back against that notion:

...the electron has no path or position so the notion “the proper time of an electron” is meaningless.

The “rest frame of the electron” never enters into the calculation (unsurprisingly, because it’s not defined).

No position means no path along which we can integrate to find the proper time. Instead we use QFT and calculate an amplitude for a decay event at various points in spacetime - "experiences time dilation" is not how I would describe that model.

You have a calculated a quantity that has the dimensions of distance over time, and you can interpret it as a speed (it's an interpretation because it is not observable even in principle and because there's no position that is changing with time) if that interpretation is helpful.
But it's a stretch to get from there to the proposition that a bound particle "experiences time dilation" (what could that phrase possibly mean?

But the state of an electron (or muon, for that matter) in a bound hydrogen atom is not classical even in approximation. So you can't just assume that the expectation value of $\mathbf{V}$, which is what appears in the quantum virial theorem, has a valid physical interpretation as the "speed" of the electron (or muon).

In response to these comments, I present a 2000 reference by Czarnecki et al. that models the decay of the bound-state of $\mu^+$ and $e^-$: Muonium Decay. (Compared to a muonic atom, muonium is a "cleaner" system to analyze since it's composed entirely of point leptons that interact via the Coulomb potential only. Indeed, observations of muonium are used to both set the parameters and test the limits of the Standard Model.) Czarnecki et al. summarize the intent of their work:

(Here I underline content that's germane to the debate in this thread.) The authors employ both Bethe-Salpeter and non-relativistic QED in their analysis, concentrating on the ground-state configuration:

and later write:

The authors go on to compare their results to those for muonic atoms:

This and the previous reference I cited provide ample evidence that at least some working physicists embrace the concept that velocity and time-dilation do indeed apply to quantum particles bound in stationary states, and so in my view, the debate is settled. Nonetheless, I would welcome literature references that support the contrary position of @Nugatory and @PeterDonis.