The Muon Lifetime Experiment and Neo-Lorentzian Interpretations of Special Relativity

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There’s actually no more difficulty in giving a neo-Lorentzian (as opposed to Minkowskian) interpretation to the time dilation of muon lifetime than to special relativity in general. Which is to say, there’s no difficulty in doing so as long as you see no problem with an empirically undetectable privileged reference frame. (And I would argue that the is, in fact, no problem with such a thing - it is both a philosophically natural postulate and even a scientifically natural one, in light of the empirically well-confirmed violations of the Bell inequalities).

To give a rough explanation of what I mean: as far as I understand it, the decay rate is something that we would theoretically determine by solving the Schrodinger equation for the quantum state. You’d start with a quantum state representing a muon, and over time it would evolve into a superposition of the muon state and various decay product states; the decay rate would be rate at which the amplitude of the muon part of the superposition diminishes. (There’s probably other ways of calculating the decay rate, but they have to give the same answer.)

The relevant fact here is that the answer you get from this calculation depends on the initial quantum state, and the quantum state for a muon at rest is different from the quantum state for a muon in motion. So, in fact, the physics predicts that the decay rate does depend on the motion of the particle. The fact that this corresponds to the answer we get from the Lorentz transformation is just a result of the Hamiltonian operator in the Schrodinger equation having a Lorentz-invariant form.

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We directly measure muon decay rate. It isn’t a theoretical decay rate, but an empirically measured decay rate.

If there is no privileged inertial frame then the decay rate should be the same in all frames of reference. If you were in a vehicle traveling at near the speed of light you would measure the same muon decay rate in the vehicle’s frame of reference as you do in a lab on the surface of the Earth. However, if someone on the surface of the Earth looked through the windows of the vehicle, would they observe the muons decaying at the same rate within the vehicle? Einstein says no. The test for this prediction is the experiment @dga471 describes. Muons are created in the upper atmosphere. We can measure their velocity, and its really high. If Einstein is wrong then they should decay before they hit the Earth, but they don’t. As Einstein predicts, the actual decay rate of those muons is slower as measured in the Earth’s frame of reference.

Why would the quantum state be different in different inertial frames? How does velocity change the laws of physics?

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I’m well aware. But generally, in order to know what those measurements tell us about nature on a deeper level, we see if we can come up with a theory that accurately predicts and explains those measurements. Notice, for example, how your interpretation of the measurements depends on the theoretical concept that the decay rate is intrinsic and independent of velocity.

The quantum state for a particle in motion is different from the quantum state for a particle at rest. The two are related by a Lorentz transformation, but they are not identical: in QFT, they are built from the vacuum state by different creation operators. (In the same way that the muon lifetimes at rest and in motion are related by a Lorentz transformation, but not identical.)

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Theoretical doesn’t mean wrong.

The equation doesn’t copy over well, but you can check the reference. Muon decay is the product of a fundamental nuclear force, the weak force, and Fermi’s coupling constant, which is a universal constant. You would have to change the fundamental laws of nature in order to get a different decay rate.

So why would velocity change the decay rate? What is the mechanism?

Since when?

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Of course not. What I’m saying is that the idea that the decay rate is independent of the velocity doesn’t come from the measurements themselves: quite the opposite. We measure muons moving relative to us to have a longer lifetime than muons at rest with respect to us; and theory predicts (both under a neo-Lorentzian interpretation as well as under a Minkowskian interpretation) that muons moving relative to us have a longer lifetime than muons at rest with respect to us. But the Minkowskian interpretation views all reference frames as being on par, and therefore tends to see the properties in the rest frame of the muon as more fundamental. Whereas the neo-Lorentzian interpretation sees the description in one reference frame as matching what actually exists at a moment in time, while descriptions in other reference frames are correct only in so far as they describe the way things appear to an observer moving relative to the privileged frame.

This refers to the proper decay rate, i.e. the decay rate as measured in a frame in which the muon is at rest. It does not refer to the decay rate of a muon moving relative to the observer.

How well versed are you in quantum mechanics and quantum field theory? In non-relativistic QM the wavefunction for a moving particle is different from the wavefunction for a stationary particle. It’s a little more complicated in QFT, but the operators that create a moving particle from the vacuum state are different from the operator that creates a stationary particle.

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What is a rest frame? The Earth is spinning. The Earth is also orbiting around the Sun. The Sun is orbiting around the Milky Way. The Milky Way is orbiting around the gravitational center of the Virgo Supercluster. The Virgo Supercluster is moving away from other galaxies. So how do you define a rest frame in this system?

Better yet, Earth’s orbital speed is changing throughout the year because it doesn’t have a perfectly circular orbit. Do you think these measurements will change as the velocity of the Earth changes? Should we observe a different speed of light here on Earth depending on what time of year it is?

If an observer were moving at the same velocity as the muon as it hurtled through Earth’s atmosphere, what decay rate would they measure and why?

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A rest frame of a object is an inertial frame in which the object has zero velocity, and an inertial frame is (roughly speaking) a reference frame where Newton’s first law holds. But from the rest of your paragraph it seems what you actually want to ask is how we can define a privileged reference frame. And the answer to that (in the neo-Lorentzian interpretation) is easy: the privileged frame is the one that describes what actually exists in the changing 3D universe at a moment in time. The privileged frame is a theoretical postulate of the neo-Lorentzian interpretation, and it turns out to be empirically undetectable, but this doesn’t invalidate the concept.

(In the Minkowskian interpretation, of course, that definition doesn’t work because there is no ontologically privileged way slicing the 4D universe into a 3D universe changing over time. But that doesn’t mean the definition doesn’t work in its own context.)

Nope, and nothing I’ve written implies that we should.

The observer would measure the proper decay rate, and (from the perspective of a reference frame moving with the earth) the reason is that the observer’s clocks are slowed down by the same factor as the muon’s decay rate.

And the reason that both the muon’s decay rate and the observer’s clocks are slowed down is simply due to the dynamical effects of the (Lorentz-invariant) laws of physics.

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Isn’t that just a shared inertial frame?

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So what is an inertial frame? This is how Wiki defines it:

Therefore, a spaceship travelling at a constant velocity of 100 km/s with respect to the Earth could be defined as an inertial frame, correct? From everything I have read, Newton’s First Law applies to a spaceship with a constant velocity and can be defined as an inertial frame in Newtonian physics. Do you agree?

Objects moving at a constant velocity with respect to each other actually exist in the universe.

So the speed of light should be the same for every observer no matter what their relative velocity is?

In other words, relativity.

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Yes, that is literally what I said an inertial frame is.

This is a complete non-sequitur. I think we’re on different pages here.

The measured speed of light is, yes.

Yes… the neo-Lorentzian interpretation is an interpretation of special relativity. It’s not like its some non-relativistic theory. Time dilation (e.g. slowing of all physical processes) and length contraction occur for objects in motion relative to the privileged frame, as a causal consequence of the laws of physics. In any other inertial frame, you have apparent time dilation and length contraction for objects in motion relative to that frame, as can be shown from the effects of time dilation and length contraction on the moving observer’s instruments.

I’m sensing something of a disconnect between us in this conversation, so allow me to try and clarify my position. Your original comment in Jim’s thread was this:

My main intent was to point out that, in fact, the muon decay rate does depend on velocity, that this is exactly what we see in the muon lifetime measurements, and that the mechanism for the change in decay rate is nothing other than the usual laws of physics operating on a moving system rather than a stationary system.

In retrospect I didn’t even need to mention the neo-Lorentzian interpretation to make this point - it can be validly made from any reference frame. (It is just that I’m used to invoking it in relation to the privileged reference frame of the neo-Lorentzian theory.) Time dilation and length contraction are not just transformations that result from changing your reference frame. They are also, in the perspective of any one reference frame, physical effects that happen to moving objects, explicable by reference to the laws of physics described in that frame. See John Bell’s paper, “How to Teach Special Relativity”.

Time dilation, in particular, is (again, from the perspective of a single reference frame) the slowing down of physical processes for systems in motion, compared to systems at rest. And (in principle, because QFT is complicated and they generally do not compute the quantum state directly) it can be seen for the muon in how the quantum state for a moving muon evolves differently than the quantum state for a stationary muon.

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My main point is that the observed decay rate is dependent on relative velocity between the muon and the observer. Do we agree on this point?

I would imagine the evolution is the same. You get the same particles going in and out of the Feynman diagram. The only thing that would change is probability of where you would expect a decay to happen.

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Yes, absolutely.

This is incorrect! Incoming and outgoing lines on Feynman diagrams are labeled by their momenta - there absolutely is dependency on the motion of the muon. Moreover, if the evolution between the original muon state and the decayed state was independent of the velocity of the muon, the theory would predict that the decay rate would not vary with the velocity, in direct contradiction to SR and to observations.

Relativity isn’t something that is tacked on to QFT by applying Lorentz transformations to the output of calculations (as if QFT calculations only give the proper decay rate, and we have to add the time dilation by hand). Rather, it is woven right into the dynamics of the theory.

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How would differences in momentum change the evolution of the particles?

Again, that would be relative velocity. If you are travelling with the muon it decays at the non-relativistic rate.

Let’s say there are two observers. One observer is on the ground and is measuring the decay rate of the muons streaming down through the atmosphere. The other observer is in a vehicle flying next to the muon and that observer measures the decay rate (assuming they don’t crash through the Earth). The two observers measure different decay rates for the same set of muons. Wouldn’t I use the Lorentz transformations to predict the difference in the observed decay rates for the two observers?

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In two ways: it would change the momentum of the decay products (obviously), and it would change the decay rate (the rate at which the amplitudes in the superposition shift from being focused on the original muon state to the decay products). It does this because the Hamiltonian acts on the different quantum states differently.

You keep asking variations on “how does the motion of the muon affect the evolution of the quantum state”; I really don’t know what else to tell you other than that’s what happens when you apply the Schrodinger equation to the quantum state.

Yeah, but we’re not travelling with the muon. Allow me to quote myself:

I’ll also reiterate: Bell’s “How to Teach Special Relativity” is illuminating here.

You could. You could also do the QFT calculations to predict the decay rates. The observer standing on the earth would do that starting with the quantum state for a moving muon (since the muon is moving relative to their reference frame), and the calculation would spit out some answer T1. The observer in the muon’s rest frame would start with the quantum state for a stationary muon (since the muon is at rest in their reference frame, and the earth - squashed into an ellipsoid by length contraction - is hurtling towards them at some noticeable fraction of the speed of light), and they get an answer T0.

Now, T0 is the proper decay rate, and T1 is time-dilated from T0 by the appropriate factor. In that sense, T1 can be related to T0 by a change in reference frame. But it is also the case that T1 is nothing more than the result of the laws of physics operating on the moving muon state. And it is different from T0, which is the result of the laws of physics operating on the stationary muon state. The decay rate does depend on the motion of the muon.

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I guess I don’t view that as a change in the evolution of the particle. I am thinking more about the interactions that produce the muon and the particles the muon decays into. Those stay the same.

But if we were, we would see a shorter decay time. That’s the point.

It depends on the motion of the muon with respect to the observer. That’s the important bit.

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The interactions that the muon participates in and the decay channels that result from them are all part of the Hamiltonian operator that goes into the Schrodinger equation to evolve the quantum state.

Whether that is the important bit depends on how one interprets special relativity. My initial remark was simply this: the muon lifetime observations are as open to the neo-Lorentzian interpretation as any other special relativistic phenomena.

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Here’s a really interesting (if incredibly obscure, for some people) Wikipedia article giving an overview of several different alternative formulations of SR which have been proposed, some of which are empirically equivalent, some of which are not:

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