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Experiments may only measure gauge-invariant, coordinate-independent quantities

I have finally found the time needed to see every page among the 12+7+16 pages of the Japanese papers on the Earth's contribution to muon's \(g-2\) and it's clear that if I had opened all the papers before I wrote my first fast blog post about it, the post wouldn't have been written because the papers are childish and ludicrously wrong. Let me start with superficial observations about their style and background.



First, the authors are clearly not professional particle physicists. You won't find any Feynman diagrams – or the words "loop", "Feynman", "diagram", for that matter – in any of the three papers. Well, particle physicists would generally agree that you need Feynman diagrams – and probably multiloop ones – to discuss the muon's magnetic moment at the state-of-the-art precision.




Instead, we clearly read papers by 2-3 Japanese men who tell us: Look how completely stupid and pretentious particle physicists have been with their Feynman digrams, loop integrals, all this difficult garbage. We can just use some high school and undergraduate physics and find something more interesting. The Japanese men use this simple enough toolkit because they're basically experimenters and their toolkit could be enough.

Well, that outcome would be juicy, indeed, but it's unlikely – and in this case, they haven't beaten the professional techniques by those of the average Joe. People often have other agendas than science – they would love if experimenters trumped theorists in doing theoretical calculations; or if African physicists or women in science trumped the white men. Well, most of the time, they just don't. More generally, you shouldn't put ideological agendas and a wishful thinking above the scientific evidence.




Their calculation of the extra effect that is supposed to cure the muon anomaly is intrinsically a classical, undergraduate calculation and their result is something like\[

\mu_{\rm m}^{\rm eff} \!\simeq\!
(1\!+\!3\phi/c^2)\,\mu_{\rm m}

\] which says that they muon's magnetic moment should be adjusted by a correction proportional to the gravitational potential \(\phi\). And they substitute the Earth's gravitational potential. Well, if I had paid attention to that sentence in the first abstract, I wouldn't have written my TRF blog post, I think. It's silly.

First, as some people have mentioned, if some new effect were proportional to the gravitational potential, it would be the contribution to the potential from the Sun, and not Earth, that would dominate. Please, do this calculation with me. The gravitational potential is equal to \(GM/R\), up a to a sign.

The Sun is heavier (higher \(M\)) but further from us (greater \(R\)) than the center of our planet. Which celestial body wins? Well, the mass ratio is\[

\frac{M_{\rm Sun}}{M_{\rm Earth}} = \frac{2\times 10^{30}\,{\rm kg}}{6\times 10^{24}\,{\rm kg}} \approx 330,000.

\] Similarly,\[

\frac{R_{\rm Sun-Earth}}{R_{\rm Earth}} = \frac{150\times 10^{6}\,{\rm km}}{6,378\,{\rm km}} \approx 23,500.

\] So the mass ratio and distance ratio are \(330,000\) and \(23,500\), respectively. Clearly, the mass ratio dominates, and the gravitational potential from the heavier Sun is therefore larger by a factor of \(330,000/23,500\approx 14\). To claim that an effect is proportional to the gravitational potential, to consider the Earth, but to neglect the Sun is just wrong.

(One could also discuss the contributions to the potential from the location of the Solar System within the galaxy and similar things. The Solar System's speed within galaxy is 230 km per second, 7.5 times higher than the Earth's orbital speed 30 km per second. Because the gravitational potential \(\phi\sim v^2/2\) for circular orbits, the galactic contribution is some 55 times greater than the Sun's. Well, this 55 is probably just a rough estimate that would be accurate if the potential's profile in the galaxy were \(1/r\) which it's not. And by the way, if the dependence on the Sun's gravitational potential mattered, there would be seasonal variations of the muon's magnetic moment and lots of other weird effects.)

Note that the Sun won only because we considered \(GM/R\). If we had a higher integer power of \(R\) in the denominator, the Earth would win because the huge distance of the Sun would "matter" twice or thrice. The effect proportional to \(GM/R^2\) would be proportional to the gravitational acceleration – i.e. the acceleration of the lab bound to the Earth's surface relatively to the freely falling frame. And \(GM/R^3\) is the scaling of the tidal forces (accelerations), and those are some components of the Riemann curvature tensor.

So the Earth could win if we considered gravitational accelerations or tidal forces (=curvatures in GR). Those would be exactly the situations that would be plausible because the claim that an effect depends on the acceleration of the lab or non-uniformity of the gravitational field is compatible with the equivalence principle. (But those corrections would be far too tiny to be seen experimentally.) But the dependence of the muon's magnetic moment on the gravitational potential violates the equivalence principle.

The people defending these Japanese papers tend to say "there can be lots of terms, mess, blah blah blah," and we can surely get the desired terms (they claimed to cancel the 3.6-sigma deviation with an unexpected precision of 0.1 sigma, another point that should raise your eyebrows). However, the very point of the equivalence principle – and other big principles and symmetries of modern physics – is that Nature is not that messy. She's clean if you grab Her by Her pußy properly and some facts hold regardless of the complexity of any experiments.

Haters of physics love to dismiss physicists' knowledge – the world is so complex and mysterious, anyway (here I am basically quoting that young woman from the humanities who has sometimes asked for the translations to English). Well, much of the progress in science is all about the invalidation of these pessimistic claims and physicists have gotten extremely far in that process, indeed. Lots of things have been demystified and decomplexified – the hopeless complexity is just apparent, on the surface, and physicists know how to look beneath the surface even if mortals can't. They can still make lots of very precise and correct statements about things, even seemingly complicated things.

For example, the equivalence principle says that if you perform an experiment inside a small enough and freely falling lab which has no windows, the results don't allow you to figure out whether you're in a gravitational field or not. If the ratio of the electron's and muon's magnetic moments depended on your being near Earth, you could say whether you're near the Earth inside that lab, and the equivalence principle would be violated. That's it.

Aside from their complete misunderstanding the equivalence principle, neglecting the Sun which is actually dominant, and failing to discuss any Feynman diagrams at all, the papers also misstate the ratio of the electric and magnetic fields in the muon \((g-2)\) experiments by three orders of magnitude, and do other things – see phenomenologist Mark Goodsell's comments about that, plus his argumentation why the effect has to cancel. A fraction of these problems is enough to conclude that the Japanese papers are just some mathematical masturbation that you shouldn't read in detail because it's a waste of time. After some time you spend with the paper, you will become almost certain that they just tried to find a formula that numerologically agrees with the muon anomaly. Gravitational potentials looked attractive to them, they have used them, and they cooked (more precisely, kooked) a rationalization afterwards. Well, this research strategy is driven by random guesses and a wishful thinking and it's not surprising that it fails most of the time.

(I agree with phenomenologist Mark Goodsell, however, who says that this "inverse" approach is often right because research is a creative process. Of course I often try to guess the big results first and then complete the "details", too. One just needs to avoid fooling himself.)

But I want to return to the title. Defenders of the paper, like TRF commenter Mike, tell us that experiments usually measure lots of coordinate-dependent effects etc. Well, this is the fundamental claim that shows that Mike – and surely others – completely misunderstands the meaning of the equivalence principle and gauge symmetries in modern physics.

The meaning of coordinate redefinitions in the general theory of relativity is that those are the group of local, gauge transformations of that theory and only "invariant" quantities that are independent of these transformations may be measured! Only the invariant quantities are "real". That's really the point of the adjectives such as "invariant". That's how gauge transformations differ from any other transformation of observable quantities. They're transformations that are purely imaginary and take place in the theorist's imagination – or, more precisely, in the intermediate stages of a theorist's calculation. But the final predictions of the experiments are always coordinate-independent.

In particular, rigid rulers and perfect clocks may only measure the proper distances and proper times. It's totally similar with the coordinate dependence of other quantities beyond distances and times – and with all other gauge invariances in physics.

The case of electromagnetism is completely analogous. Electrodynamics has the \(U(1)\) gauge symmetry. The 4-potential \(A_\mu\) transforms as \[

A_\mu \to A_\mu + \partial_\mu \lambda

\] which is why it depends on the choice of the gauge – or changes after a gauge transformation defined by the parameter \(\lambda(x,y,z,t)\). That's why experiments can't measure it. The choice of \(\lambda\) is up to a theorist's free will. You can choose any gauge you want and experimenters can't measure what you want. They measure the actual object you're thinking about; they don't measure your free will.

On the contrary, the electromagnetic field strength doesn't transform,\[

F_{\mu\nu}\to F_{\mu\nu}.

\] That's why electric and magnetic fields at a known point in the spacetime may be measured by experiments – their magnitude may appear on the displays. Also, the Araronov-Bohm experiment may measure the integral \(\oint A_\mu dx\) over a contour surrounding a solenoid modulo \(2\pi\). That seemingly depends on \(A_\mu\) but the contour integral of \(A_\mu\) modulo \(2\pi\) (in proper units) is actually as invariant as \(F_{\mu\nu}\). After all, \(\oint A_\mu dx=\int F\cdot dS\) is the magnetic flux through the contour. And that's why it can be measured (even if the particle avoids the "bulk" of the contour's interior) – by looking at the phase shift affecting the location of some interference maxima and minima.

The case of coordinate dependence is completely analogous. One can choose coordinates in many ways in GR – they're like the choice of gauges in electromagnetism. But the experiments can't measure artifacts of a choice of coordinates. They can only measure real effects – effects that may be discussed in terms of gauge-invariant (including coordinate-independent) concepts. For example, when LIGO sees seismic noise, the seismic noise isn't just an artifact of someone's choice of coordinates. The seismic activity is a genuine time dependence of the proper distances between rocks inside Earth – and the "time" in this sentence could be defined as some proper time, e.g. one measured by clocks attached to these rocks, too.

I must add a disclaimer. Coordinates may be defined as some proper distances and proper times based on some real-world objects. For example, you may specify a place in continental Europe (at any altitude up to kilometers) as the place's proper distance from a point in Yekaterinburg, in Reykjavik, and Cairo. Three proper distances \((s_Y, s_R, s_C)\) are a good enough replacement for the Cartesian \((x,y,z)\) – one must be careful that such maps aren't always one-to-one, however (for example, points above and below the plane of the triangle have the same value of the three proper distances).

So we could be worried that the three rulers – that measure proper distances from these three cities – are a counterexample to my statement about the coordinate independence of measurable quantities. Experiments directly measured some coordinates. But that's only because we had to define the coordinates to be the proper distances in the first place! So the first step in an experiment, the measurement of the proper distances of an object in Europe from the three cities, shouldn't even be considered a measurement yet. It should be considered a calibration. Experiments that probe laws of physics – e.g. a proposed effect that depends on the place of Europe – may only begin once you measure something else on top of the three proper distances! And note that when coordinates are equal to three proper distances and experiments measure these three numbers for an object, it's still true that "experiments only measure invariant quantities". The quantities are both invariant and (someone's particular chosen) coordinates.

So Mike uses lots of the right words from the physicist's toolkit but the deep statements are just wrong. He completely misunderstands the equivalence principle and gauge symmetries – principles that underlie much of modern physics and allow us to be sure about so many things despite the apparent complexity of many situations and gadgets. One may say lots of things and write Japanese papers but they're virtually guaranteed to be wrong.

The muon \((g-2)\) anomaly hasn't been explained away. Properties of elementary particles measured inside localized labs cannot depend on the gravitational potential. And even if something depended on the gravitational potential, the Sun's contribution would be dominant. If the equivalence principle holds – and there are lots of confirmations and reasons to think it's true – the muon's magnetic moment is a universal constant of Nature so it just cannot depend on the environment (the quantities describing the local gravitational field).

One could still be worried that the experimenters who measure the muon's magnetic moment have made a mistake and they also included some corrections that depend on the gravitational field that they shouldn't, and the terms from the Japanese papers should be interpreted as the "fixes" that subtract these specious corrections. But I don't think that this worry is justified because the equivalence principle says that it's really impossible to see, in a closed lab, whether you're inside the Earth's gravitational field or in outer space.

You can only measure the acceleration (using accelerometers on your smartphone, for example) or the non-uniformities of the field (if the lab is sufficiently large). But those effects only influence the elementary particles to a tiny extent that isn't measurable. So if one believes this statement – or if he believes that the effect should be proportional to a gravitational potential – the muon \((g-2)\) experimenters cannot introduce the gravitational-potential-dependent mistake even if they wanted.

(Well, if they really wanted, they could simply add the error deliberately. "We want the muon magnetic moment to depend on the gravitational potential, so we just adjusted the readings from our apparatuses to bring you a wrong result. Our friend is a NASA astronaut and we wanted to make her and women in science more relevant, so we added her altitude to the muon's magnetic moment." Well, I don't think that they're doing it. They just ignore the gravitational field-related issues and the basic principles of physics show why this strategy is consistent and the results are actually relevant in any gravitational field. It's just wrong to "adjust" results directly measured in closed localized lab by any gravitational potentials and similar factors. This disagreement may have almost "moral" dimensions. Some people would want the experimenters to add lots of corrections which makes it "very scientific", they think. But in good science, experimenters never add corrections of the type they don't fully understand or they don't fully report – they are supposed to be comprehensible and report the readings of their displays or process them in ways that they have completely mastered and described. In this context, experimenters just never adjust any readings by any Earth's gravitational corrections and theorists understand that and why the results obtained in this way are still relevant even outside Earth's gravitational field.)



In some broad sense, the same comments apply to global symmetries, too. The special theory of relativity has the Lorentz and Poincaré symmetries. The Lorentz symmetry (a modern deformation of the Galilean symmetry) guarantees that you can't experimentally determine whether the train where you perform your experiments is moving. So if someone measured the muon magnetic moment inside a uniformly moving train, he couldn't measure any terms proportional to the train speed \(v\) even if he wanted! That's simply impossible by the symmetry – you just can't see any \(v\) or a non-trivial function of \(v\) on your experimental apparatuses' display.

You could only measure \(v\) or things that depend on \(v\) if you did something with results of experiments done both inside the moving train and inside the railway station – and on top of that, these two labs (static and moving) would have to interact or communicate with each other in some way. The muon experimenters (and their colleagues whose results needed to be relied upon) haven't done any experiment outside the Earth's gravitational field where \(\phi\approx 0\), I think, so they couldn't have made any comparison like that, which is why their results – functions of readings on their displays – just can't depend on \(\phi\). Again, the reason is analogous to the reason why the experiments done in the train cannot depend on the train speed.

These are some very general, basic, precious principles underlying modern physics. Every good theoretical physicist loves them, knows them, and appreciates them (and appreciates Albert Einstein who has really brought us this new, powerful way of thinking that's been extended in so many amazing ways). The Japanese papers are an example of ambitious work by self-confident people who don't know these rudiments of modern theoretical physics and who want to pretend that it doesn't matter. But it does matter a great deal. When it comes to the framing of the papers by many, the papers aren't anything else than just another attempted attack on theoretical physics as a discipline. Look how stupid the theoretical physicists are, using all the overly complex mathematical formalism and overlooking the obvious things that may be discussed using the undergraduate formalism. (Tommaso Dorigo makes this excited claim in between the lines and a reader of his blog makes it explicitly.) Except that the graduate school formalism is needed to discuss the magnetic moments of leptons at the state-of-the-art precision.

And after all, it's even untrue that the professional particle physicists – with their Feynman diagrams and loops etc. – are making things more complicated than they need to be and than the Japanese folks. Unlike the Japanese men and similar folks, the competent theoretical particle physicists maximally utilize the deep principles such as symmetries that greatly simplify situations and calculations. It's really the Japanese men who dedicate dozens of pages to the calculation of an effect that is known to be zero by a straightforward symmetry-based argument!

In other words, if you don't know the tools of mathematics and modern theoretical physics, you're bound to spend most of your time by being lost inside seemingly complicated stuff and expressions that the competent folks immediately see to equal zero – or another simple value (and you're almost guaranteed to do lots of errors in that context). Competent theoretical physicists still spend lots of time with complicated stuff, but that's because everything that could be have been simplified has been simplified.

If you want to be a good or at least decent theoretical physicist, you just need to learn those methods and principles and you need to learn them properly. If you think that experiments "usually measure gauge-variant or coordinate-dependent quantities", you have learned almost nothing from modern physics yet.

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