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Myths about the Planck scale distortions

I just read a rather frustrating conversation between Sabine Hossenfelder and Moshe Rozali at Backreaction. Moshe, an associate professor at UBC in Canada, is trying to explain, in a typical Moshe-like excessively polite way, that Sabine, a postdoc, is making a rather elementary mistake that makes pretty much all papers that she has ever written nonsensical.

He tries to encourage her to be less sloppy. Sabine is apparently convinced that an angel guarantees that she can't be wrong and refuses to understand very clear, indisputable, and sometimes really transparent statements and arguments, indicating that she thinks that she is "teaching" Moshe new things. ;-(

Discovering new physics by misprints

Sabine believes in all kinds of unmotivated and almost certainly impossible distortions associated with the Planck scale. And she is not the only one, for that matter: there are many people similar to Giovanni Amelino-Camelia or Jack Ng. One textbook example is a "lower bound on the wavelength" (see the comments under her article) that she seems to consider as a fact. Needless to say, there can't exist any lower bound on the wavelength. Why? Because any wavelength is related to any other wavelength by a boost - by a Lorentz transformation that is a symmetry of Nature (and of quantum gravity, at least in the flat space superselection sectors of the Hilbert space).

The fact that there can't exist things such as a "lower bound on the wavelength" has been known at least since 1905 when Einstein realized that one inertial observer will see a different frequency than another observer who is moving with respect to the first one. Have you heard of time dilation or the Doppler shift? Because both observers have the same rights, according to the special relativistic bill of rights, both values of the frequency or the wavelength are equally good, too.




Special relativity has become a pillar of general relativity and general relativity is one of the components of a theory of quantum gravity. You can't just throw away the insights of 1905. They are alive and kicking. In particular, it is impossible for one value of the wavelength to be allowed and another value of the wavelength to be forbidden.

Lorentz symmetry might be broken but there doesn't really exist a glimpse of evidence or motivation why it should be broken. Such a breaking might be studied as a possibility, to some extent, but it would cause many problems, it would most likely return us philosophically before 1905, and the people who say that such a breaking is needed in quantum gravity are clearly wrong because string theory satisfies all the required properties of quantum gravity but it also respects the Lorentz invariance in flat space.

This symmetry doesn't mean that the curved geometry including the backreaction has a Lorentz isometry. It means that one can create a Lorentz-boosted configuration (including the curved geometry) for every initial configuration and they obey the same laws. If one curved spacetime solves the equations, the boosted spacetime solves them, too. Moreover, the boost doesn't change the asymptotic geometry of the flat space which is why the symmetry acts inside the same superselection sector - a fact that makes the symmetry really useful.

Quantum gravitational limits on the minimum length apply to the internal structure of bound states (you can't probe the internal architecture of matter too accurately if your probes are extended) but they cannot apply to the wavelength of one photon or graviton. The very same comments are true about the uncertainties. If there are inherent Planckian uncertainties in the measurement of distances, they apply to the internal geometry of a physical system but not to the center-of-mass position of the whole physical system embedded into flat space - an observable that behaves just like it behaved in non-gravitational physics.

The latter point is also easily seen: the center-of-mass degrees of freedom are complementary to the total momentum (that can have any values, by Lorentz invariance). If you simply Fourier-transform the wavefunction in the momentum variables (that must exist), you obtain the wavefunction in the position variables, just like you always did. Gravity doesn't change anything about this procedure. The objects themselves include the curved space around their mass distributions but this is just a detailed description of the identity of these objects. The symmetry doesn't disappear.

Vague, universal symbols

I was trying to understand why these trivial points are so enormously difficult for many people who claim to be interested in quantum gravity. My best explanation is that they interpret symbols in a sloppy way. What do I mean?

Take a sexy female alternative physicist with a male voice, as a beloved TRF reader has pointed out ;-), for example Dr Louise Riofrio. What is her most important equation? Well, it is GM=tc^3. Now, many readers haven't heard about this famous equation but let us study its meaning and the psychology behind it. :-)

The first comment I would say is that Louise - much like most amateur physicists - use symbols for observables without paying too much attention what these observables actually mean. So when she writes "M", it is supposed to be a universal "M" (mass) across the Universe. In the same way, "t" is a universal time that is also accepted everywhere. Louise doesn't have to explain her formula, its origin, and its interpretation in detail: the formula itself is a holy word.

Most readers might know that the universal quantities appearing in her formula don't exist. There are many different masses in the world (for example the mass of an elephant or a cow), many different times (each observer in relativity has a different way to measure times, and she can still measure the duration between any two events she likes), and moreover, the laws of physics are local and describe particular objects or regions of space so you should better avoid relations between the mass of the whole Universe and other "global" quantities because they are almost guaranteed to be violated. Even if you are lucky and your private relationship is not violated, it is not too useful because it would only relate or predict one or two quantities in a Universe with effectively infinitely many degrees of freedom.

Now, if you could explain the previous paragraph to Dr Louise Riofrio, she would suddenly understand not only why her equation is wrong but why her whole method of thinking and "discovering new physics" is wrong. I guess you won't succeed in this magnificent pedagogical task. ;-)

The case of Sabine Hossenfelder is pretty much isomorphic but letters like "G, M, t, c" are replaced by "delta x, E, p". Once again, in Sabine's setup, there is a universal "delta x" in the Cosmos, much like a universal energy "E" or momentum "p". And they must be subjects to similar, Hossenfelderian laws of physics. What the laws can be if you have three or four letters to play with?

Well, there can be a relationship between "E" and "p". But the simple one, the relativistic dispersion relation, has already been found by Einstein so Sabine and similar fans of physics inevitably decide to write a different one. They must modify the old one. Because they call themselves "quantum gravity physicists", they write down a random formula incorporating the Planck scale.

Just to be sure, rationally speaking, there is absolutely no reason, evidence, or justification for someone to modify the relativistic dispersion relation. All the alternative formulae are unmotivated and, as far as we can say today, probably incompatible with a consistent theory resembling general relativity in an appropriate limit. All known advanced theories to describe anything in the world are compatible with the good old dispersion relation and all useful or promising modified relations we have seen may be interpreted in terms of a spontaneous symmetry breaking, starting from the old ones.

And to write new formulae just in order to find a new place where your Planck scale could occur is just plain silly; however, certain people are always more eager to incorporate random misprints and distortions into old equations than to consider the possibility that some of those could be necessary and exact. If you want to find something new, true, and important about theoretical physics, you should better look elsewhere. There are hundreds of other places where you could look and if you don't know them, it's too bad because you are really imagining that physics is about the "E=mc^2" written on the T-shirt. It's not.

The story about "delta x" is analogous. What laws can Sabine write down that would involve this universal "delta x"? Well, the symbol is similar to the symbols in quantum mechanics where the uncertainty principle is one of the most famous principles. But in quantum gravity, one can write a simpler inequality involving the Planck scale, namely "delta x > Planck length".

Suddenly, you can write eleven or more of (crappy, absurd, and virtually identical) papers about the "minimum length".

Now, the inequality is a fair qualitative idea that was written down decades ago, that you can keep in mind, and that you can "derive" by dimensional analysis. And it can have a serious meaning if you are extremely careful about the interpretation of "delta x". But that's not what Sabine wants to do. She wants this inequality to be a general law of physics that applies to all quantities that might deserve the symbol "delta x".

Needless to say, such a hypothesis is simply wrong.

The center-of-mass position of a physical system doesn't have any uncertainties of this kind. Objects that are embedded into an asymptotically flat (or other well-defined) space are "rooted" at infinity so tightly that uncertainties simply disappear. As explained above, the internal geometry can be and probably is uncertain but the center-of-mass position can't be limited by a new uncertainty principle. The center-of-mass degrees of freedom are the canonically dual quantities to the total momentum and the latter certainly does exist.

Analogously, new things appear when the energy exceeds the Planck energy but you must carefully interpret the term "the energy" in this sentence. For example, it can mean the total center-of-mass energy in a collision (and you start to create black holes when the energy is higher than that). If you only talk about the energy of one object or the energy measured in an arbitrary reference frame, there is clearly no new effect or transition (i.e. nothing special) near the Planck value, as guaranteed once again by the Lorentz symmetry.

But the new Einsteins usually don't care. They think that Einstein's main contribution is "E=mc^2" and they try to offer something analogous. Louise teaches us about her "GM=tc^3", Sabine teaches us "E = randomfunction(p, Planck scale)" and "delta x > Planck length", and both of them - and, frankly speaking, dozens of others - seem to think that this puts them at Einstein's or string theorists' level.

(Ms Hossenfelder declared on her blog that she has forgotten about her writings about random modifications and deformations of dispersion relations. See e.g. hep-th/0510245 where three variations of the silly idea, namely GUP, DSR, MDR, are even linked with each other. Ten more related papers can be accessed from the previous link.)

However, both Einstein as well as serious physicists today are using and must be using symbols very carefully. There is no room for a universal "E" or universal "p" or universal "delta x". In fact, both Einstein (the real one) and the contemporary serious physicists have been working, are working, and must be working with hundreds of various energies "E" and momenta "p" and uncertainties "delta x" and they must never forget how the particular quantities are defined and distinguished from each other - even if they have the same dimension.

The Devil is usually in the details.

I am afraid that this cognitive step - and, more generally, rigor - is just way too difficult for certain people. With their assumptions about the meaning of observables and the universal sketch of a task for new physicists, it is pretty much inevitable that they must end up with another incorrect equation or inequality relating ill-defined symbols. Do you want to help them in their research? What Louise or Sabine want you to tell them is how they should modify the shape of the wooden earphones.

Louise would be excited if you told her about a new relationship between "G" and "M" - for example, she was thrilled when I translated her equation into the Planck units :-) (M=t) - and Sabine would think that you are a constructive physicist if you proposed another but equally absurd "vacuum" dispersion relation between "E" and "p" or if you derived a new incorrect consequence of her incorrect assumptions about "E", "p", or "delta x".

It is unfortunately much harder to explain them that they are doing something really stupid with the very method how they use their brains. It is much harder to liberate them from their extremely naive and narrow mental boxes. It is much harder to explain them that they could be perhaps making variations of the same elementary mistake all the time. It is hard to convince them that if they tried a little bit harder and restricted their excessive arrogance and unjustifiable self-confidence, they could perhaps learn some new things and fix their old misunderstandings.

And that's the memo.

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