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Trans-Planckian gravity: Bambi-Freese paradox

Cosimo Bambi and Katherine Freese argue that there exists a general paradox in quantum gravity. It goes as follows:

There exist trans-Planckian, massive particles in quantum gravity, for example extremal black holes. Their lifetime is much greater than the Planck time, too. Consequently, the loops in Feynman diagrams involving these massive particles will make it possible to violate the baryon or lepton number for long periods of time in pretty much every theory of quantum gravity.
If you actually think a little bit rigorously about these things, their conclusion turns out to be, of course, completely wrong. Let us look at these things in detail.

Generalized uncertainty principle

First of all, you can see that the authors are complete outsiders in quantum gravity. Pretty much every sentence is slightly - or more than slightly - wrong. For example, at the beginning, they discuss the generalized uncertainty principle(s). A punch line of theirs is that "Delta x" is always greater than the Planck length.

This is a popular simplified assertion that we often use to describe the effect of quantum gravity on distances but one certainly cannot interpret it in the way that Bambi and Freese do. Their main problem here is that they use the same symbols for many different things and they are able to confuse themselves.

When they say that "Delta X" is greater than the Schwarzschild radius of an object, they clearly talk about "Delta X" that is the internal radius of an object. On the other hand, they seem to use "Delta X" as the uncertainty of the position of the center of mass at the end. The internal radius and the uncertainty of the position are very different things.

If you understand quantum mechanics of the Hydrogen atom, what would you think about a person who confuses the radius of the electron with its position uncertainty in the Hydrogen atom i.e. with the Bohr radius? A quantum gravity expert obviously thinks the same thing about someone who does the same childish mistake as Bambi and Freese.

In the literature, most of the talk about the "minimum length" in quantum gravity is a vague sloppy babbling of incompetent people who don't have enough imagination to ever understand how Nature actually solves these things - think about profoundly and permanently confused authors such as Ng, Amelino-Camelia, Hossenfelder, and dozens of others. In reality, something new is indeed going on at the Planck scale, but to assume that
  1. it must be possible to talk about distances even in this regime and
  2. all the distances must always be strictly greater the Planck scale

is a double naivite. If you actually look at any consistent realization of quantum gravity - and we have quite many setups to do so, including AdS/CFT, Matrix theory, perturbative string theory, and more informal descriptions of quantum gravity involving effective field theory - you will see that the "generalized uncertainty principle" in the strict Bambi-Freese sense is certainly wrong. Sometimes we cannot talk about distances at all because it is no longer a good degree of freedom. But even when we can, there are cases in which it can be sharply defined.

More concretely, you will see that in contradiction to their statements, the position of the center of mass of a black hole - or another object - can be arbitrarily sharp. It is, in fact, a consequence of Lorentz invariance. In particle physics inside the Minkowski space, one must always be allowed to construct particle states with well-defined momentum vectors. And it is always possible to define a complementary operator of the center-of-mass position that can always have its eigenstates.

It is true that the internal structure of an object cannot be shorter than a Planck scale - but in specific vacua with additional couplings or radii, you must be very careful which Planck scale you talk about. However, this comment certainly doesn't mean that every quantity whose dimension is that of length must be greater than the Planck scale. Quite on the contrary.

Even if you are a bizarre, speculative alternative physicist who thinks that the reality is described by an entirely different theory of quantum gravity than the theory we still call "string theory", you must agree that string theory provides us with one or many realizations (depending how you count) of a physical system that satisfies all the consistency criteria expected from a theory of quantum gravity. So it is certainly sufficient to falsify many "general" statements about quantum gravity as long as they are incorrect.

And the statement about the uncertainty of the center-of-mass position in every theory of quantum gravity is certainly incorrect. Consider, for example, Matrix theory. It is easy to construct eigenstates of the operators Trace(X_i) and falsify the Bambi-Freese "generalized uncertainty principle".

Even more problematic is their statement about the "minimum time" associated with a massive object because it combines the confusion discussed above - the incorrect identification of the internal structure with the center-of-mass position - with some frequently used incorrect interpretations of the "energy-time uncertainty relation".

Their three worries

Still on page one, they list three possible worries resulting from their sloppy thinking:

  1. long-lived virtual macroscopic objects
  2. light particles with trans-Planckian energies
  3. super-Planck heavy processes contributing to B or L violation

It almost looks like a homework exercise from a textbook, showing three serious conceptual errors that a student may make. The solution is, of course, that

  1. all generic trans-Planckian heavy objects in quantum gravity that have at least a marginal right to be considered elementary are black holes
  2. in effective field theory, the energy can only be integrated up to a cutoff, and the cutoff should never exceed the Planck energy for very general reasons
  3. because of the first point, the only heavy particles that would deserve to occur in loops of Feynman diagrams are black holes but I will argue below that we know for sure that any contribution of theirs that could cause B or L violation should be erased because they are not really elementary particles or because of other reasons.

OK, let me assume that the reader is familiar with basics of quantum field theory so he or she knows that their point (2) was really silly: any result that you obtain by integrating the loops over energies much greater than the Planck scale is clearly unphysical and shows that you didn't renormalize things correctly or you misunderstood what a cutoff is.

The remaining points are (1) and (3) and they are really equivalent.

Loops of big objects

If your theory admits composite objects, such as the Hydrogen atom in the Standard Model, should they be allowed to run in the loops of Feynman diagrams? Under normal circumstances, the answer is, of course, No. Feynman diagrams represent a perturbative expansion of a well-defined theory with well-defined fields at the tree level - and this set doesn't include the Hydrogen atom. If you derive the Feynman rules e.g. for QED properly, the Hydrogen atoms are simply not running in the loops. Period. Electrons and quarks (or protons, in an effective theory wit hadrons) are. Hydrogen atoms in loops would amount to double-counting.

But is there some sense in which large objects can become virtual particles? Perhaps - but one must be more careful.

Heavy elementary objects

In all consistent definitions of quantum gravity we have, generic extremely heavy elementary particle species can always be interpreted as microstates of black holes. To be sure, the term "quantum gravity" in the previous sentence is equivalent to "string theory" because "consistent theories of quantum gravity we have" and "string theory" is the very same thing. But I deliberately used the term "quantum gravity" because the conclusion has nothing whatsoever to do with strings per se. The rule applies to the vacua (and their descriptions) which contain no perturbative strings, too. It is almost certainly a general fact about quantum gravity.

There are many similar facts about quantum gravity that have been demonstrated in so many inequivalent ways and in so many complementary descriptions that they have become a part of our general understanding of physics. If you learn them, you will be able to answer most of the questions and fix most of the errors in the papers by all these confused quantum gravity outsiders. These qualitative questions - about the transitions between the light and heavy objects, about the uncertainty relations at the Planck scale etc. - have simply been answered for quite some time.

For example, as long as your particles collide with energies lower than the Planck scale, the low-energy effective field theory description is legitimate. However, trans-Planckian collisions inevitably start to produce black holes. Black holes may be viewed as very heavy elementary particles. But general relativity including event horizons becomes much better zeroth approximation to understand physics of these heavy "elementary particles" than perturbative Feynman diagrams: general relativity is weakly coupled while the Feynman diagrams are strongly coupled in this regime. You are almost guaranteed to get wrong answers if you only consider the "elementary particle" picture and only pick a few loop diagrams.

I think that every competent person would agree with me even though we can't offer any rigorous and universal proof of the assertion, especially because some of the words (including the term "quantum gravity") are not rigorously defined. But I want to emphasize that our understanding of these general insights about quantum gravity is completely independent of our - so far incomplete - knowledge of the right vacuum, background, or compactification of string/M-theory that describes the world around us.

Why is it independent? Simply because these qualitative results hold in all of them. 10^{500} is not high enough number to create this kind of diversity that would allow us to violate the general rules of quantum gravity.

Loops of black holes

So do the extremal black holes inevitably lead to fast processes violating the baryon or lepton number? No, they don't. Just follow Feynman's method to think about other people's ideas: keep a specific enough example of the situation that the other person talks about in your head. If you do so, you will clearly see that the general conclusion by Bambi and Freese is incorrect.

For example, pick any known semi-realistic vacuum of string theory. Much simpler and more symmetric vacua would be enough, too. Try to follow the Bambi-Freese arguments and check which of them work and which of them don't work in your particular example. Once again, even if you believe crazy speculations that there exists another consistent theory of quantum gravity that has nothing to do with string theory, you will be able to falsify their general conclusions because 10^{500} counterexamples are enough. ;-)

Even your one counter-example is enough.

So when will their arguments fail? As I have said, you will see that their comments that the center-of-mass position of an object cannot have eigenstates will be wrong in your picture. But concerning their main argument leading to the baryon or lepton number violation, you will simply be unable to reproduce it.

Why? Because string/M-theory instructs you to use different rules to compute scattering amplitudes than Bambi and Freese do. In the correct rules, you never explicitly include loops of large black holes. For example, in perturbative string theory, you sum Feynman diagrams that look like Riemann surfaces. They can be informally interpreted as a summation over all possible particle species that arise as vibrational patterns of a string. But again, this translation is inaccurate because it would lead you to different regions of the moduli space of Riemann surfaces that you should use for integrals.

The rules of string theory naturally generalize the rules of a quantum field theory with infinitely many species but they are not quite identical.

When your calculation is finished, you will see that very large black holes "in loops" cannot cause any very large violation of the baryon and lepton numbers. Instead, you will see that string theory - and quantum gravity - suppresses the processes induced by quantum gravity effects by the Planck scale. For example, for a baryon-number-violating dimension-six operator O, the expected induced term in the low energy Lagrangian will be

  • O / M_{Planck}^2

which is compatible with observations (at least so far), except for theories with a low gravity scale where the potential problem is appreciated by the model builders. There will never be an amplification of such a term by a very large life expectancy of a very large black hole running in the loop. This is simply not how quantum gravity works. And I would say that quantum field theory doesn't work in this fashion either. The argument of the Bambi-Freese type would look naive even in the context of ordinary, non-gravitational quantum field theory.

Instead of vague pseudo-arguments based on shaky assumptions and speculative generalizations of Feynman rules, we can use the actual, established, tested rules of quantum gravity to see that the "paradox" definitely doesn't occur.

It is illegitimate to use the Feynman rules including black holes loops and even if we succeeded to translate the stringy calculation into a calculation similar to the Bambi-Freese setup, string theory would urge us to either eliminate some regions of the moduli space or include exponentially decreasing coupling constants or eliminate some species from the loops altogether. Perhaps, it would lead to cancellations that Bambi and Freese didn't expect.

I don't know which answer is correct because I don't know how to reorganize the correct calculation in the Bambi-Freese fashion, including the black hole loops. But I don't need to know these things to be sure about the answer to the main question: the existence of large black holes simply doesn't cause any arbitrarily strong violation of the baryon or lepton numbers. While it would be interesting to have a field-theoretical reorganization of some calculations that includes black holes in the loops, physicists are not obliged to find one. They can believe that it doesn't exist: I tend to believe it doesn't exist. However, if Bambi and Freese use such a speculative framework for their argument, they are expected to actually construct such a framework.

Everyone who has learned some string theory knows these things. But still, some results of string theory look mysterious - even though we know for sure that they are correct - because of arguments similar to those by Bambi and Freese.

Matrix theory: large gravitons

For example, in Matrix theory, the gravitons are represented as large bound states of some particles called D0-branes. The internal size of these bound states increases as a power of N, either as N^{1/3} or N^{1/9}. At any rate, if you send N to infinity which you really should, the bound states representing gravitons become astronomically large in Planck units. Nevertheless, these large clouds, even if their centers are a micron away from each other which means that the clouds are almost perfectly overlapping, will avoid any interaction with each other.

It is very surprising but we have some general words that explain "why" it is so. Supersymmetry cancellations could be a part of the answer. On the other hand, we probably need some inherently stringy explanation that doesn't rely on supersymmetry. The most typical comment we say to explain "why" the interactions between the clouds are negligible - a comment whose variations apply to many similar situations in string theory - is that most of the degrees of freedom responsible for the large internal radius are connected with very high frequencies and their effects on the slow degrees of freedom therefore cancel very accurately.

Again, it is important to understand that this mystery doesn't mean any uncertainty about the answer. The answer is certainly that the clouds' interactions are almost zero. The confusion is a psychological one and it is difficult to formulate it accurately enough so that it becomes a sharp paradox. There exist many situations in string theory where the theory shows that it is smarter than we are and it is able to achieve things that we would a priori consider impossible - including all the dualities, mirror symmetry, holography, interpolations between theories with different numbers of dimensions, and so on.

We can eventually understand the miracles, too. But it is true that Nature is often faster than we are. However, the goal of physics is not to paint ourselves as geniuses and fool and humiliate Mother Nature. The goal is to understand Nature.

Finally, I want to emphasize that even if our understanding of the Universe is not complete, there are many qualitative insights about quantum gravity that will never change because they are independent of the kind of ignorance that remains. The qualitative behavior of particles and black holes near the Planck energy and beyond has been understood. It is time for all authors of good and tolerable papers to learn these things.

And that's the memo.

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snail feedback (1) :

reader dan said...


I am using this space, if your moderators will allow me, to announce
to your readers and whoever else surfs on by, the start of the
international ''Vaclav Klaus Climate Joke Awards'' here:

They "honor" (sic) people who say stupid things about the climate
crisis. The awards are satire. Then again, maybe they aren't satire.
The page is up now and running and we are accepting nominations via
the comment section throught the year, anytime you spot a good quote,
send it in.


PS: if you want to do a separate blog one day on the awards blog,
please feel free to do so.