## Saturday, March 03, 2007

### Steven Weinberg vs obsolete physics ideas: torsion

A reader has pointed out the following exchange between Steven Weinberg and Friedrich Hehl in the new issue of Physics Today. Prof Hehl has written down some theories of gravity with an extra torsion tensor. They have no relevance for experiments and they have nothing to do with the most important advances in physics of the last 100 years or so.

Prof Hehl offers some scientifically unsubstantiated statements or loud assertions why the torsion tensor is needed or why it is special, citing some physically irrelevant sources from the early 1920s. Weinberg of course gives the only answer that a sane physicist can give: the torsion tensor is just another tensor field - one that isn't needed for any symmetry, consistency, or beauty - so if there is no experimental reason why it should be added, and surely there is no such reason today, it won't be added. Period.

Weinberg as a relativistic heretic

The origin of this controversy goes back to the 1970s. Weinberg's textbook on general relativity was very modern - and oriented towards the interpretation of general relativity as a part of the effective quantum field theory - as it presented the metric tensor as another field in spacetime whose local symmetry happens to coincide with the diffeomorphism symmetry but it is just a technical detail: interacting spin-two fields simply must have gauge symmetries that reproduce diffeomorphisms. Because of that, we can interpret the whole theory as a theory of curved space but we don't have to: the metric tensor may also be viewed as another field living in the Minkowski spacetime or, equivalently - by symmetries - any other spacetime that you might imagine to be your starting point. You don't need to know the words "curved space" to calculate the predictions of general relativity.

A certain group of people in cosmology has reacted just like religious bigots and they wanted Weinberg to "retract" these statements whose validity is completely obvious to anyone who has any idea how field theory - especially quantum field theory - works. However, the deeper you penetrate into the community of the loop-quantum-gravity-like pseudoscientists and their fans, the less clear these things are to them. Weinberg has never retracted but I think that it is fair to say that these loud irrelevant fourth-class scientists have intimidated Weinberg into silence which is kind of scary.

Unification vs segregation

Those people think exactly in the opposite way than a theoretical physicist should. Theoretical physicists want to unify the laws of Nature. They want to understand an ever greater set of phenomena using theories with an ever smaller number of independent assumptions and parameters. Gravity is a manifestation of something that we can call spacetime geometry - but all of physics may be viewed as a manifestation of some "generalized geometry". There is no fundamental gap between gravity and other fields. There is only one world whose parts constantly interact. Any attempt to separate the world into two parts - geometry and matter - is bound to be an approximation or worse. All of these objects in field theory are just some tensors that are coupled according to some rules.

In fact, string theory shows that the metric tensor field and the matter fields arise in the very same way from more fundamental ingredients.

What's important for these interactions is whether they respect some crucial symmetries and whether they lead to self-consistent predictions that are finite and whether these predictions can be successfully compared to experiments. We also want the number of independent parameters - the total number of all coefficients of terms that can be added without modifying the symmetries - to be as small as possible so that the theory's predictivity is as large as it can be.

Regardless of words, the most general interactions between your tensors must be considered

Everything else is just religious nonsense. You may try to guess other principles or ideas how the theory should look like that can lead you to the right theory if you're lucky. But they don't have to. You can't consider your own idiosyncratic beliefs to be an argument for your approach before any other material evidence - either theoretical or experimental - for your theory appears.

What about the torsion? Torsion is a hypothetical part of the Christoffel connection that is antisymmetric in the lower two indices. In conventional general relativity, the symmetric connection is derived from the metric tensor and its torsion is simply zero. This is the grand theory that has been successfully tested. The three-form H-field in many vacua of string theory may be viewed as some kind of torsion. It's because the conditions for an unbroken supersymmetry include the term proportional to the H-field in a way that is analogous to the old papers that discussed torsion together with spinors.

Other fields in string theory

But if you don't know this "torsion" jargon, you don't lose anything. The two-form B-field and its exterior derivative, the H-field, are just other examples of fields in the effective field theory. They have some couplings and some gauge symmetries and string theory predicts all of them, up to field redefinitions that can, of course, always be made. It is somewhat misleading to use the word "torsion" because we can't really say that all objects are affected universally by the background fields. It is more usual that we interpret the H-field as a generalization of the electromagnetic field than a kind of a torsion tensor. And we have good reasons to do so.

For example, charged objects are also influenced by non-gravitational gauge fields. In the presence of matter, it is no longer true that the geometry knows everything about the natural motion of objects in a general situation. We need other fields, too. Once we accept that there are other fields, we must consider the most general set of rules controlling these degrees of freedom that are consistent with the given symmetry and consistency principles. In particular, the torsion is just another tensor and it is not true that its couplings are completely determined. All contractions of indices etc. are legitimate a priori.

The statements that the dogmatic torsion is necessary because of [some incoherent principle] are completely dumb. Torsion is not necessary simply because the theories we have don't include any torsion, they are self-consistent, and they moreover agree with experiments. It is plausible that a more complete theory would predict new fields but these fields must be massive, otherwise they would contradict observations. For example, the three-form H-field in four-dimensional string-theoretical vacua may be Hodge-dualized to a one-form which is a gradient of a scalar field called the universal axion. This particle may or may not exist but it must be massive, otherwise it would induce new forces that are not observed.

Irrational pressures

At any rate, the idea that there are some additional aesthetic conditions in field theory that tell you that you should include fields that are otherwise clearly unnecessary or conditions that tell you that you shouldn't allow some interactions of some fields just because you want to use some name for these fields is analogous to astrology. Nothing like that can be used in science. Such new ideas could only become valid if you showed that they are necessary for some kind of new symmetry, or that they must arise from an underlying high-energy theory. At the sociological level, I am flabbergasted how the people who understand physics and contributed to physics at a rate below 0.1% of Steven Weinberg are self-confident when they try to intimidate him.

Einstein's flawed attempts

In the last decades of his life, Einstein used to think about many unified theories. He thought that only gravity and electromagnetism were real: everything else was supposed to miraculously emerge from the approach. So he has tried all the silliest theories you can imagine - for example, an asymmetric metric tensor whose antisymmetric part describes F_{mu nu}. Torsion was another example. The greatest mistake of Einstein was his inability to accept the probabilistic nature and predictions of quantum mechanics. But the unjustified attempts to "extend" the metric tensor in order to cheaply include electromagnetism may be viewed as the second greatest blunder of his life.

For example, if we imagine that the metric tensor is not symmetric, we are still allowed to split it into the symmetric and antisymmetric part. These two parts can be treated separately: they can have different interactions. If you treat them separately, you are still able to satisfy all principles of your field theory. The Lagrangian is locally Lorentz-symmetric and the full action is diffeomorphism invariant if you do it right. An action written in terms of an asymmetric tensor could "look" shorter than a general action describing the action for the symmetric part and the antisymmetric part but Nature never cares whether something "looks" shorter. For example, the action of eleven-dimensional supergravity is not really "short" but it is the most symmetric gravitational low-energy field theory that exists. It is symmetry and rigidity, not the length, that matters in physics. The crackpots won't ever get this point.

The same comment applies to torsion. If you consider an asymmetric Christoffel connection, you are still allowed to break it into pieces, i.e. irreducible representations of the Lorentz group or "GL(4)", and to add different interactions for these pieces. For diffeomorphism invariance, the symmetric part will be equivalent to what you get from a metric tensor, and the antisymmetric part is just another tensor field. There can't be any natural unification here. If your action looked simple in terms of an asymmetric metric tensor or an asymmetric connection, it would be a pure coincidence. You would still have to consider all possible deformations of this theory - in which the interactions of the parts differ - to be equally valid candidates to describe reality.

Horizons and the geometric intuition

Is there something in GR that you can't derive by assuming that the metric tensor is just another tensor field on some background - e.g. the Minkowski background? Well, GR predicts the existence of spacetime topologies and causal diagrams that differ from the Minkowski spacetime. Are they possible? Well, almost certainly. But still, their existence is compatible with the interpretation of the metric tensor as another field. The geometric intuition just gives you a good tool to deal with some singularities: for example, you may find that the black hole horizon is a coordinate singularity and you can continue your physical laws to the interior of the black hole. You can see that there is nothing special happening near the black hole event horizon.

But this conclusion also follows from a careful analysis of field redefinitions that are helpful to understand physics near the black hole horizon. These field redefinitions are nothing else than diffeomorphisms, and by making the geometry look smooth near the event horizon, you obtain a natural hypothesis what should happen when you cross the horizon: namely nothing. Experimentally speaking, we're not quite sure. We will never be sure unless the whole planet falls into a black hole which is not the best collective career move.

It can still be true that you die when you hit the black hole horizon. But the required laws would violate locality and causality - principles whose precise form is influenced by non-zero values of the spin-two tensor that we happen to call the metric. These principles are valuable. The dogma about the existence of torsion is not an independently valuable physical principle and Weinberg has always been 100% right when he rejected irrational arguments to include such "principles" into science.

And that's the memo.

Update: Dean of crackpots

I was also told that the dean of crackpots has written about this exchange. The dean himself offers several characteristically absurd comments attempting to paint Steven Weinberg as the owner of extreme opinions. Steven Weinberg is one of the people who have defined the mainstream of particle physics for more than 30 years.

In the discussion, some people including Sean Carroll and Moshe Rozali correctly say that one must include all terms in the Lagrangian that are consistent with given symmetries. The dean himself argues that "he understands the effective field theory philosophy", but in order to instantly show that he doesn't, he says that he is unconvinced because quantum field theories should be valid at all energies. QCD is and N=8 SUGRA may also be, so why not. Well, he's just too limited.

Whether or not these theories are well-behaved in the UV can't change the fact that new physics must surely enter at the conventional 4D Planck scale or earlier, for example because our world includes gravity. Our world can't be a pure QCD as the famous apple demonstrates. With gravity, all these theories are only effective field theories. Even in the case of N=8 SUGRA, the supergravity description itself is clearly incomplete non-perturbatively because it can't reproduce poles from the black hole intermediate states.

In the debate with Sean Carroll at the beginning of the debate, the dean shows that he clearly doesn't understand that the torsion is just another tensor and its couplings are not determined. It's just amazing how incredibly ignorant this person is - a person that has been chosen by dozens of journalists to talk big about physics.

Crackpot Tony Smith tries to spin some - already bizarre - statements by Paul Ginsparg who has conjectured that Steven Weinberg has "renounced his views". The similarity of their language with the medieval catholic bigots is clearly causing them no pain whatsoever. As an argument supporting the opinion that Weinberg has "renounced" his views, they say that Weinberg likes extra dimensions in string theory which are geometrical in nature. Well, that's nice that they are geometrical but the low-energy field theory in 10D is just another field theory with some tensors, and so is its decomposition in the form of the four-dimensional effective field theories. In all cases, it is Weinberg's rules of physics that are important, not pre-conceived opinions about "geometry".

All of string theory may be viewed as a certain generalization of geometry. The real question is how exactly the right generalization works. ;-) There's no doubt that string theory has already refined our notions about geometry - by topology-changing transitions, mirror symmetry, T-duality and so on. If we want to answer the question what is the right form of geometry in Nature, we must isolate the right physics arguments and calculations instead of attaching silly stickers "geometric - good" and "non-geometric - bad" to different ideas. If you choose any set of axioms or ideas that are called "geometry" at a given moment, you are never guaranteed that Nature is going to satisfy them. The previous sentence has been proved many times in the history of physics. It is She who decides, not you.

Another "wise man" called Eugene Stefanovich argues that Weinberg also has non-orthodox views on quantum field theory because he starts his derivation of the theory from particles which makes fields less fundamental. Last semester I have largely followed Sidney Coleman's QFT I notes that start from particles, too. What's exactly non-orthodox about it? All these concepts - including fields and particles - are ultimately parts of the overall picture. There is no God-given algorithm telling you what you should start with when you learn or teach these things. Any attempt to pretend that such a God-given algorithm exists is religious bigotry, not science. Every particle physicist who thinks that particles are not important in particle physics is deeply confused. Moreover, even if Coleman and Weinberg were the only two physicists who followed this approach, which they're not, it would no longer be a fringe pedagogical direction.

Why do we neglect higher-derivative terms

Peter Woit also completely misunderstands why we neglect higher-derivative terms in various theories such as the Dirac theory or general relativity. He argues that we must start with minimal couplings and boldly make predictions to avoid being not even wrong. But this approach is the obsolete perspective of the 1920s. Today, a physicist who understand her field would certainly not argue in this way. The reason why the higher-derivative terms (e.g. higher powers of curvature in general relativity) are not that important is that they are higher-dimension operators whose effects decrease faster as you go to longer distances: every derivative adds a 1/L factor to the typical size. The operators with many additional derivatives are called irrelevant perturbations and it is the most relevant ones that dominate the long distance physics. You can always choose sufficiently long distance scale so that the irrelevant operators will become as unimportant as you wish. There is no other rational justification to eliminate the higher-derivative terms - in fact, one can't completely eliminate them at all without contradicting the rules of the renormalization group flow. Even if the higher-derivative operators were absent at one scale, you generate them if you flow into another scale. They can't be absent universally.

Because Peter Woit argues that one should study "simple" theories of this kind because they are "beautiful" proves that his sense of "beauty" is based on ideas that have been known to be inconsistent with the laws of quantum mechanics for more than 30 years and he clearly can't understand anything important from the last 30+ years of particle physics. Beauty can no longer be measured in this obsolete Woitian way. It is no longer possible to truncate theories in this way. There is nothing special about the "minimal" theories he likes to think about. At the quantum level, one can't really define such minimal theories at all.

There's just far too much organized influence terrorizing people in science. Whenever your results or conclusions of your work disagree with a sufficiently large group of ignorants, they will attack you personally in the worst possible ways and hire unwise journalists who do the same in the media. They will present the fact that your results reject their preconceptions as your moral flaw.

I think that it has become extremely unpleasant to be a part of institutionalized science, and I am looking forward to be away from the focus of these intellectual bottom-feeders who exist not only on Not Even Wrong and who enjoy a silent approval by many of the leftist officials in the Academia.