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The trouble with Krauss' criticism of TOE has published a rant by a fanatic named Lawrence Krauss,

The Trouble with Theories of Everything.
Most of it is an attempt to explain the important idea that most theories we use in physics are effective theories that are optimized for certain scales – for phenomena whose typical energy or typical length belongs to a certain fuzzily defined interval. It would be great if someone were explaining this important point, an unsung scientific revolution, as Krauss correctly calls it.

But he stops short of doing it right – and goes well beyond that, too. The insights about the renormalization groups aren't the goal of his tirade. Instead, they are just some new tools in Krauss' lame attacks against state-of-the-art theoretical physics.

Just to be sure, his text makes it clear that he doesn't quite know what he's talking about even when it comes to the scale-dependence of our theories itself – before it is expected to get any controversial. First, he tries to answer the question "At which level do the effective theories and the scale dependence become relevant?" His answer is:
The advent of quantum mechanics changed everything. When quantum mechanics is combined with relativity, it turns out, rather unexpectedly in fact, that the detailed nature of the physical laws that govern matter and energy actually depend on the physical scale at which you measure them.
But this is just bullšit. Quantum mechanics has changed many things but the importance of effective theories – the dependence of the right description on the scale – didn't start in quantum mechanics. The importance of a scale in many theories of Nature has nothing to do with quantum mechanics.

It obviously and demonstrably exists in classical, i.e non-quantum, physics as well. Even if he read the first sentence of the definition of an effective field theory at Wikipedia, he could be reminded about the wrongness of his statement. Effective field theories exist in statistical physics – even classical statistical physics – too.

This is particularly obvious because models of classical statistical physics are sometimes mathematically equivalent to quantum field theories. The partition sum in classical statistical physics is a sum over "spatial" configurations of the degrees of freedom and they may be a similar set – or even an isomorphic one – as the spacetime histories in the quantum path integral. Classical systems are some of the best toy models to learn the renormalization group – the ideas about the scale dependence. See e.g. this book about percolation. The critical behavior and scale invariance may be found there, too. And so on.

Similarly, the emergence of effective field theories and the scale dependence doesn't depend on relativity, either. We have effective field theories for non-relativistic systems, too. We must just be careful about the separate treatment of the length scales and time scales (or momentum scales and energy scales). They're inequivalent things. Well, in the truly "Galilean invariant" theories, a relationship between them gets restored but it's a different one. In classical mechanics, the kinetic energy is \(E=p^2/2m\) so the energy scales like the squared momentum. This differs from the relativistic \(E=pc\) but there is still a relationship between the energy and momentum scales.

It's pretty unpleasant to read texts like Krauss' because they're never quite right. It resembles some important physics but it's never done quite right. You know that the writer didn't quite know what he was talking about. Well, Feynman had the same experience with the books he was ordered to judge by their covers; search for "universally lousy" over there.

But let me get to the main question, namely whether a true theory that works at all scale does exist or can exist. Krauss writes:
We know of no theory that both makes contact with the empirical world, and is absolutely and always true.
Well, you don't because you're just a bunch of loud and obnoxious physics bashers. But what's more important is that the state-of-the-art theoretical physicists do know a theory that both makes a contact with the empirical world and is – almost certainly – absolutely and always true. It's called string theory. Later in his tirade, Krauss admits that people have lots of evidence that the previous two sentences are true but at the beginning, he prefers to pretend that he has never heard of string theory at all.

Near the end, he writes
This theory, often called superstring theory, produced a great deal of excitement among theorists in the 1980s and 1990s, but to date there is not any evidence that it actually describes the universe we live in.
This is absolutely ludicrous. The amount of evidence that string theory is right is strictly greater than the amount of evidence that quantum field theory is right. Because string theory may be shown to reduce to quantum field theory with all the desired components and features at the accessible scales, it follows that they're indistinguishable from the empirical viewpoint. That's why Krauss' statement is exactly as ludicrous as the statement that there exists no evidence backing quantum field theory.

Needless to say, his statement is even more ludicrous than that because the evidence backing string theory is actually more extensive than the evidence supporting quantum field theory. Unlike renormalizable quantum field theory that bans gravity, string theory predicts it. And one may also mention all the Richard-Dawid-style "non-empirical" evidence that string theory is correct.

OK, near the end, where he already admits that there exists a theory that makes his initial statements about non-existence wrong if the researchers in the subject are right, we read:
While we don’t know the answers to that question [whether there is a theory that is valid without limitations of scales], we should, at the very least, be skeptical. There is no example so far where an extrapolation as grand as that associated with string theory, not grounded by direct experimental or observational results, has provided a successful model of nature. In addition, the more we learn about string theory, the more complicated it appears to be, and many early expectations about its universalism may have been optimistic.
The claim that "there has been no theory not grounded by direct experimental or observational results that has provided a successful model of Nature" is clearly wrong. Some ancient philosophers – and relatively modern chemists – have correctly guessed that the matter is composed of atoms even though there seemed to be no chance to see an individual atom or determine its size, at least approximately.

But it was correctly "guessed", anyway.

Another example: When he was completing his equations, Maxwell added his new correction term – pretty much the only one he has added (the rest of his work on the equation was "unification of previous results"). Changing electrical fields induce magnetic fields swirling around them – a mirror image of Faraday's insight. Maxwell didn't have any direct experimental evidence relevant for that term at that time. He used consistency or symmetry to get convinced that it had to be there. And be sure, it's there.

Also, both special and general theory of relativity were constructed without any direct experimental or observational results. The Morley-Michelson experiment could have "slightly" invalidated the previous statement except that Einstein has always claimed that he wasn't aware of that experiment at all and it played no role in his derivations. Similarly, general relativity was constructed by purely theoretical methods. The perihelion precession of Mercury was nice but it was just a "by the way" observation that Einstein noticed. This anomaly was in no way the "soil" in which the research of Einstein that produced GR was "grounded".

Also, people could have "guessed" the theory that the beta-decay is caused by the virtual W-bosons and Z-bosons – at the time when they could only produce energies in particle collisions that were an order of magnitude too low for the production of such new particles. But the theoretical constraints in the construction of the electroweak theory are so severe that we simply don't have a choice. The beta-decay exists, there have to be charged currents, and from the rate of the beta-decay which is very slow, one may determine the mass of the W-boson which is high.

I could go on and on and on. The requirement that one needs to ground the research in "direct experimental or observational results" is absolutely preposterous and if unhinged fanatics like Krauss had been imposing this dogma throughout the history of science, about 1/2 of the breakthroughs would have become impossible or they would demand centuries of extra waiting or more.

The constraints filtering candidate theories of quantum gravity are much stronger than those in the electroweak theory which is why we may deduce certain conclusions about physics that is even further on the energy axis from the "current experiments" than the W-bosons were from the experiments when people began to understand that the beta-decay is due to such heavy particles.

Undeformability of string theory

But the essential questions for his tirade are
  • Whether string theory may be an effective field theory approximating another theory.
  • Whether it only works up to a certain scale and than breaks down.
The evidence is overwhelming that the answers to both questions are No.

Let me remind you of some of the basic relevant facts. In quantum field theory, you may include some new "correction", a new operator in the Lagrangian with a tiny coefficient, e.g. a high-energy operator. Or you may introduce some new heavy particles that are hard to produce. These modifications of the original theory will slightly change the predictions. The deformed theory may remain compatible with all the experiments, within the error margins, if the deformations are small enough.

We know that this is simply not possible in string theory. This statement isn't hype or a TV commercial or a wishful thinking. It is an easy-to-see technical result, at least perturbatively, and it is mentioned in a subset of basic string theory textbooks. In perturbative string theory, we define the whole spacetime dynamics using an auxiliary theory on the two-dimensional world sheet. This theory has to be scale-invariant – conformal – for consistency. The Weyl (scaling) transformations on the world sheet have to be a gauge symmetry in the covariant description because they're needed to decouple all the physical modes of the two-dimensional graviton field on the world sheet, and so on. It's easy to show that string theory amplitudes extracted from a non-conformal world sheet theory will be inconsistent. Unphysical modes, negative probabilities, divergences, all these inconsistencies and others will immediately emerge.

But the space of deformations of a conformal field theory is finite-dimensional. There are only a finite number of parameters in the 2D CFT that you may change or deform to stay consistent. These deformations are actually in one-to-one correspondence with the moduli spaces, roughly speaking the configuration spaces for the scalars in the spacetime etc. Every deformation you can make is a solution. And the "string theories" with differently deformed 2D CFT dynamics are actually equivalent. They are the same theory. You may always get the string theory from the "deformed world sheet dynamics" as a state in the string theory resulting from the "undeformed world sheet dynamics". It is a state in which some particular particles – which demonstrably existed in the undeformed theory – got condensed. One may pinpoint the right particles to condense: there is a one-to-one correspondence between the particles in the spacetime and the operators on the world sheet (the so-called vertex operators) that you may add to the world sheet action.

There is only one (super)string theory. This is a hugely nontrivial claim which is nevertheless compatible with all the thousands of papers full of insights that we know about string theory.

String theory won't break at too short length scales

Also, unlike (some) effective quantum field theories, string theory demonstrably doesn't break at length scales shorter than a finite cutoff. Its range of validity is unlimited in this sense. This is also a totally technical, and easily understandable, result.

First, note that perturbative string theory is a theory that is designed to produce the exact results at energies comparable to the string scale \(1/\ell_{\rm string}\), or the length scale comparable to the string length \(\ell_{\rm string}=\sqrt{\alpha'}\). So when the energies are comparable to the string scale, we get all the nice exact formulae, like the Veneziano amplitude, and all the exact corrections.

But string theory doesn't break down when the energies are higher than the string scale, either. Or when the distances are shorter than the string scale. To mention a trivial example of the latter, consider T-duality. When a spatial dimension is compactified on a circle of radius \(0.01\times \ell_{\rm string}\), the physics is absolutely equivalent to physics on a circle of radius \(100\times \ell_{\rm string}\). The inverse radius gives the same physics as the original one. So it's not more mysterious.

When the distances gets really short, even in comparison with the small value of the coupling constant \(g_{\rm string}\) or some of its important positive powers, the perturbative expansions in string theory may become problematic. 20 years ago, people could have been afraid that string theory wasn't well-defined in the "even more extreme" conditions. But in the mid 1990s, Matrix theory and AdS/CFT became the first two full definitions of physics of string/M-theory that works totally universally, at all energy scales etc. There is no realistic doubt today that while something could have gone wrong, ill-defined, or ambiguous at shorter length scales or higher energies, it doesn't.

If the coupling constant is of order one, strings are "thick" and "strongly interacting" and their states morph into black hole microstates. But the qualitative connections remain firmly in place. There are T-duality-like relationships between short-distance physics and long-distance physics – which are more generally known as the UV-IR (ultraviolet-infrared) connections. When a black hole is large, the curvature near the event horizon is low, and that's why the low-energy, low-curvature theory is sufficient for high masses, too. Because string theory contains gravity (and black holes), the observation in the previous sentence actually implies that the same laws that describe low-mass, low-energy phenomena unavoidably constrain (and, in fact, fully determine) the character and behavior of all the high-energy excitations, too.

So you know, there is a conceivably defensible room for the belief that string theory could turn out to be an inadequate description of the Universe although I think that such a negative view has been made extremely unlikely by now. But physics bashers like Krauss try to go well beyond these doubts: they are trying to question or even deny some completely well-established technical features of string/M-theory. And that's enough for them not to be credible scientists or sources of information or inspiration about the recent, contemporary, or future physics.

While arguing that a TOE shouldn't exist, Krauss wrote
But if we expect our theories to be complete, that means that before we can have a theory of anything, we would first have to have a theory of everything—a theory that included the effects of all elementary particles we already have discovered, plus all the particles we haven’t yet discovered! That is impractical at best, and impossible at worst.
He may talk in his characteristically arrogant way and formulate sentences in ways that want the listeners to agree that something is unlikely. But it's still true that there is nothing unlikely about it. Indeed, if you want a theory of everything, you need a theory of everything – something that is able to predict all the new things that may be predicted in the future. "We" are confident that we know what the theory is (even if we don't fully understand what the theory says about every question we would like to be answered). The claim that this possibility is "impossible" has been totally debunked and whether a theory of everything is "impractical" for answering some questions is totally irrelevant when we try to answer the essential question whether it exists and whether it's the exact theory of the Universe. A theory of everything is surely impractical for answering most questions that are too down-to-Earth but there's nothing unexpected or wrong about this fact.

It should be obvious to everybody that any conclusion that it can't exist boils down to someone's circular or otherwise sloppy reasoning and anti-scientific fanaticism. The final theory clearly can exist and the evidence has not only strengthened unbelievably that it does exist, but also that we know what the theory is. In fact, if you fixed some logical bugs in Krauss' argumentation, you might see that arguments similar to his strongly indicate that the final theory – in whose existence you may believe, like many others – probably isn't a particular quantum field theory. You may arrive at this conclusion because ordinary particular quantum field theories may always be deformed etc. so by their essence, they are effective or incomplete theories. You need "something like" string theory for a final theory and string theory is the only thing we know that is "something like" string theory.

By the way, Krauss largely repeats the now-standard delusions of numerous other mediocre physics-bashing pseudointellectuals. In this blog post, I haven't discussed this sentence yet:
In addition, the more we learn about string theory, the more complicated it appears to be, and many early expectations about its universalism may have been optimistic.
String theory's becoming "more complicated" only means that we are understanding the theory more finely than we did before, and it is an unquestionably a good thing. When Columbus saw the American continent for the first time, it could have looked like a triangle or a tetrahedron etc. But as the people began to live there, they were getting familiar with every mountain, every bay, every baywatch, and so on. The structure of America is rich, like many things in the real world.

Krauss pretends that our knowing the rich structure of string theory is a bad sign. But it is a complete nonsense. He is confusing a complicated theory that someone randomly and artificially constructed – probably because he wanted to circumvent the empirical falsification of his simpler theory – with the complicated structure of a theory that is fully determined. If there are many complicated aspects of a theory that remains unique, like string theory, it simply means that we understand many more predictions than we did before.

We are not creating an awkward theory with lots of extra knobs. The theory objectively exists as a mathematical structure and is unique. If we ask about the detailed properties of all the mathematically possible theories that behave as what was known as type IIA superstring theory since the 1970s, we still find out that there is only one (consistent) solution and all the answers may be completely determined. The theory seems to have features that you could name "many new knobs" – like different shapes of the hidden dimensions. It may allow a spectrum of branes with complicated unique rules about which branes can terminate on which other branes, and it allows fluxes etc. but all these things are totally determined by theoretical considerations.

It's exactly the fact that we haven't had any choice to change the structure of the knobs – and we haven't used any new empirical data to "adjust" the structure of knobs – that make string theory so powerful. All the "knobs" are being imposed upon us by the mathematical consistency and even when all these features we only understood in recent decades are properly taken into account, we still get a theory that is compatible with all the observations and experiments that have ever been made, within our current abilities to falsify theories. This is why it's so amazing: the rich stucture of the theory is determined by mathematical constraints but it nevertheless remains totally compatible with totally independent, empirical constraints.

Lawrence Krauss fundamentally misunderstands all these important things. He misunderstands modern science, he misunderstands our epoch. It's a question what really motivates him. I guess that he's been born as an obscene jerk who just can't live without spitting at things that are considered important by others. He may be accidentally right – but not always right – when he does it to religious beliefs. But he is almost totally wrong when he tries to bash modern physics.

He is just a semi-educated, semi-intelligent, ideologically blinded left-wing sourball and one should never confuse someone's being an unfriendly individual who spits on everything with someone's being a balanced skeptical scientist. These are different things because an honest skeptical scientist is also being skeptical about his own claims he makes whenever he tries to be as obnoxious an aßhole as possible. Krauss is never skeptical about those which is one of the reasons why he inevitably ends up writing texts whose majority is worthless rubbish.

The same issue of features an essay by "Mad Max" Tegmark, Life is a Braid in Spacetime. I haven't read it yet.

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