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Gravity from spin-two gauge invariance

Yesterday, a blogging critic of physics argued that physics is a corrupt science that has no credibility. The author of these statements is an "improved" version of Peter Woit and Lee Smolin.

The latter individuals only argue that physics has no credibility if it requires brains that are stronger than their brains while the improved crackpot, let me call him Swolin Plus, also includes all of experimental physics, among other things, to his list of blasphemies.

What arguments does the recent crackpot use? Well, they're not really arguments. He misinterprets another posting by Christine Dantas, a woman from Latin America, who was impressed by a 2004 paper by a somewhat irrational physicist named T. Padmanabhan.

I remember that paper very well because I was asked about it and studied it in detail back in 2004: see also my text on sci.physics.research. We have also exchanged some e-mails with Prof Deser, one of the main people who have contributed to proofs of various theorems discussed in this text. Prof Deser was obviously irritated by the apparent inability of T. Padmanabhan to understand any rational arguments and proofs after a long chain of e-mail exchanges between these two Gentlemen.

Higher-spin gauge invariances

Perturbative string theory may be used to show that massless particles can only have spins 0, 1/2, 1, 3/2, 2. This conclusion follows from an analysis of the energy of various harmonic oscillators included in the string that contribute to the mass of the resulting particle. This conclusion beautifully agrees with facts about gauge invariance that may be derived using spacetime arguments.

If you consider any semirealistic physical system, it reduces to quantum fields at long distances - fields that are able to create particles. Because of the rotational symmetry, these particles may be classified according to their spin. For spins equal to 0 or 1/2, one only creates states of positive norms (think about the Klein-Gordon and Dirac fields). However, for spin 1 and higher, there are inevitably negative-norm states in the Hilbert space created by the simplest version of these quantum fields. For example, the time-like component of a 4-vector field creates states whose norm has the opposite (negative) sign than the space-like components of the same field.

Positive probabilities require a certain kind of beauty

Indefinite Hilbert spaces are unacceptable because they would lead to predictions of negative probabilities. Nature can't afford such bizarre things: negative probabilities would make Her mentally sick. She has one choice only: to decouple all the states with the wrong norm - and perhaps some additional states. If these wrong particles are decoupled, they don't interact with the rest of physics and become harmless.

Such a decoupling implies an infinite amount of accidents that are equivalent to a symmetry. In fact, it is a gauge symmetry. The adjective "gauge" means that the parameters of the transformations are functions of spacetime coordinates.

I prefer this quantum discussion because our world is quantum, after all. In a classical world, the inconsistencies caused by the absence of a gauge symmetry wouldn't be that terrifying - there are no negative probabilities in a classical theory because there are no probabilities there at all. However, they would imply some kind of instability. Let me return to the quantum setup because it is the right one anyway.

Because negatively-normed states must be eliminated for all fields with spin 1 or higher, we must find a corresponding gauge invariance. At the linearized level - where you only keep the quadratic parts of the actions - you essentially know what the gauge invariance for a higher-spin field must be. For spin-1 fields, you obtain

$\delta A_\mu = \partial_\mu \lambda$
while the higher-spin fields have additional subscripts of $A_\mu$. The parameters of the gauge transformations $\lambda$ receive some Lorentz vector indices, too, and the prescription for the infinitesimal transformation above may be supplemented with a few terms that differ by a permutation of the Lorentz vector indices. Half-integral fields also need a spinor index and gamma matrices may occur on the right-hand side.

When you analyze what kinds of gauge invariance are able to make all the dangerous negative-norm states harmless, you will find out that at the linearized level, the spin-1 fields need the electromagnetic gauge invariance described above, spin-3/2 fields require a linearized version of local supersymmetry, and the simplest spin-2 fields - symmetric tensors - force you to impose the following gauge invariance:
$\delta h_{\mu\nu} = \partial_\mu v_\nu + \partial_\nu v_\mu.$
We will roughly understand later why it is not too interesting to consider higher-spin fields. Also, you should notice that differential forms of rank p always generate spin-1 particles only (as opposed to spin p, for example). The maximum value of $J_{xy}$ equals one (in any spacetime dimension) because in a complex basis of $J_{xy}$ eigenstates, you are not allowed to repeat the indices $x+iy$ because of the antisymmetry of the tensor. Of course, massless differential forms are allowed (and frequent) in string theory, too, and it is no contradiction: their spin equals 1.

Both actions and the transformations may (or must) be deformed

The quadratic actions are not too interesting because they don't lead to any interactions. If you want to create a general enough, interesting, interacting theory, you must allow cubic and higher terms in the action. Also, you must be general and allow non-linear terms in the fields and the transformation parameters as your recipe for the gauge transformations: the first major problem that Paddy wasn't able to understand, after months of studying this simple problem.

Because gauge transformations form a group and groups are rather constrained, rare mathematical objects, there are not too many ways how you can add non-linear terms to the gauge transformation. For spin-1 fields, you inevitably end up with a Yang-Mills theory if you impose other physical requirements. For spin-3/2 fields, you end up with supergravity theories and an inclusion of gravitational spin-2 fields is forced upon you as a bonus. More generally, for spin-2 fields, the only mathematically meaningful non-linear modification of the transformation expressed above is
$\delta h_{\mu\nu} = \nabla_\mu v_\nu + \nabla_\nu v_\mu$
or something that is related to it by a field redefinition. The nabla derivative above includes the usual term with the Christoffel connection calculated from the metric
$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}.$
The original linearized gauge group that was isomorphic to some kind of $U(1)^\infty$ is deformed into the diffeomorphism group. The resulting class of theories with gauge-invariant actions that are allowed in this framework is equivalent to general relativity written in terms of variables $h_{\mu\nu}$ instead of $g_{\mu\nu}$. But it doesn't matter how you write things and what conventions or variables you use. What matters is physics: a fact that the broader loop quantum gravity community will never be able to understand due to their intrinsic mental limitations.

The tensor that is coupled to the field $h_{\mu\nu}$ must be conserved and it turns out that only if you choose anything else than the stress-energy tensor, the conservation of such a current will constraint the theory so severely that the interactions become impossible. That's why the spin-two fields are coupled to stress energy tensor in all interesting theories.

For the symmetric tensors i.e. spin-2 fields, you will see that deforming the Abelian gauge invariance into the diffeomorphism group is de facto the only way to get interesting interacting theories and the gauge invariance constraint leads to the same class of theories as the diffeomorphism symmetry usually dictated by general relativity - because it is really the same thing. Extra terms can be classified according to their dimensions - including the number of derivatives - as particle physicists like to do.

The trivial solution is just a limit

In fact, the "other" solution, namely to keep the Abelian gauge symmetry, is not a genuine new solution. The Abelian gauge symmetry for spin-2 fields we started with - one with partial derivatives instead of the covariant derivatives - is a contraction of the diffeomorphism group or, in other words, a limit of the diffeomorphism group for a vanishing Newton's constant.

All other examples of theories with tensorial gauge symmetries require an infinite number of massless fields and their interplay (see "massless higher-spin theories") - and they are thus not too physical.

You might think that it is sensible to treat theories with a vanishing Newton's constant - but non-vanishing higher-derivative terms - as a special class. But the renormalization flows really discourage you from this kind of thinking because if Newton's constant is vanishing at one energy scale, it is typically not vanishing at another scale. The "subclass" mentioned previously can't really be separated from the diffeomorphism-symmetric theories in a renormalization-flow-invariant way.

The same analysis of possible non-linear deformations implies our previous conclusions for other values of the spin: Yang-Mills theories are the only nice new theories for spin-1 fields, supergravity is the only nice solution for the spin-3/2 case, and there are no interesting theories of massless particles with spin greater than two. Moreover, even massive spin-1 theories are constrained: the gauge symmetry must always be broken by some kind of Higgs mechanism.

In string theory and quantum gravity in general, there are many massive particles with much higher spins but the quantum field description becomes unusable in all these cases. These massive high-spin fields depend on the whole infinite tower of string states, including the massless ones, and the solutions of string theory are the only new loopholes of the otherwise strict rules of local effective quantum field theories with gauge invariances.


There are robust and reliable spacetime arguments constraining possible spins of massless fields and the form of their interactions derived from gauge symmetries. These conclusions agree with the known types of massless spectra in various vacua of string theory even though these facts are derived very differently in string theory than they are derived in effective field theory.

Paddy's paper has had two parts that were both wrong. One part was wrong because
  • he didn't realize that one must allow non-linear terms in the gauge transformations which is why he missed the "bulk" of this class of theories, namely theories equivalent to generally covariant theories (gravitational theories) - the real meat of this whole enterprise
  • he didn't realize that the class he considered, one with Abelian gauge invariances, is a special, measure-zero subset of the previous generally covariant class that can't be separated in a renormalization-flow-invariant way
  • moreover some of the "new" gauge-invariant actions he proposed were not really gauge-invariant because of technical errors in his calculations

At any rate, all this stuff shows that Paddy hasn't really understood the old papers to start with. What he wrote is a summary of his confusion, not a path to a new result in science. But you can see that wrong papers are immediately abused by all kinds of crackpots and science-haters, and because most journalists in the present era are also science-haters, to one extent or another, a variant of these anti-physical sentiments is often sold to the general public.

That's not too good. Needless to say, the situation with crackpots such as Peter Woit and Lee Smolin is analogous. The content of all criticisms of science by these people - especially their fanatical anti-mathematical sentiment - is a pure reflection of their own severe intellectual limitations and 100% of support of this crap in the society is created by combining the screams of people who are equally stupid and dishonest as Smolin and Woit or even more stupid and dishonest.

A sensible reader of newspapers and blogs will never pay too much attention to millions of ignorants and morons who have no idea about the subject. You have always a much higher chance to find the correct answer if you ask one person who actually knows what he is talking about.

And that's the memo.

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