## Sunday, February 01, 2009

### Computational universe vs Lorentz symmetry

Moshe Rozali wrote a very sane text about the importance of Lorentz symmetry for the search for the fundamental laws of Nature:
The Universe is probably not a quantum computer
I agree with every word he wrote. He says that many people who are following the physics blogosphere want to believe that their area of expertise is actually sufficient to find a theory of everything.

So Seth Lloyd of the quantum computing fame wants to believe that the world is a quantum computer. Robert Laughlin wants to imagine that quantum gravity is an example of the fractional quantum Hall effect. Other people have their own areas of expertise, too. Peter Woit wants to believe that a theory of everything can be found by mudslinging and defamations while Lee Smolin wants to believe that the same theory can be found by selling caricatures of octopi to the media (following some subtle and not so subtle defamations, too).

Moshe Rozali correctly tells them that if they are going to ignore the Lorentz symmetry, a basic rule underlying special relativity, they are almost guaranteed to fail. Lorentz symmetry is experimentally established and even if it didn't hold quite accurately, it holds so precisely that a good theory must surely explain why it seems to work so extremely well in the real world.

Moreover, the state-of-the-art theories of the world are so constrained - i.e. so predictive - exactly because they are required to satisfy the Lorentz symmetry. Because of this symmetry, quantum field theories only admit a few marginal or relevant deformations. If you assume that they make sense up to extremely high energy scales, you may accurately predict all of their low-energy physics as long as you know a few important parameters. Such a "complete knowledge" of physics in terms of a few parameters would be impossible in non-relativistic theories.

String theory is even more constrained than quantum field theory: it has no adjustable dimensionless non-dynamical parameters whatsoever. In some sense, you may view string theory as a tool to generate privileged quantum field theories with some massless spectrum and infinitely many very special, selected massive fields with completely calculable interactions. So all the Lorentz constraints that apply to quantum field theory can do the analogous job in string theory, too.

However, in string theory, the character of Lorentz symmetry is even more direct. The very short distance physics of string theory is pretty much guaranteed to respect the Lorentz symmetry. Whenever you look at regions that are much smaller than all the curvature radii of a D+1-dimensional spacetime manifold, the dynamics of a closed string reduces to a collection of D+1 free scalars on the worldsheet which manifestly preserves the Lorentz symmetry. And one can show that the interactions respect it, too.

Open strings may violate the Lorentz symmetry spontaneously, for a nonzero B-field or a magnetic field on the brane, and one can enumerate a couple of related ways to spontaneously break the Lorentz symmetry with the presence of branes and their worldvolume fields. But none of these pictures ever hides the fact that the fundamental theory behind all these possibilities is Lorentz-invariant.

There's a lot of confusion in the public about the fate of the Lorentz symmetry in general relativity. Be sure that the Lorentz symmetry is incorporated into the very heart of general relativity.

General relativity generalizes special relativity; it doesn't deny it. General relativity can be defined as any collection of physical laws that respect the rules of special relativity (including Lorentz invariance) in small enough regions of spacetime - regions that can, however, be connected into a curved manifold. All breaking of Lorentz symmetry in general relativity can always be viewed as a spontaneous breaking by long-distance effects and configurations.

In fact, even in spacetimes with a lot of curved regions - such as spacetimes with many neutron stars or even black holes - one can use the tools of special relativity in many contexts: either in very small regions that are much smaller than all the curvature radii, or in regions that are much larger than stars and black holes. In the latter description, the stars and black holes may be viewed as local point masses or tiny disturbances that follow the laws of relativistic mechanics at much longer distances, anyway.

So if someone completely neglects Lorentz invariance, the player that became so essential in 1905, he shouldn't be surprised if theoretical physicists simply ignore him or her. It is not necessary for a theory to be Lorentz-invariant from the very begining. But a theory only starts to be interesting as a realistic theory of our world after one proves that Lorentz invariance holds exactly (or almost exactly).

I am personally convinced that theories that try to break Lorentz invariance by small effects are not well-motivated. But even if I insist on the things that have been established only, the "at least almost accurate" Lorentz symmetry that has been demonstrated is an extremely powerful constraint on any theory. If you invent a random theory for which no reason why it should be Lorentz-invariant is known, it is extremely likely that the Lorentz symmetry doesn't work at all and the theory is therefore ruled out.

There are actually approaches to string theory that are not manifestly Lorentz-invariant. For example, the BFSS matrix model, or M(atrix) theory, is a 0+1-dimensional quantum field theory - a U(N) gauge theory with 16 supercharges. You can also say that it is a quantum mechanical model with many degrees of freedom organized into large Hermitean matrices. It resembles non-relativistic quantum mechanics, with some extra indices and a quartic potential.

There is no a priori reason to think that such a seemingly non-relativistic theory - whose symmetry actually includes the Galilean symmetry known from non-relativistic physics - should be Lorentz-invariant. Except that one can defend and "effectively prove" this relativistic symmetry by arguments based on string dualities. Although it can't be completely obvious from the very beginning, the original BFSS matrix model describes a relativistic 11-dimensional spacetime of M-theory. But the relevance of the matrix model for M-theory only began to be studied seriously when arguments were found that these two theories were actually equivalent.

You simply can't expect your non-relativistic model to be equally interesting for physicists if you don't have any evidence that your model respects Lorentz invariance - or if it even seems very likely that it cannot respect it. Physicists would be foolish to treat your theory on par with QED or the BFSS matrix model because it seems excessively likely that your theory can't agree with some of the basic properties of the spacetime we know.

In the discussion below Moshe's article, Giotis argues that the Lorentz symmetry cannot be preserved at high energies because "it is broken by gravity". Well, it's definitely not, at least not according to the descriptions (effective field theory; string theory) that we are using these days. More generally, symmetries in effective quantum field theory tend to hold at high energies/temperatures and they are broken at low energies/temperatures while Giotis rotates this rule upside down.

To make things worse, Moshe tells Giotis that "what he is saying is fine". Well, it's clearly not as Moshe could have noticed by observing Giotis' profound dissatisfaction.

Finally, Jacques Distler "sees a tension" between the importance of Lorentz invariance on one side and the emergent character of space on the other side. Well, Jacques, be sure that the only tension is that these conditions are constraining. The real statement about these two facts is that spatial geometry is emergent and whenever it emerges and is much longer than the fundamental scale, the Lorentz symmetry mixing these spatial dimensions and time must hold.

This also answers Jacques' question about the number of dimensions: Lorentz invariance must hold for all dimensions that are larger than the fundamental scale. We will look at similar issues momentarily.

Emergence and the role of Lorentz symmety in the grand scheme of things

The comments above should be completely uncontroversial. But let me add a few more speculations.

Because space is emergent in string theory, the Lorentz symmetry - a symmetry linking space and time - has to be emergent, too. This symmetry of special relativity is telling us that things can't move faster than light in the newly emergent geometry. What is this constraint good for? Is Nature trying to tell us something deeper than that?

Well, I am confident that special relativity is important for life as we know it because motion is very helpful for animals and the equivalence of all inertial frames is the simplest (and maybe the only plausible) method for Nature to guarantee that the very motion won't kill the animals. Imagine that you would feel any motion - you would probably vomit all the time and die almost instantly. ;-)

The Lorentz symmetry and the Galilean symmetry were the two most obvious realizations of the equivalence of all inertial frames that Nature could choose from, and She chose the Lorentz symmetry because it treats space and time more democratically than the Galilean symmetry. (I could probably construct more robust anthropic arguments even though they would probably not be based on the motion of animals only - simply because the low value of "v/c" for animals indicates that the finiteness of "c" is not necessary for life itself.)

But in the previous two paragraphs, we were talking about the 3+1 large dimensions of spacetime only. String theory has additional dimensions that can emerge in various ways and that are dual to each other - and the Lorentz symmetry applies to all these dimensions as long as they become larger than the curvature (and compactification) radii. In some sense, that's quite shocking.

Imagine that you study T-duality. Compactify a string theory on a small, substringy circle of radius R. There will be a lot of winding modes of the closed strings - an approximate continuum of them. As you know, physics will be equivalent to physics on a circle of radius 1/R, in string units, and the winding modes will become momentum modes. If R is much smaller than one, 1/R is much longer than one.

So on the T-dual circle of radius 1/R, Lorentz symmetry (and the speed limit given by the speed of light) must hold. Imagine that you translate this symmetry into the language of the winding modes: it will look very awkward and its consequences, such as the speed limit, will sound surprising if not miraculous. If you construct Fourier series out of your winding modes, there will be a universal speed limit: wouldn't it be shocking if you knew nothing about T-duality? You could probably rediscover T-duality if you happened to derive the speed limit first.

Nevertheless, all these symmetries have to hold. Of course, once you prove that physics is equivalent to a "weakly coupled string/M-theory" of a known type on a large enough manifold, the Lorentz invariance becomes indisputable and proven. But I feel that even before you prove such things, string theory must "know" about the required Lorentz symmetry that applies to all geometries that can "emerge" in a given physical picture.

I am not sure whether this constraint is vacuous or non-trivial for geometries whose size is comparable to the fundamental scale. But I still feel that the Lorentz symmetry in "all conceivable emergent geometric pictures" may imply a lot of new inequalities - and perhaps even identities. Can you tell me what they are or prove that they don't exist?