# Lorentz Violation and the Second Law

Lubos has a posting today about how Lorentz violating theories of quantum gravity seem to imply violations to the second law of thermodynamics:

http://motls.blogspot.com/2008/04/lorentz-violation-makes-perpetuum.html

I don't quite understand the arguments, but e states that the second law can be proven for any hilbert space'' which seems to imply that Lorentz violating theories don't have a good hilbert space.

First of all, can anyone explain this to me? Secondly, does anyone know why the arguments WOULDN'T apply to their favorite theory of Lorentz Violation?

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Haelfix
Theres something very fishy about this paper's arguments, and its been nagging me for a couple of days (this coming from a big believer in lorentz invariance at all scales). There is at first glance a rather liberal use of exchanging macroscopic and microscopic arguments along with quantum and classical. Anyway:

The first problem is that its not clear what you mean by having the 2nd law hold microscopically, or in other words how to define entropy for quantum systems. Its usually done within the density matrix approach, and directly ties into quantum information theory and entaglement etc. But I gather there are conceptual issues with this notion, which is why you have disagreement in the literature about how best to define it. But assume for now that what they do is valid, and any quantum system of N objects maximizes the probability for quantum entropy (Call it X) to increase and taking the classical limit is smooth as X becomes the usual stat mech entropy. Good!

Now, I can write down a renormalizable field theory that trivially violates lorentz invariance to a small degree. Coleman and Glashow did just that, and characterized exactly how many deformations are allowed. The theories they write down, it seems to me, fully respect any microscopic notion of entropy.

Now take one of those field theories, and include gravity and then UV complete it in some way so as to make a theory of quantum gravity. Assume such a thing exists!

The argument essentially implies then that somewhere between the inclusion of gravity and the UV completion of the theory, somehow we have broken the criteria on the maximization of X. --I don't see how thats possible, even in principle.

I must admit two things: I am unfamiliar with the work of Glashow and Coleman, and that I am pretty ignorant about how the second law of thermodynamics gets incorporated into a quantum field theory. On the first count, I'd like to hear a synopsis of that work, or have a reference to something I could (in principle, at least) read. As to entropy in QFT, naively it's all a phase space argument---i.e. the reason that taus decay to electrons and muons with equal probability is because the final states are all more or less massless, for example.

But this is based on defining a good hilbert space of in'' states and out'' states, so that you can write down a unitary S matrix. Lubos' comment that the second Law can be proved for any hilbert space seems to me to support this.

Now take one of those field theories, and include gravity and then UV complete it in some way so as to make a theory of quantum gravity. Assume such a thing exists!
I'll have to think about this last part for a bit longer. Is this hypothetical UV completion Lorentz violating?

Haelfix
Try the wiki on Von Neuman entropy or the density matrix formalism.

But yea, by hypothesis we want the uv completion to be lorentz violating as well. Of course, its very likely no such thing exists, but then the argument is irrelevant.

Fra
Second law => bound on relative change

I wonder if someone knows of anyone that has speculated about a generalised form of the second law as implying a generalized kind of bounded relative change. So that one can see the bound of information progagation more or less as a consequence of a generalized second and and thus in a probabilistic setting, meaning that we see the second law as a statistical expectation and this should imply that similar fluctuations of the lorentz symmetry is expected simply because it's not possible to construct a confidence measure of the symmetry. In a sense one could assign a confidence in the arrow of time... so that as the scaling thins the statistics then the confidence in this arrow also drops. In that sense perhaps the entire spacetime and possible SR and GR would follow locally from statistical considerations only?

For the last year I've been thinking about this, that information geometry is the link between the second law and geometry, and that the original reasoning that leads to the second law (probabilistic reasonings based on microstructures), is the same reasoning that should induce geometry. But I'm still waiting for more findings on this.

/Fredirk

Hans de Vries
Gold Member
Using a black hole with a "fuzzy" event horizon to separate higher energy particles from
lower energy particles? Aren't there simpler ways to do something like that?

-Like a prism for example which separates white light in higher and lower energy photons.
-Or a magnetic field which separates higher and lower energy charged particles.

Doesn't sound so fancy like black hole thermodynamics though......

Regards, Hans