# Statistical physics of string theory

Is there any connection between statistical physics and string theory, or a statistical interpretation of string theory, perhaps? I mean, the way electromagnetic forces and thermodynamic laws are described as emerging are all describable by statistical physics formulations. String theory accounts for gravity (I don't really know where the other two fit in), but does it do so (or is there any effort to develop a formulation) in a way reminiscent of statistical physics?

If this is a silly question, please explain why.

• I'm speaking from a standpoint of ignorance here, but I was under the impression that electromagnetic forces arise from manifestly relativistic origins. Does relativity often appear in statistical physics treatments? – S. Carnahan Jul 14 '10 at 3:53
• I didn't mean the electromagnetic force per se, but rather the form of magnetic fields. – DoubleJay Jul 14 '10 at 4:19
• DoubleJay, what are you talking about? – userN Jul 14 '10 at 4:28
• @Scott: They do. And they do. Well, they arise from a relativistic U(1) gauge theory, anyway; you can have relativity without E&M of course. And relativity definitely appears in statistical physics, because you need quantum mechanics to do statistical physics right, and you need relativity to do quantum mechanics right! But you're right that E&M isn't statistical in the same way thermodynamics is. – jeremy Jul 14 '10 at 4:49
• Why? I don't see anything in there beyond basic undergrad E&M? Or am I missing something here? – jeremy Jul 14 '10 at 15:08

What you seem to be thinking isn't really statistical physics, but effective field theory (EFT). Loosely, EFTs take a more fundamental theory, average something out, and give you a field theory that works in some domain of applicability.

Electromagnetism in matter is an EFT because it takes electromagnetism in vacuum with lots of point sources (atoms, molecules, electrons, ...) and, because we can't solve E&M with $10^{23}$ sources exactly, we average over them, making some simplifying assumptions, and get a new theory. In this case, E&M in matter looks almost exactly like E&M in vacuum, but with some extra undetermined constants ('permittivity', etc.) which are determined by 'microscopic' physics, but we usually just measure because it's very hard to calculate them. (It looks just like in vacuum by construction--you can make less aggressive assumptions and get nonlinear theories that apply, e.g., in very high frequency fields.)

Similarly, classical E&M in vacuum is an EFT of quantum electrodynamics, where we average out the quantum fluctuations.

The picture of particles as "composite" particles that behave as particles that show up as single fields in a Lagrangian (e.g,. scalar mesons in a Klein-Gordon equation) are effective field theories that average out the internal degrees of freedom.

It turns out that basically every theory we know of is an EFT in some way (other than string theory, which, not being a field theory, cannot really be an EFT). In particular, the standard model is an EFT. It is believed (for good reasons, beyond the scope of this answer--read and understand any intro string theory book for a basic answer!) that the standard model is an EFT of string theory. That is, we take string theory, average out the 'stringy' degrees of freedom, and we get out the standard model.

Part of the reason we believe this is because we can get both GR and normal quantum mechanics (and various quantum field theories) in 4d in this manner from string theory. We can also get many field theories that look very much like the standard model but have different particle contents. But we can't yet get the standard model exactly out.

We do however, since we can get GR out in a limit of string theory, have a good enough description of it to start to understand stringy effects of gravity. In particular, in normal GR, it is well-known that black holes behave as "thermodynamic systems" in some sense. This was first pointed out in a famous paper by Hawking! It seemed that black holes behaved like thermodynamic systems in the sense that certain dynamical variables satisfied suspiciously identical equations to the laws of thermodynamics.

In particular, the surface area seemed to 'be' the entropy of a thermodynamic system. (For details, see: http://www.scholarpedia.org/article/Bekenstein-Hawking_entropy)

For various reasons, it was taken fairly seriously that black holes should be actual thermodynamic systems. One big reason was that it was found (also due to Hawking) that black holes should emit thermal (i.e., black body) radiation as if they were thermodynamic systems in equilibrium. The radiation emitted (Hawking radiation) was identical to radiation emitted from a blackbody at a temperature that you'd expect if the black hole was a legitimate thermodynamical system.

This was puzzling, because the entropy associated with a black hole was extremely large, but it is a well-known result (see Wald or Misner, Thorne, and Wheeler's intro GR books) that the event horizon of a black hole is featureless. In other words, any 'patch' of event horizon is identical to any other. In other words, there is only one state in classical GR: a featureless event horizon shielding a featureless singularity. (If you're concerned about other particles inside giving the black hole a large entropy, the particles inside cannot normally have an entropy high enough to account for this, and at any rate you can always consider the state after some finite (far-away) time when everyone has hit the point-singularity, which, being a point, has no structure.)

So this is a puzzle. Classical GR says there is only one state. Semi-classical GR, and classical GR assuming it's a real thermodynamic system, both say there are a gigantic number. What's going on here? What are the microscopic states?

Realizing that we have to consider microscopic states makes us see that GR is not the right picture to do this. GR is the already averaged theory. Just like a naive picture of a gas in terms of its extensive quantities--temperature, pressure, volume,...--seems to tell you there is only one state: a gas described by (T,P,V,...).

It turns out, string theory (and no other theory!) lets us calculate the entropy of black holes by counting microscopic states! The difficulty is that, unlike a gas where the states are |particle(x,y,z,spin,....)>, the string states are really screwed up looking states that have absolutely no classical analog (and I don't understand well enough to describe simply). But we can count them, and when we superimpose them we get a classical black hole plus quantum corrections, when we average them to get an EFT we get a classical black hole, and when we count them we get its entropy!

So, yes, there is a connection between strings and statistical physics. That connection is averaging over stringy states gives you an EFT, and one we can calculate is when that EFT is GR, and we can use it to calculate black hole entropy by counting stringy states explicitly.

Edit: References included due to discussion in the comments:

• "Exploring Black Holes" by Taylor and Wheeler. Basic black hole stuff, including discussion of shell coordinates (coordinates of observer at fixed $r=r_0$ away from the black hole) which show $dr_{\text{shell}}/dt_{\text{shell}} \rightarrow c$ as $r\rightarrow 2M = r_{\text{horizon}}$, and who observes things in what systems.

• "General Relativity" by Wald and "Quantum Field Theory in Curved Spacetime" by Wald. Intermediate to advanced level discussion of black holes, horizons, black hole entropy, and black hole thermodynamics in the first. In the second, more formal qft in curved spacetime, Hawking radiation, and entropy.

• "The Thermodynamics of Black Holes" by Wald (http://arxiv.org/abs/gr-qc/9912119). A review-level article on thermodynamics of black holes, which contains a more lengthy discussion of the quantum mechanics, statistical mechanical aspects, as well as various approaches to solving the "entropy problem." Many of the references in this paper are also very good.

• "Black Holes as Effective Geometries" by Balasubramanian, de Boer, El-Showk, Messamah (http://arxiv.org/abs/0811.0263). A research article showing the possible calculating the entropy in a way that lets us understand the geometry of the individual microstates explicitly. Difficult, but the intro parts should be reasonably accessible.

• Great answer. The hidden agenda of my question was to find out if string theory can provide the same inspirations for fields like machine learning and sociophysics that statistical physics (or EFT) models have. And the answer is, as far as I can infer from your answer - perhaps, but they would be much more weird and involved by far than anything out there currently. Fascinating. – DoubleJay Jul 14 '10 at 4:40
• A finite amount of far-away time will pass when the last photon of an in-falling observer leaves the event horizon. But the more correct way to rephrase what I said is to say that there is an associated (finite) amount of far-away 'relaxation time' after which any disturbances caused by in-falling matter die down, and the black hole looks identical to an 'eternal' black hole solution. This time is of order the size of the black hole, so this usually happens very quickly. – jeremy Jul 14 '10 at 15:04
• I don't understand this answer. Of course there is an intersection between statistical physics and string theory: consider systems with lots of strings (like black holes) and do statistics! But that doesn't mean there is any deep connection between the way string theory quantizes gravity and statistical mechanics. – Clay Cordova Jul 14 '10 at 20:06
• The entire community of theoretical physics disagrees with that... The stringy calculations of black hole entropy caused a lot of interest in the theory community exactly because of this! There is a deep connection in the same way quantum statistical mechanics is deeply related to macroscopic thermodynamics. Classically, there was absolutely no reason at all to expect GR had anything at all to do with a picture like this; that it does is really stunning! The expected picture before strings, was in terms of canonical QG, and looked quite different (and could not produce the correct answer). – jeremy Jul 15 '10 at 1:06
• Jeremy, I agree that there are deep statistical physics problems that are related to black holes. I'm a student of Vafa so believe me I appreciate black hole microstate counting. That being said, the fact that black holes are thermodynamic entities does not mean that gravity is statistical in string theory. As I explained in my answer below, the graviton persists in the spectrum to arbitrarily high energies so gravity is not statistical even if some objects which gravitate, like black holes, are. – Clay Cordova Jul 15 '10 at 18:58