I am coming back to you with a new format: Questions. Let me start with two questions I have been thinking about recently but that I don't know a good answer to.
't Hooft limit of classical field equations
The 't Hooft limit leads to important simplifications in perturbative QFT and is used for many discoveries around AdS/CFT, N=4 super YM, amplitudes etc etc. You can take it in its original form for SU(N) gauge theory where its inventor realized you can treat N as a parameter of the theory and when you do perturbation theory you can do so in terms of ribbon Feynman diagrams. Then a standard analysis in terms of Euler's polyhedron theorem (discrete version of the Gauss-Bonnet-theorem) shows that genus g diagrams come with a factor 1/N^g such that at leading order for large N only the planar diagrams survive.
The argument generalizes to all kinds of theories with matrix valued fields where the action can be written as a single trace. In a similar vain, it also has a version for non-commutative theories on the Moyal plane.
My question is now if there is a classical analogue of this simplification. Let's talk the classical equations of motion for SU(N) YM or any of the other theories, maybe something as simple as
d^2/dt^2 M = M^3 for NxN matrices M. Can we say anything about simplifications of taking the large N limit? Of course you can use tree level Feynman diagrams to solve those equations perturbatively (as for example I described here), but is there a non-perturbative version of "planar"?
Can I say anything about the structure of solutions to these equations that is approached for N->infinity?
Path Integral Entanglement
Entanglement is the distinguishing feature of quantum theory as compared to classical physics. It is closely tied to the non-comutativity of the observable algebra and is responsible for things like the violation of Bell's inequality.
On the other hand, we know that the path integral gives us an equivalent description of quantum physics, surprisingly in terms of configurations/paths of the classical variables (that we then have to take the weighted integral over) which are intrinsically commuting objects.
Properties of non-commuting operators can appear in subtle ways, like the operator ordering ambiguity how to quantize the classical observable x^2p^2, should it be xp^2x or px^2p or for example (x^2p^2 + p^2x^2)/2? This is a true quantization ambiguity and the path integral has to know about it as well. It turns out, it does: When you show the equivalence of Schroedinger's equation and the path integral, you do that by considering infinitesimal paths and you have to evaluate potentials etc on some point of those paths to compute things like V(x) in the action. Turns out, the operator ambiguity is equivalent to choosing where to evaluate V(x), at the start of the path, the end, the middle or somewhere else.
So far so good. The question that I don't know the answer to is how the path integral encodes entanglement. For example can you discuss a version of Bell's inequality (or similar like GHZ) in the path integral language? Of course you would have to translate the spin operators to positions .
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7 comments:
Regarding the first topic:
1/N plays the role of ``Planck's constant'', i.e. it's the quantization parameter. So the limit N->oo is the semi-classical limit.
Maybe this
E. G. Floratos, J. Iliopoulos and G. Tiktopoulos,
``A NOTE ON SU(infinity) CLASSICAL YANG-MILLS THEORY,''
Phys. Lett. B217 (1989) 285
would be relevant?
Two interacting particles 1,2 are in general entangled - you see this in the path integral Z as the fact that it does not factorize as Z1*Z2.
But I guess I am missing the point of your question...
Of course they don't factorise but that you already have for correlations (which can be classical). I am asking about entanglement which manifests itself in violations of inequalities that are obeyed classically.
You can compute correlation functions in the path integral formalism. There you do have a notion of locality. So you could characterize entanglement by studying how these correlation functions behave. And you do have a notion of non-classical behavior, since you can compute these correlation functions with the ``classical'' measure or by taking the corrections into account-perturbatively (e.g. loop expansion) or non-perturbatively (e.g. lattice regularization).
I finally got hold of the Floratos etal paper. They build on Jens Hoppe's way of viewing area preserving diffeos of a membrane (they write S^2 but in fact any two dimensional surface will do) as some N->infinity limit of SU(N). They argue that that this allows them to interpret the gauge indices as fourier decomposition of two additional coordinates. I don't see that this would in any way be related to planarity of feynman diagrams.
Actually you can think of planar Yang-Mills in exactly the way you described above, i.e. at infinite N the path integral is exactly determined by a single saddle point, which is given by a set of 4 infinity by infinity matrices, which are usually dubbed master field, however as far as I can tell nobody has so far managed to make any use of that or determine any properties of these.
The original source for this are some old lecture notes by Witten called "The 1 / N Expansion In Atomic And Particle Physics" , which you can find here:
http://www-lib.kek.jp/cgi-bin/img_index?8002242
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