Discrete Time and Kleinian Structures in Duality Between Spacetime and Particle Physics by Lawrence B Crowell

Ed Witten probably has a lot better things to do than peruse the FQXi essays. The guy is utterly brilliant. He writes very long papers, where he is one of the few people who write 60 page papers I am willing to read. For the most part I prefer not to read papers over 25 pages in length. Of course one point in writing these essays is to put at least one ball into the basket so that I might get a bit of attention --- Warhol's 15 minutes of fame.

Cheers LC

The clear problem with GEM idea in general is these are an intermixing of internal and external symmetries in ways that are problematic.

The Einstein field equations are similar to Maxwell's, and in the post^3 Newtonian form are identical (modulo a factor of 2) to Maxwell's equations. The post^2 form is similar to electric and magnetic statics. The magnetic field analogue is similar in some ways to magnetism in EM theory. This is usually what is meant by gravito-magnetic field,

Cheers LC

The difficulty remains if the field theory is abelian, as in U(1) QED, or if it is nonabelian in the YM sense. This comes from the problem of characterizing the symmetries of the S matrix if there is a mass. Review the Dirac equation and the mass-gap in the two solutions which separate by a gap p ~ m on the momentum light cone.

The S-matrix acts on shift the state or momentum state of a particle. A state with two particle states |p, p'> is acted upon by the S matrix through the T matrix

S = 1 - i(2π)^4 δ^4(p - p')T

So that T|p, p'> != 0. For zero mass plane waves scatter at almost all energy. The Hilbert space is then an infinite product of n-particle subspaces H = (x)_nH^n (here (x) = "otimes" or Cartesian product). As with all Hilbert spaces there exists a unitary operator U, often U = exp(iHt), which transforms the states S acts upon. U transforms n-particle states into n-particle states as tensor products. The unitary operator commutes with the S matrix

SUS^{-1} = [1 - i(2π)^4 δ^4(p - p')T]U[1 + i(2π)^4 δ^4(p - p')T^†]

= U + i(2π)^4 δ^4(p - p')[TU - UT^†] + [(2π)^4 δ^4(p - p')]^2(TUT^†).

By Hermitian properties and unitarity it is not difficult to show the last two terms are zero and that the S-matrix commutes with the unitary matrix. The Lorentz group then defines operator p_μ and L_{μν} for momentum boosts and rotations. The S-matrix defines changes in momentum eigenstates, while the unitary operator is generated by a internal symmetries A_a, where the index a is within some internal space (the circle in the complex plane for example, and we then have with some

[A_a, p_μ] = [A_a, L_{μν}] = 0.

This is a sketch of the infamous "no-go" theorem of Coleman and Mundula. This is what prevents one from being able to place internal and external generators or symmetries on the same footing.

The way around this problem is supesymmetry. The generators of the supergroup, or a graded Lie algebra, have 1/2 commutator group elements [A_a, A_b] = C_{ab}^cA_c (C_{ab}^c = structure constant of some Lie algebra), plus another set of graded operators which obey

{Q_a, Q_b} = γ^μ_{ab}p_μ,

which if one develops the SUSY algebra you find this is a loophole which allows for the intertwining of internal symmetries and spacetime generators. One might think of the above anti-commutator as saying the momentum operator, as a boundary operator p_μ = -iħ∂_μ which has a cohomology, where it results from the application of a Fermi-Dirac operator Q_a. Fermi-Dirac states are such that only one particle can occupy a state, which has the topological content of d^2 = 0. This cohomology is the basis for BRST quantization.

This is why most physicists who work on this stuff take supersymmetry seriously. It is also one reason why many schemes which purport to derive gravitation or unify gravitation with EM in some elementary was can be subject to strong questions. Of course supersymmetry remains a hypothetical, though some signatures of it have been detected. We will have to wait for the LHC to yield such results before anything is conclusive.

Cheers LC

I have attached a rough draft that is somewhat sketchy, on more mathematical detail with the physics here.

Cheers LCAttachment #1: AdSqubit.pdf

The discrete nature of time might be compared to a numerical ordering. I carry this further to consider a discrete structure in general.

Cheers LC

The discussion was on fractal structure. I have a bit of a crazy idea on convolving Sloan Digital Sky Survey (SDSS) data with CMB data. After Planck has done its survey I might do this. It will be interesting to see if the Hausdorff dimensions of the two structures are equal or related in some way.

The discrete group structure and cosets defines a Schottky set. The partitions of the manifold, or regions partitioned by Jordan curves, have a self similar structure, or a Moebius group realization, which makes it a Cantor type of set. The Cantor set, or "comb partition" of the interval, is an elementary form of a fractal geometry. Mandelbrot in fact found a self similar form of noise in transmission lines which had a Cantor set structure within any time interval.

If this structure is present in quantum gravity and the quantum wave function of a cosmology, then it should persist through the inflationary epoch and have signatures on the large scale. This discrete fractal structure might be the source for non-Gaussian structures in the CMB.

Cheers LC

After the LHC starts up at full steam in 2012 data on SUSY should start coming in. I think it likely SUSY will be found by 2015. The Higgs particle is more difficult to separate from the noise and it might take a bit longer. I think that by the end of this decade we will have MSSM data and will be working on more sublte issues of AdS~CFT and so called soft black holes with heavy ion data.

Cheers LC