Monday, July 12, 2010

Gravity, Two Times, Six Dimensions

For those interested in theories with two times, such as Itzhak Bars' S-theory, Waldron et al. have written an exciting new paper on a D=6 description of D=4 physics.

Gravity, Two Times, Tractors, Weyl Invariance and Six Dimensional Quantum Mechanics

Fefferman and Graham showed some time ago that four dimensional conformal geometries could be analyzed in terms of six dimensional, ambient, Riemannian geometries admitting a closed homothety. Recently it was shown how conformal geometry provides a description of physics manifestly invariant under local choices of unit systems. Strikingly, Einstein's equations are then equivalent to the existence of a parallel scale tractor (a six component vector subject to a certain first order covariant constancy condition at every point in four dimensional spacetime). These results suggest a six dimensional description of four dimensional physics, a viewpoint promulgated by the two times physics program of Bars. The Fefferman--Graham construction relies on a triplet of operators corresponding, respectively to a curved six dimensional light cone, the dilation generator and the Laplacian. These form an sp(2) algebra which Bars employs as a first class algebra of constraints in a six-dimensional gauge theory. In this article four dimensional gravity is recast in terms of six dimensional quantum mechanics by melding the two times and tractor approaches. This "parent" formulation of gravity is built from an infinite set of six dimensional fields. Successively integrating out these fields yields various novel descriptions of gravity including a new four dimensional one built from a scalar doublet, a tractor vector multiplet and a conformal class of metrics.


Lawrence B. Crowell said...

This is based on a post I sent to vixrablog.

A 12 dimensional theory with two times has some possibilities, but there are a number of hurdles which must be overcome. To see some of this it is necessary to look at string theory as it stands. Something occurred to me yesterday which I have been pondering for a while. Maybe Kneemo and Philip can shed some light on this. The relevant dimensions are 26, 16, 11, and 10. The ten dimensional supersymmetric theory is 10 dimensions, 9 space plus time, which defines the 10 dimensional Supersymmetric Yang Mills (SYM) theory. This theory is a nongravitational theory which describes open strings, and the reason for 10 dimensions is that the Virasoro algebra has an anomaly cancellation property in this number of dimensions. The theory is not entirely satisfactory, for it is not renormalizable outside of compactification. We jump to the other end of these number to 26, which is the dimensionality of the bosonic string, where again this cancels anomalies at this dimension. I will avoid discussing the matter of the mathematics here, for that would digress too far into Virasoro algebras. However, these are vital to understand much about string theory. The relationship between the 26 and 10 dimensions involves the number of supercharges (charges which define supersymmetric fields), which contains 8 charges plus their superpairs --- 16 in total. This involves an interesting relationship between Clifford algebras and the Cayley numbers 1, 2, 4, 8, whereby if you add two to these you get 3, 4, 6, and 10. For a Cayley number n the supersymmetric theory is an so(n+1,1) group action on the Cayley or Moufang plane (the subspace where Trace(V) = 0 and we can define a density matrix in quantum mechanics). Again there is some machinery here which I will avoid. The paper by Dray and Manogue breaks some of this mathematics out. The Freudenthal Triple System (FTS) defines a 3-cycle which constructs a 10 dimensional theory. So given the Cayley number n = 8 (for supercharges) the theory is a CL_{9,1} = R[16](+)R[16]. (here the term (+) means oplus.

Now what about our 11 dimensions? That comes about from a 4-cycle. The FTS is due to an automorphism on the Jordan matrix algebra which defines a sum, trace of quadratic elements and a determinant. As with vector spaces the determinant of a matrix gives its eigenvalues. We can go one bit higher, a hyperdeterminant. The 3-cycle is a rule on fields (ψ*ψ)ψ =/= 0 then ψ(ψ*ψ)ψ = 0, which is a cohomology. The 4-cycle takes this into a spinor-vector rule with the product , which involves an antisymmetric system of elements which appear to define a hyperdeterminant. The result is this defines a Clifford algebra CL_{10,1} = R[32]. So this in a rule of thumb is where we add 3 to each of the Cayley numbers. I refer you to a paper by Baez and Huerta for this how 3 and 4 cycles determine Clifford algebras on Cayley numbers, with Clifford dimensions 3, 4, 7, and 11..

Lawrence B. Crowell said...


So what we have is a nice system in 10 dimensions, which is dual to something in 16 = 8+8 dimensions. In group theory this is SO(10) and E_8xE_8, where the last part is the infamous heterotic string, or closed string which carries 24 field elements that contain the “graviton.” The SO(10) is our more well behaved (well except for renormalization) open string theory (eg type II) which describes things like the nuclear interaction. We also have this 4-cycle stuff, which pops us up one dimension and completes in some low energy approximation this thing we call M-theory. Within this structure for N = 4 supersymmetry the AdS/CFT theorem may be derived. This says the isometries of the boundary of an AdS spacetime contains the conformal structure of a quantum field theory. The structure of this appears more generally involved with the relationship between the 3 and 4-cycles, or equivalently the determinant (F_4 automorphism) and the hyperdeterminant. There is also some deep topological relationship between them. I think that the paper “Freudenthal triple classifcation of three-qubit entanglement” Borsten,1 Dahanayake, M. J. Duff_,H. Ebrahim, W. Rubens is important in this regards and a comparison with the recent paper by Duff.

So what about this business of two times? Well the AdS spacetime is a spacetime which is a hyperboloid solution in a higher dimension and two times. So this takes us into something involving F-theory. I. Bars has an interest in demonstrating that this additional time direction is not something which “flaps in the breeze” or that can be ignored, but plays a direct role in M-theory and strings. There are variations on this theme,with additional dilator or axion fields, or new degrees of freedom for these with respect to the type II string. This involves the trace of the 24 elements of the heterotic part which contains the dilaton field and popping it up one dimension. Most of the paper involves how this theory can break down into the SYM in 9 + 1 dimension, and the rest of the 3-cycle determined Clifford algebras. This can be seen in the paper by Bars & Kuo[/link]. The start of the paper centers around what appears to be a gauge fixing condition on the Majorana fermion fields of the theory. This from my early reading of things reduces the theory by throwing out field theoretic information, which might make the theory somewhat artifactual.

The idea is worth consideration of course. All the above that I set up above sits in one dimension higher, where AdS spacetimes are found from a reduction of a dimension, and there is no physics usually considered with this.

Cheers LC