1. Supersymmetry and Lorentz Invariance as Low-Energy Symmetries
in a Fundamental Statistical Theory
Roland E. Allen and Seiichirou Yokoo
Physics Department, Texas A&M University
Motivations
(1) A Euclidean path integral in quantum physics is equivalent to a partition function in statistical physics. This suggests that a fundamental description of Nature should start with some sort of statistical picture. The true picture is likely to be richer than the one presented here, which may be only a starting point. On the other hand, the present model already works quite well in many respects.
(2) It would to nice to have a truly fundamental theory -- which explains the origins of
Lorentz invariance gravity gauge fields and their symmetry
supersymmetry fermionic fields bosonic fields
quantum mechanics spacetime.
The present program is thus more ambitious than, e.g., superstring theory, which simply postulates most of these aspects of Nature. At the same time, the present picture involves the familiar concepts of grand unification, supersymmetry, higher dimensions, and topological defects.
APS-DPF2006, JPS2006, and Pacific Region particle physics October 31, 2006

2. Successes
 Standard Model fermions and their sfermion partners are automatically coupled to
SO(N) gauge bosons. The same is true of the Higgs and Higgsino fields.
Lorentz invariance is obtained as a low-energy symmetry, with the correct coupling of Standard Model fermions and sfermions etc. to the gravitational vierbein.
A primitive form of supersymmetry emerges, which can be reformulated to yield standard supersymmetry.
At very high energy there is a violation of CPT as well as Lorentz invariance.
The third and fourth points are the main new features of the present talk.
Bosonic and fermionic quantum fields, and spacetime, emerge from a microscopic statistical description.
There is automatically a cutoff at a length scale comparable to the Planck length.

3. Two more points which are not really successes, but which are interesting:
Both the gauge boson fields and the gravitational field emerge as collective modes of the more fundamental fermions and their superpartners.
There is an arbitrariness in the length scale of the internal space which may have anthropic implications.

4. Not yet accomplished
The Einstein-Hilbert action for the gravitational field, the Yang-Mills action for the gauge fields, and the analogous terms for the gaugino and gravitino fields are assumed to arise from a response of the vacuum that is analogous to the Landau diamagnetic response of a metal.
(If this assumption is correct, it still does not solve the cosmological constant problems, since the response of the vacuum would consist of the Einstein-Hilbert action plus a term that would have the form of a cosmological constant.)
 We have also not derived Yukawa couplings, scalar boson mass terms, and gaugino masses, so it is necessary to assume that these terms arise from radiative corrections or other mechanisms not treated here.

11. Now let us consider the issue of CPT invariance.
Since Lorentz invariance => CPT invariance,
CPT violation => Lorentz violation.
But Lorentz violation does not necessarily imply CPT violation.
However, it turns out that the present theory predicts CPT violation, as well as Lorentz violation, at very high energy.
The reason that the Lorentz-violating term in the action is also odd under CPT.

15. Note the minus sign in the Lorentz-violating term for left-handed fields. It is this sign that gives rise to CPT violation (at very high energy) in the present theory.
In fact, when the usual P, T, and C operations are applied to the 4-component Dirac field, the Lorentz-violating term is found to be odd.

16. New Results Presented Here
The primitive supersymmetry of our theory leads to standard supersymmetry at low energy.
The Lorentz-violating term in the action also violates CPT invariance.