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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
(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
quantum mechanics
fermionic fields
gauge fields and their symmetry
bosonic fields
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
 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.
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
There is an arbitrariness in the length scale of the internal space which
may have anthropic implications.
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 EinsteinHilbert 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.
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.
Note the minus sign in the Lorentz-violating term for lefthanded 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 4component Dirac field, the Lorentz-violating term is found to be
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.