Transcript GENERIC RELATIVITY

GENERIC RELATIVITY
David Ritz Finkelstein
Georgia Institute of Technology
FQXi
Reykjavic, 2007.07.22
Thanks
• Heinrich Saller (Heisenberg Institute)
• Andrei Galiautdinov, James Baugh, Mohsen
Shiri-Garakani (Georgia Tech)
• Ruis Vilela Mendes (Lisbon)
• Tchavdar Palev (Sofia)
• Many others
Concept of generic structure
• (= Structurally stable, regular, or rigid
structure; as opposed to singular structure)
• A structure is generic if it is isomorphic to all
the structures in some neighborhood.
• A Lie algebra is generic iff it is semisimple.
Every Lie algebra is a limit of generic Lie
algebras (I. E. Segal).
• Generic: Lorentz, unitary.
Generic principle
• Structure tensors are found by
measurement, which has errors.
Singular structures are articles of faith,
not experiment.
• Therefore physical Lie
algebras must be generic.
(Almost said by Segal 1951)
Infinities
• Structural instability  dynamical instability,
infinities.
• The Heisenberg Lie algebra has an
essentially unique useful faithful irreducible
representation (FIR) and it is infinitedimensional.
• Semisimple Lie algebras have infinite spectra
of useful FIRs in which every observable has
discrete bounded spectrum.
• Structural stability permits dynamical stability
and finiteness.
Levels of aggregation
• Field level F:
f(x)
• Event level E :
x =  dx
• Differential level D: dx
• Levels are related by set theory:
Classical levels give rise to classical
levels
Canonical quantization
• Makes the Lie algebra of commutation
relations less singular by the replacement
qp-pq = 0  qp-pq = ih
alg(q,p,0)  h(1)=alg(q, p , i)
• Reduces the singularity of Level F, not E or D.
• Leaves the theory singular
Generic quantization
1. Make the Lie algebra of commutation
relations generic by a small change in the
structure tensor
2. Choose a finite-dimensional representation
“near” the singular limit.
•
Examples:
h(1)  so(3) or so(2,1)
h(4)  so(6) or so(5,1) or …
Fermi statistics  Clifford statistics
Quantum set theory
• Specialize to the quantifier
P: V  Cliff V
corresponding to the power set functor P.
Dim Pn R = 1, 2, 4, 16, 65536, 265536, …
Multiquantification
• If V is the ket space of one fermion then
Grass V is the ket space of many fermions,
• Grass2 V is the ket space of many sets of
many fermions, …
• Similarly for the ket spaces Bose V, Bose2V,
etc. of Bose multistatistics.
• Functors like Grass and Bose are quantifiers
of quantum theory, analogous to the power
set of classical set theory.
Generic relativity
• Generic quantization of general relativity
• Must regularize alg( xm, gmn(x), Dm(x))
• Usual postulates Dg = 0 = Torsion are
singular. General covariance is singular. Any
regular theory must have torsion and graviton
rest mass (and photon rest mass).
• It would be disappointingly trivial if the
graviton and photon rest masses happen to
be so small they don’t even affect cosmology,
but this cannot be excluded a priori.
Origin of generic metric
• “… in a discrete manifoldness, the ground
of its metric relations is given in the
notion of it, while in a continuous
manifoldness, this ground must come
from outside…”
Riemann
• But a Lie group is the ground of its own metric
relations, the Killing form. This is regular for a
regular group.
• The Minkowski metric is the singular limit
of the so(6) Killing form: as
Generic fields
• The usual field construction f(x) fails for noncommutative x.
• But field theory is actually many-quantum
theory. The generic form of this concept is
straightforward, based on P.
• Field variables are involutors (=
creation/annihilation operators) defined on a
lower level and represented on the field level
F.
N. Bohr, Causality and
Complementarity (1935)
• “On closer consideration, the present formulation of
quantum mechanics in spite of its great fruitfulness
would yet seem to be no more than a first step in the
necessary generalization of the classical mode of
description … (W)e must be prepared for a more
comprehensive generalization of the complementary
mode of description which will demand a still more
radical renunciation of the usual claims of so-called
visualization.”