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.

• Singular: Galileo, translation, Poincaré, Diffeomorphism, commutative, Heisenberg, Bose statistics, Fermi statistics, …

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 infinite-dimensional.

• 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

aBose statistics Palev statistics

[a, a*]=I [p, q]=r, etc.a=p+iq

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

Poincaré so(4,2),Minkowski Killing

Generic fields

• The usual field construction f(x) fails for non-commutative 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.”