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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.

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GENERIC RELATIVITY

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  1. GENERIC RELATIVITY David Ritz Finkelstein Georgia Institute of Technology FQXi Reykjavic, 2007.07.22

  2. Thanks • Heinrich Saller (Heisenberg Institute) • Andrei Galiautdinov, James Baugh, Mohsen Shiri-Garakani (Georgia Tech) • Ruis Vilela Mendes (Lisbon) • Tchavdar Palev (Sofia) • Many others

  3. 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, …

  4. 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)

  5. 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.

  6. 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

  7. 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

  8. Generic quantization • Make the Lie algebra of commutation relations generic by a small change in the structure tensor • 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 a Bose statistics  Palev statistics [a, a*]=I  [p, q]=r, etc. a=p+iq

  9. 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, …

  10. Multiquantification • If V is the ket space of one fermion then Grass V is the ket space of many fermions, • Grass2V is the ket space ofmanysets 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.

  11. 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.

  12. 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

  13. 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.

  14. 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.”

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