PDF Ebook Finite Quantum Relativity
To synthesize gravity theory and quantum theory we extrapolate the common element of the relativistic lines of thought of Einstein and Heisenberg. Both substantially simplified and regularized some of the fundamental groups of physics. Extrapolating this process leads to a theory with a simple group. This requires one to modify the commutation relations for both general relativity and the standard model, but the change can be as small as desired. As an unexpected reward, all observables of the simple theory have finite spectra. The main singular non-simple group today is the Heisenberg group. The least change that simplifies it replaces each canonical pair p,q of the usual singular quantum theory by an SO(3) triplet ?p, ?q, ?r with large quantum number l and replaces Planck’s constant by an action variable ?r . The resulting finite quantum theory unifies space-time coordinates, energy-momentum variables, and dynamical variables in a way that preserves exact Lorentz invariance.
For exercise we first simplify a toy linear harmonic oscillator. It becomes a rotator with high fixed angular-momentum quantum number l. In the Heisenberg quantum theory all oscillators are isomorphic, but the finite quantum theory has distinct classes of soft, medium, and hard oscillators. Medium oscillators approach the familiar ones of the singular limit, but soft and hard oscillators have negligible zero-point energy and grossly violate equipartition and the Heisenberg uncertainty relation. Their energy is almost entirely kinetic for soft oscillators and potential for hard. Similarly, in the Heisenberg quantum dynamics all times are alike but in the finite quantum dynamics any time variable has distinct early, middle and late eras. This breaks invariance under time translation except as a singular limit, but exact conservation of energy is still possible.
The simple commutation relations can be represented in a real Clifford algebra. Clifford algebra over the binary field 2 = {0,1} is a useful algebraic classical logic more expressive than Boolean algebra. We interpret complex Clifford algebra correspondingly as quantum logics. The hierarchy of Clifford algebras strengthens the usual Hilbert-space quantum kinematics as set theory strengthens Boolean logic.
Time-energy interconversion becomes conceivable in the simple finite theory. On dimensional grounds the power release possible is on the order of the Planck power (10 51 W).
Contents
1 Group simplicity, regularity, and stability
- 1.1 Lorentz group
1.2 Heisenberg group
1.3 Segal algebra
1.4 Singularities of singular groups
1.5 Darwinian selection of regular groups
1.6 Finiteness of simple theories
1.7 Regular Hessians
1.8 Flattening and flexing
1.9 Absolutes and radicals
1.10 Thought travel
1.11 S and XS variables
1.12 The system interface
1.13 Measurement under gravity
1.14 Other approaches
2 The action variable
- 2.1 Universality of the action variable
2.2 Simplifying statistics
2.3 Atomic process hypothesis
2.4 Simplifying dynamics
2.5 Simplifying the diffeomorphism group
2.6 Action physics
3 Quantum dynamics
- 3.1 Dynamical group
3.2 Chronometrical group
3.3 Io actions
4 Finite quantum harmonic oscillator
- 4.1 Stationary harmonic oscillator
4.2 Quantum internal space
4.3 Oscillator Hamiltonian
4.4 Soft, medium, and hard oscillators
4.5 Dynamical harmonic oscillator
4.6 Timed operators
4.7 Flexing the timed algebra
4.8 Commutation relations
4.9 Flexing the dynamics
5 Concepts of Clifford algebra
- 5.1 Definition: Clifford ring
5.2 Unification
5.3 Unified unification
5.4 Quantum unification
5.5 Natural Clifford algebras
5.6 Signatures
5.7 Baugh numbers
5.8 Order cut-off
6 Cliffordian logic
- 6.1 Classical Clifford-algebraic logic
6.2 Classical simplex
6.3 Classical binary exponential
6.4 Quantum Clifford-algebraic logic
6.5 Quantum simplex
6.6 Quantification
6.7 Quantum simplicial space-time
7 Cliffordian gravity
- 7.1 Clifford manifold
7.2 Simplifying the canonical group. I
7.3 Simplifying canonical Newtonian mechanics
7.4 Simplifying gravity
8 Summation
9 Acknowledgments
10 Appendices
- 10.1 Appendix: Kochen-Specker-Schmidt theorem
10.2 Appendix: SO(2,1)
10.3 Appendix: Flexings and flattenings
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PDF Ebook Finite Quantum Relativity
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