Warner A. Miller Florida Atlantic University

Transcription

1 Constructing the Scalar Curvature at an Event in a Discrete Spacetime Warner A. Miller Florida Atlantic University Miami 2008 Quantum Gravity remains one of the primary challenges to physics We search for a coupling between a history of a discrete quantum system and a corresponding simplicial spacetime geometry. (Delta Interaction, Computational Universe, Causal Dynamical Triangulations, Spin Foam)

2 Why Regge Calculus? If nature is indeed fundamentally discrete; built out of a finite number of elementary quantum phenomena. Then one may hope that by studying the discrete representations of the most beautiful geometric theory of nature we know, gravitation, one may be able to glean some of the fundamental features of the discretization that may yield way points to a true understanding of the basic building blocks of nature. We are not aware of a more pure, geometricallybased discrete model of gravity than Regge Calculus. The laws of gravitation appear to be encoded locally on the lattice in with less complexity than in the continuum. 2

3 Geometrodynamics The curvature of spacetime tells matter how to move; and in turn, matter tells spacetime how to curve G µν }{{} Curved Spacetime Geometry = 8π T µν }{{} Matter and F ields The Quantum Computational Universe The curvature of spacetime yields information flow and in turn the computational history tells space how to curve h v ɛ h δa h δg ab = 8πT ab V ab }{{} Computational M atter 3

4 Regge Calculus (RC) In Regge Calculus the spacetime geometry is represented by a simplicial lattice. The discrete geometry is built of internally flat 4- dimensional triangles (simplices). Hilbert Action I = (4) R d (4) V proper Regge-Hilbert Action I R = 8π triangle hinges, h ɛ h A h I C = h v C θ v 4

5 Scalar Curvature Invariant δa α = R α β µν A β (u µ v ν u ν v µ ) R β ν = R µβ µν R = R β β 5

6 Curvature by Parallel Transport Curvature = ( Angle V ector ) Rotates Area Circum navigated = 8 π 2 ( 4πR 2 ) = R 2 4 Rotation Bivector {}}{ e α e β R αβ µν dx µ dx ν }{{} Orentation of area Circumnavigated 6

7 Curvature in RC (2 D) 5 ɛ h = 2π i= θ i }{{} θ θ A 2 h θ 3 θ 4 θ 5 Def icit Angle Angle V ector Rotates K h = ɛ h A h 7

8 Curvature in RC (3-D) The Building block is a tetrahedron. The hinge is the edge common to the five tetrahedrons. ɛ h (3)K h = ɛ h A h 8

9 Curvature in RC (4-D) The Building block is a simplex and the hinge (h) is the triangle common to all the simplicies D =4 A h A h (4) K h = ɛ h A h 9

10 Hybrid Voronoi and Delaunay Block d (D) V proper = 2 D(D ) A h A h A co-dimension 2 version of 2 base altitude 0

11 Scalar Curvature of the Hybrid Block Curvature is concentrated on co-dimension-2 hinge, h Rotation in plane perpendicular to the hinge, h. Voronoi polygon A h is perpendicular to hinge, h Locally the Regge spacetime is an Einstein space (D)R h = D(D ) (D) K h = D(D )ɛ h A h

12 Hilbert Action I = 6π (D) Rd (D) V proper ( D(D )ɛh ) ( 2 ) 6π hinges, h A h D(D ) A h A h I R = 8π triangle hinges, h ɛ h A h Regge Action 2

13 The Reduced Hybrid Block Moving Tensors to Vertices (D) V hybrid = 2 D(D ) A h v A h v A type of Irreduciable Brillouin 4-Cell 3

14 Moving the Curvature to a Vertex: Reduced Hybrid Blocks I = 6π (D) Rd (D) V proper 6π v 6π v R v R v V v D(D ) A hva h h v 2 6π h v h 6π R h ( h R h V h ) 2 D(D ) A hva h R v = D(D ) ɛ h v A h v Q h v R h = D(D ) ɛ h h v Q h A hv A h h v A hv 4

15 Scalar Curvature Example K h = ɛ h A h

16 Shapes, Forms and Patterns A snowflake is a letter to us from the sky. in U. Nakaya, Yuki (Snow) (Poem written in937). A diamond is a letter from the depth. in F. C. Frank, Science and Tech. of Industrial Diamond, 2 (967) 9. How are we to decipher the hieroglyphics of the shapes, forms and patterns of the cosmos into a fundamental understanding of nature? A piece of the Rosetta stone may be in studying the role that Voronoi and Delaunay lattices play in the description of nature. 6

17 Conclusions Principles of general relativity applied directly to the lattice geometry Provides a true finite representation of the theory based on the underlying physical principles Voronoi-Delaunay duality appears to be a salient feature of Regge Calculus and yields a new fundamental hybrid building block. The underlying discrete theory appears more austere via the underlying orthogonality; however, the full theory is recovered by convergence in mean. Can the Voronoi-Delaunay structure provide a platform for complementarity in quantum gravity? 7

18 Cartan Moment-of-Rotation in RC pxpxpxpxpxpx δ (I R ) = 0 = }{{} L i L i +δl i 2 hinges, h cot(θ h )ɛ h =0 } sharing edge L {{} Regge Equation G LL V L = hinges, h sharing edge L 2 L cot (θ h) }{{} Moment Arm ɛ h }{{} Rot n Einstein Field Equation is Diagonal, and Directed along L! Identical to the variational equation! 8

19 Spacetime from a Universal Quantum Simulation.Q Computation a universal theory for discrete QM 2. Q Computers are discrete systems that evolve by local interactions 3. Every quantum system that can evolve by local interactions (e.g. lattices gauge theories) can be simulated on a Q computer If quantum gravity is; (a) discrete, and (b) a local quantum theory. Then QG too should be describable as a Q Computation. 9

20 Quantum Computational Universe Unification of QM and GR based on Quantum Communication U φ + φ + + U = φ φ U = P 0 }{{} P +e iφ P }{{} P φ φ φ φ 20

21 Important Points and Future Work 4-Dimensions (null tetrad). CU posses a causal structure (similar to Causal Sets) CU possess additional, internal degs of freedom (qubits) (unlike Causal Sets) Back Reaction (metric fluctuations track computational matter) Local phases & local energies are positive (bound on density of quantum ops) Preliminary CU results (homogeneity -> inflation -> non-inflationary period). Existence of smallest length scales. Although no cosmological constant term is explicit, however slowly-varying cosmological terms can be generated by quantum dynamics. Explore the relation of elementary particle physics and the standard model to quantum computation. 2