1. Causal Space-Time on a Null Lattice with Hypercubic Coordination Martin Schaden Rutgers University - Newark Department of Physics

2. I.Introduction Causal Dynamical Triangulation (CDT) of Ambjørn and Loll Polynomial GR without metric? Topological Lattices and Micro-Causality II.Topological Null Lattice Discretization of a causal 2,3,and 4 dimensional manifold Spinor Invariants III. The Manifold Constraint Construction of the Topological Lattice Theory (TLT) for vierbeins for spinors IV. Outlook Lattice GR analytically continued to an SU L (2) x SU R (2) theory with SU(2) structure group? Outline

3. Intro: Causal Dynamical triangulation (CDT) A B C Critical Continuum Limit ? local # d.o.f. ? t Ambjørn,Jurkiewicz, Loll 2010 Too restrictive ?

4. Null Lattice  Light-like signals naturally foliate a causal manifold: The (future) light cones of a spatial line segment (in 2d), a spatial triangle (in 3d) and of a spatial tetrahedron (in 4d) intersect in a unique point: 1+1 d 2+1 d 3+1 d  Each spatial triangle (tetrahedron) maps to a point on a spatial (hyper-)surface with time-like separation - if each vertex is common to 3 (4) otherwise disjoint triangles (tetrahedrons), the mapping between points of two timelike separated spatial surfaces is 1to1 !!!! - the (spatial) coordination of a spatial vertex therefore is 6 (12) for 2 (3) spatial dimensions. In d=2 (3)+1 dimensions the spatial triangulation is hexagonal (tetrahedral) and fixed! LENGTHS ARE VARIABLE: LATTICE IS TOPOLOGICALLY (HYPER)CUBIC ONLY

5. The Null Lattice in 1+1 and 2+1 dimensions 1+1 d 2+1 d 3+1 d

6. Spatial triangulation Hexagonal in 2d Coordination=6 Tetrahedral in 3d Coordination=12 Linear in 1d Coordination=d(d+1)=2 The tetrahedra in this triangulation of spatial hypersurfaces in general are neither equal nor regular!

7. Hilbert-Palatini GR with null co-frames Co-frame, so(3,1) curvature is a 2-form is a 1-form constructed from an SL(2,C) connection 1-form First order formulation of electromagnetism: First order formulation for a scalar and a spinor

8. The Metric and a Skew Spinor Form Local (spatial) lengths on this lattice are given by the scalar product : with the SL(2,C) invariant skew-symmetric tensor and the invariant skew product The latter satisfies algebraically:

9. Invariants and Observables Basic SL(2,C) invariants Closed loops: Open strings: Most local examples: Observables are real scalars of SL(2,C) invariant densities. Some are: 4-volume Hilbert-Palatini term Holst term

10. 10 The Lattice Hilbert-Palatini Action volumeHilbert-PalatiniHolst

11. Invariant Measures and Partial Localization The integration measures for the spinors and SL(2,C) transport matrices are dictated by SL(2,C) invariance: or The SL(2,C) invariant measure for the tetrads is: Parameterizing: The invariant measure of SL(2,C) matrices: Must factor the infinite volume of the SL(2,C) structure group! Fix SL(2,C)/SU(2) boosts and localize to the compact SU(2) subgroup of spatial rotations by requiring: Physically, this condition implies that one can find a local inertial system in which 4 events on the forward light cone are simultaneous. This local inertial system always exists and is unique up to spatial rotations.

12. 12 Invariant Measures and Partial Localization

13. The Manifold Condition

14. The Manifold Condition for Coframes For the lattice to be the triangulation of a manifold, distances between events that are common to two inertial systems (“atlases”) must be the same:

15. The eBRST and TFT of the Manifold Condition The manifold condition can be enforced by a local TLT with an equivariant BRST. The action of this TLT is: The additional fields of the TLT can in fact be integrated out and give the triangle Inequality constraints:

16. The Manifold Condition for Spinors

17. The Manifold TFT for Spinors

18. 18 The Manifold TFT measure

19. Analytic Continuation to SU L (2) x SU R (2) 19 Analytic continuation of the partially localized model: & spatial triangle inequalities

20. Conclusion 20  Causal description of discretized space-time on a hypercubic null lattice with fixed coordination.  Local eBRST actions that encode the topological manifold condition  eBRST localization to compact SU(2) structure group of spatial rotations.  Analytic continuation to an SU L (2) x SU R (2) lattice gauge theory with “matter”.  The partially localized causal model is logarithmically UV-divergent only? Sorry: Causality may be irrelevant in the early universe (BICEP2)