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Causal Space-Time on a Null Lattice with Hypercubic Coordination

Causal Space-Time on a Null Lattice with Hypercubic Coordination. Martin Schaden Rutgers University - Newark Department of Physics. Outline. Introduction Causal Dynamical Triangulation (CDT) of Ambjørn and Loll Polynomial GR without metric? Topological Lattices and Micro-Causality

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Causal Space-Time on a Null Lattice with Hypercubic Coordination

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

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

  3. Intro: Causal Dynamical triangulation (CDT) • Foliated Gravity as QM of space histories (Ambjørn and Loll): • Foliated space-time with spatial submanifolds at fixed temporal separation. Triangulation of spatial manifolds with fixed length “a”, but varying coordination number. • Can be “rotated” to Euclidean space, but contrary to Euclidean QG corresponds to a subset of causal manifolds only. • There exist 3 distinct phases: A, B and C that depend on the coupling constants A t Critical Continuum Limit ? local # d.o.f. ? C B Too restrictive ? • Ambjørn,Jurkiewicz, Loll 2010

  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: 3+1 d 1+1 d 2+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 3+1 d 1+1 d 2+1 d

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

  7. First order formulation for a scalar and a spinor Hilbert-Palatini GR with null co-frames First order formulation of electromagnetism: Upon shifting the auxiliary field , Co-frame, is a 1-form is a 2-form so(3,1) curvature constructed from an SL(2,C) connection 1-form Introducing a basis of anti-hermitian 2x2 matrices , the co-frame 1-form corresponds to the anti-hermitian matrix, on the link . The separation between these nodes is light-like or null Null links are described by bosonicspinors.

  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: The for are the 6 spatial lengths of the tetrahedron whose sides are . Note that these vectors are all given in the inertial system at. The lengths are invariant under local SL(2,C) transformations (). Since two vectors for which d.o.f. per site:spinors: 2x2x4-6[sl(2,C)]=10, 10-4 phases=6 metric d.o.f. skew form: 2x6-2constraints=10,10-4 =6 only | related to metric

  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. The Lattice Hilbert-Palatini Action volume Hilbert-Palatini Holst Letting the Hilbert-Palatini action is proportional to Is the only dimensionless coupling of the lattice model. Without cosmological constant - no critical limit! In units , critical and thermodynamic limits coincide if the average 4-volume of the universe is fixed ( is the Lagrange multiplier).

  11. The integration measures for the spinors and SL(2,C) transport matrices are dictated by SL(2,C) invariance: Invariant Measures and Partial Localization or Parameterizing: The SL(2,C) invariant measure for the tetrads is: 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: The invariant measure of SL(2,C) matrices: is the (non-compact) SL(2,C) Haar measure: 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. Invariant Measures and Partial Localization The partially gauge-fixed lattice measure with residual compact structure group SU(2) is: where is the 3-volume of the (spatial) tetrahedron formed by and It is related to the 4-volume by: This partial gauge fixing is local and ghost-free. It is unique and the determinant may be evaluated explicitly. The residual structure group is compact SU(2). The lattice measure itself is non-compact, but the partial localization removes the non-compact flat parts of the measure. [The UV-divergence of the residual lattice model at small is logarithmic only, and can be regulated with .]

  13. The Manifold Condition By the procedure outlined above, any causal manifold can be discretized on a topological null-lattice with hypercubic coordination. Example: Minkowski space-time with null coframes, Independent of the node (scaled and Lorentz-transformed frames here are equivalent). The Minkowski metric (distances) in these coordinates is: and , preserving orientation. BUT: NOT EVERY orientation-preserving configuration of null-frames on a lattice with hypercubic orientation represents a discretized manifold !! Since the #d.o.f. is correct the additional conditions are topological and do not alter the #d.o.f.

  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: Backward null coframe In inertial system at For a fixed event the six spatial lengths therefore are the sides of a spatial tetrahedron (the one formed by 4 events in the backward light cone of ). They can be assembled to a tetrahedron if and only if they satisfy triangle inequalities. We thus arrive at the condition that a configuration of (forward) null frames on this lattice is the discretization of a manifold if and only if:

  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: with real bosonic Lagrange multiplier fields enforcing 10 constraints on 16 (backward) tetrads , 16 anti-ghosts and , an equal number of 16 ghosts and 6 bosonic topological ghosts (ghost#=2) and an equal number of topological antighosts (ghost#=-2) and an additional bosonic(2,C) symmetry. The TLT therefore has 10+16+6+6-6((2,C))=32 bosonicd.o.f. and 16+16=32 fermionicd.o.f. , or a total of 0 d.o.f. . The eBRST 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 The manifold condition for the spinors is: or where the 6 real phases are not all independent, because . Consider the U4(1) transformations: and choose the to satisfy: Choosing and denoting These relations decouple and can be solved (modulo b.c.): manifold condition

  17. The Manifold TFT for Spinors The TFT of the manifold condition is obtained by using it to fix (part of) the U4(1). BRST: () ()

  18. The Manifold TFT measure One can integrate out the real bosonic fields as well as the fermionic ghosts to arrive at the TLT contribution to the measure: with and where Note: Observables of the lattice theory do NOT DEPEND on the parameter , but lattice configurations generally are triangulations of causal manifolds for only! The dependence on a particular “direction” used in the construction of the TLT has disappeared (as it must).

  19. Analytic Continuation to SUL(2) x SUR(2) Analytic continuation of the partially localized model: Respects the residual SU(2) symmetry and at gives a Euclidean (positive semidefinite) lattice measure of an theory with a diagonal structure group. ` with & spatial triangle inequalities

  20. Conclusion • 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 SUL(2) x SUR(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)

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