Causal Space-Time on a Null Lattice with Hypercubic

CoordinationMartin SchadenRutgers University - Newark

Department of Physics

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 SUL(2) x SUR(2) theory with SU(2)

structure group?

Outline

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

BC

Critical Continuum Limit ?

local # d.o.f. ?

t

Ambjørn,Jurkiewicz, Loll 2010

Too restrictive ?

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

The Null Lattice in 1+1 and 2+1 dimensions

1+1 d2+1 d

3+1 d

Spatial triangulation

Hexagonal in 2dCoordination=6

Tetrahedral in 3dCoordination=12

Linear in 1dCoordination=d(d+1)=2

The tetrahedra in this triangulation of spatial hypersurfaces in general are neither equal nor regular!

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

Introducing a basis of anti-hermitian 2x2 matrices , the co-frame 1-formcorresponds to the anti-hermitian matrix,

on the link . The separation between these nodes is light-like or null

Null links are described by bosonic spinors .

First order formulation of electromagnetism:

Upon shifting the auxiliary field ,

First order formulation for a scalar

and a spinor

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 ().

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

Since two vectors for which

Invariants and ObservablesBasic SL(2,C) invariantsClosed loops:Open strings:

Most localexamples:

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).

Invariant Measures and Partial LocalizationThe 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:

is the (non-compact) SL(2,C) Haar measure:

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.

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:

12

Invariant Measures and Partial Localization

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 .]

𝑝 .𝑔 . 𝑓 .→

𝜏4(𝒏)𝑉 (𝒏)

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.

−ℓ𝝁𝝂𝟐 =¿

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.

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:

−𝐸𝜇(𝒏)∙𝐸𝜈 (𝒏)=ℓ𝜇𝜈2 =−~𝐸𝜇 (𝒏′) ∙~𝐸𝜈(𝒏′)

Backward null coframe In inertial system at

For a fixed event the six spatial lengths 0<ℓ𝜇𝜈

2 (𝒏′ )=−𝐸𝜇 (𝒏′−𝜇−𝜈)∙𝐸𝜈(𝒏′−𝜇−𝜈)

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:

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 bosonic d.o.f. and 16+16=32 fermionic d.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 triangleInequality constraints:

The Manifold Condition for SpinorsThe 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 denotingThese relations decouple and can be solved (modulo b.c.):

manifold condition

The Manifold TFT for SpinorsThe 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).

Analytic Continuation to SUL(2) x SUR(2)

19

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

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 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)