causal space-time on a null lattice with hypercubic coordination
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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)