Transcript Varieties of Causal Sets

Varieties of Causal Sets
Jose L Balduz Jr
Department of Physics
Mercer University
APS Meeting April 2007
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Outline: Varieties of Causal Sets
What Are Causal Sets? Why?
Standard definition (partial ordering). Motivation.
Allowed structures in standard causal sets.
Generalized Causal Sets
Transitive link as physical nonlocal connection.
Geodesic core and halo. Time dilation.
Missing link, spacelike link, causal loop link.
1.
Spacelike Links
Generating causal sets from hypergraphs.
Allowed structures. Duality.
Time and distance measures; the light cone.
2.
Causal Loops
The (forward and backward) Global Ordering.
Construction. Invariant skeleton.
Loops and their causal shadows.
3.
Nonlocal Links
A conductance model for effective distance, with probabilitistic halo.
Distance shrinkage and inversion.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Outline: Varieties of Causal Sets
Causal Set
Generalized or “Loopy” Causal Set
Geodesic Core and Halo
Global Ordering of Core Classes
Global Core Skeleton
Loopy Causal Sets and Global Structure
Loopy Skeleton with Shadows
Classification of (Loopy) Causal Sets
Conductance Model of Propagation on Graphs
Model 1: Uncorrelated Links + Graphs
Model 2: Correlated links w. Partition Function + Graphs
Conclusions
Appendix: Causal Set Connected Cores for N = 2, 3, 4 and 5
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
CAUSAL SET
This is a (locally finite) set of points with a pair-wise causal
relation (one-way, time-like) that obeys certain rules:
•The relation is transitive.
•There are no closed loops.
* If one also assigns a “volume” to each point, then in the limit of
many points this is equivalent to a general-relativistic spacetime
manifold. To see this consider a “sprinkling” or uniform
scattering of the points onto the manifold…
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Generalized or “Loopy” Causal Set (Causet)
For an ordered graph,
the rules for causal links are more relaxed.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Generalized or “Loopy” Causal Set (Causet) (2)
Additional types of allowed triangles:
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Generalized or “Loopy” Causal Set (Causet) (3)
Reinterpretation of causal triangles:
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Geodesic Core & Halo
• A link is geodesic iff it cannot be replaced by a longer causal
path. Apply this only to causets without loops.
• Multiple geodesic paths from A to B can be interpreted as
examples of time dilation. (See Twin Paradox.)
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Geodesic Core & Halo (2)
•Any non-loopy causet can be uniquely reduced to its geodesic
core by deleting any non-geodesic links. A number of (nonloopy) causets can in turn be generated from this core by adding
any number of non-geodesic links. These NG links form a “halo”
around the core, and they affect propagation on the causet.
• This classifies all non-loopy causets by core equivalence class.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Global Ordering of Core Classes
Irrespective of halo, any core can be ordered uniquely
in the forward and backward directions.
Points in each forward (backward) level or “raft” have all past (future) points
located in earlier (later) rafts and are not linked to each other.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Global Ordering of Core Classes (2)
For each point, the time indices (t+, t-) give the lengths of
maximal (past, future) geodesics passing through that point.
In this case, the sum is  6.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Global Core Skeleton
• On the (t+, t-) plane, a dot is occupied if the intersection of the
t+ raft and the t- raft is not empty.
• The same skeleton is
shared by different cores,
which may have different
numbers of points.
Skeleton is thus also an
equivalence class of cores
and (non-loopy) causets.
• For any point in a given
dot (shown as ), some
dots lie in its future, some
in its past, others in its
present.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Loopy Causal Sets and Global Structure
If loops are present, the above construction fails.
• Some points are in the future of a loop, so they cannot
be assigned a t+.
• Some points are in the past of a loop, so they cannot be
assigned a t-.
• Some points are in between loops, so they have neither
t+ nor t-.
• Other points are causally separate from any loop, so they
have both t+ and t-.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Loopy Causal Sets and Global Structure (2)
The loopy residues are the future and past shadows of the set of points
touching loop links. If all the loop links are removed,
there results a unique core and halo. Therefore even all loopy causets
can be classified by core class and skeleton class.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Loopy Skeleton with Shadows
Presence of loops creates future and past shadows.
There are four types
of points:
• PF: accessible
from both the past
and the future.
• Some
causal
paths avoid the
loopy region.
• P: accessible only
from the past.
• F: accessible only
from the future.
• Ø: not causally
accessible.
• Other paths vanish
into or emerge from
the loops.
Once all loop links are removed, all points can be assigned (t+, t-)
and a global order skeleton uniquely determined.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Classification of (Loopy) Causal Sets
Starting from any loopy causet we can find its core:
1) Remove all loop links (causal loops)  Unique core and halo;
2) Remove all halo links (non-local, non-geodesic)  Unique core.
Every loopy causet has unique:
• Loops
• Halo
• Core
(local order, geodesics)
• Skeleton
(global order)
Conversely,
we can generate any loopy causet
by starting from a core and reversing the process.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Conductance Model of Propagation on Graphs
(Resistance: R ~ length ~ delay time)
• For simple graphs, this mimics well propagation time
of a Schroedinger field pulse.
• Multi-segment paths are long.
•Direct connections shorten distance.
• We extend this to causets without loops, as long as there is
no current “backflow” against the correct causal flow.
 Halo links shorten the interval (distance) between points,
relative to their interval as calculated with core only.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Conductance Model of Propagation on Graphs (2)
(Resistance: R ~ length ~ delay time)
• We consider only strands (linear cores) of N links.
We calculate the effective length R(N) and the
effective Planck length R(N)/N: RPlanck.
• For each potential non-local “shortcut” we define
a cost function and a Boltzman factor:
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Toy Models for Effective Length (Resistance)
Model 1: Uncorrelated Links
• Model assumptions:
1) Normal (core) links have resistance R(1) = R1 ≡ 1.
2) Halo links are all present, but they have resistance
Rn = e+nb > 1, since b > 0.
As T → 0 (b → ∞) we recover the core, with Rn ≡ N.
• Results (Also see graphs.)
1) RPlanck = R(N)/N decreases with N.
2) For sufficiently small N and large kT,
R(N) is inverted: i < j but R(i) > R(j), and
there is a sub-Planck regime: R(N) < R(1) = 1.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Toy Models for Effective Length (Resistance)
Model 1: Uncorrelated Links (2)
Approximate formula (Ignored some edge effects):
The effective Planck scale gets bigger
as the system gets smaller.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Toy Models for Effective Length (Resistance)
Model 2: Correlated Links w. Partition Function
• Model assumptions:
1) All links have the same resistance R1 ≡ 1.
2) The effective strand length is calculated as an average over all
possible haloes.
3) For each halo, its “energy” is calculated as the sum of energies
(cost functions as above) for all possible shortcuts made
available by the halo. This includes shortcuts that use more than
one halo link.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
Toy Models for Effective Length (Resistance)
Model 2: Correlated Links w. Partition Function (2)
• Results (Also see graphs.), limited to N = 1, 2, 3.
1) RPlanck = R(N)/N decreases with N.
2) For sufficiently small N and large kT,
R(N) is inverted: i < j but R(i) > R(j).
3) but there is no sub-Planck regime:
R(N) > R(1) = 1 always.
Still, the effective Planck scale gets bigger
as the system gets smaller.
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
CONCLUSIONS
• Extension of causal set to simple ordered graph (loopy causet)
expands the range of possible causal global structure, including
causal shadowing, possibly related to black holes and such…
•All loopy causets can be classified and generated by
Skeleton/Core/Halo/Loops in a unique way. (Born-Oppenheimer
analogy: Nuclei + Electrons  Core + Halo & Loops)
• Inclusion of non-local (non-geodesic) links serves to decrease
the effective length of a strand in a non-scaling way, so we may
have
TPlanck(macro) = TPlanck  10-43 sec
but TPlanck(micro) = 1/ETeV(weak)  10-28 sec?
(Solve Hierarchy Problem?)
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University
APS Meeting April 2007 Jacksonville
Jose L Balduz Jr, Mercer University