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Time as Unfolding of Process
David Rideout
Department of MathematicsUniversity of California, San Diego
Time in Quantum Gravity 13 March 2015
Time as Unfolding of (a Quantum) Process
David Rideout
Department of MathematicsUniversity of California, San Diego
Time in Quantum Gravity 13 March 2015
Plan of this Talk
1 Introduction to Causal Sets
2 Dynamics of Causal Sets
3 Results
4 Quantum Causal Set Dynamics via Action Integral
5 Summary and Conclusions
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Outline
1 Introduction to Causal Sets
2 Dynamics of Causal Sets
3 Results
4 Quantum Causal Set Dynamics via Action Integral
5 Summary and Conclusions
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
G. ’t Hooft, “Quantum gravity: a fundamental problem and some radical ideas",in Recent developments in gravitation, Cargèse Summer School Lectures 1978
take lattice discretization seriously, as model for gravitymetric tensor difficult to define on lattice, while respecting covariancesimpler, invariant concept: causal orderingin discrete language, partial orderingthis partial order defines the latticeconsistency with causal ordering of continuum defines correspondencebetween discrete and continuumproposes that the early universe will emerge from a single discreteelement, followed by a tree
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Taketani stages of theory construction
substance of theory:
"what really exists?"
perihelion precession of Mercury
what should phenomena theory explain?
"equations of motion for substance"
spacetime manifold
Einstein’s equations
Aristotelian ’gravity’
Galilean ’gravity’
Special Relativity
General Relativity
Quantum Gravity
dynamics
phenomenology kinematics
(deflection of light by the Sun)
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Causal Sets: Fundamentally Discrete GravityBased upon two main observations:
Richness of causal structureNeed for discreteness
Properties of discrete causal order ≺:
irreflexive (x 6≺ x)transitive (x ≺ y and y ≺ z ⇒ x ≺ z)locally finite (|{y|x ≺ y ≺ z| <∞)
x
y
Some definitions:
relation & linkchain & antichain, height & widthcausal interval or order intervalmaximal & minimal elements
x
y
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Spacetime Manifold as Emergent StructureThe continuum approximation
Embedding – order preserving map φ : C → (M, g)
x ≺ y ⇔ φ(x) ≺ φ(y) ∀x, y ∈ CFaithful embedding (‘Sprinkling’):
“preserves number – volume correspondence”Spacetime does not possess structure at scalessmaller than discreteness scale
∃ faithful embedding =⇒ (M, g) approximates C
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tim
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space
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Clock Time: Timelike Distance
Timelike geodesic is extremal chronological curve
Lorentzian signature =⇒ longest curve
Length L of longest chain between x and y
d(x, y) := L
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Clock Time: Timelike DistanceLength L of longest chain between x and y
d(x, y) := L
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Clock Time: Timelike DistanceLength L of longest chain between x and y
d(x, y) := L
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Brightwell & Gregory (Phys. Rev. Lett. 66: 260-263(1991)) state:
L(ρV )−1/d → md as ρV →∞
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Outline
1 Introduction to Causal Sets
2 Dynamics of Causal Sets
3 Results
4 Quantum Causal Set Dynamics via Action Integral
5 Summary and Conclusions
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
What do we mean by Dynamics?Hamiltonian evolutionLagrangian ActionSum over historiesClassical:
P (E) =∑
γ∈Ep(γ)
Quantum:A(γ) = eiS(γ)/~
P (E) =∑
q
|∑
γ∈E;γ(T )=q
A(γ)|2
Quantum Measure
P (E) = D(E,E)
D(E1, E2) = “∑
γ∈E1,γ′∈E2
ei(S(γ)−S(γ′))/~δ(γ(T ), γ′(T ))”
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Simplest Dynamical Law: ‘Typical’ Objects
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Simplest Dynamical Law: ‘Typical’ Objects
Sequence of many coin flips.Which is the ‘typical’ sequence?
1 HTHHHHTTHHTHTHHTTTTTHTHHTTTHHHHTTTHHHTH
2 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
3 THTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHT
4 HTHTTHTTTHTTTTHTTTTTHTTTTTTHTTTTTTTHTTT
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Typical Graphs
Typical Causal Sets
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1
7 810
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75 6
9 10 8
11 13
16
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22
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23
25
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29
30 31
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
‘Entropy Crisis:’ Dynamical Emergence of theContinuum
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
tim
e
space
2N lnN continuua vs. 2N2/4 Kleitman-Rothschild orders
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Sequential Growth DynamicsDR, R Sorkin
Grow causal set, ‘one element at a time’, beginning with empty setStochastic (Markov) processProbabilities based upon three principles
‘Internal temporality’ (Causet grows only to the future)
Discrete general covariance
Bell causality
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Sequential Growth DynamicsDR, R Sorkin
Grow causal set, ‘one element at a time’, beginning with empty setStochastic (Markov) processProbabilities based upon three principles
‘Internal temporality’ (Causet grows only to the future)
Discrete general covariance
Bell causality
The poset of finite causal sets
q3
q = 10
2
2
3
3
q
q
q
q
q
3
33
2
1
2
q4
2
2
2
2
3
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Sequential Growth DynamicsDR, R Sorkin
The poset of finite causal sets
q3
q = 10
2
2
3
3
q
q
q
q
q
3
33
2
1
2
q4
2
2
2
2
3
Pr(Cn → Cn+1) ∝$−m∑
k=0
($ −mk
)tk+m
Infinite sequence of free parameters (‘coupling constants’)tn ≥ 0
‘Transitive percolation’ dynamics tn =(
p1−p
)n
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Outline
1 Introduction to Causal Sets
2 Dynamics of Causal Sets
3 Results
4 Quantum Causal Set Dynamics via Action Integral
5 Summary and Conclusions
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Time as Unfolding of (a Quantum) Process:Results‘Pre-Quantum’ :
Cyclic cosmology with evolving ‘coupling constants’Gives rise to deSitter like early universeGives rise to ‘internal time’ within Complex Networks(e.g. Internet)
Quantum :
When do the Kleitman-Rothschild causets dominate?(in volume-time, n)Are almost all histories roughly time-reversalsymmetric?Is the dynamics able to escape from theKleitman-Rothschild super-exponential dominance?Is there current observational evidence hinting atquantum cosmology of this form?
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Structure of Transitive Percolation
Completely homogeneous : future of an element isindependent of anything spacelike to itDue to random fluctuations, however, appearsinhomogeneous in time
����������������������������������������������
����������������������������������������������
������������������
������������������
"post"
‘Originary’ dynamics subsequent to post : Eachnewborn element must connect to at least one otherelementUniverse expands to volume ∼ 1/p
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Cosmic Renormalization
Growth dynamics formally Markovian,because entire past history is taken ascurrent state, however has long memory‘Cosmic renormalization’:Can describe growth of subsequent cyclesas new (originary) dynamics, withrenormalized parameters (tn)
Transitive percolation is unique fixed pointof cosmic renormalizationAttractive fixed point, no cycles (pointwiseconvergence)Known that tn = (α/ lnn)n, α ≥ π2/6contains infinite number of posts.
(tn)
(t̃n)
( ˜̃tn)
(Denjoe O’Connor, Xavier Martin, DR, Rafael Sorkin)(Avner Ash, Patrick McDonald, Graham Brightwell)
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Originary PercolationRandom tree era
limit p� 1
originary — each elt chooses exactly one ancestor simple model of random treeexponential expansionfuture of every element itself originary percolation=⇒ causal set is 0+1 dimensional at smallestscalesnot exactly spacetime manifold of GR
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Originary Percolation
N = 16, p = 0.2
Early Universe of Growth Dynamics
1
10
100
1000
5 10 15 20 25 30 35
max N♦2+1 dS
3+1 dS
4+1 dS
maxN
♦+1
L
` = 6.81± .72m = 1.926± .023
M. Ahmed and DR, Phys.Rev.D 81, 083528 (2010)arXiv:0909.4771 [gr-qc]
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Network Science
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Network Science
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Network Science
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Network Science
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Outline
1 Introduction to Causal Sets
2 Dynamics of Causal Sets
3 Results
4 Quantum Causal Set Dynamics via Action Integral
5 Summary and Conclusions
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Path Integral for Gravity
Lorentzian functional integral for gravity
Z =
∫
(M,g)eiSEH[(M,g)]/~
Lorentzian ‘path’ sum over causal sets
Z =∑
C
eiSEH[C]/~
Restrict sum to fixed (finite) cardinality−→ Fixed spacetime volume ∼ unimodular gravityNeed expression for SEH[C]
Discrete �: Towards an Expression for SEH[C]Discrete D’Alembertian operator (R. Sorkin)
�(2)φ(x) =4
`2
−1
2φ(x) +
∑
y∈L1
−2∑
y∈L2
+∑
y∈L3
φ(y)
where
Li = {y ∈ C|y ≺ x and N♦(y, x) = i− 1}x
L1
L2
L3
L2
Discrete �: Towards an Expression for SEH[C]
Discrete D’Alembertian operator (R. Sorkin)x
L1
L2
L3
L2
In 4d: (Benincasa-Dowker)
�(4)φ(x) =4√6`2
−φ(x) +
∑
y∈L1
−9∑
y∈L2
+16∑
y∈L3
−8∑
y∈L4
φ(y)
High density `→ 0 limit
xL1L2L3L4
High density `→ 0 limit
xL1L2L3L4
Einstein-Hilbert action for Causal Sets(Benincasa-Dowker PRL Jan 2010))
In curved spacetime:
lim`→0〈�(4)φ(x)〉 =
(�− 1
2R(x)
)φ(x)
�(4) (-2) gives Ricci scalarUse to write Einstein-Hilbert action for causal set
S(4)EH [C] = O(1)(N(C)−N1(C)+9N2(C)−16N3(C)+8N4(C))
where Ni(C) = |{x, y ∈ C|N♦(x, y) = i− 1}|Expression for path sum for causal sets, appropriateto 4d:
Z =∑
C∈Ceiβ̃(N(C)−N1(C)+9N2(C)−16N3(C)+8N4(C))
Einstein-Hilbert action for Causal Sets(Benincasa & Dowker PRL Jan 2010)
Expression for (4d) path sum for causal sets:
Z =∑
C∈CeiSEH[C]/~ =
∑
C∈Ceiβ̃(N(C)−N1(C)+9N2(C)−16N3(C)+8N4(C))
0
5
1
2
3 4
N = 6N1 = 6N2 = 0N3 = 1
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Generalized ‘Wick Rotation’
Usual approach is to perform Wick rotation t→ it
Alternative: Analytically continue coefficient β̃ 7→ iβCasts sum into thermodynamic partition function
Z =∑
C∈Ce−β(N(C)−N1(C)+9N2(C)−16N3(C)+8N4(C))
‘Euclidean’ sum, can be analyzed numericallyusing Metropolis Monte Carlo techniques
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Markov Chain Monte Carlo on Causal Sets(J. Henson, DR, R. Sorkin, S. Surya)
Markov Chain: random walk on set of ‘states’,governed by mixing matrix MTheorem: If M satisfies
ErgodicityDetailed balance
Pr(C1)Pr(C1 → C2) = Pr(C2)Pr(C2 → C1)
then, independently of initial state, at late timesprobability to visit state C is Pr(C)
Metropolis Monte Carlo over (naturally labeled)partial orders (x ≺ y =⇒ x < y)Found two moves which satisfy these conditions−→ Use uniform mixture of two movesTransitivity — must enforce non-local constraint onrelationsDefine link xC y: x ≺ y and {z|x ≺ z ≺ y} = ∅
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Relation Move
0
1
2
0
2
1
If xC y: Remove single relationx ≺ yIf x ⊀ y and form critical pair(past(x) ⊆ past(y) andfut(y) ⊆ fut(x)):Insert single relation x ≺ yElse do nothing
L1
L2 L3
V
chain
Lambda
antichain
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Link Move
0
1
2
0 1
2
If xC y: Remove all relations fromincpast(x) to incfut(y),save those required by transitivityvia other elementsIf x ⊀ y, and @ links from incpast(x)to incfut(y):Insert all relations from incpast(x)to incfut(y)
Else do nothing antichain
L1
L3
L2
chain
V
Lambda
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
~→∞ Limit~→∞ =⇒ β = 0 Uniform measure on samplespaceCausal sets on up to 16 elements enumeratedexplicitlyKleitman-Rothschild theorem (Trans. AMS 1975)
0
7 9111213 1719202122
2325 26 27 282930 2431
1
8 1018
2
14 15 16
3 4 56
Non-locality / long range interaction How big is big?In the remainder of this paper we will adoptthe convention that any inequality or otherstatement about functions of n will bemeant to be true only for all n sufficientlylarge, where how large depends on thestatement. This will be a convenience sincethere are so many such statements below.
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Height distribution for n ≤ 9
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9
fraction o
f n-o
rders
n
1
2
3
4
5
6
7
8
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
0 1 2 3 4 5 6 7 8 9
fra
ctio
n o
f n
-ord
ers
n
1
2
3
4
5
6
7
8
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Height distribution for n ≤ 82
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80
fra
ctio
n o
f n
-ord
ers
n
1
2
3
4
5
6
7
8
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Height distribution for n ≤ 82 (logscale)
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
0 10 20 30 40 50 60 70 80
fra
ctio
n o
f n
-ord
ers
n
1
2
3
4
5
6
7
8
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Mean height for n ≤ 82
1.5
2
2.5
3
3.5
4
4.5
0 10 20 30 40 50 60 70 80
me
an
he
igh
t
n
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Cardinality of Level 2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
fra
ctio
n o
f n
-ord
ers
(cardinality of level 2) / n
72737679
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Number of minimal and maximal elements
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
-30 -20 -10 0 10 20 30 40
fra
ctio
n o
f 5
8-o
rde
rs
|min||max|
|max| - |min|
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Ordering Fraction
0
0.002
0.004
0.006
0.008
0.01
0.012
0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42
fra
ctio
n o
f 5
8-o
rde
rs
r
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Mean ordering fraction for n ≤ 82
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0.5
0 10 20 30 40 50 60 70 80
me
an
ord
erin
g f
ractio
n
n
3/81/3
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Escape from KR orders
0.0001
0.001
0.01
0.1
1
1 2 3 4 5 6 7 8
fra
ctio
n o
f 6
4-o
rde
rs
height
beta = -0.057beta = 0
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Tension with ΛCDM Concordance Model
“Current measurements of the low and highredshift Universe are in tension if we restrictourselves to the standard six parameter modelof flat ΛCDM.”
[Wymann, Rudd, Vanderveld, Hu, arXiv:1307.77152(2 Jan 2014)]
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Tension with ΛCDM Concordance Model
0.5 1.0 1.5 2.0z
- 2
0
2
4
6
8
Ρ L H z LΡ L H 0L
[I. Jubb, F. Dowker, private communication, based onarXiv:1407.5405 [gr-qc]]
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Outline
1 Introduction to Causal Sets
2 Dynamics of Causal Sets
3 Results
4 Quantum Causal Set Dynamics via Action Integral
5 Summary and Conclusions
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Time as Unfolding of (a Quantum) Process‘Pre-Quantum’ :
Cyclic cosmology with evolving ‘coupling constants’Gives rise to deSitter like early universeGives rise to ‘internal time’ within Complex Networks(e.g. Internet)
Quantum :
When do the Kleitman-Rothschild causets dominate?(in volume-time, n) For some n > 100 perhaps.Are almost all histories roughly time-reversalsymmetric? No!Is the dynamics able to escape from theKleitman-Rothschild super-exponential dominance? Yes!Is there current observational evidence hinting atquantum cosmology of this form? Perhaps!
Time as Unfoldingof (a Quantum)
Process
D Rideout
Introduction toCausal SetsPreliminaries
Definitions
Correspondence withContinuum
Clock Time
Dynamics ofCausal SetsWhat is Dynamics?
Uniform Distribution
Sequential GrowthDynamics
ResultsTransitive Percolation
Originary Percolation
Network Science
Quantum CausalSet Dynamics viaAction IntegralMarkov Chain Monte Carloon Causal Sets / PartialOrders
Results
Onset of AsymptoticRegime
Finite β
Observational Cosmology
Summary andConclusions
Escape from KR orders
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2 3 4 5 6 7 8 9
fra
ctio
n o
f o
rde
rs
height
n = 67; beta = -0.045n = 64; beta = -0.057