time as unfolding of (a quantum) process · time in quantum gravity 13 march 2015. time as...

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

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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



3 Results




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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


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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)


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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


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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

e

space

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0.9

0.92

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0.96

0.98

1

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

tim

e

space


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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


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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

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1


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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

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-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 →∞


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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 ))”


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Simplest Dynamical Law: ‘Typical’ Objects


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Simplest Dynamical Law: ‘Typical’ Objects

Sequence of many coin flips.Which is the ‘typical’ sequence?

1 HTHHHHTTHHTHTHHTTTTTHTHHTTTHHHHTTTHHHTH

2 HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

3 THTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHT

4 HTHTTHTTTHTTTTHTTTTTHTTTTTTHTTTTTTTHTTT


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Typical Graphs

Typical Causal Sets

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7 810

20

16

25

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9 11

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30

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911 13 172021 22 23 2426

28 30 2731 29

1

810 16 1825

2

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12 15

4

14

5 67

0

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4 3

75 6

9 10 8

11 13

16

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181920

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30 31


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‘Entropy Crisis:’ Dynamical Emergence of theContinuum

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2N lnN continuua vs. 2N2/4 Kleitman-Rothschild orders


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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


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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

qq

3

33

2

1

2

q4

2

2

2

2

3


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The poset of finite causal sets

q3

q = 10

2

2

3

3

q

q

q

q

q

qq

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


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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?


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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


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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)


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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


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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]


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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

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


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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


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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} = ∅


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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


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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


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~→∞ 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.


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Height distribution for n ≤ 9

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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

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f n

-ord

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n

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8


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Height distribution for n ≤ 82

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8


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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

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n

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8


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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


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Cardinality of Level 2

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0.5

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

fra

ctio

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f n

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(cardinality of level 2) / n

72737679


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Number of minimal and maximal elements

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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|


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Ordering Fraction

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0.002

0.004

0.006

0.008

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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

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r


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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


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Escape from KR orders

0.0001

0.001

0.01

0.1

1

1 2 3 4 5 6 7 8

fra

ctio

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f 6

4-o

rde

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height

beta = -0.057beta = 0


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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)]


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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]]


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Outline



3 Results




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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!


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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

time as unfolding of (a quantum) process · time in quantum gravity 13 march 2015. time as...

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