Welcome toThe Propagation of Light in Causal Set Theory Spring 2009 Jake Askeland, Jonathan Baptist,...

ThePropagationof Light inCausal Set

Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Welcome

Welcome to

CAMCOS Reports Day

Spring 2009


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

The Propagation of Light in Causal Set Theory

Spring 2009

Jake Askeland, Jonathan Baptist, Miranda Braselton,David von Gunten, Douglas Mathews, Duncan McElfresh,

Cheuk Wong

In collaboration with NASA Ames Research Center, Dr. Jeff ScargleCAMCOS Advisor: Slobodan Simić


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Outline

1 Introduction

2 Space-time and Light

3 Causal Sets

4 Feynman

5 Our Discrete Model

6 Johnston

7 Goal and Results

8 Computation

9 The Future


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Introduction: Relativity

Einstein discovered Relativity.

Relativity is a physical theory that explains how gravityworks.

Relativity gives good results at large distances.

It describes how planets move and why solar systems areshaped the way they are.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Introduction: Quantum Mechanics

Quantum Mechanics describes how tiny things interact.

It describes how atoms move and are structured.

Quantum Mechanics gives nondeterministic results, sopredictions of events are in probabilities.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Introduction: Theories in Conflict

Quantum Mechanics and Relativity don’t make sensetogether.

They are inconsistent with each other, and yet, accuratelydescribe real events in our universe.

For the last 50 years, physicists have searched for a newtheory that unifies these two.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Introduction: Gamma Ray Bursts


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Introduction: Gamma Ray Bursts

Gamma rays are made of particles of light, called photons,at their highest energies.

Bursts happen about once per day, in places all over theuniverse.

Gamma Ray Bursts are the brightest thing we know of,lighting up the entire universe!

Scientists discovered gamma rays with higher energy tooklonger to arrive.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Introduction: FERMI Gamma Ray Space Telescope

FERMI launched June 11, 2008 to investigate thisphenomenon.

FERMI collects data on photons that traveled billions oflight years.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Introduction: Speed(s) of Light?

Is the speed of light really constant?

The data from FERMI may suggest the speed of a photonis dependent on its energy.

Causal Set Theory(Sorkin, et al.) is a physical theory thatmay provide an explanation for this.

Causal Set Theory is a theory intending to unify QuantumMechanics and Relativity.

As such, we can only compute the probabilities of events.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Introduction: Our Model

We use a mathematical framework based on Causal SetTheory.

Our goal was to determine if the speed of light is constant.

Comparing the model with the FERMI data could result innew understanding.

If the model fits, great! If it doesn’t, great?

Let’s take a closer look at what we have done thissemester...


Theory

Introduction

Space-timeand Light

Space-time

Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Space-time

Definition

Space-Time is a mathematical model of the universe combiningspace and time.

Any event that occurs in the universe has a specificlocation in space-time

The space-time we observe has 3 spatial dimensions andone temporal dimension

Thus Space-Time has 4 dimensions, or 3+1 D


Theory

Introduction

Space-timeand Light

Space-time

Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

1+1D Space-time


Theory

Introduction

Space-timeand Light

Space-time

Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Space-Time Representation of CAMCOSPresentation Day


Theory

Introduction

Space-timeand Light

Space-time

Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Proper Time

Time passes differently for particles with different velocities

proper time is τ =√

t2 − x2

τ is proper time, time as percieved by the blue particle

t is time percieved by the red particle

x is distance away from the red particle


Theory

Introduction

Space-timeand Light

Space-time

Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Nature of Light

Definition

Light is electromagnetic radiation of any wavelength

The light spectrum is made up of the differentwavelengths of light from radio to gamma

The smallest packet of light is called a photon


Theory

Introduction

Space-timeand Light

Space-time

Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Light Spectrum

Increasing energy =⇒


Theory

Introduction

Space-timeand Light

Space-time

Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Speed of Light

When a photon travels through a medium, such as air,water or glass, the speed of light is slowed

Einstein postulated that photons travel at a constantspeed, c ≈ 3 · 108m/s in vacuumOne of the goals of this project is to explore the possibilitythat light in a vacuum does not, in fact, travel at aconstant rate, but actually varies in speed depending onthe energy of the photon.


Theory

Introduction

Space-timeand Light

Space-time

Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Light Cone

Definition

The light cone (L) of a point P is the set of all points to andfrom which a particle can reach or be reached while traveling atthe speed of light or less.

��

��

��

Time

Space

P

S

Q

RCan influence

connected to P

Not causally

Can be influenced

event P

by event P


Theory

Introduction

Space-timeand Light

Space-time

Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Light Cone

Shows where a particle may travel to in space with respectto time

��

��

��

Time

Space

P

S

Q

RCan influence

connected to P

Not causally

Can be influenced

event P

by event P


Theory

Introduction

Space-timeand Light

Space-time

Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Light Cone

The boundaries of the cones show the path of a photontraveling at the speed of light

The slope of these boundaries are ± the speed of light

��

��

��

Time

Space

P

S

Q

RCan influence

connected to P

Not causally

Can be influenced

event P

by event P


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Theory of Causal Sets

Initiated by Sorkin(1987):

Attempts to unify Relativity and Quantum Mechanics

An alternative to string theory

Space-time is discrete

Gives up the properties of a continuous universe


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Definition of a Causal Set

Definition

A causal set, S , is a discrete set equipped with a relation ofpartial order.

The discrete set is spacetime.

The partial order describes whether a point P caninfluence point Q in spacetime.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Discrete Space-time Analogy

Film analogy:

Like the universe, a film is perceived to be continuous

Each second of film consists of multiple frames

Fast sequencing produces the illusion of continuous motion

Similarly, discrete space-time is perceived as continuous


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Space-time Is Discrete

No distance, area, volume, etc. in the usual sense.

In discrete space-time, these are determined by thecombinatorics of the set.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Graphical Representation of a Causal Set

Easy to represent as graphs of points and arrows betweenthem:

The vertices are points in space-time

The edges, or arrows, are transitions between the points

Time

Space1

3

2

4


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Causal Relations

Causal relations and events obey intuitive laws of physics:

No going backwards in time

No self-causing events (no loops)


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Light Cone With Causal Relations

��

��

��

1

3

2

4

Space

Time

S

RCan influence

connected to P

Not causally

Can be influenced

event P

by event P


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Feynman PathSum

All Paths

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Feynman Path Sum

Feynman plus path = Feynman Path Sum


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Feynman PathSum

All Paths

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Quantum Mechanics and Feynman Path Integral

Quantum Mechanics works differently from ClassicalMechanics.

Feynman Path Integral is one of the main approach toQuantum Mechanics.

In our model, we applied something similar to FeynmanPath Integral (Feynman Path Sum) to Causal Set Theory.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Feynman PathSum

All Paths

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Probability Amplitude

In Quantum Mechanics, one of the main idea regardingparticles is probability amplitude.

We can find the probability of a particle traveling frompoint A to B by taking the absolute square of a complexnumber.

This complex number is called the probability amplitude,denoted by Ψ.

prob(A B) = |Ψ|2


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Feynman PathSum

All Paths

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Feynman Path Sum

Feynman (1948) developed Feynman Path Integral as anapproach to Quantum Mechanics. Since Causal Set Theorydeals with discrete space time, we will be using the discreteversion of Feynman Path Integral, Feynman Path Sum instead.

Definition

Path Sum is the summation of the probability amplitudes of allpossible paths for a particle to travel from point A to point B.

prob(A B) =

∑γ

Ψ(γ)

2

where γ runs over all possible paths from A to B


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Feynman PathSum

All Paths

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Feynman Path Sum, cont.

The probability amplitudes for all paths that go from pointA to point B are added together.

The probability of a particle moving from point A to pointB is the absolute square of the path sum:

prob(A B) =

∑γ

Ψ(γ)

2



Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Feynman PathSum

All Paths

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Feynman Path Sum, cont.

prob(A B) =

∑γ

Ψ(γ)

2



Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Feynman PathSum

All Paths

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

What Are All Paths?

Following Feynman’s approach:

All paths include paths that “go backwards” in time

All paths include paths that move outside the light cone.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

AdjacencyMarticies

AmplitudeMatrix

PropagatorMatrix

Johnston

Goal andResults

Computation

The Future

Representing the Causal Set

To visualize the causal set mathematically we create anadjacency matrix using the causal connections


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

AdjacencyMarticies

AmplitudeMatrix

PropagatorMatrix

Johnston

Goal andResults

Computation

The Future

Amplitude Matrix Definition

The amplitude matrix contains the probability amplitudesfor single jumps

The entries are defined by Ψ = e iS where

S = S [Energy, Starting Point, Ending Point]

S is called the action and is pretty much proper time


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

AdjacencyMarticies

AmplitudeMatrix

PropagatorMatrix

Johnston

Goal andResults

Computation

The Future

Amplitude Matrix Example

The entry in row 1, column 2 contains the probabilityamplitude, e iS1 , for a single jump from point 1 to point 2.

A =

0 e iS1 e iS2 e iS3

0 0 0 e iS4

0 0 0 e iS5

0 0 0 0


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

AdjacencyMarticies

AmplitudeMatrix

PropagatorMatrix

Johnston

Goal andResults

Computation

The Future

Amplitude Matrix Example

The probability amplitude of a photon moving from point 1 to4 using all paths is Ψ = e i(S3)

γ(1,4)+ e i(S1+S4)γ(1,2)·γ(2,4)

+ e i(S2+S5)γ(1,3)·γ(3,4)

.

A =

0 e iS1 e iS2 e iS3

0 0 0 e iS4

0 0 0 e iS5

0 0 0 0


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

AdjacencyMarticies

AmplitudeMatrix

PropagatorMatrix

Johnston

Goal andResults

Computation

The Future

Path Sums Using Matrices

Independently implemented by CAMCOS Spring2008/Day, Johnston 2008

Each nth power of A gives the probability amplitude of apath of length n

We take the infinite sum of the amplitude matrix to createthe propagator matrix K

K = A + A2 + . . . = (I − A)−1 − I


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

AdjacencyMarticies

AmplitudeMatrix

PropagatorMatrix

Johnston

Goal andResults

Computation

The Future

Path Sums Using Matrices Example

Applying this formula to our example gives the following:

K = A + A2 + ... =

0 e iS1 e iS2 Ψ0 0 0 e iS4

0 0 0 e iS5

0 0 0 0

Where Ψ = e i(S3) + e i(S1+S4) + e i(S2+S5).

In practice our matrices are 800 by 800 so Ψ becomescrazy


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

AdjacencyMarticies

AmplitudeMatrix

PropagatorMatrix

Johnston

Goal andResults

Computation

The Future

Intermission

Intermission


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

AdjacencyMarticies

AmplitudeMatrix

PropagatorMatrix

Johnston

Goal andResults

Computation

The Future

Summary

What you learned so far:

Physics background

Causal Set Theory

Feynman Path Sum

What‘s to come:

Previous work

Our project goals

Results

Computational implementation of our model

Further research


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

The Future

Previous works on Causal Set Theory

The work of Johnston (Imperial College) has some similaritieswith ours. However:

His method is different (e.g., he doesn’t allow all paths inhis calculation).

His goal is different (to solve the Klein-Gordon equation).

We implemented Johnston’s model, but we found that hisapproach to Feynman Path Sum and Causal Set does not seemto be suited to examining the question of whether the speed oflight is constant.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Previous Work

Goal

Results

Computation

The Future

CAMCOS 2008

In spring 2008, CAMCOS built a simulated environment forexperiments.

Discrete spacetime

Feynman path sums

Matrix models

This is what we do:

1 Define how our photons should move - different definitionsof S - the action.

2 Calculate the speed of our photons, by observing wherethey land on the detector.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Previous Work

Goal

Results

Computation

The Future

CAMCOS 2008

Using this framework, many different definitions of S wereused.

Unfortunately, results were inconclusive.

Experiments showed that photons traveled at manyspeeds, but did not ”prefer” standard c .


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Previous Work

Goal

Results

Computation

The Future

What We Want

We want our experimental results to agree with FERMI’sobservations.

Light usually travels at 2.99x108 meters per second.

But a photon’s speed can vary slightly, depending onits energy.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Previous Work

Goal

Results

Computation

The Future

What We Want

Figure: An Ideal Probability Vs. Velocity Graph

In this experiment, 25% of all photons arrived at the detectorwith speed c .

The peak indicates that most photons still travel at standard c .


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Previous Work

Goal

Results

Computation

The Future

Example Results

Figure: Probability vs. Velocity for Energy Scaling A


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Previous Work

Goal

Results

Computation

The Future

Example Results

Figure: Probability vs. Velocity for Energy Scaling B


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Previous Work

Goal

Results

Computation

The Future

Example Results

Figure: Probability vs. Velocity for Energy Scaling C


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Previous Work

Goal

Results

Computation

The Future

Example Results

What you just saw:

There is a distinct peak.

We have not yet observed that this peak changes with aphoton’s energy.

When we ran our experiment with Johnston’s approach,but did not see a conclusive peak.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Previous Work

Goal

Results

Computation

The Future

Project accomplishments

Tried different several different formulas for S .

Improved our experiment runtime by a factor of 10.

Began using a histogram to represent our probabilities.

Observed consistent results with many differentconfigurations.

Changed the code to allow for 2 and 3 space dimensions.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation

ComputationalAnalysis

The Future

Computational Analysis

Why is it hard to understand and describe these probabilities?

Our universe is really big,

There are a lot of points in space-time a photon can be.

So “All paths” means a lot of calculations.

Imagine ψ = e i(S3) + e i(S1+S4) + e i(S2+S5) + ... for morepoints than you can imagine!


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation


The Future


In a single run of our model universe:

We restrict ourselves to 800 space-time points.

That’s still 640,000 matrix entries in each of 4 matrices.

We then compute tens of millions of paths!

But at least our ψ is manageable...


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation


The Future


ψ =e i(S1) + e i(S1+S4) + e i(S2+S5) + e i(S8+S3) + e i(S8+S1)+e

i(S8+S3)+

e i(S2) + e i(S4+S7) + e i(S7+S8) + e i(S9+S2) + e i(S4+S3)+ei(S4+S2)+













e i(S2) + e i(S9+S2) + e i(S3+S0) + e i(S1+S4) + e i(S6+S6)+ei(S9+S5)+...


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation


The Future

Comparison to previous models

Previous work on this idea has been done without experimentalmodeling:

The run-time of naive models would be unbounded, theycould continue for lifetimes, or just hours.

Thus, there would be no bounds on the size or quantity ofoutput, so data would be unwieldy.

And so, of course, one would have no idea whether light inthe model tends to travel at what we call the speed oflight.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation


The Future

Comparison to CAMCOS 2008

At 800 points in a single run of the experimental model:

The run completes in 38.1 seconds.

Files outputted during the run total 14.7 megabytes.

A distribution of velocities and their probabilities isunrecognizable or unreproducible.

If we amassed results from this model it would take over 2hours and nearly 3 gigabytes of hard drive space to getenough data for a reasonable histogram (200 runs).


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation


The Future

Computational Analysis: Our model

At 800 points in a single run of our experimental model:

A run completes in 4.12 seconds.

We append only the data we need to a file of about 5kilobytes.

Our model can aggregate results,

So we can see a consistent velocity/probability distributionwhich appears to be Gamma*

It takes 13.8 minutes and about 1 megabyte of hard drivespace for the same 200-run histogram.

*A Gamma distribution has a skew to the left based on a shapeparameter.


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResults

Computation


The Future


Comparison @ 800 pts:

Year Run time Output size Velocity distribution

< 2008 N/A N/A None2008 38.1 seconds 14.7 mbytes Unrecognizable2009 4.12 seconds 5 kbytes Humanly recognizable


Theory

Introduction

Space-timeand Light

Causal Sets

Feynman

Our DiscreteModel

Johnston

Goal andResu

Welcome toThe Propagation of Light in Causal Set Theory Spring 2009 Jake Askeland, Jonathan Baptist,...

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