)

Causal Set Phenomenology

Fay Dowker

Imperial College, London

o

I Plan [

1. Introducing the Causal Set (Definition and

Hypothesis)'tHooft; Myrheim; Bombelli, Lee, Meyer and Sorkin

2. Effects of discreteness on the continuum

(We need evidence)FD, Henson, Sorkin

A. A model of detector response to scalarfield source

B. A model'of massive particles

3. Could high energy cosmic rays be eVI­dence?

4. Conclusions

J

I Causal Sets I

A causal set (or causet for short) is a set ofelements, C , with a binary relation, -<, called

<

"precedes" which satisfies the fOllowing ax-Ioms:

1. Tra nsitivity: if x -< y and y -< z then x -< z,

\:Ix, y, z E C;

2. Non-circularity: if x -< y and y -< x thenx == y \:Ix,y E C (non-circularity);

3. Local finiteness: for any pair of fixed ele­

ments x and z of C, the set {ylx -< y -< z} ofelements lying between x and z is finite.

Of these axioms, the first two say that C IS

a part ia II y ord ered set 0r poset and the t hi rdmakes the set discrete.

2

I The Causal Set Hypothesis I

The deep structure of spacetime is a causalset

The discrete elements of the causal set are

related to each other only by the partial or­dering that corresponds to a microscopic no­tion of before and after, and the continuumnotions of length, space and time arise onlyas approximations at large scales. Just asordinary matter appears smooth and contin­uous on large scales but is really made ofatoms, so it is proposed spacetime appearscontinuous to us but is fundamentally dis­crete.

-----

3

The number of causal set elements in a re­

gion gives rise in the continuum approxima­tion to what we experience as the spacetimevolume of the region (in fundamental unitsclose to Planck units) and the order givesrise in the continuum approximation to thespacetime causal order.

Powerful theorems (by Hawking, Malamentand Levichev) in continuum causal analysistell us that the volume and causal structure

are enough to recover the full geometry andthis means that a causal set can contain enoughinformation to be able to be well-approximatedby a continuum spacetime.

+

A nice way to represent a causal set is bydrawing its Hasse diagram. For example:

-

FUTURE

.W PAST

Elements are vertices and relations are edges.Element x precedes ("is in the past of") y

and y precedes z. Transitivity implies thatx precedes z so that relation does not needto be drawn in. The irreducible relations are

called links. Element w is unrelated to anyother.

In a causal set that could be our visible uni­verse there would be of the order of 10240

elements with a correspondingly large andcomplex web of relations.

[Phenomenological effects of discreteness I

Assuming that causal set theory is right, thespacetime we observe around us is only anapproximation to deeper level of reality thatis a causal set. One way to investigate po­tential effects of that underlying discretenessis to make models of matter, fields, par­ticles etc. on the background of a causalset that could have our observed spacetime(Minkowski space for definiteness) as an ap­proximation.

Question: What could that causet be?

Rough Answer: A causet that arises by dis­cretising Minkowski spacetime "faithfully".

Detailed Answer: A causet that is produced(with high probability) by a Poisson processof "sprinking" elements at random into Minkowskispace so that the mean number of elementsfalling in any region is given by the volumeof the region (in near-Planck units). Thesprinkled elements are endowed with the or-der relations induced by the Minkowski causalorder.

---------.-

A Poisson distribution of points in 1+1Minkowski

•

IMPORTANT POINT: This distribution is

Lorentz invariant - it is uniform in any frame.Causal sets are Lorentz invariant.

[Claim: fundamental spacetime discreteness+ Lorentz invariance leads more or less uniquelyto causal sets Henson]

7

I A. Model of source-detector response I

Consider the following setup

---~

p

wotlJ..l~~af. $OlLV(.Q. J

ci~e (Joff\1 (J. ~JIt55 SctJ o-r

Co f\J T IN lA.U.M

In the continuum, we model the output Fof the detector as the integral of the field'svalue over the region V, i.e.

F == Iv d4y ¢(y)

¢(y ) == q .Jp ds G (x ( s ), y )

where P is the worldline of the source, q isthe charge of the source and s is proper timealong P.

G(x, y) is the retarded Green's function andis equal to

G(x, y) == 211f8(lx - y12) if y E J+ (x)o otherwise

_1_8(yO - xO - r)41fr '

where r is the spatial distance from x to y.

A short calculation shows that the field ¢(y)

is roughly constant over the detector regionand is equal to q/41rR as expected.

In a causet model of this same situation a

sprinkling Os of 4D Minkowski space replacesthe continuum background. The source isnow modelled as the subset, P, of the causetwhich lies in a tube of unit Planck cross sec­tion - in the rest frame of the detector ­centred on the continuum worldline of thesource.

The detector region D is replaced by the set15 of all elements that were sprinkled intothat region of spacetime. A scalar field is afunction from the elements of the causet tothe Reals.

\0

We need a causet analogue of the retarded

Green's function G(e'i, e,j) which is zero unlessej is "on the past light cone of e/'. Thereis such a thing [Daughton, Pullin, Salgado,Sorki n]:

~ W~~\jr,)..y-

G(e'i, ej) == V6/121r if ei -< ej and the relationbetween them is a link i.e. there's no otherelement between them. And it's zero other­

wise. This works because the past links froma given element are distributed like this:

o

o

II

o

\2..

The output of the detector in this discreti­sation is F a sum over all elements sprinkledinto the detector region:

F== Le.Ei5¢(ej)J

¢(ej) == q LeiEP G( ei, ej)

where q is the charge.

This is the total number of links between

elements in ,F and in 15 and it is somethingwe know how to estimate because we know

the statistics of the sprinkling.

The expected value of the field at ej is

(ePeej)) == qv~"i127r times the expected num­ber of links between ej and elements sprinkledinto region P.

To calculate this, imagine P broken up intoI

tiny volumes, d4x. The probability that there'llbe an element in d4x at x is proportional tod4x and the probability that it will be linked

to ej at point y is the probability that theinterval between x and y is empty of sprin­kled points. That probability is e-l(x,y) wherelex, y) is the spacetime volume of the inter­val.

t3 I

Thus we need to calculate

and it gives

so the deviation from the continuum result is

truly tiny. But what about fluctations aboutthe mean?

The statistical nature of the correspondencebetween the continuum and the causal setwill mean that there are always deviationsfrom mean values (a crucial ingredient in thederivation of the measured value of the cos­mological constant from causal set theory)

\

The question is, how large will they be? Usu-ally they are very small (e.g. 10-120 for thecosmological constant). And so it turns outin this case.

In general, if the mean of some quantity isa large number N (here N is the number oflinks between two subsets of a causet) andthere are no correlations, then the standarddeviation is VN and the IIsignal to noise ra­tio" goes like VN4t.

For our link counting, there are some mildcorrelations to be taken into account, butthe bound on the s.d. is still roughly VN.For a detector of nuclear size with a timeresolution of 10-15s and R == 1M Parsec itmeans a signal to noise ratio of 1015.

That's good agreement: TOO GOOD!!

I B. Swerves I

The fOllowing is a model (due to Henson) ofa massive particle moving on a causal set.The idea is that a particle cannot follow anexact straight line geodesic because of theunderlying discreteness but it does the best itcan do and still be approximately Markovian.

We call the effect swerves, after Lucretius:

"The atoms must a little swerve at times

- but only the least, lest we should seemto feign motions oblique, and fact refute usthere."

lb

Swerves occur when a particle tries to movealong a straight line geodesic but cannot be­cause the underlying reality is discrete.

Consider a sprinkling into Minkowski space­time (1+1 dimensions are shown, really it is3+1)

•

17

• ••

•

• ••

At each stage: particle moves from the el­ement, en, where it is to one in its causalfuture, minimising the change in its momen­

tum, within a proper time Tf. Repeat.

Choosing Tf to be large compared with thePlanck scale means that the swerving effectis small: the change in momentum is smallat each step. But over many steps it can addup.

Due to the randomness of the sprinkling, themomentum is going on a Lorentz invariantrandom walk on the mass shell. In the hy­drodynamics limit of a large number of stepsthis is described by a "Brownian motion",governed by a diffusion equation on the massshell:

This is the unique (up to a free parameter)Poincare invariant, relativistically causal dif­fusion on M4 x H3. The free parameter isthe diffusion constant. It is an Ornstein­

Uhlenbeck type stochastic process: a diffu­sion in momentum that drives a secondaryprocess in spacetime.

Although the discrete model was rather adhoc (a nd aIso depends on the conti nuum, soit can't be fundamental) the uniqueness ofthis diffusion process means that any discretemodel that is Lorentz invariant and causal

will end up giving the same continuum model.

Dudley 1965; FD, Henson, Sorkin

1.0

No conservation of energy:

Swerves produce a statistical acceleration ofparticles: an initial distribution that is peakedat zero energy in some frame will spread sothat later the distribution has support at highenergies in that frame. Is there any evidencethat particles do get spontaneously acceler­ated?

No conservation of energy:

Swerves produce a statistical acceleration ofparticles: an initial distribution that is peakedat zero energy in some frame will spread sothat later the distribution has support at highenergies in that frame. Is there any evidencethat particles do get spontaneously acceler­ated?

1.0

I Cosmic Rays I

We have been cOllecting data on high en­ergy cosmic rays for many decades and thisis what it looks like:

[Page 6 of Anchordoqui et alhttp://arxiv.org/a bs/hep-ph/0206072]

2.1

from·

,

104

>I- .L:.L.,

Q) 0 102(I)

~(I) INE

~','-../ -1

x 10:)G::

-4

10

-7

10

-10

10

-13

10

-16

10

~r-

-1910

-22

10

-25

10

Fluxes of Cosmic Rays

(I par'ticle per' m'-second)

Knee(I particle per m'-year)

FIG, 1: Compilation of measurements of the differential energy spectrum of CRs. The dotted

line shows an E-3 power-law for comparison, Approximate integral fluxes (per steradian)are also shown [18].

Their origin is a still a mystery. In the reviewpaper cited before the following were listedas just some of the proposed sources:

Supernovae explosions, Large scale Galac­tive wind termination shocks, Pulsars, ActiveGalactic Nuclei, BL Lacertae, Spinning su­permassive black holes, Large scale motionsand related shock waves resulting from struc­ture formation, Relativistic jets produced bypowerful radiogalaxies, Electric polarisationfields in plasmoids produced in planetoid im­pacts on neutron star magnetospheres, Mag­netars, Starburst galaxies, Magnetohydrody­namic wind of newly formed strongly mag­netized neutron .stars, Gamma ray burst fire­balls, Strangelets, Hostile aliens with a bigcosmic ray gun.

Interestingly, the mechanism for the accel­eration of the particles within this myriadof sources is the same: Fermi acceleration

which is a statistical acceleration of chargedparticles scattering off random magnetic fieldirregularities: like swerves except that swervesoccur in the vacuum.

So could swerves be responsible for the ac­celeration required to produce a cosmologicalbackground of high energy protons, say, thatcould account for the cosmic ray data?

Unfortunately, no. The Lorentz invarianceof the process means that a proton that al­ready has energy l018eV, say, in the cosmicframe, sees the lifetime of the universe asonly about ten years in its rest frame. Ifthe diffusion constant is such that this pro­ton has a reasonable chance of doubling itsenergy in the lifetime of the universe then itwould be so big that boxes of hydrogen gasin the lab would spontaneously heat up at ameasureable rate.

r

Speculations

So this simple model of swerves can't explainthe cosmic ray data. But the data just criesout for a universal, cosmological accelerationmechanism and it seems worth trying to im­prove on the model.

One improvement would be to make it quan­tum mechanical. We don't yet have the sameoverarching framework for quantum randomwalks that we do in the classical case. But re­

sults from quantum information theory sug­gest that in contrast to classical diffusion inwhich the variance of a Gaussian distribution

grows as t,· in a quantal diffusion the vari­

ance would increase as t2 [Carlos Perez ].If we can argue that hydrogen in the lab isclassical whereas protons in space are quan­tal, we may be able to finesse the laboratoryconstraints on the diffusion constant and pro­duce a model that can provide a cosmologicalacceleration mechanism for cosmic rays.

I Conclusion I

The concrete kinematics of causal set the­

ory allows us to do phenomenology. Lorentzinvariance is a key feature in all models.

A simple model of source-detector couplingshows that a scalar field on a causal set back­

ground can reproduce continuum results verywell - TOO WELL!

The model of swerves has new and poten­

tially measureable effects on particle propa­gation.

Perhaps quantum swerves can produce a cos­

mological background of high energy cosmic

ray primaries to match the observed flux. Highenergy cosmic rays would then be the Brown­

ian motion of our age: the phenomenon thatin the future will convince us of the funda­

mental discreteness of spacetime itself.

2b