An introduction to Causal sets

Joe Henson: Causal sets

An introduction to Causal sets

1: Discreteness without symmetry breaking


Discovery : a case study

Lord Kelvin’s approach:

How did we discover the properties of matter at the atomic scale?

• Hydrodynamics was the best developed understanding of matter;• Mathematically natural from Kelvin’s perspective;• A conservative generalisation of current theories;• Attractive properties on paper;• Wrong.

“For the only pretext seeming to justify the monstrous assumption of infinitely strong and infinitely rigid pieces of matter, the existence of which is asserted as a probable hypothesis by some of the greatest modern chemists in their rashly-worded introductory statements, is that urged by Lucretius and adopted by Newton—that it seems necessary to account for the unalterable distinguishing qualities of different kinds of matter.”

Atoms really are vortices in some aether.

Lord Rayleigh’s approach:

By noting that Kanchenjunga could be seen “fairly bright” from Darjeeling, Rayleigh estimated 1/β to be 160km and thus obtained a value for Avagadro’s constant as around 4 x 10^23 !

Discovery : a case study

Make a simple modelling assumption and look for observable consequences.

Modelling atoms as small perfectly reflective spheres gives:

• Minimal (but not totally generic) assumption.• Natural physically, not a mathematical generalisation of current theory

or what seems natural based on mathematics.• Allows calculation.• Leads towards correct theory.


Main points for this series

• It is possible – but not easy – to discretize spacetime while preserving symmetries in the continuum approximation.

• This provides an interesting foil for arguments about “generic phenomenology of QG”.

• It also led to the only successful prediction that has ever come out of QG.

• Investigations of possible dynamics for causal sets present unique challenges but also give some reasons for hope.

Planck Scale Problems


Only a particle of mass greater than m can probe distances less than . But it is only at scales >> that the effects of the particle on spacetime are negligible.

Possibilities: at the Planck scale…•geometrical properties become fuzzy or uncertain;•Geometrical concepts are inadequate;•Spacetime is discrete.

Several “clues” from current theory (infinities in GR, QFT, BH entropy) suggest that the replacement should be discrete. Are there well-motivated ways to model the effects of spacetime discreteness?

Further Considerations


• GR is most naturally treated as a spacetime rather than space + time theory, but cannonical quantisation requires the latter, leading to considerable problems.

• Many types of discretisation also break the symmetries of GR.

• A Lorentzian Gromov-Hausdorff type distance is hard to define.

• With these facts in mind we consider discretising spacetime: is there a long list of possibilities, or are models restricted by simple principles like symmetry preservation?


Continua as approximationsE.g. consider a classical non-relativistic particle. If we had reason to believe that time was discrete, how could we recover continuous trajectories from discrete?

Here, we might say that C approximates A if C is within some given distance of the linear interpolation B of A.

BA C

Discrete Structures Continua

Discrete/continuum correspondence ~

’

~ must be surjective;If ~ ’ and ~ ’ then ’ and ’ must be physically indistinguishable.

Quantum complications?


If the underlying dynamics is quantum, shouldn’t we be concerned with approximations between classical continua and quantum sums of histories?

But note: in a semi-classical state, the path integral is dominated by paths that approximate the classical path being “tracked”:

Measurements in accordance with the classical approximation are very “course-grained” and, by definition, do not measurably disturb the state.

If you have a sum of many paths that don’t approximate the classical one, you can’t magically get back the right classical measurement results!

Conclusion: some of the individual paths in the history space must approximate to continua, so we still need a discrete/continuum correspondence.

Spacetime as approximation


Along these lines we might imagine many discretisations of spacetime + fields.

Euclidean geometry could be approximated by a piecewise-linear version as before.

Technical problem: define distance between Lorentzian manifolds?Killer problem: No way any such approximation of Minkowski can be Lorentz invariant!


An intriguing result

Given the causal structure and the conformal factor of a manifold with Lorentzian metric, one can recover the dimension, differential structure, topology and metric of that manifold.

Causal structure + volumes = geometry

Taking this causal structure as fundamental, we arrive at a simple way of discretising spacetime.

We can define a relation between points in a Lorentzian manifold such that x y if x is to the past of y. The causal structure of a manifold is this relation up to symmetries.


The causal order of a spacetime is a partial order < on the set of points C, meaning:

Causal sets

To get a discrete version of this, we add:

Order Causal structure

Number Volume


Continua as approximations

Discrete Structures Continua

Discrete/continuum correspondence

Binary relation ~ ’

~ must be surjective;If ~ ’ and ~ ’ then ’ and ’ must be physically indistinguishable.

’


We must recover the spacetimes of GR as approximations to some of these causal sets. When does a spacetime (M,g) approximate to (C, )?

Spacetime as approximation

If (C, ) is the partial order on some set of points in M

which is the order induced by the causal order of (M,g), we say that

(C, ) is “embeddable” in (M,g).

It is “faithfully embeddable” if that set of points could have arisen, with relatively high probability, from “sprinkling”:

This ensures that, for large regions, n ≈ ρV

This defines the discrete/continuum approximation.


Recovering GeometryGiven a causal set, how do we work out the properties of its continuum approximation (and if it has one)?

E.g: If (M,g) ~ (C, ) , what dimension does M have?

In an interval of Minkowski, the fraction of pairs of points that are causally related is a function of the dimension. E.g. in 2D half of all pairs of points are related, and in 3D it’s less. Reversing this relation gives a dimension estimator:

“Manifoldlike” causal sets have integer valued, matching dimension estimators.Similar results for lengths, topology, etc…


Lorentz invariance?

“Is this discrete structure Lorentz invariant?”

Bad question: can only talk about continuum symmetries when there is a continuum! What we rally want to know is:

“Does the discrete structure, in and of itself, serve to pick out a preferred direction in the approximating continuum?”

A fertile ground for phenomenology, and a problem for most discrete structures. Does discreteness imply Lorentz violation?

Similar example: in the continuum approximation, a sphere of glass is rotationally invariant but a sphere of crystal is not. (NB: it is not the transformations of the microscopic configuration we are worried about here).


A hint for Lorentz invariance

The Poisson process under boosts


Causal sets

This distribution is invariant under all volume preserving transformations. In Minkowski this includes Lorentz transformations.

So, does the discrete structure, in and of itself, serve to pick out a preferred direction in the approximating continuum?

Theorem: There is no “equivariant” map between outcomes of sprinklings and directions in Minkowski.

1)

2) Causal information is Lorentz invariant.

3) Outcomes of the sprinkling process do not pick a direction.

Lorentz Invariance: a theorem


But there is no uniform probability distribution on this non-compact group, so that can be no such map D. So a sprinkling picks out no direction.

Illustrating the theorem


In Euclidean space, sprinkling are rotationally invariant, but a direction from the marked point can be defined to the nearest sprinkled point:

In Lorentzian space, for any finite distance from the marked point, there is an infinite volume closer to the marked point, which must contain sprinkled elements.

Corrolaries:•No direction from a sprinkling without marked point;•No set of directions from a sprinkling;•No finite graph from a sprinkling.


Lorentz invariance!

Random discreteness saves symmetry


Conclusion

• Causal set discreteness is the only known way to make a “fuzzy” structure that approximates to Minkowksi space at large scales in a fully Lorentz-invariant way.

• Good symmetry properties are hard to come by and so this is a restrictive, simple principle to build from, like Rayleigh’s.

• Next: what can this tell us about possible deviations from standard theory?



2: consequences of spacetime discreteness

Applying Causal Sets

• Causal set discreteness provides the only known way to “fuzz” spacetime at small scales while preserving symmetries at the continuum level. This impacts on important questions…

• Are there generic signals of Planck-scale spacetime fuzziness? Does Lorentz Invariant discreteness have specific signals? Atomic Matter : attenuation of light, dispersion, scattering, defects…

• Other compelling, general expectations of what a quantum theory of discrete spacetime would predict? Predicting the cosmological constant.



Fields on causal sets

It has been suggested that any fuzziness in distance measurements will cause loss of coherence of light from distant sources.

Is discreteness/fuzziness consistent with observation?

The problem is also relevant to dynamics. How do we recover effective locality from causal sets?

But this line of reasoning does not accord with Lorentz symmetry. Can we put a field on a causal set to test this?


Scalar fields on MinkowskiWe need to make some approximation to the local, Lorentz invariant D’Alembertian operator. A lattice provides an easy way to recover locality, but breaks Lorentz invariance. On the other hand, the Lorentz invariant causal set discretisation makes it more difficult to recover locality.

On a light-cone lattice:

A weighted sum of field values at a finite set of “near neighbours”.

But in a truly Lorentzian discretisation, there can be no such finite set.

E.g.: how many “links” to a given element are there in a sprinkling of Minkowski?


Lorentz invariance or locality: a choice

D

D’

Consider a sprinkling of Minkowski.

If there is a non-zero probability of a near neighbour of x being sprinkled into region D…

There is an equal probability in D’.

x

Thus there must be an infinite amount of near neighbours, however they are defined.


Approximating Green’s functionsEquivalently to the d’Alembertian, the field theory can be defined by the Green’s function of the d’Alembertian:

In 4-dimensional Minkowski space, the Retarded Greens’ function is given by

A delta function on the future light-cone of x.

This function is defined using purely causal information.


Approximating Green’s functionsWe have seen that the links from one element “hug the light-cone”. Consider following function on pairs of causal set elements:

In the limit of dense sprinkling (with suitable normalisation) this function goes to the delta-function on the future light-cone, G(x,y).

This can be used to define the propagation of a scalar field on the causal set:

This method has some problems, but can be used to give a model of a scalar field propagating from source to detector. This helps us to see whether causal set discreteness is consistent with the coherence of light travelling over long distances, and gives an example of Lorentz invariant discrete dynamics.


A Model of PropagationWe can model propagation from a small source to a distant detector and compare the standard model with the causal set model. We define the signal F as follows:

In the continuum we are finding the measure of the set of pairs of points (one in source, one in detector) that are null related.


A Model of Propagation

DetectorSource

Discrete version:

R


A Model of PropagationIn the causal set case, to find the detector signal we counted the number of links between the source and detector region for a typical causal set approximating to Minkowski space. The result is the same, with negligible corrections.

Spacetime “fluctuations” → loss of coherence

The signal varies with the strength of the source just as in the continuum. No significant random or systematic effects come in, e.g. to change the phase of a propagating wave.

I.e. no Lorentz violation, no loss of coherence.

The cosmological constant problem


The cosmological constant:

• If the cosmological constant comes from the zero-point energy of QFT, shouldn’t it be 1?

• In other words, why is there an approximately flat manifold at all?

• If it’s not 1 why isn’t it 0?• Is it a coincidence that has only just become

significant?


The cosmological constant problem

A hint for a possible solution:

Is the cosmological constant a product of discreteness and random/quantum fluctuations?

A condensed state analogy


Consider a (square) membrane embedded in 3D space with metric , extrinsic curvature K and intrinsic curvature H:

Thermal fluctuations will impart a an energy of to each mode on the membrane up to molecular cutoff , and thus we expect a surface tension of order 1 in dimensionless units.

BUT: not all actually existing membranes have such high surface tension! Some have a low, fluctuating surface tension with 0 mean value of

!

A condensed state analogy


Fluid membranes are made of amphiphilic molecules (like soap), which have low solubility. At high enough concentration, the surfacereaches a critical density of molecules and wrinkles rather than taking on a higher density. The free energy density on the membrane has a minimum:

And the surface tension (conjugate to the total area) is therefore zero.

Unimodular gravity and


Unimodular gravity is a gravity theory in which the volume element is held constant but the rest of the diffeos – the unimodular group – are allowed.

Classically this is equivalent to GR: for any co-ordinate patch there is a co-ordinate systems for which |g|=1, and then the actions agree.

In invariant language, in unimodular gravity the total volume is a physical constant. Implementing this as a constraint on the variation of g, the cosmological constant is now a Lagrange multiplier:

Einstein Unimodular

The argument


Ingredient 1 (unimodular gravity): in QG, and V will be conjugate variables like E and t are in standard QM. So there is an uncertainty relation:

Ingredient 1 (causal sets): there is an intrinsic uncertainty in continuum volumes because they are not fundamental:

So we have an expression for the uncertainty in


THIS IS THE ONLY SUCCESSFUL PREDICTION FROM QUANTUM GRAVITY.

EVER.What scientists do: take note of successful heuristic predictions and develop on them!

Modelling everpresent


Can we further test these ideas?

We don’t know a QG theory in which and V are conjugate and V has “sqrt(N)” fluctuations.

Stochastic toy version of argument:

Try implementing a stochastically fluctuating in finite difference approx:

Modelling everpresent


Consider a homogeneous, isotropic universe: how do we add the fluctuating ?

We could throw away the acceleration equation and substitute a fluctuating or try a linear combination of the two equations. This is not GR but perhaps may model the causal set idea of an uncertain . Does the toy argument pan out?

Modelling Everpresent


The cosmological constant does track the matter energy density in this mode as expected (this result is independent of the exact ansatz used).



This work only scratches the surface of event the toy model.

• What effect does the addition of anisotropy and inhomogeneity have?

• Still need to model the actual observations of accord with what we see, within this model.

• Without a quantum gravity theory, is there other hueristic reasoning that can be used for modelling beyond the stochastic level?

• Condensed state analogs?• Develop quantum causal set dynamics.

Conclusions


THIS IS THE ONLY SUCCESSFUL PREDICTION FROM QUANTUM GRAVITY.

EVER.

However it has not received the attention it deserves, and as a result there are lots of avenues for interesting research and chances for further predictions.



3: Indications for causal set dynamics

A formidable task

• Aim: a theory with QFT on curved spacetime and GR as limits.

• Two ideas: formal generalisation or theory construction from physical principles.

• For causal sets, there are many problems to tackle either way (non-manifoldlike causal sets, recovering locality, no Wick rotation…)


An entropic problem?


Almost all causal sets are KR orders:

This looks nothing like a manifold according to our discrete/continuum correspondence, sprinkling. A local dynamics could not suppress such a large entropy.

But: the number of possible relations scales like this too; an action on causal sets cannot be local in the sense used above.

Questions:•Do we always see these kinds of posets dominating in toy dynamics?•sprinklings make sense physically – are they also natural mathematically?

Their number

Classical Sequential Growth


A quantum SOH can be seen as a generalization of a stochastic theory; we can test the principle approach in a stochastic setting.

Sequential growth: the causal set, starting from one element, is “grown” by randomly adding elements to the future (or spacelike) to existing elements:

0

1

1

2

2

2

Defining all the “transition probabilities” gives a probability measure on infinite causal sets.

“Percolation” is a well studied model of this type in which the new element is related to any given past element with probability p before transitive closure is taken.

Classical Sequential Growth


Physical Principles:•general covariance= labeling invariance, e.g.:

•Bell causality: the ratios of probabilties of two transitions does not depend on elements spacelike to the two possible new elements.

P( ) = P( )21

3 4

21

4 3

Percolation obeys these rules. In fact (almost) all processes that obey these rules have infinitely many elements with no spacelike elements, and “flow towards” percolation over many bounces.

Typical percolated posets look nothing like KR orders. They also have a continuum limit in a sense, but not the right one.

Defining the path integral


Instead we might try to learn something from a generalised path integral

1. How to do analytical continuation?2. What is the action? How do we approximate any local operator

on a sprinkled causal set?3. How to suppress non-manifoldlike causal sets? Limit set of

causal sets summed over/dynamics?

A simple 2D example“Quantum gravity” in 2D is a trivial theory. Consider a path integral over all spacetime intervals, of trivial topology, in 2D. The path integral reduces to

Reducing again to the case of fixed volume (like the “unimodular restriction” in 4D), all that matters is the measure. We now discretise this path integral.

VG

S 8

1)()(

iSeDZ

A simple 2D exampleConsider partial orders on some set of objects, for example the integers.

1

2

3

1

2

3=

1

2 3

The intersection of two partial orders on the same set has only the relations shared by both. The intersection of two linear orders is called a 2D order.Claim: the set of 1D and 2D orders contains all of the causal sets that can be embedded into Minkowski space, which includes everything that can be embedded in any topologically trivial 2D causal interval.

A simple 2D example

21

3

46

5

54

63

12

54

32

1

6

To see this, think of the two liner orders and light-cone co-ordinates. A point in the in causal future of another in 2D Minkowski if both its co-ordinates are greater Same rule as 2D orders.

In 2D all of the metric is in the conformal factor, so varying the density of embedding gives faithful embeddings for every metric.

A simple 2D exampleAny 2D order can be embedded in Minkowski space, and conversely any finite number of points that can be embedded in Minkowski space, with their causal order, are a 2D order.

So all finite sprinklings into 2D Minkowski are included in this set. Also, in 2D, all metrics on the interval are diffeomophic to a flat metric multiplied by a position dependent factor:

Sprinklings in these spaces are also included, so we have discretisation s of all 2D intervals of trivial topology.

A simple 2D exampleThis gives an natural way to define a limited class of causal sets that contains all 2D intervals.

)(~ DZ orders 2D

1~Z

Is this easily-defined restriction enough? There are still non-manifoldlike causets in the sum. What kind of causal set dominates?

As the original histories were diffeomophism classes, the discrete ones are relabelling classes, or “unlabelled causets”.

A little technology


The uniform measure on unlabelled causal sets (i.e. isomorphism classes of causal sets) of size N is called U(N).

Another measure of interest is the measure on (labelled) causal sets is P(N). This comes from putting uniform measure on two linear order (i.e. two permuntations of {0,…,N}) and intersecting them.

Also, it is not hard to see that the random causet obtained by sprinkling N points into an Interval of Minkowski space is the same as that obtained from P(N).

Putting these two facts together, we see that causets that are faithfully embeddable into Minkowski dominate the sum as N -> infinity.

Hard to compare to other 2D quantum geometry models. Can we extend this to a path integral between an initial and final closed boundary? Characterise causal sets that can be embedded on cylinders?

Study the magnitudes of fluctuations about flat space. No fluctuations if we send “Planck scale” to 0.

Is something similar true in higher dimension? Conjecture: faithful embeddings (with flat metric?) are always more numerous than general embeddings, for any Lorentzian manifold.

2D model: Open questionsIntriguing result, but in a very limited model.

In general


In higher dimensions, we need an expression fro the Einstein Hilbert action.

How do we approximate any local operator at all?

A non-local, causal “d’Alembertian”


Back to scale fields on sprinkled causal sets.

In the lattice we had a weighted sum of the field values at the nearest neighbours to define d’Alembertian. Can be “average this out” over the hyperboloid without divergences? E.g. in 2D:

Averaging over sprinklings, this will define an approximation to the D’Alembertian as a weighted integral over the past light cone. Remarkably, it does not diverge.

L_i : all elements y such that there are only i elements causally between x and y.


A potential problemTake e.g. the expression for a constant field. The approximation is a weighted integral over the whole past light cone, and the integrand has significant non-zero values over an infinite volume. Can we avoid divergances due to contributions from “far down the light-cone”?

We can rewrite this integral:

Crucially, some “large” terms in I do not contribute to J:


A potential problemLet us assume that the field is of compact support, and slowly varying in some frame (for now), and work in that frame. We are in 2D, so in light-cone co-ords:

Work in units where k >> 1.

Using the previous equations, which get rid of terms of order k^ -1 and k^-2, we see that the contribution to J(k) is small -- of order k^-3. This shows how the above relations help to regain approximate locality. Thus the only significant contribution is from region 3:

Curved space


Consider a sprinkling into curved space where the appears on a scale larger than the support of the field (which is >> the discreteness scale). We can apply the same operator. If we recover a local operator at all, dimensional considerations give

A straightforward but tedious calculation using Riemann normal co-ordinates shows the constant to be a half for dimensions < 8.

By applying this operator to the constant field we can then read off the curvature, and thus the EH action:

Where N_i is the number of intervals containing i+1 elements .

Current directions


The action is a sum of the number of certain sizes of intervals with some parameters. If we analytically continue these we can get a real path integral to play with.

Monte Carlo simulations with causal sets are being developed and we now see that thermalisation is practical.


Conclusions• The causal set offers a discretisation of

spacetime consistent with the symmetries we observe.

• It led to a prediction which deserves to be followed up.

• While there are intriguing hints, a good dynamics has yet to be found.

• There are lots of open problems in kinematics, dynamics and phenomenology.


WhY?


HOW?


Where?


HOW?


A potential problem

We need to expand the field:

The constant term gives a constant contribution. Let us examine the uv term


A potential problem

What is quantum theory?

Probability for a sequence of measurement outcomes :

Can be thought of as a function from subsets of a history space Ω to the reals.

What is essential?Consider the probability for a an outcome of a sequence of measurements, e.g. double slit:

A = “Particle went through slit 1 and ended up at point x”.B = “Particle went through slit 2 and ended up at point x”.A or B = “Particle ended up at point x.”

However,

A non-local, causal “d’Alembertian”


For a sprinklings in flat space:

Definition depends only on the causal set structure. Applying it to sprinklings into curved spacetimes, a calculation shows that


A potential problemTake e.g. the expression for a constant field. The approximation is a weighted integral over the whole past light cone, and the integrand has significant non-zero values over an infinite volume. Can we avoid divergances due to contributions from “far down the light-cone”?

We can rewrite this integral:

Crucially, some “large” terms in I do not contribute to J:

The state of the art


This gives an ansatz for a “path integral”

Analytically continuing parameters a,b,c,d can give a real path integral.

Problems: •Solve analytically?•Can we efficiently sample space of causal sets?•If so how can be have any confidence in analytically continuing back?

The Problems• Our criterion for manifoldlikeness is physically nice, but can I be made to arise naturally in some theory?• We lack a natural way to characterise manifoldlikeness.• Manifoldlike causets form a vanishingly small subset of causal sets.• For a “traditional” – local – dynamics this represents a problem.

• Similar problems for any theory of QG – getting configurations to dominate that look smooth at large scales.

An introduction to Causal sets

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