An introduction to Causal sets

1. An introduction to Causal sets 1: Discreteness without symmetry breaking Joe Henson: Causal sets

2. Discovery : a case study How did we discover the properties of matter at the atomic scale? Lord Kelvin’s approach: Atoms really are vortices in some aether. “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.” • 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. Joe Henson: Causal sets

3. Discovery : a case study Lord Rayleigh’s approach: Make a simple modelling assumption and look for observable consequences. Modelling atoms as small perfectly reflective spheres gives: 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 ! • 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. Joe Henson: Causal sets 100 years later, it is not the atomicity of matter which is in question, but theatomicity of spacetime. Again, we are equipped with a discrete kinematical picture, but lack a definitive dynamics. Can we make progress by proceeding in the spirit of Rayleigh’s calculation?

4. 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. Joe Henson: Causal sets

5. 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? Joe Henson: Causal sets

6. 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? Joe Henson: Causal sets

7. Continua as approximations E.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. A B C Discrete Structures Discrete/continuum correspondence ~ Continua g ’ g ~ must be surjective; If g ~ g1’ and g ~ g2’then g1’ and g2’ must be physically indistinguishable. Joe Henson: Causal sets

8. 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. Joe Henson: Causal sets

9. 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. But we live in a Lorentzian world! Technical problem: define distance between Lorentzian manifolds? Killer problem: No way any such approximation of Minkowski can be Lorentz invariant! Joe Henson: Causal sets

10. An intriguing result 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. 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. Joe Henson: Causal sets

11. Causal sets The causal order of a spacetime is a partial order < on the set of points C, meaning: To get a discrete version of this, we add: Order  Causal structure Number  Volume Joe Henson: Causal sets

12. Continua as approximations Discrete Structures Continua g ’ g Discrete/continuum correspondence Binary relation g ~ g ’ ~ must be surjective; If g ~ g1’ and g ~ g2’then g1’ and g2’ must be physically indistinguishable. Joe Henson: Causal sets

13. Spacetime as approximation We must recover the spacetimes of GR as approximations to some of these causal sets. When does a spacetime (M,g) approximate to (C, )? 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. Joe Henson: Causal sets

14. Recovering Geometry Given 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… Joe Henson: Causal sets

15. Lorentz invariance? A fertile ground for phenomenology, and a problem for most discrete structures. Does discreteness imply Lorentz violation? “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?” 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). Joe Henson: Causal sets

16. A hint for Lorentz invariance The Poisson process under boosts Joe Henson: Causal sets

17. Causal sets So, does the discrete structure, in and of itself, serve to pick out a preferred direction in the approximating continuum? 1) This distribution is invariant under all volume preserving transformations. In Minkowski this includes Lorentz transformations. 2) Causal information is Lorentz invariant. 3) Outcomes of the sprinkling process do not pick a direction. Theorem: There is no “equivariant” map between outcomes of sprinklings and directions in Minkowski. Joe Henson: Causal sets

18. 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. Joe Henson: Causal sets

19. 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. Joe Henson: Causal sets

20. Lorentz invariance! Random discreteness saves symmetry Joe Henson: Causal sets

21. 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? Joe Henson: Causal sets

22. An introduction to Causal sets 2: consequences of spacetime discreteness Joe Henson: Causal sets

23. 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. Joe Henson: Causal sets

24. Fields on causal sets Is discreteness/fuzziness consistent with observation? It has been suggested that any fuzziness in distance measurements will cause loss of coherence of light from distant sources. But this line of reasoning does not accord with Lorentz symmetry. Can we put a field on a causal set to test this? The problem is also relevant to dynamics. How do we recover effective locality from causal sets? Joe Henson: Causal sets

25. Scalar fields on Minkowski We 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? Joe Henson: Causal sets

26. Lorentz invariance or locality: a choice x Consider a sprinkling of Minkowski. If there is a non-zero probability of a near neighbour of x being sprinkled into region D… D There is an equal probability in D’. D’ Thus there must be an infinite amount of near neighbours, however they are defined. Joe Henson: Causal sets

27. Approximating Green’s functions Equivalently 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. Joe Henson: Causal sets

28. Approximating Green’s functions We 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. Joe Henson: Causal sets

29. A Model of Propagation We 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. Joe Henson: Causal sets

30. A Model of Propagation R Source Detector Discrete version: Joe Henson: Causal sets

31. A Model of Propagation In 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. 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. Spacetime “fluctuations” → loss of coherence Joe Henson: Causal sets

32. 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 L has only just become significant? Joe Henson: Causal sets

33. The cosmological constant problem A hint for a possible solution: Is the cosmological constant a product of discreteness and random/quantum fluctuations? Joe Henson: Causal sets

34. A condensed state analogy Consider a (square) membrane embedded in 3D space with metric g, extrinsic curvature K and intrinsic curvature H: Thermal fluctuations will impart a an energy of ~ T 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 ! Joe Henson: Causal sets

35. 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. Joe Henson: Causal sets

36. Unimodular gravity and L 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. Einstein Unimodular 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: Joe Henson: Causal sets

37. The argument Ingredient 1 (unimodular gravity): in QG, L 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 L: Joe Henson: Causal sets

38. THIS IS THE ONLY SUCCESSFUL PREDICTION FROM QUANTUM GRAVITY. EVER. What scientists do: take note of successful heuristic predictions and develop on them! Joe Henson: Causal sets

39. Modelling everpresent L Can we further test these ideas? We don’t know a QG theory in which L and V are conjugate and V has “sqrt(N)” fluctuations. Stochastic toy version of argument: Try implementing a stochastically fluctuating L in finite difference approx: Joe Henson: Causal sets

40. Modelling everpresent L Consider a homogeneous, isotropic universe: how do we add the fluctuating L? We could throw away the acceleration equation and substitute a fluctuating L, or try a linear combination of the two equations. This is not GR but perhaps may model the causal set idea of an uncertain L. Does the toy argument pan out? Joe Henson: Causal sets

41. Modelling Everpresent L The cosmological constant does track the matter energy density in this mode as expected (this result is independent of the exact ansatz used). Joe Henson: Causal sets

42. Modelling Everpresent L Joe Henson: Causal sets

43. Modelling Everpresent L 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 L 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. Joe Henson: Causal sets

44. 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. Joe Henson: Causal sets

45. An introduction to Causal sets 3: Indications for causal set dynamics Joe Henson: Causal sets

46. 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…) Joe Henson: Causal sets

47. An entropic problem? Almost all causal sets are KR orders: Their number 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? Joe Henson: Causal sets

48. 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: 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. 2 2 1 0 1 2 Joe Henson: Causal sets

49. 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. 3 4 4 3 P( ) = P( ) 1 2 1 2 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. Joe Henson: Causal sets

50. Defining the path integral Instead we might try to learn something from a generalised path integral How to do analytical continuation? What is the action? How do we approximate any local operator on a sprinkled causal set? How to suppress non-manifoldlike causal sets? Limit set of causal sets summed over/dynamics? Joe Henson: Causal sets