1 Questions and Motivations 2 Non-locality in quantum gravity Particle model

1. Quantum gravity and the origin of quantum theoryLee SmolinPerimeter Institute for Theoretical Physics 1 Questions and Motivations 2 Non-locality in quantum gravity Particle model Particles from non-locality Lattice model (if time) Non-locality on astrophysical scales (if time) 7 Conclusions Work with Fotini Markopoulou gr-qc/0311059 and in preparation

2. Prelude:

3. Prelude: Anton says that quantum unpredictability reflects events which happen without a sufficient reason. Indeed, no reason can be found within a nucleus to tell us when it will decay, or within a photon to tell us whether or not it will pass a polarizer.

4. Prelude: Anton says that quantum unpredictability reflects events which happen without a sufficient reason. Indeed, no reason can be found within a nucleus to tell us when it will decay, or within a photon to tell us whether or not it will pass a polarizer. If we still demand a sufficient reason, we must look outside the system, to a more detailed description of its relations with the rest of the universe.

5. Prelude: Anton says that quantum unpredictability reflects events which happen without a sufficient reason. Indeed, no reason can be found within a nucleus to tell us when it will decay, or within a photon to tell us whether or not it will pass a polarizer. If we still demand a sufficient reason, we must look outside the system, to a more detailed description of its relations with the rest of the universe. Perhaps there we will find Chris’s ZING.

6. We have made progress in quantum gravity, but the foundational problems are unsolved • The measurement problem is still unsolved after 80 years. • It gets worse when the observer is inside the system

7. We have made progress in quantum gravity, but the foundational problems are unsolved • The measurement problem is still unsolved after 80 years. • It gets worse when the observer is inside the system • Attempts to make sense of quantum cosmology are not • convincing, to us.

8. We have made progress in quantum gravity, but the foundational problems are unsolved • The measurement problem is still unsolved after 80 years. • It gets worse when the observer is inside the system • Attempts to make sense of quantum cosmology are not • convincing, to us. • In addition there is the problem of constructing real 4d • observables measurable by observers in the universe.

9. LQG has not resolved the foundational problems of quantum cosmology • The measurement problem is still unsolved after 80 years. • It gets worse when the observer is inside the system • Attempts to make sense of quantum cosmology are not • convincing, to us. • In addition there is the problem of constructing real 4d • observables measurable by observers in the universe. • And there is the problem of proving that general relativity • emerges as the low energy limit. There are indications, • but no proof.

10. Technical issues with the low energy limit • There are problems defining the sum over spin foams • Sums over complex numbers seldom converge. • In QM and QFT on backgrounds, this is resolved by making use • of Euclideanization to make sums convergent. But without a • background metric any Euclideanization prescription is arbitrary • and may break diffeomorphism invariance. • Any renormalization group must renormalize sums over spin • foams, not just single spin foams

11. There are also difficulties recovering smooth manifolds • as a low energy limit of spin foam models. • Originally spinnets and spin foams were embedded. • It was observed that the formulation is much simpler if • we drop the embedding. • Fotini’s causal spinnet histories • CLKR’s reduction of spin foams to group field theory. • Crane et al: Without manifold conditions, dual triangulation • gives rise to manifold with conical singularities. • Perhaps these are the matter degrees of freedom? • Inverse locality problem: local in the coarse graining derived • from averaging over spin foams may not be strictly local in • the microscopic level.

12. The inverse problem for discrete spacetimes: Its easy to approximate smooth fields with discrete structures.

13. The inverse problem for discrete spacetimes: Its easy to approximate smooth fields with combinatoric structures. But generic graphs do not embed in manifolds of low dimension, preserving even approximate distances. ? Those that do satisfy constraints unnatural in the discrete context,

14. If locality is an emergent property of graphs, it is unstable: G: a graph with N nodes that has only links local in an embedding (or whose dual is a good manifold triangulation) in d dimensions.

15. If locality is an emergent property of graphs, it is unstable: G: a graph with N nodes that has only links local in an embedding (or whose dual is a good manifold triangulation) in d dimensions. Lets add one more link randomly. Does it conflict with the locality of the embedding?

16. If locality is an emergent property of graphs, it is unstable: G: a graph with N nodes that has only links local in an embedding (or whose dual is a good manifold triangulation) in d dimensions. Lets add one more link randomly. Does it conflict with the locality of the embedding? d N ways that don’t.

17. If locality is an emergent property of graphs, it is unstable: G: a graph with N nodes that has only links local in an embedding (or whose dual is a good manifold triangulation) in d dimensions. Lets add one more link randomly. Does it conflict with the locality of the embedding? d N ways that don’t. N2ways that do. Thus, if the low energy definition of locality comes from a coarse graining of a combinatorial graph, it will be easily violated in fluctuations.

18. New hypothesis: The conceptual and technical obstacles to understanding the low energy limit are related • Perhaps quantum theory cannot be fully made sense of • because it is an approximation to a deeper theory. • Perhaps the quantum sums over spin foams cannot • be defined because quantum mechanics emerges • from spin foams rather than the reverse. But, by the experimental disproof of the Bell inequalities: Any deeper theory must be non-local. • Perhaps the non-local theory is based on combinatorial histories like spin foams. • This idea is the subject of this talk.

19. The basic idea: • The fundamental theory is combinatorial and deterministic or stochastic. It is based on an evolving graph like a spin network. • The low energy theory recovers spacetime as an averaged description, with the graph embedded in it. • Because the coarse grained notion of locality incompletely represents the fundamental notion, there are stray edges, connecting nodes that are far away in the coarse grained notion of locality. • Statistical mechanics for the whole system, plus reasonable conditions, implies quantum mechanics for subsystems. • The stray non-local links are the missing • hidden variables.

20. We have been studying models of non-locality in discrete spacetime models such as LQG: A regular lattice or weave with a random distribution of non-local links. P(n,m)= probability that nodes n and m are connected.

21. We have been studying models of non-locality in discrete spacetime models such as LQG: A regular lattice or spinnet with a random distribution of non-local links. P(n,m)= probability that nodes n and m are connected.

22. We have been studying models of non-locality in discrete spacetime models such as LQG: A regular lattice or spinnet with a random distribution of non-local links. P(n,m)= probability that nodes n and m are connected. P<<1 These models have conflicting macro and micro notions of locality

23. We have found so far four applications of such a conflict • between micro and macro locality: • Hidden variables theories of quantum mechanics gr-qc/0311059 PRD 04 • matter fields from gauge fields + non-locality • large macroscopic corrections to the low energy limit (MOND-like effects) • Cosmological implications (microscopic derivation of bi-metric or VSL theories)

24. We have found so far four applications of such a conflict • between micro and macro locality: • Hidden variables theories of quantum mechanics gr-qc/0311059 PRD 04 • matter fields from gauge fields + non-locality • large macroscopic corrections to the low energy limit (MOND-like effects) • Cosmological implications (microscopic derivation of bi-metric or VSL theories)

25. Quantum mechanics has been understood as a low energy approximation to the statistical mechanics of non-local systems. • Approaches that use matrix models: • Steve Adler • Artem Starodubtsev • ls • Local variables are invariants, such as eigenvalues • Off diagonal elements carry non-local information • Quantum mechanics for unitary invariants is derived from • certain limits of statistical mechanics for matrix models. • Approach based on stochastic differential equations: Nelson • Brownian motion on configuration space • Conservation of an average energy • gives solutions to Schrödinger's equation.

26. In this talk we make use of Nelson’s method. Matrix methods are more powerful, and are under investigation.

27. The basics of Brownian motion on configuration space. probability density: probability current: Conservation: Evolution Forward: Backward: Current velocity: Osmotic velocity:

28. Nelson’s assumptions: 1) There is a universal Brownian motion. 2) Energy is conserved, on the average • H, the average energy,is a function only of, r, va and position. • H is positive definite. • H is invariant under time reversal invariance. • H is invariant under rotations • H is local, so it is of the form • H contains only those terms that dominate in the low velocity and long long wavelength limit. The only density in the problem is r, hence: Under time reversal: Hence:

29. < Energy > conservation alsoimplies the Nelson/Newton’s law: where 3d assumption: vanishing vorticity: Thus S exists such that

30. From Nelson’s assumptions: 1) There is a universal Brownian motion. 2) Energy is conserved, on the average 3) Vanishing vorticity We have the conservation of Now choose: : The conservation of H implies:

31. Problem: Nelson’s proof is not constructive. • We provide constructive, but approximate. • realizations of Nelson’s conditions. • 1 Particle mechanics • Lattice gauge theory (in progress) These are inspired by the structures in LQG

32. The particle model

33. Assume that the low, low (non-relativistic) limit of QG satisfies: • There is a flat space of dimension d. • There are N particles in that space. • They satisfy Newton’s laws in the limit hG= l2 -> 0 with some V. • To leading order in hG there are corrections that come from • the failure of the low energy approximation to coincide with • locality in the underlying theory. • These residual non-localities are • described by a graphGon N nodes. • More detailed assumptions: • The average distance between particles, L>>l • n, the valence ofG. 1 << n << N. • The graphGis randomly distributed in space. • Time evolution of G can be neglected.

34. Matrices: Q adjacency matrix of G. Qija= Qij = 1 if i and j connected, 0 if not Xa = diagonal Xaij = dijxa Combine the matrices

35. Matrices: Q adjacency matrix of G. Qija= Qij = 1 if i and j connected, 0 if not Xa = diagonal Xaij = dijxa Combine the matrices • The basic result: • Consider the eigenvalues lai of Ma. • Assume that some regions of the universe are “hot” • so that for those degrees of freedom nx does not vanish. • Assume the average nx is large, so nnx l4/L4 is O(1). • n will be the number of connections to hot nodes. • Then the evolution of the probability distribution for the l’s • is given to leading order by the Schrödinger equation..

36. How Nelson’s conditions are satisfied: Condition 1: Every degree of freedom feels a universal Brownian motion. Dyson’s theorem: If some elements of a matrix undergo Brownian motion, so do the eigenvalues: In our case: The off diagonal elements are constant. Some xa’s fluctuate, with average diffusion constant nx Consider a l whose x is cold (nx=0).

37. How Condition 2 is satisfied: The condition is: Compute: where From the assumption that Newton’s laws are satisfied to leading order we know that Because to zeroth order the l’s trace the x’s The stochastic derivatives are defined by

38. The correction due to non-local terms is: The ratio of the correction to the classical acceleration is order Under reasonable physical conditions the […] is O(1). Hence, to leading order we have shown,

39. Showing condition 3: • Assume that the probability currents for the x’s • are curl free. • A similar calculation shows that to leading order the • same is true of the probability currents for the l’s

40. Open issues: • What happens to higher order? Are there • non-linear corrections? • Does coherence decay in time? • Nelson’s dynamics is not quite equivalent to Schrödinger for configuration spaces with • non-trivial topology. (Because eiS is not • guaranteed to be a phase.) • But averaged energy is still conserved. Do we • then get mixed states?

41. But, If LQG really unifies gravity and QM, shouldn’t it automatically tell us about unifying the rest of physics? F. Markopoulou, ls, hep-th/05???

42. A spin network with a non-local link 1/2

43. A network with a non-local link Add a loop in the fundamental rep, N, of G. (1/2,N) Couple to Yang-Mills means add labels, a rep r, of gauge group G= SU(N) on each link, similarly for nodes.

44. It looks to a local observer like a spin 1/2 particle in the fundamental rep. of SU(N). A network with a non-local link labeled (j=1/2, r= fundamental) (1/2,N)

45. It looks to a local observer like a spin 1/2 particle in the fundamental rep. A network with a non-local link labeled (j=1/2, r= fundamental) (1/2,N) So we naturally get fermions, and unlike SUSY in the fundamental representation of any gauge fields.

46. So a small amount of non-locality is nothing to be afraid of. A spinnet w/ non-local links looks just like a local spinnet with particles.

47. So a small amount of non-locality is nothing to be afraid of. A spinnet w/ non-local links looks just like a local spinnet with particles. But this implies that the dynamics and interactions of matter fields are already determined by the dynamics of the gravity and gauge fields. Could this work? Model: trivalent spinnets (2+1) with local moves. Markopoulougr-qc/9704013

48. Relation between fermion and gravity dynamics: pure gravity amplitude i k i k Aijnklm n m j l j l Let the i=1/2 line be non-local k i k A1/2jnklm n m j l j l This is a propagation amplitude for a fermion k Y k A1/2jnklm Y n m j l j

49. Lets look at this in detail: 1 Y 1 A1/2 1/2 1/2111 Y 1/2 1 1/2 1 1/2 1 The standard LQG fermion amplitude has the form: 1 1 Y Y F[1] 1/2 1/2 1 1 1 We have to do this twice to reproduce the pure gravity move: F[1]2 = A1/2 1/2 1/2111 j

50. Interactions come from moves that are local microscopically, but non local macroscopically: A spin-1 boson: 1/2 1/2 B 1 1/2 1/2