Germ á n Sierra IFT@UAM-CSIC in collaboration with P.K. Townsend and J. Rodr íguez-Laguna

1. Space, time and Riemann zeros Germán Sierra IFT@UAM-CSIC in collaboration with P.K.Townsend andJ. Rodríguez-Laguna 1st-i-Link workshop, Macro-from-Micro: Quantum Gravity and Cosmology, 24-27 June 2013, CSIC, Madrid

2. A mathematical problem Riemann hypothesis (1859): the complex zeros of the zeta function all have real part equal to 1/2

3. A physical approach Polya and Hilbert conjecture (circa 1910): There exists a self-adjoint operator H whose discrete spectra is given by the imaginary part of the Riemann zeros, This is known as the spectral approach to the RH The problem is to find H: the Riemann operator

4. His girlfriend? Richard Dawking

5. Outline • The Riemann zeta function • Hints for the spectral interpretation • H = xp model by Berry-Keating and Connes • Landau version of the Connes’s xp model • Analogue of Hawking radiation (M. Stone) • The xp model à la Berry-Keating revisited • Extended xp models and their spacetime interpretation • Xp and Dirac fermion in Rindler space • Contact with Riemann zeta

6. Based on: “Landau levels and Riemann zeros” (with P-K. Townsend) Phys. Rev. Lett. 2008 “A quantum mechanical model of the Riemann zeros” New J. of Physics 2008 ”The H=xp model revisited and the Riemann zeros”, (with J. Rodriguez-Laguna) Phys. Rev. Lett. 2011 ”General covariant xp models and the Riemann zeros” J. Phys. A: Math. Theor. 2012 ”An xp model on AdS2 spacetime” (with J. Molina-Vilaplana) arXiv:1212.2436.

7. The zeta function: Rosetta Stone in Maths Zeta(s) can be written in three different “languages” Sum over the integers (Euler) Product over primes (Euler) Product over the zeros (Riemann) Importance of RH: imposes a limit to the chaotic behaviour of the primes If RH is true then “there is music in the primes” (M. Berry)

8. Quick look to the zeta function

9. Counting non trivial zeros The number of Riemann zeros in the box is given by Smooth (E>>1) Fluctuation

10. Riemann function • Entire function • Vanishes only at the Riemann zeros • Functional relation

11. Sketch of Riemann’s proof Jacobi theta function Modular transformation

12. Support for a spectral interpretation of the Riemann zeros Selberg’s trace formula (50´s): Classical-quantum correspondence similar to formulas in number theory Montgomery-Odlyzko law (70´s-80´s): The Riemann zeros behave as the eigenvalues of the GUE in Random Matrix Theory -> H breaks time reversal Berry´s quantum chaos proposal (80´s-90´s): The Riemann operator is the quantization of a classical chaotic Hamiltonian Berry-Keating/Connes (99): H = xp could be the Riemann operator

13. Berry´s quantum chaos proposal (80´s-90´s): the Riemann zeros are spectra of a quantum chaotic system Analogy between the number theory formula: and the fluctuation part of the spectrum of a classical chaotic Hamiltonian Dictionary: Idea: prime numbers are “time” and Riemann zeros are “energies”

14. The Berry-Keating/Connes model: • In 1999 Berry and Keating proposed that the 1D classical Hamiltonian H = x p, when properly quantized, may contain the Riemann zeros in the spectrum • The Berry-Keating proposal was parallel to Connes adelic approach to the RH. These approaches are semiclassical (heuristic)

15. The classical H = xp model The classical trayectories are hyperbolae in phase space Unbounded trayectories Time Reversal Symmetry is broken ( )

16. Berry-Keating regularization Planck cell in phase space: Number of semiclassical states Agrees with number of zeros asymptotically (smooth part)

17. Connes regularization Cutoffs in phase space: Number of semiclassical states As spectrum = continuum - Riemann zeros

18. Are there quantum versions of the Berry- Keating/Connes “semiclassical” models?

19. Are there quantum versions of the Berry- Keating/Connes “semiclassical” models? • Quantize H = xp • Quantize Connes xp model • Quantize Berry-Keating model Continuum spectrum Landau model Hawking radiation Rindler space

20. Quantization of H = xp Define the normal ordered operator in the half-line H is a self-adjoint operator: eigenfunctions The spectrum is a continuum On the real line H is doubly degenerate with even and odd eigenfunctions under parity

21. Brief review of the Landau model Lagrangian of a 2D charge particle in a uniform magnetic field B in the gauge A = B (0,x): Classically, the particle follows cyclotronic orbits: Cyclotronic frequency: QM: Landau levels Highly degenerate

22. Wave functions of the Lowest Landau Level n=0 (LLL) Degeneracy : magnetic length

23. Effective Hamiltonian of the LLL Projection to the LLL is obtained in the limit Define In the quantum model Degeneracy of LLL

24. Landau model and Riemann zeros (with P.K. Townsend) Add an electrostatic potential xy Normal modes: cyclotronic and hyperbolic

25. In the limit only the lowest Landau level is relevant Effective Lagrangian: Effective Hamiltonian

26. Quantum derivation of Connes semiclassical result In the limit the eigenfunctions are confluent hypergeometric fns even odd

27. Matching condition on the boundaries (even functions) For odd functions 1/4 -> 3/4 Taking the log In agreement with the Connes formula

28. Problem with Connes xp • Spectrum becomes a continuum in the large size limit • No real connection with Riemann zeros Go back to Berry-Keating version of xp and try to make it quantum

29. An analogue of Hawking radiation in the Quantum Hall Effect M. Stone 2012 Hall fluid at filling fraction 1 with an edge at x=0 2DEG There is a horizon generated by Velocity at the edge In the limit proyect to the LLL edge Edge excitations are described by a chiral 1+1 Dirac fermion

30. Eigenfunctions Confluent Hypergeometric Parabolic cylinder A hole moving from the left is emitted by the black hole and leaves a hole inside the event horizon

31. Probability of an outgoing particle or hole is thermal Corresponds to a Hawking temperature with a surface gravity given by the edge-velocity acceleration

32. Berry-Keating xp model revisited (with J. Rodriguez-Laguna 2011) xp trajectory Classical trayectories are bounded and periodic

33. Quantization Eigenfunctions

34. H is selfadjoint acting on the wave functions satisfying Which yields the eq. for the eigenvalues parameter of the self-adjoint extension Periodic Antiperiodic Riemann zeros also appear in pairs and 0 is not a zero, i.e

35. In the limit First two terms in Riemann formula Not 7/8

36. Berry-Keating modification of xp (2011) xp =cte Same as Riemann but the 7/8 is also missing

37. Concluding remarks in Berry- Keating paper We are not claiming that our hamiltonian H has an immediate connection with the Riemann zeta function. This is ruled out not only by the fact that the mean eigenvalue density differs from the density of Riemann zeros after the first terms, but by a more fundamental difference in the periodic orbits. For H, there is a single primitive periodic orbit for each energy E; and for the conjectured dynamics underlying the zeta function, there is a family of primitive orbits for each ‘energy’ t, labelled by primes p, with periods log p. This absence of connection with the primes is shared by all variants of xp.

38. “All variants” of the xp model GS 2012 General covariance: dynamics of a massive particle moving in a spacetime with a metric given by U and V action: metric: curvature:

39. : spacetime is flat Change of variables to Minkowsky metric spacetime region Rindler coordinates Boundary : accelerated observer with

40. Dirac fermion in Rindler space (GS work in progress) Solutions of Dirac equation Boundary conditions Reproduces the eq. for the eigenvalues

41. Lessons: • xp model can be formulated as a relativistic field theory • of a massive Dirac fermion in a domain of Rindler spacetime • is the mass and is the acceleration of the boundary • energies agree with the first two terms of Riemann formula • provided Where is the zeta function?

42. Contact with Riemann zeta function Use a b-field with scaling dimension h (h=1/2 is Dirac) Solve eqs. of motion and take the high energy limit Similar to first term in Riemann’s formula Provided

43. Boundary condition on B Riemann zeros as spectrum !!

44. Conclusions and suggestions • The xp model is a promissing candidate to yield a spectral interpretation of the Riemann zeros • Connes xp -> Landau xy model -> Analogue of Hawking radiation in the FQHE (Stone). No connection with Riemann zeros. • Berry-Keating xp -> Dirac fermion in Rindler space • Conjecture: b-c field theory in Rindler space with some additional ingredient may yield the final answer • Where are the prime numbers in this construction? • Is there a connection between Quantum Gravity and Number theory?

45. Thanks for your attention