Germán Sierra in collaboration with P.K. Townsend and J. Rodríguez-Laguna 1st-i-Link workshop, Macro-from-Micro: Quantum Gravity and Cosmology, June 2013, CSIC, Madrid

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

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, The problem is to find H: the Riemann operator This is known as the spectral approach to the RH

Richard DawkingHis girlfriend?

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

Based on: “Landau levels and Riemann zeros” (with P-K. Townsend) Phys. Rev. Lett “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 ”General covariant xp models and the Riemann zeros” J. Phys. A: Math. Theor ”An xp model on AdS2 spacetime” (with J. Molina-Vilaplana) arXiv:

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)

The number of Riemann zeros in the box is given by Smooth (E>>1) Fluctuation

Riemann function - Entire function - Vanishes only at the Riemann zeros - Functional relation

Sketch of Riemann’s proof Jacobi theta function Modular transformation

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

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”

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)

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

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

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

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

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 Rindler space Hawking radiation

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

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

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

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

Add an electrostatic potential xy Normal modes: cyclotronic and hyperbolic (with P.K. Townsend)

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

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

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

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

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

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

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

(with J. Rodriguez-Laguna 2011) Classical trayectories are bounded and periodic xp trajectory

Quantization Eigenfunctions

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

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

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

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. Concluding remarks in Berry- Keating paper

“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:

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

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

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?

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

Boundary condition on B Riemann zeros as spectrum !!

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?

Thanks for your attention