Germán Sierra

Instituto de Física Teórica UAM-CSIC, Madrid

Tercer Encuentro Conjunto de la Real Sociedad Matemática Española

y la Sociedad Matemática Mexicana, Zacatecas, México, 1-4 Sept 2014

Riemann hypothesis (1859):

the complex zeros of the zeta function

all have real part equal to 1/2

€

ς sn( ) = 0, sn ∈ C → sn =1

2+ i En, En ∈ ℜ, n ∈ Z€

ς(s)

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,

€

H ψ n = En ψ n ⇒ En ∈ ℜ ⇒ RH : True

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• The xp model à la Berry-Keating revisited• Extended xp models and their spacetime interpretation• xp and Dirac fermion in Rindler space• …….

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)Nucl. Phys. B. 2013

”The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime” J. of Phys. A 2014; arxiv: 1404.4252

€

ς(s) =1

n s, Re s >1

1

∞

∑

Zeta(s) can be written in three different “languages”

Sum over the integers (Euler)

Product over primes (Euler)

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ς(s) =1

1− p−s, Re s >1

p= 2,3,5,K

∏

Product over the zeros (Riemann)

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ς(s) =π s / 2

2(s −1)Γ(1+ s /2)1−

s

ρ

⎛

⎝ ⎜

⎞

⎠ ⎟

ρ

∏

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

€

0 < Re s <1, 0 < Ims < E

is given by

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NR (E) = N(E) + N fl (E)

N(E) ≈E

2πlog

E

2π−1

⎛

⎝ ⎜

⎞

⎠ ⎟+

7

8+ O(E−1)

N fl (E) =1

πArgς (

1

2+ i E) = O(log E)

Smooth(E>>1)

Fluctuation

Riemann function

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ξ(s) ≡ π −s / 2 Γs

2

⎛

⎝ ⎜

⎞

⎠ ⎟ς(s)

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ξ 1

2− iE

⎛

⎝ ⎜

⎞

⎠ ⎟=ξ

1

2+ iE

⎛

⎝ ⎜

⎞

⎠ ⎟

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

€

ξ

€

ξ(s) =ξ(1 − s)

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:

€

N fl (E) = −1

π

1

m pm / 2sin m E log p( )

m=1

∞

∑p

∑

and the fluctuation part of the spectrum of a classical chaotic Hamiltonian

€

N fl (E) =1

π

1

m2sinh(mλα /2)sin m E Tα( )

m =1

∞

∑α

∑

Dictionary:

€

Periodic trayectory (α ) ↔ prime number (p)

Period (Tα ) ↔ log p

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

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x(t) = x0 e t , p(t) = p0 e−t , E = x0 p0

Time Reversal Symmetry is broken ( )

€

x → x, p → −p

The classical H = xp model

Unboundedtrayectories

Berry-Keating regularization

Planck cell in phase space:

€

x > l x, p >l p , h = l x l p = 2π (h =1)

Number of semiclassical states

Agrees with number of zeros asymptotically (smooth part)

€

Nsc (E) ≈E

2πlog

E

2π−

E

2π+

7

8

Connes regularization

Cutoffs in phase space:

€

x < Λ, p < Λ

Number of semiclassical states

As spectrum = continuum - Riemann zeros

€

Λ→ ∞€

Nsc (E) ≈E

2πlog

Λ2

2π−

E

2πlog

E

2π+

E

2π

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

xp model revisited

Dirac in Rindler

Define the normal ordered operator in the half-line

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H0 =1

2(x ˆ p + ˆ p x) = −i(x

d

dx+

1

2)

€

φE (x) =1

x1/ 2−i E

H is a self-adjoint operator: eigenfunctions

The spectrum is a continuum

€

E ∈(−∞,∞)

€

0 < x < ∞

Quantization of H = xp

On the real line H is doubly degenerate with even and oddeigenfunctions under parity

€

φE+(x) =

1

x1/ 2−i E , φE

− (x) =sign(x)

x1/ 2−i E

€

H = x p +l p

2

p

⎛

⎝ ⎜

⎞

⎠ ⎟, x ≥ l x

(with J. Rodriguez-Laguna 2011)

Classical trayectories arebounded and periodic

xp trajectory

€

l x

The semiclassical spectrumagrees with the smooth zeros

Quantization

€

H = x ˆ p +l p

2

ˆ p

⎛

⎝ ⎜

⎞

⎠ ⎟ x

Eigenfunctions

€

x −1/ 2

H is selfadjoint acting on the wave functions satisfying

Which yields the eq. for the eigenvalues

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ϑ =0 ↔ En, − En , E0 = 0

ϑ = π ↔ En, − En , E0 ≠ 0

parameter of the self-adjoint extension

Riemann zeros also appear in pairs and 0 is not a zero, i.e

€

ς(1/2) ≠ 0

€

ϑ =π

€

ϑ

Periodic

Antiperiodic

In the limit

€

E /l x l p >>1

€

n(E) ≈E

2π hlog

E

l x l p

−1 ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟+

1

2+ O(1/ E)

€

l x l p = 2π h

€

n(E) ≈E

2π hlog

E

2π h−1

⎛

⎝ ⎜

⎞

⎠ ⎟+

1

2+ O(1/ E)

Not 7/8

Agrees withfirst two termsin Riemann formula

Berry-Keating modification of xp (2011)

€

H = x +l x

2

x

⎛

⎝ ⎜

⎞

⎠ ⎟ p +

l p2

p

⎛

⎝ ⎜

⎞

⎠ ⎟, x ≥ 0

€

ˆ H = x +l x

2

x

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

ˆ p +l p

2

ˆ p

⎛

⎝ ⎜

⎞

⎠ ⎟ x +

l x2

x

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

xp =cte

Same as Riemann but the 7/8 is also missing

These two models explain the smooth part of the Riemann zeros but not the fluctuations. The reason is that theseHamiltonians are not chaotic, and do not contain isolatedperiodic orbits related to the prime numbers.

In fact any conservative 1D Hamiltonian will have all its orbitsperiodic.

Geometric interpretation of the general xp models

GS 2012

The action corresponds to a relativistic massive particle moving ina spacetime in 1 +1 dimensions with a metric given by U and V

€

l p = maction:

metric:

curvature:

The trayectories of the xp model are geodesics

€

H = x p +l p

2

p

⎛

⎝ ⎜

⎞

⎠ ⎟→U = V = x →R(x) = 0 : spacetime is flat

Change of variables to Minkowsky metric

€

ds2 = ημν dx μ dxν

spacetime region

€

x ≥ l x, − ∞ < t <∞

€

l x€

U

€

l x

Rindler spacetime

€

ρ ≥l x, − ∞ < φ < ∞

Boundary : world line of accelerated observer with

€

a =1/l x

€

U

€

ds2 = dρ 2 −ρ 2 dφ2

x 0 = ρ sinhφ, x1 = ρ coshφ

Rindler coordinates

The black hole metric near the horizon can be approximated by the Rindler metric -> applications to Quantum Gravity

Dirac fermion in Rindler space (GS 2014)

Boundary condition

The domain U is invariant under shifts of the Rindler time

€

φ

Equation of motion

€

i∂φ χ = HR χ , χ =χ −

χ +

⎛

⎝ ⎜

⎞

⎠ ⎟

Rindler Hamiltonian

€

HR =−i(ρ ∂ρ +1/2) m ρ

m ρ i(ρ ∂ρ +1/2)

⎛

⎝ ⎜

⎞

⎠ ⎟

€

−ie iϑ χ− = χ+ at ρ = l x

We recover the spectum of the x(p+1/p) model

€

χ±(ρ,φ) = e−i E φ miπ / 4 K1/ 2± i E (m ρ )

The boundary condition gives the equation for the eigenvalues

€

e iϑ K1/ 2−i E (m l x ) − K1/ 2+i E (m l x ) = 0

Summary:

- xp model can be formulated as a relativistic field theoryof 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

€

1/l x

€

l p

€

l xl p = 2π h

Where are the primes?

Moving mirrors and prime numbers:

Moving mirror: is a mirror whose acceleration is constant

€

l n

First mirror is perfect (n=1)

Mirrors n=2,3,…. areSemitransparent (beam splitters)

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l n = l 1 n

Light ray trayectory

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(ρ1,φ1) → (ρ 2,φ2)

Periodic trajectory: boundary -> n^th mirror -> boundary

This is the time measured by the observer’s clock (=propertime)

Double reflection: 1 –> n1 -> 1 -> n2 -> 1

Triple reflection

prime numbers correspond to unique paths characterized by observer proper times equal to log p

Spacetime implementation of Erathostenes sieve

harmonic mirror array

€

l n = l 0 en / 2

Proper times for reflection and double reflection

Dirac action with delta function potentials

We now take a massless fermion and add delta function interactionslocalized in the position of the mirrors

Depends on three reflection coefficients

Between the mirrors the fermion moves freely with the Hamiltonian

The delta functions give the matching conditions of the wave functions

The Hamiltonian with these BC’s is selfadjoint

In the n^th interval

Define

The BC’s imply

Eigenvalue problem

Scalar product

Solutions for which the norm is finite (discrete spectrum) or normalizablein the Dirac delta sense (continuous spectrum)

A semiclassical approximation

Recursive solution

Assume that

Harmonic model

If

If the state is not normalizable (missing in the spectrum)

€

ϑ ≠0, π

If the state is normalizable in the Dirac sense

The harmonic model can be solved exactly

States in the continuum

The Riemann model

Moebius function

If the spectrum is a continuum

€

σ >1/2

In the limit when En is a Riemann zero one finds

€

σ → 1/2

Recall

Diverges as unless we choose

€

σ → 1/2

Norm of the state

In the semiclassical limit the zeros appear as eigenvalues of the Rindler Hamiltonian, but this requires a fine tuning of the parameter

€

ϑUnder certain assumptions one can show that there are not zerosoutside the critical line -> Proof of the Riemann hypothesis

• We have formulated the H = x(p+ 1/p) in terms of a Dirac fermion

in Rindler spacetime. This gives a new interpretation of the Berry-Keating parameters

• To incorporate the prime numbers we have formulated a new model

based on a massless Dirac fermion with delta function potentials.

• We have obtained the general solution and in a semiclassical limit we found a spectral realization of the Riemann zeros by choosing a potential related to the prime numbers.

• The construction suggests a connection between Quantum Gravity and Number theory.

€

l x , l p

Muchas gracias por su atención