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  (s)
all have real part equal to 1/2

1
 sn   0, sn  C  sn   i E n ,
2
E n  , n  Z

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 n  E n n  E n    RH : True
This is known as the spectral approach to the RH
The problem is to find H: the Riemann operator
His girlfriend?
Richard Dawking
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
Zeta(s) can be written in three different “languages”

Sum over the integers (Euler)
 (s)  
1
Product over primes (Euler)
 (s) 


p 2,3,5,
1
, Re s  1
s
n
1
, Re s  1
s
1 p
 s 
Product over the zeros (Riemann)  (s) 

1 
2(s 1)(1 s/2)    

 s/ 2
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  Im s  E
is given by

Smooth
(E>>1)
Fluctuation
N R ( E )  N(E )  N fl ( E )
 7
E 
E
N( E ) 
1  O(E 1 )
log
 8
2 
2
1
1
N fl (E )  Arg  (  i E )  O(logE )

2
Riemann
(s)  

s/ 2
function
s 
  (s)
2 

- Entire function

- Vanishes only at the Riemann zeros
- Functional relation
(s)  (1  s)


1
2


1
2


  iE    iE 
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:

1
N fl (E)    
sinm E log p
m/2
 p m1 m p
1
and the fluctuation part of the spectrum of a classical chaotic Hamiltonian

Dictionary:


1
N fl (E)   
sinm E T 
  m 1 m2sinh(m /2)
1
Periodictrayectory( )  primenumber(p)
Period(T )
 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 H = xp model
The classical trayectories are hyperbolae in phase space
t
x(t)  x0 e , p(t)  p0 e , E  x0 p0
t
Unbounded
trayectories

Time Reversal Symmetry is broken (
x x, pp
)
Berry-Keating regularization
Planck cell in phase space:
x  l x , p l p , h  l x l p  2  (h 1)

Number of semiclassical states
N sc (E) 
E
E
E 7
log


2
2 2 8
Agrees with number of zeros asymptotically (smooth part)
Connes regularization
Cutoffs in phase space:
x  , p  

Number of semiclassical states
As
 
E
2
E
E
E
N sc (E) 
log

log

2
2 2
2 2
spectrum = continuum - Riemann zeros
Are there quantum versions of the BerryKeating/Connes “semiclassical” models?
Are there quantum versions of the BerryKeating/Connes “semiclassical” models?
- Quantize H = xp
- Quantize Connes xp model
-Quantize Berry-Keating model
Continuum spectrum
Landau model
xp model revisited
Dirac in Rindler
Quantization of H = xp
Define the normal ordered operator in the half-line
1
d 1
ˆ
ˆ
H0  (x p  p x)  i(x  )
2
dx 2
0 x 
H is a self-adjoint operator: eigenfunctions

E (x) 
1
x1/ 2i E

E (,)
The spectrum is a continuum
 line H is doubly degenerate with even and odd
On the real
eigenfunctions under parity
 (x) 

E

1
x
1/ 2i E
,
sign(x)
 (x)  1/ 2i E
x

E
 l 2p 
H  x p  , x  l
p 

(with J. Rodriguez-Laguna 2011)
x

xp trajectory
Classical trayectories are
bounded and periodic
The semiclassical spectrum
agrees with the smooth zeros
lx
Quantization
 l 2p 
H  x pˆ   x
pˆ 


Eigenfunctions
x 1/ 2

H is selfadjoint acting on the wave functions satisfying
Which yields the eq. for the eigenvalues

parameter of the self-adjoint extension
  0  En,  En, E0  0
    En,  En, E0  0

Periodic
Antiperiodic
Riemann zeros also appear in pairs and 0 is not a zero, i.e

 
 (1/2)  0
In the limit
E /l x l
p
1
E 
E

n(E) 
log

2  h  l x l

l xl

Agrees with
first two terms
 formula
in Riemann

p
p
 1
1
 2  O(1/ E)

 2 h
 1
E 
E
n(E) 
1  O(1/ E)
log
2  h  2  h  2
Not 7/8
Berry-Keating modification of xp (2011)
 l 2x  l 2p 
H  x  p  , x  0
x 
p 

 l 2x 1/ 2  l 2p  l 2x 1/ 2
Hˆ  x   pˆ  x  
x  
pˆ 
x 

Same as Riemann but the 7/8 is also missing
xp =cte
These two models explain the smooth part of the Riemann
zeros but not the fluctuations. The reason is that these
Hamiltonians are not chaotic, and do not contain isolated
periodic orbits related to the prime numbers.
In fact any conservative 1D Hamiltonian will have all its orbits
periodic.
Geometric interpretation of the general xp models
GS 2012
The action corresponds to a relativistic massive particle moving in
a spacetime in 1 +1 dimensions with a metric given by U and V
l
action:
p
m
metric:

curvature:
The trayectories of the xp model are geodesics
 l 2p 
H  x p  U  V  x R(x)  0
p 

Change of variables to Minkowsky metric
: spacetime is flat
ds2   dx dx

U
spacetime region
x  l x,    t  
l



x
Rindler spacetime
Rindler coordinates
ds  d   d
2
2
2
U
2
x 0   sinh, x1   cosh
l
x
 l x ,      
Boundary : world line of accelerated observer with

a 1/l x
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)

The domain U is invariant under shifts of the Rindler time
Equation of motion
 
i    HR ,    
 

Rindler Hamiltonian

Boundary condition

i( 1/2)

m
HR  

m
i( 1/2)

ie i   
at   l
x


We recover the spectum of the x(p+1/p) model
 (,)  e
i E  mi  / 4
K1/ 2i E (m )
The boundary condition gives the equation for the eigenvalues
ei K1/ 2i E (ml x )  K1/ 2i E (ml x )  0
Summary:
- xp model can be formulated as a relativistic field theory
of a massive Dirac fermion in a domain of Rindler spacetime
l
p
is the mass and 1/l
x
is the acceleration of the boundary
- energies agree with the first two terms of Riemann formula
provided



l xl p  2 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,…. are
Semitransparent (beam splitters)
l
n
l 1 n
Light ray trayectory
(1, 1 )  (2 ,  2 )

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 interactions
localized 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
Eigenvalue problem
In the n^th interval
Define
The BC’s imply
Scalar product
Solutions for which the norm is finite (discrete spectrum) or normalizable
in the Dirac delta sense (continuous spectrum)
A semiclassical approximation
Recursive solution
Assume that
Harmonic model
If
If   0,  the state is not normalizable (missing in the spectrum)

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
 1/2
In the limit


the spectrum is a continuum
 1/2 when En is a Riemann zero one finds
Recall
Diverges as
Norm of the state
 1/2 unless we choose

In the semiclassical limit the zeros appear as eigenvalues of the Rindler
Hamiltonian, but this requires a fine tuning of the parameter 
Under certain assumptions one can show that there are not zeros
outside 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 BerryKeating parameters l x , l p
• 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.
Muchas gracias por su atención