Transcript Space, time and Riemann zeros (Madrid, 2013)

Germán Sierra
IFT@UAM-CSIC
in collaboration with
P.K. Townsend and J. Rodríguez-Laguna
1st-i-Link workshop, Macro-from-Micro: Quantum Gravity and Cosmology,
24-27 June 2013, CSIC, Madrid
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
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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. 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.
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
N fl (E ) 
1

Arg  (
1
 i E )  O(log E )
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 
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:

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
Periodic trayectory ( )  prime number (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)  x 0 e , p(t)  p0 e , E  x 0 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
Continuum spectrum
Landau model
Hawking radiation
-Quantize Berry-Keating model
Rindler space
Quantization of H = xp
Define the normal ordered operator in the half-line
1
d 1
H 0  (x pˆ  pˆ 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

On the real line H is doubly degenerate with even and odd
eigenfunctions under parity
 (x) 

E

1
x
1/ 2i E
,
sign(x)
 (x)  1/ 2i E
x

E
Lagrangian of a 2D charge particle in a uniform magnetic field B
in the gauge A = B (0,x):
2
2 





 dx
dy
e B dy
L       
x
2 dt  dt   c dt
Classically, the particle follows cyclotronic orbits:

Cyclotronic frequency:
c 
QM: Landau levels
Highly degenerate
 
eB
c
1 
E n   c n  , n  0,1,
 2 
Wave functions of the Lowest Landau Level n=0 (LLL)
1
i ky y (xky )2 / 2
 (x, y)  1/ 4 1/ 2 e e
 Ly

Lx  Ly

Degeneracy
NL 
Lx Ly
2
2
 N

c
: magnetic length
eB
Effective Hamiltonian of the LLL
Projection to the LLL is obtained in the limit
c    0
2
2 





 dx
dy
e B dy
e B dy
L       
x  LLLL  
x
2 dt  dt   c dt
c dt

Define
p
y
2
 LLLL  p

In the quantum model

LLLL  p
dx
 HLLL 
dt
dx
dt
x, p  i
HLLL  0
Degeneracy of LLL

(with P.K. Townsend)
Add an electrostatic potential xy
2
2 





 dx
dy
e B dy
L       
x e x y
2 dt  dt   c dt
Normal modes: cyclotronic and hyperbolic

eB
eB
c 
cosh  ,  h  i
sinh 
c
c

2 c 2 
sinh( 2 ) 

e B 2 

In the limit
eB
c
c 
, h  i
c
B
 c   h
only the lowest Landau level is relevant
Effective Lagrangian:


LLLL  p
dx
 h x p
dt
Effective Hamiltonian


H LLL   h x p
Quantum derivation of Connes semiclassical result
1  2 
H
px  py 

2  
2
2 
x   e  x y
 
In the limit  c   h the eigenfunctions are confluent hypergeometric fns

even

odd

1 i E 1 (x  i y) 2 
 (x, y)  e
M 
, ,

2
2
4 2 2

3 i E 3 (x  i y) 2 

x 2 / 2 2
 E (x, y) (x  i y) e
M 
, ,

2
4
2
2
2



E
x 2 / 2
2
Matching condition on the boundaries (even functions)
 E (x,L) e i L x /
1 i E 
 
 2 i E
2
4 2  L
 E (L, x) 
1
2 
1 i E 
2
   
4 2 
For odd functions 1/4 -> 3/4

Taking the log
E
L2
N(E) 
log
 1 N(E)
2
2
2
In agreement with the Connes formula

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
M. Stone 2012
Hall fluid at filling fraction 1
with an edge at x=0
2DEG
There is a horizon generated by
V xy
Velocity at the edge

In the limit
edge
v edge 
 c   h

eB
y
proyect to the LLL
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
 l 2p 
H  x p  , x  l
p 

(with J. Rodriguez-Laguna 2011)
x

xp trajectory
Classical trayectories are
bounded and periodic
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

First two terms
in Riemann formula


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
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.
“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
l
action:
metric:

curvature:
p
m

 l 2p 
H  x p  U  V  x R(x)  0
p 

: spacetime is flat
Change of variables to Minkowsky metric
ds2   dx  dx
spacetime region

x  l x ,   t 
U
Rindler coordinates
ds2  d 2   2 d 2
x 0   sinh , x1   cosh 
 l x ,      
l

Boundary : accelerated observer with

x
a 1/l x
Dirac fermion in Rindler space
Solutions of Dirac equation
Boundary conditions
Reproduces the eq. for the eigenvalues
(GS work in progress)
Lessons:
- 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 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
1
h
4
 7
E 
E
n(E) 
1  O(1/ E)
log
2  h  2  h  8

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