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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
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
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. 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.
€
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)
€
ς(s) =1
1− p−s, Re s >1
p= 2,3,5,K
∏
Product over the zeros (Riemann)
€
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
€
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
€
ξ(s) ≡ π −s / 2 Γs
2
⎛
⎝ ⎜
⎞
⎠ ⎟ς(s)
€
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
€
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?
- 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
€
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
Lagrangian of a 2D charge particle in a uniform magnetic field Bin the gauge A = B (0,x):
€
L =μ
2
dx
dt
⎛
⎝ ⎜
⎞
⎠ ⎟2
+dy
dt
⎛
⎝ ⎜
⎞
⎠ ⎟2 ⎡
⎣ ⎢
⎤
⎦ ⎥−
e B
c
dy
dtx
Classically, the particle follows cyclotronic orbits:
€
ωc =e B
μ c
Cyclotronic frequency:
QM: Landau levels
€
En = hωc n +1
2
⎛
⎝ ⎜
⎞
⎠ ⎟, n = 0,1,K
Highly degenerate
Wave functions of the Lowest Landau Level n=0 (LLL)
€
ψ(x, y) =1
π 1/ 4 Ly1/ 2
e i ky y e−(x−ky )2 / 2
€
Lx ×Ly
Degeneracy
€
NL =Lx Ly
2π l 2= NΦ
€
l =hc
e B: magnetic length
€
LLLL = pdx
dt− HLLL ⇒ HLLL = 0
Effective Hamiltonian of the LLL
Projection to the LLL is obtained in the limit
€
ωc → ∞ ⇔ μ → 0
€
L =μ
2
dx
dt
⎛
⎝ ⎜
⎞
⎠ ⎟2
+dy
dt
⎛
⎝ ⎜
⎞
⎠ ⎟2 ⎡
⎣ ⎢
⎤
⎦ ⎥−
e B
c
dy
dtx → LLLL = −
e B
c
dy
dtx
Define
€
p =h y
l 2→ LLLL = p
dx
dt
In the quantum model
€
x, p[ ] = i h
Degeneracy of LLL
Add an electrostatic potential xy
Normal modes: cyclotronic and hyperbolic
€
ωc =e B
μ ccoshϑ , ωh = i
e B
μ csinhϑ sinh(2ϑ ) =
2λμc 2
e B2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
L =μ
2
dx
dt
⎛
⎝ ⎜
⎞
⎠ ⎟2
+dy
dt
⎛
⎝ ⎜
⎞
⎠ ⎟2 ⎡
⎣ ⎢
⎤
⎦ ⎥−
e B
c
dy
dtx − e λ x y
(with P.K. Townsend)
In the limit
€
ωc >> ωh
Effective Lagrangian:
€
LLLL = pdx
dt− ωh x p
only the lowest Landau level is relevant
€
ωc ≈e B
μ c, ωh ≈ i
λ c
B
Effective Hamiltonian
€
HLLL = ωh x p
Quantum derivation of Connes semiclassical result
€
H =1
2μpx
2 + py +h
l 2x
⎛
⎝ ⎜
⎞
⎠ ⎟2 ⎡
⎣ ⎢
⎤
⎦ ⎥+ e λ x y
€
ωc >> ωhIn the limit the eigenfunctions are confluent hypergeometric fns
€
φE+(x,y)∝ e−x 2 / 2l 2
M1
4+
i E
2,1
2,(x − i y)2
2l 2
⎛
⎝ ⎜
⎞
⎠ ⎟
φE− (x,y)∝(x − i y) e−x 2 / 2l 2
M3
4+
i E
2,3
2,(x − i y)2
2l 2
⎛
⎝ ⎜
⎞
⎠ ⎟
even
odd
Matching condition on the boundaries (even functions)
€
φE+(x,L) =e i L x / l 2
φE+(L, x) →
Γ1
4+
i E
2
⎛
⎝ ⎜
⎞
⎠ ⎟
Γ1
4−
i E
2
⎛
⎝ ⎜
⎞
⎠ ⎟
L2
2l 2
⎛
⎝ ⎜
⎞
⎠ ⎟
−i E
=1
Taking the log
In agreement with the Connes formula
€
N(E) =E
2πlog
L2
2πl 2+1− N(E)
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
€
V = λ x y
Hall fluid at filling fraction 1with an edge at x=0
There is a horizon generated by
edge
€
vedge =λ
eBy
Velocity at the edge
In the limit proyect to the LLL
€
ωc >> ωh
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 holeand 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
€
H = x p +l p
2
p
⎛
⎝ ⎜
⎞
⎠ ⎟, x ≥ l x
(with J. Rodriguez-Laguna 2011)
Classical trayectories arebounded and periodic
xp trajectory
€
l x
H is selfadjoint acting on the wave functions satisfying
Which yields the eq. for the eigenvalues
€
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
First 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
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 ina spacetime with a metric given by U and V
€
l p = maction:
metric:
curvature:
€
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
Rindler coordinates
€
ρ ≥l x, − ∞ < φ < ∞
Boundary : accelerated observer with
€
a =1/l x
€
U
€
ds2 = dρ 2 −ρ 2 dφ2
x 0 = ρ sinhφ, x1 = ρ coshφ
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 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 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
€
h =1
4
€
n(E) ≈E
2π hlog
E
2π h−1
⎛
⎝ ⎜
⎞
⎠ ⎟+
7
8+ O(1/ E)
• 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?