Germán Sierra Instituto de Física Teórica UAM-CSIC, Madrid Tercer Encuentro Conjunto de la Real...
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Transcript of Germán Sierra Instituto de Física Teórica UAM-CSIC, Madrid Tercer Encuentro Conjunto de la Real...
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
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)
€
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
xp model revisited
Dirac in Rindler
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
€
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
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
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)
€
l n = l 1 n
Light ray trayectory
€
(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
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
Scalar product
Solutions for which the norm is finite (discrete spectrum) or normalizablein the Dirac delta sense (continuous 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 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