H = xp and Riemann Zeros Johar M. Ashfaque The Riemann hypothesis states that the complex zeros of ζ(s) lie on the critical line 1 Re(s) = 2 that is the non-imaginary solutions En of 1 ζ + iEn = 0 2 are all real. George Pólya suggested a physical approach to prove the Riemann hypothesis. Also David Hilbert is associated with this conjecture, although there is no evidence he conjectured it. The conjecture simply states that if ρn =
1 + itn 2
are the non-trivial zeroes of the Riemann zeta function, then the tn ’s correspond to the eigenvalues of a hermitian operator. Properties of the Riemann operator that are suggested by the quantum analogy. Lets call this operator H. 1. H has a classical counterpart (the Riemann dynamics) corresponding to the Hamiltonian flow on a phase space. 2. The Riemann dynamics is chaotic that is unstable and bounded. 3. The Riemann dynamics does not have a time-reversal symmetry. 4. The Riemann dynamics is homogeneously unstable. 5. The classical periodic orbits of the Riemann dynamics have periods that are independent of energy E and given by multiples of logarithms of prime numbers that is Tm.q = m log q (m = 1, 2, ...; q prime) and associated actions are Sm,q = Em log q. 6. The Maslov phases associated with the orbits are also peculiar: they are all π. 7. The Riemann dynamics possesses complex periodic orbits (instantons) whose periods are Tcomplex,m = imπ. 1
8. For the Riemann operator, leading-order semi-classical mechanics is exact: 1 ζ + iE 2 is a product over classical periodic orbits without corrections. 9. The Riemann dynamics is quasi-one-dimensional.
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Semi-classical Level Counting
For any classically bound Hamiltonian Hcl (x, p) in one dimension, the number of quantum levels with energy less than E the counting function is N (E) =
A(E) + ... h
where ... denotes higher order terms in terms of of Planck’s constant ~ = A(E) is the phase space area under the contour
h 2π
and
Hcl (x, p) = E. With
1 + iEn ζ 2
=0
there is an immediate problem that the classical motion is not bound so A is infinite. The system needs to be regularized. The simplest regularization is to truncate x and p by extending the Planck cell with sides lx and lp and area h = lx lp so that A becomes finite. This makes the system quantum mechanically quasi-one-dimensional. Thus Z E/lp dx 1 E E − lp − lx + ... N (E) = h x lp lx E E log − 1 + 1 + ... = h h The constant (sub-leading) term should be modified by the Maslov phase. To guess this we note that the closed phase-space contour which turns by −2π the extra term in the counting function is + 12 . For 1 ζ + iEn = 0 2
2
the turn is + π2 so the extra term should be − 81 . Choosing units such that ~ = 1 that is h = 2π gives E E 7 N (E) = log − 1 + + ... 2π 2π 8 This is precisely the asymptotic form of the smoothed counting function for the Riemann zeros namely θ(E) Nsm (E) = +1 π where 1 1 E + iE θ(E) = − log π + = log Γ 2 4 2 correct to terms that do not vanish as E → ∞.
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The Hamiltonian: H = xp
Consider the operator d . dx To use this operator for a proof of the Hilbert-Pólya conjecture, and therefore of the Riemann hypothesis, this operator has to be Hermitian. But this operator is not Hermitian since Z ∞ dψ hψ, Hψi = −i~ ψ ∗ x dx dx Z ∞lx ∗ dψ x 6= i~ ψdx dx lx = hHψ, ψi H = xp = −i~x
for general ψ. But an operator which is Hermitian and corresponds to the Hamiltonian H = xp. Consider the simple symmetric operator 1 d 1 xp + px = −i~ x + . H= 2 dx 2 To find the eigenfunction corresponding to the eigenvalue E of this operator requires to solve Hψ = Eψ.
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Substituting the operator into this equation gives d 1 −i~ x + ψ = Eψ dx 2 i~ dψ = E+ ψ −i~x dx 2 dψ E 1 −x = + ψ dx i~ 2
ψ
x
1 E
ln = − ln
− ψ0 i~ 2 x0 ψ0 ψ(x) = 1 iE x 2− ~ x0
and for general ψ, this operator is Hermitian.
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