[Paper Review] The Scaling Hamiltonian

Alain Connes, Caterina Consani|arXiv (Cornell University)|Oct 31, 2019

Spectral Theory in Mathematical Physics13 references4 citations

TL;DR

This paper proposes a novel operator-theoretic semi-local framework to address Weil's positivity and the Riemann Hypothesis, linking the Berry-Keating Hamiltonian to spectral realizations of zeta zeros via quantized calculus. It resolves apparent conflicts between absorption and emission spectra in semiclassical approximations by showing the minus sign manifests through Maslov phases, and demonstrates that Li's attempted proof fails due to a non-bounded function not in the Hardy space, highlighting the necessity of global Poisson formulas for correct normalization.

ABSTRACT

We first explain the link between the Berry-Keating Hamiltonian and the spectral realization of zeros of the Riemann zeta function of the first author, and why there is no conflict at the semi-classical level between the "absorption" picture of A. Connes and the semiclassical "emission" computations of M. Berry and J. Keating, while the minus sign manifests itself in the Maslov phases. We then use the quantized calculus to analyse the recent attempt of X.-J. Li at proving Weil's positivity, and understand its limit. We then propose an operator theoretic semi-local framework directly related to the Riemann Hypothesis.

Motivation & Objective

• To reconcile the semiclassical absorption and emission pictures in the context of the Berry-Keating Hamiltonian and the spectral realization of zeta zeros.
• To analyze X.-J. Li's attempt to prove Weil’s positivity using quantized calculus and identify its fundamental flaw.
• To establish a semi-local trace formula framework that captures finite parts in Riemann-Weil explicit formulas via canonical Hilbert space operators.
• To argue that global Poisson formulas are essential for correct normalization of principal values, which local approaches neglect.
• To propose a conjecture that the semi-local framework can serve as a viable path toward proving Weil’s positivity, grounded in operator theory and geometric insights.

Proposed method

• Uses the quantized calculus to analyze Li’s cutoff procedure, particularly focusing on the function φ(z) and its failure to be bounded in the half-plane Re(z) > 1/2.
• Applies the semi-local trace formula on L²(X_S), where X_S is the quotient of the finite product of local fields Q_v for v ∈ S by the multiplicative group Q_S^*, forming a canonical Hilbert space.
• Analyzes the scaling Hamiltonian H = PQ in the context of the Heisenberg commutation relations, showing its generator is x∂x + 1/2, with the minus sign appearing in Maslov phases.
• Demonstrates that the Poisson formula underlies the semi-local framework by constructing Schwartz functions vanishing at zero and whose Fourier transforms also vanish at zero, using differential operators like D_u² + D_u.
• Extends the construction of Sonine space functions to the semi-local setting, showing that differential operators commuting with the Fourier transform preserve the required support and vanishing conditions.
• Uses the scaling action on L²(R) as a model for the global system, showing how the semi-local picture approximates the adèle class space and reveals the role of local normalization in principal values.

Experimental results

Research questions

• RQ1 Why is there no conflict between the absorption spectrum in [10] and the emission spectrum in [1,2] at the semiclassical level, despite differing physical pictures?
• RQ2 How does the minus sign in the Hamiltonian manifest in the semi-classical limit, and why does it not vanish in the cutoff procedure of [1,2]?
• RQ3 Why does X.-J. Li’s approach to proving Weil’s positivity fail, and what role does the function φ(z) play in this failure?
• RQ4 Can a semi-local framework alone achieve Weil’s positivity, or is a global Poisson formula necessary for correct normalization of principal values?
• RQ5 How does the geometric structure of the Scaling Site emerge from the semi-local operator-theoretic framework, and what is its conceptual significance?

Key findings

• The apparent conflict between absorption and emission spectra is resolved at the semiclassical level because the minus sign reappears in Maslov phases, preserving consistency despite different physical interpretations.
• Li’s approach would have succeeded if the function φ(z) belonged to the Hardy space H^∞ on the half-plane Re(z) > 1/2, but φ(z) is meromorphic and unbounded there, invalidating the proof.
• The semi-local trace formula on L²(X_S) provides a canonical construction of Hilbert space operators that capture the finite parts in the Riemann-Weil explicit formulas.
• The global Poisson formula is essential for correct normalization of principal values; neglecting local contributions (e.g., by truncating primes) leads to incorrect normalization, as seen in the failure to preserve the 2logΛ term from [10].
• The differential operator D_u² + D_u, defined by D_u(f)(x) = x f'(x), commutes with the Fourier transform and generates Schwartz functions vanishing at zero and whose Fourier transforms also vanish at zero, enabling construction of Sonine space functions.
• The adjoint operator Δ₂ = D_u² - D_u reveals the tropical structure of the Scaling Site, linking operator-theoretic constructions to arithmetic geometry.