Related papers: Phase operator on $L^2(\mathbb{Q}_p)$ and the zeroes of Fisher and Riemann

Phase operator on $L^2(\mathbb{Q}_p)$ and the zeroes of Fisher and Riemann

URL: http://arxiv.org/abs/2102.13445v1
Date: Fri, 26 Feb 2021 13:02:37 GMT
Title: Phase operator on $L^2(\mathbb{Q}_p)$ and the zeroes of Fisher and Riemann
Authors: Parikshit Dutta and Debashis Ghoshal
Abstract summary: We show that a phase operator' conjugate to it can be constructed on a subspace $L(mathbbQ_p)$ of $L(mathbbQ_p)$. We discuss how to combine this for all primes to possibly relate to the zeroes of the Riemann zeta function.
Score: 0.0
License: http://creativecommons.org/licenses/by/4.0/
Abstract: The distribution of the non-trivial zeroes of the Riemann zeta function, according to the Riemann hypothesis, is tantalisingly similar to the zeroes of the partition functions (Fisher and Yang-Lee zeroes) of statistical mechanical models studied by physicists. The resolvent function of an operator akin to the phase operator, conjugate to the number operator in quantum mechanics, turns out to be important in this approach. The generalised Vladimirov derivative acting on the space $L^2(\mathbb{Q}_p)$ of complex valued locally constant functions on the $p$-adic field is rather similar to the number operator. We show that a `phase operator' conjugate to it can be constructed on a subspace $L^2(p^{-1}\mathbb{Z}_p)$ of $L^2(\mathbb{Q}_p)$. We discuss (at physicists' level of rigour) how to combine this for all primes to possibly relate to the zeroes of the Riemann zeta function. Finally, we extend these results to the family of Dirichlet $L$-functions, using our recent construction of Vladimirov derivative like pseudodifferential operators associated with the Dirichlet characters.

Related papers

Klein-Gordon oscillators and Bergman spaces [55.2480439325792]
We consider classical and quantum dynamics of relativistic oscillator in Minkowski space $mathbbR3,1$. The general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic functions on the K"ahler-Einstein manifold $Z_6$.
arXiv Detail & Related papers (2024-05-23T09:20:56Z)
Quantum charges of harmonic oscillators [55.2480439325792]
We show that the energy eigenfunctions $psi_n$ with $nge 1$ are complex coordinates on orbifolds $mathbbR2/mathbbZ_n$. We also discuss "antioscillators" with opposite quantum charges and the same positive energy.
arXiv Detail & Related papers (2024-04-02T09:16:18Z)
Vacuum Force and Confinement [65.268245109828]
We show that confinement of quarks and gluons can be explained by their interaction with the vacuum Abelian gauge field $A_sfvac$.
arXiv Detail & Related papers (2024-02-09T13:42:34Z)
More on symmetry resolved operator entanglement [0.0]
We focus on spin chains with a global $U(1)$ conservation law, and on operators $O$ with a well-defined $U(1)$ charge. We employ the notion of symmetry resolved operator entanglement (SROE) introduced in [PRX Quantum 4, 010318 (2023) and extend the results of the latter paper in several directions. Our main results are: i) the SROE of $rho_beta$ obeys the operator area law; ii) for free fermions, local operators in Heisenberg picture can have a SROE that grows logarithmically in time or saturate
arXiv Detail & Related papers (2023-09-07T21:58:18Z)
Rigorous derivation of the Efimov effect in a simple model [68.8204255655161]
We consider a system of three identical bosons in $mathbbR3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$.
arXiv Detail & Related papers (2023-06-21T10:11:28Z)
Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
arXiv Detail & Related papers (2022-06-17T17:55:43Z)
Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis [0.0]
We find a correspondence between the distribution of the zeros of $zeta(z)$ and the poles of the scattering matrix $S$ of a physical system. In the first we apply the infinite-components Majorana equation in a Rindler spacetime and compare the results with those obtained with a Dirac particle. Here we find that, thanks to the relationship between the angular momentum and energy/mass eigenvalues of the Majorana solution, one can explain the still unclear point for which the poles and zeros of the $S$-matrix exist always in pairs.
arXiv Detail & Related papers (2021-08-17T19:52:15Z)
Holomorphic family of Dirac-Coulomb Hamiltonians in arbitrary dimension [0.0]
We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is, operators on the half-line of the form $D_omega,lambda:=beginbmatrix-fraclambda+omegax&-partial_x.
arXiv Detail & Related papers (2021-07-08T11:48:57Z)
Models of zero-range interaction for the bosonic trimer at unitarity [91.3755431537592]
We present the construction of quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered.
arXiv Detail & Related papers (2020-06-03T17:54:43Z)
Bistochastic operators and quantum random variables [0.0]
We considerintegrable functions $Xrightarrow mathcal B(mathcal H)$ that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type.
arXiv Detail & Related papers (2020-04-30T12:45:54Z)

This list is automatically generated from the titles and abstracts of the papers in this site.