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arXiv:1602.03212

Bohr's correspondence principle in quantum field theory and classical renormalization scheme: the Nelson model

Zied Ammari, Marco Falconi

Published Feb 9, 2016• 10 citations
arXivHTMLPDF

Abstract

In the mid Sixties Edward Nelson proved the existence of a consistent quantum field theory that describes the Yukawa-like interaction of a non-relativistic nucleon field with a relativistic meson field. Since then it is thought, despite the renormalization procedure involved in the construction, that the quantum dynamics should be governed in the classical limit by a Schrödinger-Klein-Gordon system with Yukawa coupling. In the present paper we prove this fact in the form of a Bohr correspondence principle. Besides, our result enlighten the nature of the renormalization method employed in this model which we interpret as a strategy that allows to put the related classical Hamiltonian PDE in a normal form suitable for a canonical quantization.

DOI
10.48550/arxiv.1602.03212

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In the mid Sixties Edward Nelson proved the existence of a consistent quantum field theory that describes the Yukawa-like interaction of a non-relativistic nucleon field with a relativistic meson field. Since then it is thought, despite the renormalization procedure involved in the construction, that the quantum dynamics should be governed in the classical limit by a Schrödinger-Klein-Gordon system with Yukawa coupling. In the present paper we prove this fact in the form of a Bohr correspondence principle. Besides, our result enlighten the nature of the renormalization method employed in this model which we interpret as a strategy that allows to put the related classical Hamiltonian PDE in a normal form suitable for a canonical quantization.
Since the development of quantum mechanics it has been natural to analyze the connection between classical and quantum mechanical descriptions of physical systems. In particular one should expect that in some sense when quantum mechanical effects becomes negligible the system will behave like it is dictated by classical mechanics. One famous relation between classical and quantum theory is due to Ehrenfest. This result was later developed and put on firm mathematical foundations by Hepp. He proved that matrix elements of bounded functions of quantum observables between suitable coherents states (that depend on Planck's constant h) converge to classical values evolving according to the expected classical equations when h goes to zero. His results were later generalized by Ginibre and Velo to bosonic systems with infinite degrees of freedom and scattering theory. In this thesis we study the classical limit of Nelson model, that describes non relativistic particles, whose evolution is dictated by Schrödinger equation, interacting with a scalar relativistic field, whose evolution is dictated by Klein-Gordon equation, by means of a Yukawa-type potential. The classical limit is a mean field and weak coupling limit. We proved that the transition amplitude of a creation or annihilation operator, between suitable coherent states, converges in the classical limit to the solution of the system of differential equations that describes the classical evolution of the theory. The quantum evolution operator converges to the evolution operator of fluctuations around the classical solution. Transition amplitudes of normal ordered products of creation and annihilation operators between coherent states converge to suitable products of the classical solutions. Transition amplitudes of normal ordered products of creation and annihilation operators between fixed particle states converge to an average of products of classical solutions, corresponding to different initial conditions.
Abstract We study the renormalized Nelson model in a semiclassical regime where the field becomes classical while the particle remains quantum. The degree of classicality is measured by a small parameter ε≪1\varepsilon \ll 1ε≪1 ε ≪ 1 . In this scaling the particle evolves on microscopic times, whereas the field exhibits nontrivial dynamics only on the macroscopic scale t=O(ε−2)t={\mathcal {O}}(\varepsilon ^{-2})t=O(ε−2) t = O ( ε - 2 ) . The natural semiclassical model is the coupled Schrödinger–Klein–Gordon (SKG) system, which encodes the time-scale separation through an explicit ε\varepsilon ε ε -dependence. Based on this scale separation in SKG, we apply the adiabatic principle to derive a new PDE for the classical field, the ε\varepsilon ε ε -free adiabatic Klein–Gordon (aKG) equation, where the field is driven by the instantaneous ground state of the particle. Our main result is a norm approximation of the Nelson dynamics by the aKG solution corrected by a quasi-free fluctuation dynamics around the classical field, generated by a renormalized Bogoliubov–Nelson Hamiltonian. As a corollary, we obtain convergence of the reduced one-body densities for both subsystems, where the fluctuation correction vanishes, thereby justifying aKG as a semiclassical Born–Oppenheimer type approximation of the renormalized Nelson model.
We consider the time evolution of the renormalized Nelson model, which describes N bosons linearly coupled to a quantized scalar field, in the mean-field limit of many particles N\gg 1 with coupling constant proportional to N^{-1/2} . First, we show that initial states exhibiting Bose–Einstein condensation for the particles and approximating a coherent state for the quantum field retain their structure under the many-body time evolution. Concretely, the dynamics of the reduced densities are approximated by solutions of two coupled PDEs, the Schrödinger–Klein–Gordon equations. Second, we construct a renormalized Bogoliubov evolution that describes the quantum fluctuations around the Schrödinger–Klein–Gordon equations. This evolution is used to extend the approximation of the evolved many-body state to the full norm topology. In summary, we provide a comprehensive analysis of the Nelson model that reveals the role of renormalization in the mean-field Bogoliubov theory.
In this paper, we develop a wave function renormalization scheme for models of non-relativistic quantum particles interacting with a quantized relativistic field, in the Hamiltonian formalism of quantum field theory. We construct the interacting Hamilton operator, in its ground state representation, for a large class of particle-field interactions; hereby addressing a number of open problems related to both ultraviolet and infrared singularities in the spin-boson and Nelson models, where the infinite wave function renormalization plays a fundamental role.
In this paper we apply the Bohr's correspondence principle to analize the asymptotic behavior of the Klein-Gordon and Dirac probability densities. It is found that in the non-relativistic limit, the densities reduce to their respective classical single-particle probability distributions plus a series of quantum corrections. The procedure is applied in two basic problems, the relativistic quantum oscillator and the relativistic particle in a box. The particle and antiparticle solutions are found to yield the same classical distribution plus quantum correction terms for the proposed limit. In the quantum oscillator case, a κ\kappaκ parameter modifies the probability distribution. Its origin is briefly discussed in terms of energy.
It has recently been shown that the classical electric and magnetic fields which satisfy the source-free Maxwell equations can be linearly mapped into the real and imaginary parts of a transverse-vector wave function which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator is physically appropriate to the free photon. The free-particle Klein-Gordon equation for scalar fields modestly extends the classical wave equation via a mass term. It is physically untenable for complex-valued wave functions, but has a sound nonnegative conserved-energy functional when it is restricted to real-valued classical fields. Canonical Hamiltonization and a further canonical transformation maps the real-valued classical Klein-Gordon field and its canonical conjugate into the real and imaginary parts of a scalar wave function (within a constant factor) which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator has the natural correspondence-principle relativistic square-root form for a free particle, with a mass that matches the Klein-Gordon field theory's mass term. Quantization of the real-valued classical Klein-Gordon field is thus second quantization of this natural correspondence-principle first-quantized relativistic Schroedinger equation. Source-free electromagnetism is treated in a parallel manner, but with the classical scalar Klein-Gordon field replaced by a transverse vector potential that satisfies the classical wave equation. This reproduces the previous first-quantized results that were based on Maxwell's source-free electric and magnetic field equations.