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arXiv:1205.4367 · Journal of Mathematical Physics

Classical limit of the Nelson model with cutoff

Marco Falconi

Published Jan 1, 2013• 33 citations
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Abstract

In this paper we analyze the classical limit of the Nelson model with cutoff, when both non-relativistic and relativistic particles number goes to infinity. We prove convergence of quantum observables to the solutions of classical equations, and find the evolution of quantum fluctuations around the classical solution. Furthermore, we analyze the convergence of transition amplitudes of normal ordered products of creation and annihilation operators between different types of initial states. In particular, the limit of normal ordered products between states with a fixed number of both relativistic and non-relativistic particles yields an unexpected quantum residue: instead of the product of classical solutions we obtain an average of the product of solutions corresponding to varying initial conditions.

DOI
10.1063/1.4775716

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In this paper we analyze the classical limit of the Nelson model with cutoff, when both non-relativistic and relativistic particles number goes to infinity. We prove convergence of quantum observables to the solutions of classical equations, and find the evolution of quantum fluctuations around the classical solution. Furthermore, we analyze the convergence of transition amplitudes of normal ordered products of creation and annihilation operators between different types of initial states. In particular, the limit of normal ordered products between states with a fixed number of both relativistic and non-relativistic particles yields an unexpected quantum residue: instead of the product of classical solutions we obtain an average of the product of solutions corresponding to varying initial conditions.
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.
We consider the Nelson model with ultraviolet cutoff, which describes the interaction between non-relativistic particles and a positive or zero mass quantized scalar field. We take the non-relativistic particles to obey Fermi statistics and discuss the time evolution in a mean-field limit of many fermions. In this case, the limit is known to be also a semiclassical limit. We prove convergence in terms of reduced density matrices of the many-body state to a tensor product of a Slater determinant with semiclassical structure and a coherent state, which evolve according to a fermionic version of the Schrödinger–Klein–Gordon equations.
We study the Classical Particle Limit of Non-Relativistic Quantum Mechanics. We show that the unitary group describing the evolution of the quantum fluctuation around any classical phase orbit has a classical limit as ħ→0 in the strong operator topology for a very large class of time independent scalar and vector potentials, which in practice covers all physically interesting cases. We also show that the mean values of the quantum mechanical position and velocity operators on suitable states, obtained by time evolution of the product of a Weyl operator centred around the large coordinates and momenta ħ −1/2 ξ and ħ −1/2 π and a fixed ħ-independent wave function, converge to the solution of the classical equations with initial data ξ, π as ħ→0 for a broad class of repulsive interactions On demontre que le groupe unitaire qui decrit l'evolution des fluctuations quantiques autour d'une orbite classique quelconque a une limite classique quand ħ→0 dans la topologie forte des operateurs, pour une vaste classe de potentiels qui couvre en pratique tous les cas physiquement interessants. On demontre que les valeurs moyennes des operateurs quantiques position et vitesse, dans des etats convenables obtenus par l'evolution temporelle du produit d'un operateur de Weyl centre autour de grandes coordonnees et de grandes impulsions ħ −1/2 ξ, ħ -1/2 π et d'une fonction d'onde independante de ħ, tendent vers la solution des equations classiques avec donnees initiales ξ, π quand ħ→0 pour une grande classe d'interactions repulsives
Some recent results on the convergence of Nelson diffusions are extended to the case of Schrödinger operators with time-dependent electromagnetic potentials. It is proven that the sequence {P n}n≥1 of measures on the canonical space of physical trajectories associated to the solutions of Schrödinger equations in Nelson’s scheme, corresponding to the sequence {(Vn,An)}n≥1 ⊆C1(R;ℛ×L2(R3)), converges in the total variation norm under the assumptions that for every fixed t the scalar potentials Vn(t) converge in ℛ, the space of Rollnik class potentials, and the vector potentials An(t) converge in Lloc∞(R;L2(R3)). In order to prove these results conditions are given under which solutions of Schrödinger equations are continuous in the (time-dependent electromagnetic) potentials in the norm of the Sobolev space H1(R3).
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.
The stochastic theory of relativistic quantum mechanics presented here is modelled on the one that has been proposed previously and that was claimed to be a promising substitute to the orthodox theory in the non-relativistic domain. So it also relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. Along the way, we introduce ad hoc hypotheses for the sole reason that they facilitate and even make possible the further development of the theory. These and the so-called Nelson equations are used to determine the drift vectors and the diffusion tensor characteristic of such a process. It is then possible to write down the Fokker-Planck and Kolmogorov equations that can be used to determine the normal and conditional probability densities. This logical sequence of steps leads to the solution of specific problems and we apply it to the special case of the free particle.