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arXiv:hep-th/0210179 · De Gruyter eBooks

New classical limits of quantum theories

S. G. Rajeev

Published Aug 16, 2004• 5 citations
arXivHTMLPDF

Abstract

Quantum fluctuations of some systems vanish not only in the limit $\hbar\to 0$, but also as some other parameters (such as $1\over N$, the inverse of the number of `colors' of a Yang-Mills theory) vanish. These lead to new classical limits that are often much better approximations to the quantum theory. We describe two examples: the familiar Hartree--Fock-Thomas-Fermi methods of atomic physics as well as the limit of large spatial dimension. Then we present an approach of the Hecke operators on modular forms inspired by these ideas of quantum mechanics. It explains in a simple way why the spectra of these operators tend to the spectrum of random matrices for large weight for the modular forms.

DOI
10.1515/9783110200010.213

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Quantum fluctuations of some systems vanish not only in the limit ℏ→0\hbar\to 0ℏ→0, but also as some other parameters (such as 1N1\over NN1​, the inverse of the number of `colors' of a Yang-Mills theory) vanish. These lead to new classical limits that are often much better approximations to the quantum theory. We describe two examples: the familiar Hartree--Fock-Thomas-Fermi methods of atomic physics as well as the limit of large spatial dimension. Then we present an approach of the Hecke operators on modular forms inspired by these ideas of quantum mechanics. It explains in a simple way why the spectra of these operators tend to the spectrum of random matrices for large weight for the modular forms.
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For a quantum observable AℏA_\hbarAℏ​ depending on a parameter ℏ\hbarℏ we define the notion ``AℏA_\hbarAℏ​ converges in the classical limit''. The limit is a function on phase space. Convergence is in norm in the sense that Aℏ→0A_\hbar\to0Aℏ​→0 is equivalent with ∥Aℏ∥→0\Vert A_\hbar\Vert\to0∥Aℏ​∥→0. The ℏ\hbarℏ-wise product of convergent observables converges to the product of the limiting phase space functions. ℏ−1\hbar^{-1}ℏ−1 times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the ℏ\hbarℏ-wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed.
In this article, we revisit the question of the validity of Hartree-Fock and random-phase approximations. We show that there is a connection between the two and while the RPA as it is known in much of the physics literature is of limited validity, there is a generalised sense in which the random phase approximation is of much wider applicability including to systems that do not possess Fermi surfaces. The main conclusion is that the Hartree-Fock approximation is a mean-field idea applied to the density operator, and the random-phase approximation is a mean-field idea applied to the number operator. The generalised RPA is used to compute single-particle properties such as momentum distribution and spectral functions. It is found that we have to go beyond the generalised RPA and include fluctuations in the momentum distribution in order to recover a nonzero imaginary part of the one-particle self energy, which is also explicitly computed, all this in any number of spatial dimensions and no bosonization is needed.
Two further examples of motion of a wave packet in external Yang–Mills fields are investigated. We find that because of the interaction with the gauge field the wave packet splits into a number of parts which remain spatially separated in the ħ →\to→ 0 limit. Therefore, these examples confirm our point of view that quantum mechanics of colored particles does not have simple classical limit.
Working within a time-dependent Hartree-Fock framework, one develops a semiclassical approximation appropriate for large systems. It is demonstrated that the standard semiclassical approach, the Thomas-Fermi approximation, is inconsistent with Hartree-Fock theory when the basic two-body interaction is short-ranged (as in nuclear systems, for example). However, by introducing a simple extension of the Thomas-Fermi approximation, one overcomes this problem. One also discusses the infinite nuclear matter problem and point out that time-dependent Hartree-Fock theory yields collective modes of the zero sound variety instead of ordinary hydrodynamic (first) sound. One thus emphasizes that one should be extremely circumspect when attempting to cast the equations of motion of time-dependent Hartree-Fock theory into a hydrodynamic-like form.