Particle–Hole Ladders
A self contained analysis demonstrates that the sum of all particle-hole ladder contributions for a two dimensional, weakly coupled fermion gas with a strictly convex Fermi curve at temperature zero is bounded. This is used in our construction of two dimensional Fermi liquids. This summary article contains the statements of the main results. The proofs are contained in the full, electronic, article. Electronic Supplementary Material: Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s00220-004-1038-2.
Singular Fermi Surfaces I. General Power Counting and Higher Dimensional Cases
We prove regularity properties of the self-energy, to all orders in perturbation theory, for systems with singular Fermi surfaces which contain Van Hove points where the gradient of the dispersion relation vanishes. In this paper, we show for spatial dimensions $d \ge 3$ that despite the Van Hove singularity, the overlapping loop bounds we proved together with E. Trubowitz for regular non--nested Fermi surfaces [J. Stat. Phys. 84 (1996) 1209] still hold, provided that the Fermi surface satisfies a no-nesting condition. This implies that for a fixed interacting Fermi surface, the self-energy is a continuously differentiable function of frequency and momentum, so that the quasiparticle weight and the Fermi velocity remain close to their values in the noninteracting system to all orders in perturbation theory. In a companion paper, we treat the more singular two-dimensional case.
The quantum particle in a box: what we can learn from classical electrodynamics
The problem of a charged particle enclosed in an infinite square potential well is analysed from the point of view of classical theory with the addition of the electromagnetic zero-point radiation field, with the aim to explore the extent to which such an analysis can contribute to enhance our understanding of the quantum behavior. First a proper treatment is made of the freely moving particle subject to the action of the radiation field, involving a frequency cutoff ωc. The jittering motion and the effective structure of the particle are sustained by the permanent action of the zero-point field. As a result, the particle interacts resonantly with the traveling field modes of frequency ωc in its proper frame of reference, which superpose to give rise to a modulated wave accompanying the particle. This is identified with the de Broglie wave, validating the choice of Compton’s frequency for ωc. For the stationary states of particles confined in the potential well, the Lorentz force produced by the accompanying field is shown to lead to discrete values for the mean speed and to an uneven probability distribution that echoes the corresponding quantum distribution. The relevance of the results obtained and the limitations of the classical approach used, are discussed in the context of present-day stochastic electrodynamics.