Narrow Results

Following the establishment of quantum mechanics around 80 years ago, many formulae that describe how atoms and molecules interact with each other and with external electromagnetic fields were derived. While such formulae provide prescriptions for calculating atomic and molecular properties many of them are inconvenient to evaluate because of the need to calculate the sums of a, large, often infinite, number of expressions constructed from various matrix elements and excitation energies. Alex Dalgarno through his major work in perturbation theory pioneered the method, now standard, in which these sums are calculated from the solutions of finite numbers of differential equations. This article is a brief summary of the method and its use.

The standard physical model of contemporary mesoscopic noise and transport consists in a phenomenologically based approach, proposed originally by Landauer and since continued and amplified by Büttiker, Imry and others. Throughout all the years of its gestation and growth, it is surprising that the Landauer–Büttiker approach to mesoscopics has matured with scant attention to the conserving properties lying at its roots: that is, at the level of actual microscopic principles. We systematically apply the sum rules for the electron gas to clarify the issue of conservation within the standard model of mesoscopic conduction. Noise, as observed in quantum point contacts, provides the vital clue.

In the light-cone representation, vacuum effects are correctly introduced into the dynamics by the inclusion of induced operators. Recently, some of the induced operators in QCD have been derived. The knowledge of these operators makes it possible to perform calculations of the propertied of QCD resonances using essentially the QCD sum rules approach; but with the inclusion of the induced operators providing the nonperturbative physics rather than the operator product expansion that is used in the standard sum rules approach.

Theoretical methods of extracting physical information from hadronic elastic scattering amplitudes are explored. These include a new sum rule yielding new mathematical bounds on cross sections which are experimentally confirmed. For hadrons made up of light quarks, a virial theorem sum rule is employed to show that the mass of a hadrons consists of half quark and half gluon energies. This sum rule is also experimentally confirmed. For Higgs particles produced from heavy mass constituents, sum rules are derived for mass shifts which have not yet been experimentally confirmed.

From sum rules in the heavy quark limit of QCD, using the non-forward amplitude, we demonstrate that if the slope ρ² = -ξ'(1) of the Isgur-Wise function ξ(w) attains its lower bound (as happens in the BPS limit proposed by Uraltsev), the IW function is completely determined, given by the function .

The analysis of the JLab proton polarization data on the ratio μ_pG_Ep(Q²)/G_Mp(Q²), together with all other existing nucleon form factor data, by the unitary and analytic model of nucleon electromagnetic structure reveals them to be consistent with all known nucleon form factor properties. Moreover, it implicates surprising non-dipole behavior of the proton electric form factor in the space-like region with the zero around the momentum transfer squared value Q² = 13GeV². Consequences of such G_Ep(Q²)-behavior on various physical quantities are briefly reviewed.

Using the fact that the neutrino mixing matrix , where U_e and U_ν result from the diagonalization of the charged lepton and neutrino mass matrices, we analyze the predictions based on the sum rules which the Dirac phase δ present in U satisfies when U_ν has a form dictated by, or associated with, discrete favor symmetries and U_e has a “minimal” form (in terms of angles and phases it contains) that can provide the requisite corrections to U_ν, so that the reactor, atmospheric and solar neutrino mixing angles θ₁₃, θ₂₃ and θ₁₂ have values compatible with the current data.