Representation of the Resonance of a Relativistic Quantum Field Theoretical Lee-Friedrichs Model in Lax-Phillips Scattering Theory

The quantum mechanical description of the evolution of an unstable system defined initially as a state in a Hilbert space at a given time does not provide a semigroup (exponential) decay law. The Wigner-Weisskopf survival amplitude, describing reversible quantum transitions, may be dominated by exponential type decay in pole approximation at times not too short or too long, but, in the two channel case, for example, the pole residues are not orthogonal, and the evolution does not correspond to a semigroup (experiments on the decay of the neutral $K$-meson system strongly support the semigroup evolution postulated by Lee, Oehme and Yang, and Yang and Wu). The scattering theory of Lax and Phillips, originally developed for classical wave equations, has been recently extended to the description of the evolution of resonant states in the framework of quantum theory. The resulting evolution law of the unstable system is that of a semigroup, and the resonant state is a well-defined function in the Lax-Phillips Hilbert space. In this paper we apply this theory to a relativistically covariant quantum field theoretical form of the (soluble) Lee model. We construct the translation representations with the help of the wave operators, and show that the resulting Lax-Phillips $S$-matrix is an inner function (the Lax-Phillips theory is essentially a theory of translation invariant subspaces). In the special case that the $S$-matrix is a rational inner function, we obtain the resonant state explicitly and analyze its particle ($V, N, \theta$) content. If there is an exponential bound, the general case differs only by a so-called trivial inner factor, which does not change the complex spectrum, but may affect the wave function of the resonant state.Comment: Plain TeX, 33 page

Distributional analysis of prospective 2009 US individual income taxes: current law and the candidates’ tax plans

The purpose of this paper is to compare the distributional characteristics of two presidential candidates’ proposed reforms to the US federal individual income tax. Using an anonymous sample of tax return data from the Brookings-Urban Institute Tax Policy Center and the Center’s simulations of 2009 tax law and the two proposals, we compare the vertical and horizontal equity of the three individual income tax regimes. Surprisingly, there is very little difference among the three proposed individual income tax regimes in terms of vertical and horizontal equity. However, when the initial effective tax rate positions and economic incomes of each pair of taxpayers are compared to the new effective tax rate positions under the two proposals, we find that the Obama proposal makes the tax system more progressive than 2009 law. This change is much more pronounced than under the McCain proposal. On the other hand, when these initial positions are compared to the two proposals viz. a viz. horizontal equity, the McCain proposed tax system is more horizontally equitable than 2009 tax law, and more horizontally equitable than the Obama proposal is when compared to 2009 tax law.Tax equity; McCain tax plan; Obama tax plan

Improving public education through strengthened local control

Education