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

Transition Decomposition of Quantum Mechanical Evolution

We show that the existence of the family of self-adjoint Lyapunov operators introduced in [J. Math. Phys. 51, 022104 (2010)] allows for the decomposition of the state of a quantum mechanical system into two parts: A past time asymptote, which is asymptotic to the state of the system at t goes to minus infinity and vanishes at t goes to plus infinity, and a future time asymptote, which is asymptotic to the state of the system at t goes to plus infinity and vanishes at t goes to minus infinity. We demonstrate the usefulness of this decomposition for the description of resonance phenomena by considering the resonance scattering of a particle off a square barrier potential. We show that the past time asymptote captures the behavior of the resonance. In particular, it exhibits the expected exponential decay law and spatial probability distribution.Comment: Accepted for publication in Int. J. Theor. Phy

Abdominal vacuum lift as an aid to diagnosing abdominal adhesions

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2006.Includes bibliographical references (leaves 18-22).The internal organs are designed to move freely and slide over one another during normal body movement. The abdominal organs, however, have a tendency to adhere to the abdominal cavity (peritoneum) and other abdominal organs after surgery or infection. These adhesions can cause pain, discomfort , inflammation, anxiety, depression, problems with conception, trouble eating, and decreased immune function. There are around 300,000 hospital admissions in the U.S. every year for patients due to adhesions.. Part of the problem is that there is no suitable method to diagnose adhesions. Recently there have been a number of studies which suggest that measuring visceral slides under ultrasound using exaggerated respiration may prove to be very promising in diagnosing adhesions non invasively. Yet there are still weaknesses in the predictive power of these procedures. For such procedures to be successfully implemented into clinical medicine and offer non invasive methods to diagnosing adhesions, they must first be able to offer higher percentage predictive values. We have worked on a number of models of an external abdominal vacuum system which we believe will increase the accuracy and predictive values of measuring visceral slides under ultrasound using exaggerated respiration.by Julius Strauss.S.B