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HORWITZ@TAUNIVM.TAU.AC.IL (or: LARRY@post.tau.ac.il)
The recently developed quantum theory utilizing the ideas and results of Lax and Phillips for the description of scattering and resonances, or unstable systems, is reviewed. The framework for the construction of the Lax-Phillips theory is given by a functional space which is the direct integral over time of the usual quantum mechanical Hilbert spaces, defined at each $t$. It has been shown that quantum scattering theory can be formulated in this way. The theory of Lax and Phillips, however, also obtains a simple relation between the poles of the $S$-matrix and the spectrum of the generator of the semigroup corresponding to the reduced motion of the resonant state. It is shown in this work that in order to obtain such a relation in the quantum mechanical case, the evolution operator must act as an integral operator on the time variable. The structure required appears naturally in the Liouville space formulation of the evolution of the state of the system. The resulting $S$ matrix is a function, as an integral operator, of $t-t'$ (i.e., homogeneous), and the semigroup is contractive. A physical interpretation of this structure may be introduced, from which we obtain a quantitative description of the expected age of a created system, and the expected time of decay of an unstable system. The superselection rule which distinguishes between the unstable system and its decay products is realized in this way. It is also shown that from this point of view, one has a natural mechanism for the dynamical mixing of the quantum mechanical states as observed by means of time translation invariant operators. In particular, this provides a model for certain types of irreversible processes, as well as for the measurement process for closed as well as open systems.
The Lee-Friedrichs model has been very useful in the study of decay-scattering systems in the framework of complex quantum mechanics. Since it is exactly soluble, the analytic structure of the amplitudes can be explicitly studied. It is shown in this paper that a similar model, which is also exactly soluble, can be constructed in quaternionic quantum mechanics. The problem of the decay of an unstable system is treated here. The use of the Laplace transform, involving quaternion-valued analytic functions of a variable with values in a complex subalgebra of the quaternion algebra, makes the analytic properties of the solution apparent; some analysis is given of the dominating structure in the analytic continuation to the lower half plane. A study of the corresponding scattering system will be given in a succeeding paper.
We analyze the correspondence between a five dimensional U(1) gauge invariant theory and four dimensional scalar QED, where the fifth dimension $(\tau)$ is an invariant parameter of evolution of the manifestly covariant one particle sector as well as for the full Fock space. The correspondence is represented by the limit in which the width of the photon mass distribution $\Delta s$ tends to zero and large $\tau$ correlations occur. In the limiting procedure, calculation of a two point diagram shows that the Pauli-Villars regularization is intrinsically related to the these long range correlations.