5,300 research outputs found
Representation of Quantum Mechanical Resonances in the Lax-Phillips Hilbert Space
We discuss the quantum Lax-Phillips theory of scattering and unstable
systems. In this framework, the decay of an unstable system is described by a
semigroup. The spectrum of the generator of the semigroup corresponds to the
singularities of the Lax-Phillips -matrix. In the case of discrete (complex)
spectrum of the generator of the semigroup, associated with resonances, the
decay law is exactly exponential. The states corresponding to these resonances
(eigenfunctions of the generator of the semigroup) lie in the Lax-Phillips
Hilbert space, and therefore all physical properties of the resonant states can
be computed.
We show that the Lax-Phillips -matrix is unitarily related to the
-matrix of standard scattering theory by a unitary transformation
parametrized by the spectral variable of the Lax-Phillips theory.
Analytic continuation in has some of the properties of a method
developed some time ago for application to dilation analytic potentials.
We work out an illustrative example using a Lee-Friedrichs model for the
underlying dynamical system.Comment: Plain TeX, 26 pages. Minor revision
Generalized Boltzmann Equation in a Manifestly Covariant Relativistic Statistical Mechanics
We consider the relativistic statistical mechanics of an ensemble of
events with motion in space-time parametrized by an invariant ``historical
time'' We generalize the approach of Yang and Yao, based on the Wigner
distribution functions and the Bogoliubov hypotheses, to find the approximate
dynamical equation for the kinetic state of any nonequilibrium system to the
relativistic case, and obtain a manifestly covariant Boltzmann-type equation
which is a relativistic generalization of the Boltzmann-Uehling-Uhlenbeck (BUU)
equation for indistinguishable particles. This equation is then used to prove
the -theorem for evolution in In the equilibrium limit, the
covariant forms of the standard statistical mechanical distributions are
obtained. We introduce two-body interactions by means of the direct action
potential where is an invariant distance in the Minkowski
space-time. The two-body correlations are taken to have the support in a
relative -invariant subregion of the full spacelike region. The
expressions for the energy density and pressure are obtained and shown to have
the same forms (in terms of an invariant distance parameter) as those of the
nonrelativistic theory and to provide the correct nonrelativistic limit
Equilibrium Relativistic Mass Distribution for Indistinguishable Events
A manifestly covariant relativistic statistical mechanics of the system of
indistinguishable events with motion in space-time parametrized by an
invariant ``historical time'' is considered. The relativistic mass
distribution for such a system is obtained from the equilibrium solution of the
generalized relativistic Boltzmann equation by integration over angular and
hyperbolic angular variables. All the characteristic averages are calculated.
Expressions for the pressure and the density of events are found and the
relativistic equation of state is obtained. The Galilean limit is considered;
the theory is shown to pass over to the usual nonrelativistic statistical
mechanics of indistinguishable particles.Comment: TAUP-2115-9
Towards a Realistic Equation of State of Strongly Interacting Matter
We consider a relativistic strongly interacting Bose gas. The interaction is
manifested in the off-shellness of the equilibrium distribution. The equation
of state that we obtain for such a gas has the properties of a realistic
equation of state of strongly interacting matter, i.e., at low temperature it
agrees with the one suggested by Shuryak for hadronic matter, while at high
temperature it represents the equation of state of an ideal ultrarelativistic
Stefan-Boltzmann gas, implying a phase transition to an effectively weakly
interacting phase.Comment: LaTeX, figures not include
Galilean limit of equilibrium relativistic mass distribution for indistinguishable events
The relativistic distribution for indistinguishable events is considered in
the mass-shell limit where is a given intrinsic property of
the events. The characteristic thermodynamic quantities are calculated and
subject to the zero-mass and the high-temperature limits. The results are shown
to be in agreement with the corresponding expressions of an on-mass-shell
relativistic kinetic theory. The Galilean limit which
coincides in form with the low-temperature limit, is considered. The theory is
shown to pass over to a nonrelativistic statistical mechanics of
indistinguishable particles.Comment: Report TAUP-2136-9
Hypercomplex quantum mechanics
The fundamental axioms of the quantum theory do not explicitly identify the
algebraic structure of the linear space for which orthogonal subspaces
correspond to the propositions (equivalence classes of physical questions). The
projective geometry of the weakly modular orthocomplemented lattice of
propositions may be imbedded in a complex Hilbert space; this is the structure
which has traditionally been used. This paper reviews some work which has been
devoted to generalizing the target space of this imbedding to Hilbert modules
of a more general type. In particular, detailed discussion is given of the
simplest generalization of the complex Hilbert space, that of the quaternion
Hilbert module.Comment: Plain Tex, 11 page
Foundations of a spacetime path formalism for relativistic quantum mechanics
Quantum field theory is the traditional solution to the problems inherent in
melding quantum mechanics with special relativity. However, it has also long
been known that an alternative first-quantized formulation can be given for
relativistic quantum mechanics, based on the parametrized paths of particles in
spacetime. Because time is treated similarly to the three space coordinates,
rather than as an evolution parameter, such a spacetime approach has proved
particularly useful in the study of quantum gravity and cosmology. This paper
shows how a spacetime path formalism can be considered to arise naturally from
the fundamental principles of the Born probability rule, superposition, and
Poincar\'e invariance. The resulting formalism can be seen as a foundation for
a number of previous parametrized approaches in the literature, relating, in
particular, "off-shell" theories to traditional on-shell quantum field theory.
It reproduces the results of perturbative quantum field theory for free and
interacting particles, but provides intriguing possibilities for a natural
program for regularization and renormalization. Further, an important
consequence of the formalism is that a clear probabilistic interpretation can
be maintained throughout, with a natural reduction to non-relativistic quantum
mechanics.Comment: RevTex 4, 42 pages; V6 is as accepted for publication in the Journal
of Mathematical Physics, updated in response to referee comments; V7 includes
final editorial correction
Relativistic mass distribution in event-anti-event system and ``realistic'' equation of state for hot hadronic matter
We find the equation of state which gives the value of
the sound velocity in agreement with the ``realistic'' equation of
state for hot hadronic matter suggested by Shuryak, in the framework of a
covariant relativistic statistical mechanics of an event--anti-event system
with small chemical and mass potentials. The relativistic mass distribution for
such a system is obtained and shown to be a good candidate for fitting hadronic
resonances, in agreement with the phenomenological models of Hagedorn, Shuryak,
{\it et al.} This distribution provides a correction to the value of specific
heat 3/2, of the order of 5.5\%, at low temperatures.Comment: 19 pages, report TAUP-2161-9
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