9,393 research outputs found
Relativistic Partial Wave Analysis Using the Velocity Basis of the Poincare Group
The velocity basis of the Poincare group is used in the direct product space
of two irreducible unitary representations of the Poincare group. The velocity
basis with total angular momentum j will be used for the definition of
relativistic Gamow vectors.Comment: 14 pages; revte
A Note on the Topology of Space-time in Special Relativity
We show that a topology can be defined in the four dimensional space-time of
special relativity so as to obtain a topological semigroup for time. The
Minkowski 4-vector character of space-time elements as well as the key
properties of special relativity are still the same as in the standard theory.
However, the new topological structure allows the possibility of an intrinsic
asymmetry in the time evolution of physical systems
Complex Energies and Beginnings of Time Suggest a Theory of Scattering and Decay
Many useful concepts for a quantum theory of scattering and decay (like
Lippmann-Schwinger kets, purely outgoing boundary conditions, exponentially
decaying Gamow vectors, causality) are not well defined in the mathematical
frame set by the conventional (Hilbert space) axioms of quantum mechanics.
Using the Lippmann-Schwinger equations as the takeoff point and aiming for a
theory that unites resonances and decay, we conjecture a new axiom for quantum
mechanics that distinguishes mathematically between prepared states and
detected observables. Suggested by the two signs of the
Lippmann-Schwinger equations, this axiom replaces the one Hilbert space of
conventional quantum mechanics by two Hardy spaces. The new Hardy space theory
automatically provides Gamow kets with exponential time evolution derived from
the complex poles of the -matrix. It solves the causality problem since it
results in a semigroup evolution. But this semigroup brings into quantum
physics a new concept of the semigroup time , a beginning of time. Its
interpretation and observations are discussed in the last section.Comment: 27 pages, 3 figure
Time Asymmetric Quantum Theory - II. Relativistic Resonances from S-Matrix Poles
Relativistic resonances and decaying states are described by representations
of Poincar\'e transformations, similar to Wigner's definition of stable
particles. To associate decaying state vectors to resonance poles of the
-matrix, the conventional Hilbert space assumption (or asymptotic
completeness) is replaced by a new hypothesis that associates different dense
Hardy subspaces to the in- and out-scattering states. Then one can separate the
scattering amplitude into a background amplitude and one or several
``relativistic Breit-Wigner'' amplitudes, which represent the resonances per
se. These Breit-Wigner amplitudes have a precisely defined lineshape and are
associated to exponentially decaying Gamow vectors which furnish the
irreducible representation spaces of causal Poincar\'e transformations into the
forward light cone.Comment: 57 pages, 6 figure
Time Asymmetric Quantum Theory - III. Decaying States and the Causal Poincare Semigroup
A relativistic resonance which was defined by a pole of the -matrix, or by
a relativistic Breit-Wigner line shape, is represented by a generalized state
vector (ket) which can be obtained by analytic extension of the relativistic
Lippmann-Schwinger kets. These Gamow kets span an irreducible representation
space for Poincar\'e transformations which, similar to the Wigner
representations for stable particles, are characterized by spin (angular
momentum of the partial wave amplitude) and complex mass (position of the
resonance pole). The Poincar\'e transformations of the Gamow kets, as well as
of the Lippmann-Schwinger plane wave scattering states, form only a semigroup
of Poincar\'e transformations into the forward light cone. Their transformation
properties are derived. From these one obtains an unambiguous definition of
resonance mass and width for relativistic resonances. The physical
interpretation of these transformations for the Born probabilities and the
problem of causality in relativistic quantum physics is discussed.Comment: 49 pages, 1 figur
Symmetry Representations in the Rigged Hilbert Space Formulation of Quantum Mechanics
We discuss some basic properties of Lie group representations in rigged
Hilbert spaces. In particular, we show that a differentiable representation in
a rigged Hilbert space may be obtained as the projective limit of a family of
continuous representations in a nested scale of Hilbert spaces. We also
construct a couple of examples illustrative of the key features of group
representations in rigged Hilbert spaces. Finally, we establish a simple
criterion for the integrability of an operator Lie algebra in a rigged Hilbert
space
Measurement of the total energy of an isolated system by an internal observer
We consider the situation in which an observer internal to an isolated system
wants to measure the total energy of the isolated system (this includes his own
energy, that of the measuring device and clocks used, etc...). We show that he
can do this in an arbitrarily short time, as measured by his own clock. This
measurement is not subjected to a time-energy uncertainty relation. The
properties of such measurements are discussed in detail with particular
emphasis on the relation between the duration of the measurement as measured by
internal clocks versus external clocks.Comment: 7 pages, 1 figur
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