8,872 research outputs found
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
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
Production-decay interferences at NLO in QCD for t-channel single-top production
We present a calculation of O(\alpha_s) contributions to the process of
t-channel single-top production and decay, which include virtual and real
corrections arising from interference of the production and decay subprocesses.
The calculation is organized as a simultaneous expansion of the matrix elements
in the couplings \alpha_{ew},\alpha_s and the virtuality of the intermediate
top quark, (p_t^2-m_t^2)/m_t^2 ~ \Gamma_t/m_t, and extends earlier results
beyond the narrow-width approximation.Comment: 33 pages, 6 Figure
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
Irreversible Quantum Mechanics in the Neutral K-System
The neutral Kaon system is used to test the quantum theory of resonance
scattering and decay phenomena. The two dimensional Lee-Oehme-Yang theory with
complex Hamiltonian is obtained by truncating the complex basis vector
expansion of the exact theory in Rigged Hilbert space. This can be done for K_1
and K_2 as well as for K_S and K_L, depending upon whether one chooses the
(self-adjoint, semi-bounded) Hamiltonian as commuting or non-commuting with CP.
As an unexpected curiosity one can show that the exact theory (without
truncation) predicts long-time 2 pion decays of the neutral Kaon system even if
the Hamiltonian conserves CP.Comment: 36 pages, 1 PostScript figure include
Gamow-Jordan Vectors and Non-Reducible Density Operators from Higher Order S-Matrix Poles
In analogy to Gamow vectors that are obtained from first order resonance
poles of the S-matrix, one can also define higher order Gamow vectors which are
derived from higher order poles of the S-matrix. An S-matrix pole of r-th order
at z_R=E_R-i\Gamma/2 leads to r generalized eigenvectors of order k= 0, 1, ...
, r-1, which are also Jordan vectors of degree (k+1) with generalized
eigenvalue (E_R-i\Gamma/2). The Gamow-Jordan vectors are elements of a
generalized complex eigenvector expansion, whose form suggests the definition
of a state operator (density matrix) for the microphysical decaying state of
this higher order pole. This microphysical state is a mixture of non-reducible
components. In spite of the fact that the k-th order Gamow-Jordan vectors has
the polynomial time-dependence which one always associates with higher order
poles, the microphysical state obeys a purely exponential decay law.Comment: 39 pages, 3 PostScript figures; sub2.eps may stall some printers and
should then be printed out separately; ghostview is o.
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