4,648 research outputs found
Reply to ``Comment on `On the inconsistency of the Bohm-Gadella theory with quantum mechanics'''
In this reply, we show that when we apply standard distribution theory to the
Lippmann-Schwinger equation, the resulting spaces of test functions would
comply with the Hardy axiom only if classic results of Paley and Wiener, of
Gelfand and Shilov, and of the theory of ultradistributions were wrong. As
well, we point out several differences between the ``standard method'' of
constructing rigged Hilbert spaces in quantum mechanics and the method used in
Time Asymmetric Quantum Theory.Comment: 13 page
The rigged Hilbert space approach to the Lippmann-Schwinger equation. Part II: The analytic continuation of the Lippmann-Schwinger bras and kets
The analytic continuation of the Lippmann-Schwinger bras and kets is obtained
and characterized. It is shown that the natural mathematical setting for the
analytic continuation of the solutions of the Lippmann-Schwinger equation is
the rigged Hilbert space rather than just the Hilbert space. It is also argued
that this analytic continuation entails the imposition of a time asymmetric
boundary condition upon the group time evolution, resulting into a semigroup
time evolution. Physically, the semigroup time evolution is simply a (retarded
or advanced) propagator.Comment: 32 pages, 3 figure
Statistical Mechanical Theory of a Closed Oscillating Universe
Based on Newton's laws reformulated in the Hamiltonian dynamics combined with
statistical mechanics, we formulate a statistical mechanical theory supporting
the hypothesis of a closed oscillating universe. We find that the behaviour of
the universe as a whole can be represented by a free entropic oscillator whose
lifespan is nonhomogeneous, thus implying that time is shorter or longer
according to the state of the universe itself given through its entropy. We
conclude that time reduces to the entropy production of the universe and that a
nonzero entropy production means that local fluctuations could exist giving
rise to the appearance of masses and to the curvature of the space
On the inconsistency of the Bohm-Gadella theory with quantum mechanics
The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum
Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom
asserts that the solutions of the Lippmann-Schwinger equation are functionals
over spaces of Hardy functions. The preparation-registration arrow of time
provides the physical justification for the Hardy axiom. In this paper, it is
shown that the Hardy axiom is incorrect, because the solutions of the
Lippmann-Schwinger equation do not act on spaces of Hardy functions. It is also
shown that the derivation of the preparation-registration arrow of time is
flawed. Thus, Hardy functions neither appear when we solve the
Lippmann-Schwinger equation nor they should appear. It is also shown that the
Bohm-Gadella theory does not rest on the same physical principles as quantum
mechanics, and that it does not solve any problem that quantum mechanics cannot
solve. The Bohm-Gadella theory must therefore be abandoned.Comment: 16 page
Rigged Hilbert Space Approach to the Schrodinger Equation
It is shown that the natural framework for the solutions of any Schrodinger
equation whose spectrum has a continuous part is the Rigged Hilbert Space
rather than just the Hilbert space. The difficulties of using only the Hilbert
space to handle unbounded Schrodinger Hamiltonians whose spectrum has a
continuous part are disclosed. Those difficulties are overcome by using an
appropriate Rigged Hilbert Space (RHS). The RHS is able to associate an
eigenket to each energy in the spectrum of the Hamiltonian, regardless of
whether the energy belongs to the discrete or to the continuous part of the
spectrum. The collection of eigenkets corresponding to both discrete and
continuous spectra forms a basis system that can be used to expand any physical
wave function. Thus the RHS treats discrete energies (discrete spectrum) and
scattering energies (continuous spectrum) on the same footing.Comment: 27 RevTex page
Distinguished trajectories in time dependent vector fields
We introduce a new definition of distinguished trajectory that generalises
the concepts of fixed point and periodic orbit to aperiodic dynamical systems.
This new definition is valid for identifying distinguished trajectories with
hyperbolic and non-hyperbolic types of stability. The definition is implemented
numerically and the procedure consist in determining a path of limit
coordinates. It has been successfully applied to known examples of
distinguished trajectories. In the context of highly aperiodic realistic flows
our definition characterises distinguished trajectories in finite time
intervals, and states that outside these intervals trajectories are no longer
distinguished.Comment: Chaos 19 (2009), 013111-1-013111-1
Tracking system analytic calibration activities for the Mariner Mars 1971 mission
Data covering various planning aspects of Mariner Mars 1971 mission are summarized. Data cover calibrating procedures for tracking stations, radio signal propagation in the troposphere, effects of charged particles on radio transmission, orbit calculation, and data smoothing
Inner products of resonance solutions in 1-D quantum barriers
The properties of a prescription for the inner products of the resonance
(Gamow states), scattering (Dirac kets), and bound states for 1-dimensional
quantum barriers are worked out. The divergent asypmtotic behaviour of the
Gamow states is regularized using a Gaussian convergence factor first
introduced by Zel'dovich. With this prescription, most of these states (with
discrete complex energies) are found to be orthogonal to each other, to the
bound states, and to the Dirac kets, except when they are neighbors, in which
case the inner product is divergent. Therefore, as it happens for the continuum
scattering states, the norm of the resonant ones remains non-calculable. Thus,
they exhibit properties half way between the (continuum real) Dirac-delta
orthogonality and the (discrete real) Kronecker-delta orthogonality of the
bound states.Comment: 13 pages, 2 figure
The Hilbert space of Chern-Simons theory on the cylinder. A Loop Quantum Gravity approach
As a laboratory for loop quantum gravity, we consider the canonical
quantization of the three-dimensional Chern-Simons theory on a noncompact space
with the topology of a cylinder. Working within the loop quantization
formalism, we define at the quantum level the constraints appearing in the
canonical approach and completely solve them, thus constructing a gauge and
diffeomorphism invariant physical Hilbert space for the theory. This space
turns out to be infinite dimensional, but separable.Comment: Minor changes and some references added. Latex, 16 pages, 1 figur
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