937 research outputs found
On -functions with bounded spectrum
We consider the class of functions in ,
whose Fourier transform has bounded support. We obtain a description of
continuous maps such that
for every function .
Only injective affine maps have this property
Conductance Phases in Aharonov-Bohm Ring Quantum Dots
The regimes of growing phases (for electron numbers N~0-8) that pass into
regions of self-returning phases (for N>8), found recently in quantum dot
conductances by the Weizmann group are accounted for by an elementary Green
function formalism, appropriate to an equi-spaced ladder structure (with at
least three rungs) of electronic levels in the quantum dot. The key features of
the theory are physically a dissipation rate that increases linearly with the
level number (and tentatively linked to coupling to longitudinal optical
phonons) and a set of Fano-like meta-stable levels, which disturb the
unitarity, and mathematically the change over of the position of the complex
transmission amplitude-zeros from the upper-half in the complex gap-voltage
plane to the lower half of that plane. The two regimes are identified with
(respectively) the Blaschke-term and the Kramers-Kronig integral term in the
theory of complex variables.Comment: 20 pages, 4 figure
Photon wave mechanics and position eigenvectors
One and two photon wave functions are derived by projecting the quantum state
vector onto simultaneous eigenvectors of the number operator and a recently
constructed photon position operator [Phys. Rev A 59, 954 (1999)] that couples
spin and orbital angular momentum. While only the Landau-Peierls wave function
defines a positive definite photon density, a similarity transformation to a
biorthogonal field-potential pair of positive frequency solutions of Maxwell's
equations preserves eigenvalues and expectation values. We show that this real
space description of photons is compatible with all of the usual rules of
quantum mechanics and provides a framework for understanding the relationships
amongst different forms of the photon wave function in the literature. It also
gives a quantum picture of the optical angular momentum of beams that applies
to both one photon and coherent states. According to the rules of qunatum
mechanics, this wave function gives the probability to count a photon at any
position in space.Comment: 14 pages, to be published in Phys. Rev.
Out of Equilibrium Thermal Field Theories - Finite Time after Switching on the Interaction - Wigner Transforms of Projected Functions
We study out of equilibrium thermal field theories with switching on the
interaction occurring at finite time using the Wigner transforms (in relative
space-time) of two-point functions.
For two-point functions we define the concept of projected function: it is
zero if any of times refers to the time before switching on the interaction,
otherwise it depends only on the relative coordinates. This definition includes
bare propagators, one-loop self-energies, etc. For the infinite-average-time
limit of the Wigner transforms of projected functions we define the analyticity
assumptions: (1) The function of energy is analytic above (below) the real
axis. (2) The function goes to zero as the absolute value of energy approaches
infinity in the upper (lower) semiplane.
Without use of the gradient expansion, we obtain the convolution product of
projected functions. We sum the Schwinger-Dyson series in closed form. In the
calculation of the Keldysh component (both, resummed and single self-energy
insertion approximation) contributions appear which are not the Wigner
transforms of projected functions, signaling the limitations of the method.
In the Feynman diagrams there is no explicit energy conservation at vertices,
there is an overall energy-smearing factor taking care of the uncertainty
relations.
The relation between the theories with the Keldysh time path and with the
finite time path enables one to rederive the results, such as the cancellation
of pinching, collinear, and infrared singularities, hard thermal loop
resummation, etc.Comment: 23 pages + 1 figure, Latex, corrected version, improved presentation,
version accepted for publication in Phys. Rev.
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
Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schroedinger evolution
In non relativistic quantum mechanics time enters as a parameter in the
Schroedinger equation. However, there are various situations where the need
arises to view time as a dynamical variable. In this paper we consider the
dynamical role of time through the construction of a Lyapunov variable - i.e.,
a self-adjoint quantum observable whose expectation value varies monotonically
as time increases. It is shown, in a constructive way, that a certain class of
models admit a Lyapunov variable and that the existence of a Lyapunov variable
implies the existence of a transformation mapping the original quantum
mechanical problem to an equivalent irreversible representation. In addition,
it is proved that in the irreversible representation there exists a natural
time ordering observable splitting the Hilbert space at each t>0 into past and
future subspaces.Comment: Accepted for publication in JMP. Supercedes arXiv:0710.3604.
Discussion expanded to include the case of Hamiltonians with an infinitely
degenerate spectru
Constructions of regular algebras
Criterion of (Shilov) regularity for weighted algebras on a
locally compact abelian group is known by works of Beurling (1949) and
Domar (1956). In the present paper this criterion is extended to translation
invariant weighted algebras with . Regular algebras
are constructed on any sigma-compact abelian group . It was proved earlier
by the author that sigma-compactness is necessary (in the abelian case) for the
existence of weighted algebras with .Comment: Submitted to Mat. Sborni
On the nonlinearity interpretation of q- and f-deformation and some applications
q-oscillators are associated to the simplest non-commutative example of Hopf
algebra and may be considered to be the basic building blocks for the symmetry
algebras of completely integrable theories. They may also be interpreted as a
special type of spectral nonlinearity, which may be generalized to a wider
class of f-oscillator algebras. In the framework of this nonlinear
interpretation, we discuss the structure of the stochastic process associated
to q-deformation, the role of the q-oscillator as a spectrum-generating algebra
for fast growing point spectrum, the deformation of fermion operators in
solid-state models and the charge-dependent mass of excitations in f-deformed
relativistic quantum fields.Comment: 11 pages Late
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