92 research outputs found
Automatic regularization by quantization in reducible representations of CCR: Point-form quantum optics with classical sources
Electromagnetic fields are quantized in manifestly covariant way by means of
a class of reducible representations of CCR. transforms as a Hermitian
four-vector field in Minkowski four-position space (no change of gauge), but in
momentum space it splits into spin-1 massless photons (optics) and two massless
scalars (similar to dark matter). Unitary dynamics is given by point-form
interaction picture, with minimal-coupling Hamiltonian constructed from fields
that are free on the null-cone boundary of the Milne universe. SL(2,C)
transformations and dynamics are represented unitarily in positive-norm Hilbert
space describing four-dimensional oscillators. Vacuum is a Bose-Einstein
condensate of the -oscillator gas. Both the form of and its
transformation properties are determined by an analogue of the twistor
equation. The same equation guarantees that the subspace of vacuum states is,
as a whole, Poincar\'e invariant. The formalism is tested on quantum fields
produced by pointlike classical sources. Photon statistics is well defined even
for pointlike charges, with UV/IR regularizations occurring automatically as a
consequence of the formalism. The probabilities are not Poissonian but of a
R\'enyi type with . The average number of photons occurring in
Bremsstrahlung splits into two parts: The one due to acceleration, and the one
that remains nonzero even if motion is inertial. Classical Maxwell
electrodynamics is reconstructed from coherent-state averaged solutions of
Heisenberg equations. Static pointlike charges polarize vacuum and produce
effective charge densities and fields whose form is sensitive to both the
choice of representation of CCR and the corresponding vacuum state.Comment: 2 eps figures; in v2 notation in Eq. (39) and above Eq. (38) is
correcte
Complete positivity of nonlinear evolution: A case study
Simple Hartree-type equations lead to dynamics of a subsystem that is not
completely positive in the sense accepted in mathematical literature. In the
linear case this would imply that negative probabilities have to appear for
some system that contains the subsystem in question. In the nonlinear case this
does not happen because the mathematical definition is physically unfitting as
shown on a concrete example.Comment: extended version, 3 appendices added (on mixed states, projection
postulate, nonlocality), to be published in Phys. Rev.
Quantum feedback with weak measurements
The problem of feedback control of quantum systems by means of weak
measurements is investigated in detail. When weak measurements are made on a
set of identical quantum systems, the single-system density matrix can be
determined to a high degree of accuracy while affecting each system only
slightly. If this information is fed back into the systems by coherent
operations, the single-system density matrix can be made to undergo an
arbitrary nonlinear dynamics, including for example a dynamics governed by a
nonlinear Schr\"odinger equation. We investigate the implications of such
nonlinear quantum dynamics for various problems in quantum control and quantum
information theory, including quantum computation. The nonlinear dynamics
induced by weak quantum feedback could be used to create a novel form of
quantum chaos in which the time evolution of the single-system wave function
depends sensitively on initial conditions.Comment: 11 pages, TeX, replaced to incorporate suggestions of Asher Pere
On classical models of spin
The reason for recalling this old paper is the ongoing discussion on the
attempts of circumventing certain assumptions leading to the Bell theorem
(Hess-Philipp, Accardi). If I correctly understand the intentions of these
Authors, the idea is to make use of the following logical loophole inherent in
the proof of the Bell theorem: Probabilities of counterfactual events A and A'
do not have to coincide with actually measured probabilities if measurements of
A and A' disturb each other, or for any other fundamental reason cannot be
performed simulaneously. It is generally believed that in the context of
classical probability theory (i.e. realistic hidden variables) probabilities of
counterfactual events can be identified with those of actually measured events.
In the paper I give an explicit counterexample to this belief. The "first
variation" on the Aerts model shows that counterfactual and actual problems
formulated for the same classical system may be unrelated. In the model the
first probability does not violate any classical inequality whereas the second
does. Pecularity of the Bell inequality is that on the basis of an in principle
unobservable probability one derives probabilities of jointly measurable random
variables, the fact additionally obscuring the logical meaning of the
construction. The existence of the loophole does not change the fact that I was
not able to construct a local model violating the inequality with all the other
loopholes eliminated.Comment: published as Found. Phys. Lett. 3 (1992) 24
Microscopic Foundation of Nonextensive Statistics
Combination of the Liouville equation with the q-averaged energy leads to a microscopic framework for nonextensive q-thermodynamics. The
resulting von Neumann equation is nonlinear: . In spite
of its nonlinearity the dynamics is consistent with linear quantum mechanics of
pure states. The free energy is a stability function for the
dynamics. This implies that q-equilibrium states are dynamically stable. The
(microscopic) evolution of is reversible for any q, but for
the corresponding macroscopic dynamics is irreversible.Comment: revte
Noncanonical quantum optics
Modification of the right-hand-side of canonical commutation relations (CCR)
naturally occurs if one considers a harmonic oscillator with indefinite
frequency. Quantization of electromagnetic field by means of such a non-CCR
algebra naturally removes the infinite energy of vacuum but still results in a
theory which is very similar to quantum electrodynamics. An analysis of
perturbation theory shows that the non-canonical theory has an automatically
built-in cut-off but requires charge/mass renormalization already at the
nonrelativistic level. A simple rule allowing to compare perturbative
predictions of canonical and non-canonical theories is given. The notion of a
unique vacuum state is replaced by a set of different vacua. Multi-photon
states are defined in the standard way but depend on the choice of vacuum.
Making a simplified choice of the vacuum state we estimate corrections to
atomic lifetimes, probabilities of multiphoton spontaneous and stimulated
emission, and the Planck law. The results are practically identical to the
standard ones. Two different candidates for a free-field Hamiltonian are
compared.Comment: Completely rewritten version of quant-ph/0002003v2. There are
overlaps between the papers, but sections on perturbative calculations show
the same problem from different sides, therefore quant-ph/0002003v2 is not
replace
Speed dependent polarization correlations in QED and entanglement
Exact computations of polarizations correlations probabilities are carried
out in QED, to the leading order, for initially polarized as well as
unpolarized particles. Quite generally they are found to be speed dependent and
are in clear violation of Bells inequality of Local Hidden Variables (LHV)
theories. This dynamical analysis shows how speed dependent entangled states
are generated. These computations, based on QED are expected to lead to new
experiments on polarization correlations monitoring speed in the light of Bells
theorem. The paper provides a full QED treatment of the dynamics of
entanglement.Comment: LaTeX, 14 pages, 2 figures, Corrected typo
Nonlocal looking equations can make nonlinear quantum dynamics local
A general method for extending a non-dissipative nonlinear Schr\"odinger and
Liouville-von Neumann 1-particle dynamics to an arbitrary number of particles
is described. It is shown at a general level that the dynamics so obtained is
completely separable, which is the strongest condition one can impose on
dynamics of composite systems. It requires that for all initial states
(entangled or not) a subsystem not only cannot be influenced by any action
undertaken by an observer in a separated system (strong separability), but
additionally that the self-consistency condition is fulfilled. It is shown that a correct
extension to particles involves integro-differential equations which, in
spite of their nonlocal appearance, make the theory fully local. As a
consequence a much larger class of nonlinearities satisfying the complete
separability condition is allowed than has been assumed so far. In particular
all nonlinearities of the form are acceptable. This shows that
the locality condition does not single out logarithmic or 1-homeogeneous
nonlinearities.Comment: revtex, final version, accepted in Phys.Rev.A (June 1998
Regularization as quantization in reducible representations of CCR
A covariant quantization scheme employing reducible representations of
canonical commutation relations with positive-definite metric and Hermitian
four-potentials is tested on the example of quantum electrodynamic fields
produced by a classical current. The scheme implies a modified but very
physically looking Hamiltonian. We solve Heisenberg equations of motion and
compute photon statistics. Poisson statistics naturally occurs and no infrared
divergence is found even for pointlike sources. Classical fields produced by
classical sources can be obtained if one computes coherent-state averages of
Heisenberg-picture operators. It is shown that the new form of representation
automatically smears out pointlike currents. We discuss in detail Poincar\'e
covariance of the theory and the role of Bogoliubov transformations for the
issue of gauge invariance. The representation we employ is parametrized by a
number that is related to R\'enyi's . It is shown that the ``Shannon
limit" plays here a role of correspondence principle with the
standard regularized formalism.Comment: minor extensions, version submitted for publicatio
Stationary Solutions of Liouville Equations for Non-Hamiltonian Systems
We consider the class of non-Hamiltonian and dissipative statistical systems
with distributions that are determined by the Hamiltonian. The distributions
are derived analytically as stationary solutions of the Liouville equation for
non-Hamiltonian systems. The class of non-Hamiltonian systems can be described
by a non-holonomic (non-integrable) constraint: the velocity of the elementary
phase volume change is directly proportional to the power of non-potential
forces. The coefficient of this proportionality is determined by Hamiltonian.
The constant temperature systems, canonical-dissipative systems, and Fermi-Bose
classical systems are the special cases of this class of non-Hamiltonian
systems.Comment: 22 page
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