137 research outputs found
Quantum Superposition Principle and Geometry
If one takes seriously the postulate of quantum mechanics in which physical
states are rays in the standard Hilbert space of the theory, one is naturally
lead to a geometric formulation of the theory. Within this formulation of
quantum mechanics, the resulting description is very elegant from the
geometrical viewpoint, since it allows to cast the main postulates of the
theory in terms of two geometric structures, namely a symplectic structure and
a Riemannian metric. However, the usual superposition principle of quantum
mechanics is not naturally incorporated, since the quantum state space is
non-linear. In this note we offer some steps to incorporate the superposition
principle within the geometric description. In this respect, we argue that it
is necessary to make the distinction between a 'projective superposition
principle' and a 'decomposition principle' that extend the standard
superposition principle. We illustrate our proposal with two very well known
examples, namely the spin 1/2 system and the two slit experiment, where the
distinction is clear from the physical perspective. We show that the two
principles have also a different mathematical origin within the geometrical
formulation of the theory.Comment: 10 pages, no figures. References added. V3 discussion expanded and
new results added, 14 pages. Dedicated to Michael P. Ryan on the occasion of
his sixtieth bithda
Transverse Invariant Higher Spin Fields
It is shown that a symmetric massless bosonic higher-spin field can be
described by a traceless tensor field with reduced (transverse) gauge
invariance. The Hamiltonian analysis of the transverse gauge invariant
higher-spin models is used to control a number of degrees of freedom.Comment: 12 pages, no figures. The general proof and the example of a spin-3
adde
A time-space varying speed of light and the Hubble Law in static Universe
We consider a hypothetical possibility of the variability of light velocity
with time and position in space which is derived from two natural postulates.
For the consistent consideration of such variability we generalize
translational transformations of the Theory of Relativity. The formulae of
transformations between two rest observers within one inertial system are
obtained. It is shown that equality of velocities of two particles is as
relative a statement as simultaneity of two events is. We obtain the expression
for the redshift of radiation of a rest source which formally reproduces the
Hubble Law. Possible experimental implications of the theory are discussed.Comment: 7 page
Effective constraint potential in lattice Weinberg - Salam model
We investigate lattice Weinberg - Salam model without fermions for the value
of the Weinberg angle , and bare fine structure constant
around . We consider the value of the scalar self coupling
corresponding to bare Higgs mass around 150 GeV. The effective constraint
potential for the zero momentum scalar field is used in order to investigate
phenomena existing in the vicinity of the phase transition between the physical
Higgs phase and the unphysical symmetric phase of the lattice model. This is
the region of the phase diagram, where the continuum physics is to be
approached. We compare the above mentioned effective potential (calculated in
selected gauges) with the effective potential for the value of the scalar field
at a fixed space - time point. We also calculate the renormalized fine
structure constant using the correlator of Polyakov lines and compare it with
the one - loop perturbative estimate.Comment: LATE
Planar Dirac Electron in Coulomb and Magnetic Fields
The Dirac equation for an electron in two spatial dimensions in the Coulomb
and homogeneous magnetic fields is discussed. For weak magnetic fields, the
approximate energy values are obtained by semiclassical method. In the case
with strong magnetic fields, we present the exact recursion relations that
determine the coefficients of the series expansion of wave functions, the
possible energies and the magnetic fields. It is found that analytic solutions
are possible for a denumerably infinite set of magnetic field strengths. This
system thus furnishes an example of the so-called quasi-exactly solvable
models. A distinctive feature in the Dirac case is that, depending on the
strength of the Coulomb field, not all total angular momentum quantum number
allow exact solutions with wavefunctions in reasonable polynomial forms.
Solutions in the nonrelativistic limit with both attractive and repulsive
Coulomb fields are briefly discussed by means of the method of factorization.Comment: 18 pages, RevTex, no figure
Comments on "Note on varying speed of light theories"
In a recent note Ellis criticizes varying speed of light theories on the
grounds of a number of foundational issues. His reflections provide us with an
opportunity to clarify some fundamental matters pertaining to these theories
Cosmological perturbations in FRW model with scalar field within Hamilton-Jacobi formalism and symplectic projector method
The Hamilton-Jacobi analysis is applied to the dynamics of the scalar
fluctuations about the Friedmann-Robertson-Walker (FRW). The gauge conditions
are found from the consistency conditions. The physical degrees of freedom of
the model are obtain by symplectic projector method. The role of the linearly
dependent Hamiltonians and the gauge variables in Hamilton-Jacobi formalism is
discussed.Comment: 11 page
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
Eikonal Approximation to 5D Wave Equations as Geodesic Motion in a Curved 4D Spacetime
We first derive the relation between the eikonal approximation to the Maxwell
wave equations in an inhomogeneous anisotropic medium and geodesic motion in a
three dimensional Riemannian manifold using a method which identifies the
symplectic structure of the corresponding mechanics. We then apply an analogous
method to the five dimensional generalization of Maxwell theory required by the
gauge invariance of Stueckelberg's covariant classical and quantum dynamics to
demonstrate, in the eikonal approximation, the existence of geodesic motion for
the flow of mass in a four dimensional pseudo-Riemannian manifold. These
results provide a foundation for the geometrical optics of the five dimensional
radiation theory and establish a model in which there is mass flow along
geodesics. Finally we discuss the case of relativistic quantum theory in an
anisotropic medium as well. In this case the eikonal approximation to the
relativistic quantum mechanical current coincides with the geodesic flow
governed by the pseudo-Riemannian metric obtained from the eikonal
approximation to solutions of the Stueckelberg-Schr\"odinger equation. This
construction provides a model for an underlying quantum mechanical structure
for classical dynamical motion along geodesics on a pseudo-Riemannian manifold.
The locally symplectic structure which emerges is that of Stueckelberg's
covariant mechanics on this manifold.Comment: TeX file. 17 pages. Rewritten for clarit
Quantization on a 2-dimensional phase space with a constant curvature tensor
Some properties of the star product of the Weyl type (i.e. associated with
the Weyl ordering) are proved. Fedosov construction of the *-product on a
2-dimensional phase spacewith a constant curvature tensor is presented.
Eigenvalue equations for momentum p and position q on a 2-dimensional phase
space with constant curvature tensors are solved.Comment: 33 pages, LaTeX, Annals of Physics (2003
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