3,316 research outputs found
On Nonlocality, Lattices and Internal Symmetries
We study functional analytic aspects of two types of correction terms to the
Heisenberg algebra. One type is known to induce a finite lower bound to the resolution of distances, a short distance cutoff which is motivated
from string theory and quantum gravity. It implies the existence of families of
self-adjoint extensions of the position operators with lattices of eigenvalues.
These lattices, which form representations of certain unitary groups cannot be
resolved on the given geometry. This leads us to conjecture that, within this
framework, degrees of freedom that correspond to structure smaller than the
resolvable (Planck) scale turn into internal degrees of freedom with these
unitary groups as symmetries. The second type of correction terms is related to
the previous essentially by "Wick rotation", and its basics are here considered
for the first time. In particular, we investigate unitarily inequivalent
representations.Comment: 6 pages, LaTe
About maximally localized states in quantum mechanics
We analyze the emergence of a minimal length for a large class of generalized
commutation relations, preserving commutation of the position operators and
translation invariance as well as rotation invariance (in dimension higher than
one). We show that the construction of the maximally localized states based on
squeezed states generally fails. Rather, one must resort to a constrained
variational principle.Comment: accepted for publication in PR
Nonpointlike Particles in Harmonic Oscillators
Quantum mechanics ordinarily describes particles as being pointlike, in the
sense that the uncertainty can, in principle, be made arbitrarily
small. It has been shown that suitable correction terms to the canonical
commutation relations induce a finite lower bound to spatial localisation.
Here, we perturbatively calculate the corrections to the energy levels of an in
this sense nonpointlike particle in isotropic harmonic oscillators. Apart from
a special case the degeneracy of the energy levels is removed.Comment: LaTeX, 9 pages, 1 figure included via epsf optio
Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta
A noncommutative geometric generalisation of the quantum field theoretical
framework is developed by generalising the Heisenberg commutation relations.
There appear nonzero minimal uncertainties in positions and in momenta. As the
main result it is shown with the example of a quadratically ultraviolet
divergent graph in theory that nonzero minimal uncertainties in
positions do have the power to regularise. These studies are motivated with the
ansatz that nonzero minimal uncertainties in positions and in momenta arise
from gravity. Algebraic techniques are used that have been developed in the
field of quantum groups.Comment: 52 pages LATEX, DAMTP/93-33. Revised version now includes a chapter
on the Poincare algebra and curvature as noncommutativity of momentum spac
Analysis of Superoscillatory Wave Functions
Surprisingly, differentiable functions are able to oscillate arbitrarily
faster than their highest Fourier component would suggest. The phenomenon is
called superoscillation. Recently, a practical method for calculating
superoscillatory functions was presented and it was shown that superoscillatory
quantum mechanical wave functions should exhibit a number of counter-intuitive
physical effects. Following up on this work, we here present more general
methods which allow the calculation of superoscillatory wave functions with
custom-designed physical properties. We give concrete examples and we prove
results about the limits to superoscillatory behavior. We also give a simple
and intuitive new explanation for the exponential computational cost of
superoscillations.Comment: 20 pages, several figure
Unsharp Degrees of Freedom and the Generating of Symmetries
In quantum theory, real degrees of freedom are usually described by operators
which are self-adjoint. There are, however, exceptions to the rule. This is
because, in infinite dimensional Hilbert spaces, an operator is not necessarily
self-adjoint even if its expectation values are real. Instead, the operator may
be merely symmetric. Such operators are not diagonalizable - and as a
consequence they describe real degrees of freedom which display a form of
"unsharpness" or "fuzzyness". For example, there are indications that this type
of operators could arise with the description of space-time at the string or at
the Planck scale, where some form of unsharpness or fuzzyness has long been
conjectured.
A priori, however, a potential problem with merely symmetric operators is the
fact that, unlike self-adjoint operators, they do not generate unitaries - at
least not straightforwardly. Here, we show for a large class of these operators
that they do generate unitaries in a well defined way, and that these operators
even generate the entire unitary group of the Hilbert space. This shows that
merely symmetric operators, in addition to describing unsharp physical
entities, may indeed also play a r{\^o}le in the generation of symmetries, e.g.
within a fundamental theory of quantum gravity.Comment: 23 pages, LaTe
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