2,355 research outputs found
On Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta
We continue studies on quantum field theories on noncommutative geometric
spaces, focusing on classes of noncommutative geometries which imply
ultraviolet and infrared modifications in the form of nonzero minimal
uncertainties in positions and momenta. The case of the ultraviolet modified
uncertainty relation which has appeared from string theory and quantum gravity
is covered. The example of euclidean -theory is studied in detail and
in this example we can now show ultraviolet and infrared regularisation of all
graphs.Comment: LaTex, 32 page
Maximal Localisation in the Presence of Minimal Uncertainties in Positions and Momenta
Small corrections to the uncertainty relations, with effects in the
ultraviolet and/or infrared, have been discussed in the context of string
theory and quantum gravity. Such corrections lead to small but finite minimal
uncertainties in position and/or momentum measurements. It has been shown that
these effects could indeed provide natural cutoffs in quantum field theory. The
corresponding underlying quantum theoretical framework includes small
`noncommutative geometric' corrections to the canonical commutation relations.
In order to study the full implications on the concept of locality it is
crucial to find the physical states of then maximal localisation. These states
and their properties have been calculated for the case with minimal
uncertainties in positions only. Here we extend this treatment, though still in
one dimension, to the general situation with minimal uncertainties both in
positions and in momenta.Comment: Latex, 21 pages, 2 postscript figure
Mimimal Length Uncertainty Principle and the Transplanckian Problem of Black Hole Physics
The minimal length uncertainty principle of Kempf, Mangano and Mann (KMM), as
derived from a mutilated quantum commutator between coordinate and momentum, is
applied to describe the modes and wave packets of Hawking particles evaporated
from a black hole. The transplanckian problem is successfully confronted in
that the Hawking particle no longer hugs the horizon at arbitrarily close
distances. Rather the mode of Schwarzschild frequency deviates from
the conventional trajectory when the coordinate is given by in units of the non local distance legislated
into the uncertainty relation. Wave packets straddle the horizon and spread out
to fill the whole non local region. The charge carried by the packet (in the
sense of the amount of "stuff" carried by the Klein--Gordon field) is not
conserved in the non--local region and rapidly decreases to zero as time
decreases. Read in the forward temporal direction, the non--local region thus
is the seat of production of the Hawking particle and its partner. The KMM
model was inspired by string theory for which the mutilated commutator has been
proposed to describe an effective theory of high momentum scattering of zero
mass modes. It is here interpreted in terms of dissipation which gives rise to
the Hawking particle into a reservoir of other modes (of as yet unknown
origin). On this basis it is conjectured that the Bekenstein--Hawking entropy
finds its origin in the fluctuations of fields extending over the non local
region.Comment: 12 pages (LateX), 1 figur
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
On Fields with Finite Information Density
The existence of a natural ultraviolet cutoff at the Planck scale is widely
expected. In a previous Letter, it has been proposed to model this cutoff as an
information density bound by utilizing suitably generalized methods from the
mathematical theory of communication. Here, we prove the mathematical
conjectures that were made in this Letter.Comment: 31 pages, to appear in Phys.Rev.
Perturbation spectrum in inflation with cutoff
It has been pointed out that the perturbation spectrum predicted by inflation
may be sensitive to a natural ultraviolet cutoff, thus potentially providing an
experimentally accessible window to aspects of Planck scale physics. A priori,
a natural ultraviolet cutoff could take any form, but a fairly general
classification of possible Planck scale cutoffs has been given. One of those
categorized cutoffs, also appearing in various studies of quantum gravity and
string theory, has recently been implemented into the standard inflationary
scenario. Here, we continue this approach by investigating its effects on the
predicted perturbation spectrum. We find that the size of the effect depends
sensitively on the scale separation between cutoff and horizon during
inflation.Comment: 6 pages; matches version accepted by PR
Uncertainty Relation in Quantum Mechanics with Quantum Group Symmetry
We study the commutation relations, uncertainty relations and spectra of
position and momentum operators within the framework of quantum group %
symmetric Heisenberg algebras and their (Bargmann-) Fock representations. As an
effect of the underlying noncommutative geometry, a length and a momentum scale
appear, leading to the existence of minimal nonzero uncertainties in the
positions and momenta. The usual quantum mechanical behaviour is recovered as a
limiting case for not too small and not too large distances and momenta.Comment: 15 pages, Latex, preprint DAMTP/93-6
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