344,828 research outputs found
The Red Queen visits Minkowski Space
When Alice went `Through the Looking Glass' [1], she found herself in a
situation where she had to run as fast as she could in order to stay still. In
accordance with the dictum that truth is stranger than fiction, we will see
that it is possible to find a situation in special relativity where running
towards one's target is actually counter-productive. Although the situation is
easily analysed algebraically, the qualitative properties of the analysis are
greatly illuminated by the use of space-time diagrams
Gauge-Invariant Differential Renormalization: Abelian Case
A new version of differential renormalization is presented. It is based on
pulling out certain differential operators and introducing a logarithmic
dependence into diagrams. It can be defined either in coordinate or momentum
space, the latter being more flexible for treating tadpoles and diagrams where
insertion of counterterms generates tadpoles. Within this version, gauge
invariance is automatically preserved to all orders in Abelian case. Since
differential renormalization is a strictly four-dimensional renormalization
scheme it looks preferable for application in each situation when dimensional
renormalization meets difficulties, especially, in theories with chiral and
super symmetries. The calculation of the ABJ triangle anomaly is given as an
example to demonstrate simplicity of calculations within the presented version
of differential renormalization.Comment: 15 pages, late
Decreasing Diagrams for Confluence and Commutation
Like termination, confluence is a central property of rewrite systems. Unlike
for termination, however, there exists no known complexity hierarchy for
confluence. In this paper we investigate whether the decreasing diagrams
technique can be used to obtain such a hierarchy. The decreasing diagrams
technique is one of the strongest and most versatile methods for proving
confluence of abstract rewrite systems. It is complete for countable systems,
and it has many well-known confluence criteria as corollaries.
So what makes decreasing diagrams so powerful? In contrast to other
confluence techniques, decreasing diagrams employ a labelling of the steps with
labels from a well-founded order in order to conclude confluence of the
underlying unlabelled relation. Hence it is natural to ask how the size of the
label set influences the strength of the technique. In particular, what class
of abstract rewrite systems can be proven confluent using decreasing diagrams
restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find
that two labels suffice for proving confluence for every abstract rewrite
system having the cofinality property, thus in particular for every confluent,
countable system.
Secondly, we show that this result stands in sharp contrast to the situation
for commutation of rewrite relations, where the hierarchy does not collapse.
Thirdly, investigating the possibility of a confluence hierarchy, we
determine the first-order (non-)definability of the notion of confluence and
related properties, using techniques from finite model theory. We find that in
particular Hanf's theorem is fruitful for elegant proofs of undefinability of
properties of abstract rewrite systems
Boundary Holographic Witten Diagrams
In this paper we discuss geodesic Witten diagrams in generic holographic
conformal field theories with boundary or defect. Boundary CFTs allow two
different decompositions of two-point functions into conformal blocks: boundary
channel and ambient channel. Building on earlier work, we derive a holographic
dual of the boundary channel decomposition in terms of bulk-to-bulk propagators
on lower dimensional AdS slices. In the situation in which we can treat the
boundary or defect as a perturbation around pure AdS spacetime, we obtain the
leading corrections to the two-point function both in boundary and ambient
channel in terms of geodesic Witten diagrams which exactly reproduce the
decomposition into corresponding conformal blocks on the field theory side.Comment: 28 pages, 4 figures, v2:included hypergeometric identities for
generic no-brane case, references added, v3:published versio
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