66,017 research outputs found
Causal loop in the theory of relative locality
Full text linkWe find that relative locality, a recently proposed Planck-scale deformation
of special relativity, suffers from the existence of causal loops. A simple and
general construction of such on-shell loop processes is studied. We then show
that even in one of the weakest deformations of the Poincar\'e group in
relative locality, causality can be violated.Comment: 5 pages, 3 figures; v3 matches the published versio
Bulk amplitude and degree of divergence in 4d spin foams
Full text linkWe study the 4-d holomorphic Spin Foam amplitude on arbitrary connected
2-complexes and degrees of divergence. With recently developed tools and
truncation scheme, we derive a formula for a certain class of graphs, which
allows us to write down the value of bulk amplitudes simply based on graph
properties. We then generalize the result to arbitrary connected 2-complexes
and extract a simple expression for the degree of divergence only in terms of
combinatorial properties and topological invariants. The distinct behaviors of
the model in different regions of parameter space signal phase transitions. In
the regime which is of physical interest for recovering diffeomorphsim symmetry
in the continuum limit, the most divergent configurations are melonic graphs.
We end with a discussion of physical implications.Comment: 25+7 pages, 10 figure
Smooth local solutions to weingarten equations and -equations
Full text linkIn this paper, we study the existence of smooth local solutions to Weingarten
equations and -equations. We will prove that, for ,
the Weingarten equations and the -equations always have smooth local
solutions regardless of the sign of the functions in the right-hand side of the
equations. We will demonstrate that the associate linearized equations are
uniformly elliptic if we choose the initial approximate solutions
appropriately
Chern-Osserman type equality for complete surfaces in R^n
Full text linkWe obtain a Chern-Osserman type equality of a complete properly immersed
surface in Euclidean space, provided the L^2-norm of the second fundamental
form is finite. Also, by using a monotonicity formula, we prove that if the
L^2-norm of mean curvature of a noncompact surface is finite, then it has at
least quadratic area growth
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