22,148 research outputs found
An Algebra of Hierarchical Graphs and its Application to Structural Encoding
We define an algebraic theory of hierarchical graphs, whose axioms
characterise graph isomorphism: two terms are equated exactly when
they represent the same graph. Our algebra can be understood as
a high-level language for describing graphs with a node-sharing, embedding
structure, and it is then well suited for defining graphical
representations of software models where nesting and linking are key
aspects. In particular, we propose the use of our graph formalism as a
convenient way to describe configurations in process calculi equipped
with inherently hierarchical features such as sessions, locations, transactions,
membranes or ambients. The graph syntax can be seen as an
intermediate representation language, that facilitates the encodings of
algebraic specifications, since it provides primitives for nesting, name
restriction and parallel composition. In addition, proving soundness
and correctness of an encoding (i.e. proving that structurally equivalent
processes are mapped to isomorphic graphs) becomes easier as it can
be done by induction over the graph syntax
Trees
An algebraic formalism, developped with V. Glaser and R. Stora for the study
of the generalized retarded functions of quantum field theory, is used to prove
a factorization theorem which provides a complete description of the
generalized retarded functions associated with any tree graph. Integrating over
the variables associated to internal vertices to obtain the perturbative
generalized retarded functions for interacting fields arising from such graphs
is shown to be possible for a large category of space-times.Comment: minor corrections, references added, no change in result
Direct computation of scattering matrices for general quantum graphs
We present a direct and simple method for the computation of the total
scattering matrix of an arbitrary finite noncompact connected quantum graph
given its metric structure and local scattering data at each vertex. The method
is inspired by the formalism of Reflection-Transmission algebras and quantum
field theory on graphs though the results hold independently of this formalism.
It yields a simple and direct algebraic derivation of the formula for the total
scattering and has a number of advantages compared to existing recursive
methods. The case of loops (or tadpoles) is easily incorporated in our method.
This provides an extension of recent similar results obtained in a completely
different way in the context of abstract graph theory. It also allows us to
discuss briefly the inverse scattering problem in the presence of loops using
an explicit example to show that the solution is not unique in general. On top
of being conceptually very easy, the computational advantage of the method is
illustrated on two examples of "three-dimensional" graphs (tetrahedron and
cube) for which other methods are rather heavy or even impractical.Comment: 20 pages, 4 figure
Lieb-Robinson bounds with dependence on interaction strengths
We propose new Lieb-Robinson bounds (bounds on the speed of propagation of
information in quantum systems) with an explicit dependence on the interaction
strengths of the Hamiltonian. For systems with more than two interactions it is
found that the Lieb-Robinson speed is not always algebraic in the interaction
strengths. We consider Hamiltonians with any finite number of bounded operators
and also a certain class of unbounded operators. We obtain bounds and
propagation speeds for quantum systems on lattices and also general graphs
possessing a kind of homogeneity and isotropy. One area for which this
formalism could be useful is the study of quantum phase transitions which occur
when interactions strengths are varied.Comment: 19 pages, 1 figure, minor modification
Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications
These notes are intended to provide a self-contained introduction to the
basic ideas of finite dimensional Batalin-Vilkovisky (BV) formalism and its
applications. A brief exposition of super- and graded geometries is also given.
The BV-formalism is introduced through an odd Fourier transform and the
algebraic aspects of integration theory are stressed. As a main application we
consider the perturbation theory for certain finite dimensional integrals
within BV-formalism. As an illustration we present a proof of the isomorphism
between the graph complex and the Chevalley-Eilenberg complex of formal
Hamiltonian vectors fields. We briefly discuss how these ideas can be extended
to the infinite dimensional setting. These notes should be accessible to both
physicists and mathematicians.Comment: 67 pages, typos corrected, published versio
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