197,789 research outputs found
Renormalization in quantum field theory and the Riemann-Hilbert problem II: the -function, diffeomorphisms and the renormalization group
We showed in part I (hep-th/9912092) that the Hopf algebra of
Feynman graphs in a given QFT is the algebra of coordinates on a complex
infinite dimensional Lie group and that the renormalized theory is obtained
from the unrenormalized one by evaluating at \ve=0 the holomorphic part
\gamma_+(\ve) of the Riemann-Hilbert decomposition
\gamma_-(\ve)^{-1}\gamma_+(\ve) of the loop \gamma(\ve)\in G provided by
dimensional regularization. We show in this paper that the group acts
naturally on the complex space of dimensionless coupling constants of the
theory. More precisely, the formula for the effective
coupling constant, when viewed as a formal power series, does define a Hopf
algebra homomorphism between the Hopf algebra of coordinates on the group of
formal diffeomorphisms to the Hopf algebra . This allows first of all
to read off directly, without using the group , the bare coupling constant
and the renormalized one from the Riemann-Hilbert decomposition of the
unrenormalized effective coupling constant viewed as a loop of formal
diffeomorphisms. This shows that renormalization is intimately related with the
theory of non-linear complex bundles on the Riemann sphere of the dimensional
regularization parameter \ve. It also allows to lift both the renormalization
group and the -function as the asymptotic scaling in the group . This
exploits the full power of the Riemann-Hilbert decomposition together with the
invariance of \gamma_-(\ve) under a change of unit of mass. This not only
gives a conceptual proof of the existence of the renormalization group but also
delivers a scattering formula in the group for the full higher pole
structure of minimal subtracted counterterms in terms of the residue.Comment: 35 pages, eps figure
Specific "scientific" data structures, and their processing
Programming physicists use, as all programmers, arrays, lists, tuples,
records, etc., and this requires some change in their thought patterns while
converting their formulae into some code, since the "data structures" operated
upon, while elaborating some theory and its consequences, are rather: power
series and Pad\'e approximants, differential forms and other instances of
differential algebras, functionals (for the variational calculus), trajectories
(solutions of differential equations), Young diagrams and Feynman graphs, etc.
Such data is often used in a [semi-]numerical setting, not necessarily
"symbolic", appropriate for the computer algebra packages. Modules adapted to
such data may be "just libraries", but often they become specific, embedded
sub-languages, typically mapped into object-oriented frameworks, with
overloaded mathematical operations. Here we present a functional approach to
this philosophy. We show how the usage of Haskell datatypes and - fundamental
for our tutorial - the application of lazy evaluation makes it possible to
operate upon such data (in particular: the "infinite" sequences) in a natural
and comfortable manner.Comment: In Proceedings DSL 2011, arXiv:1109.032
Power constructs and propositional systems
Bibliography : p. 161-176.Propositional systems are deductively closed sets of sentences phrased in the language of some propositional logic. The set of systems of a given logic is turned into an algebra by endowing it with a number of operations, and into a relational structure by endowing it with a number of relations. Certain operations and relations on systems arise from some corresponding base operation or relation, either on sentences in the logic or on propositional valuations. These operations and relations on systems are called power constructs. The aim of this thesis is to investigate the use of power constructs in propositional systems. Some operations and relations on systems that arise as power constructs include the Tarskian addition and product operations, the contraction and revision operations of theory change, certain multiple- conclusion consequence relations, and certain relations of verisimilitude and simulation. The logical framework for this investigation is provided by the deïŹnition and comparison of a number of multiple-conclusion logics, including a paraconsistent three-valued logic of partial knowledge
Wave-Function renormalization and the Hopf algebra of Connes and Kreimer
In this talk, we show how the Connes-Kreimer Hopf algebra morphism can be
extended when taking into account the wave-function renormalization. This leads
us to a semi-direct product of invertible power series by formal
diffeomorphisms.Comment: 5 pages, no figure, talk presented in the conference "Brane New World
and Noncommutative Geometry", Torino, Villa Gualino,(Italy) Octobe
Point particle in general background fields vs. free gauge theories of traceless symmetric tensors
Point particle may interact to traceless symmetric tensors of arbitrary rank.
Free gauge theories of traceless symmetric tensors are constructed, that
provides a possibility for a new type of interactions, when particles exchange
by those gauge fields. The gauge theories are parameterized by the particle's
mass m and otherwise are unique for each rank s. For m=0, they are local gauge
models with actions of 2s-th order in derivatives, known in d=4 as "pure spin",
or "conformal higher spin" actions by Fradkin and Tseytlin. For nonzero m, each
rank-s model undergoes a unique nonlocal deformation which entangles fields of
all ranks, starting from s. There exists a nonlocal transform which maps m > 0
theories onto m=0 ones, however, this map degenerates at some m > 0 fields
whose polarizations are determined by zeros of Bessel functions. Conformal
covariance properties of the m=0 models are analyzed, the space of gauge fields
is shown to admit an action of an infinite-dimensional "conformal higher spin"
Lie algebra which leaves gauge transformations intact.Comment: 21 pages, remarks on nonlinear generalization added, a mistake in the
discussion of degenerate solutions correcte
Transitive Lie algebras of vector fields---an overview
This overview paper is intended as a quick introduction to Lie algebras of
vector fields. Originally introduced in the late 19th century by Sophus Lie to
capture symmetries of ordinary differential equations, these algebras, or
infinitesimal groups, are a recurring theme in 20th-century research on Lie
algebras. I will focus on so-called transitive or even primitive Lie algebras,
and explain their theory due to Lie, Morozov, Dynkin, Guillemin, Sternberg,
Blattner, and others. This paper gives just one, subjective overview of the
subject, without trying to be exhaustive.Comment: 20 pages, written after the Oberwolfach mini-workshop "Algebraic and
Analytic Techniques for Polynomial Vector Fields", December 2010 2nd version,
some minor typo's corrected and some references adde
An algebraic Birkhoff decomposition for the continuous renormalization group
This paper aims at presenting the first steps towards a formulation of the
Exact Renormalization Group Equation in the Hopf algebra setting of Connes and
Kreimer. It mostly deals with some algebraic preliminaries allowing to
formulate perturbative renormalization within the theory of differential
equations. The relation between renormalization, formulated as a change of
boundary condition for a differential equation, and an algebraic Birkhoff
decomposition for rooted trees is explicited
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