296 research outputs found
Tensor fields of mixed Young symmetry type and N-complexes
We construct -complexes of non completely antisymmetric irreducible tensor
fields on which generalize the usual complex of
differential forms. Although, for , the generalized cohomology of
these -complexes is non trivial, we prove a generalization of the Poincar\'e
lemma. To that end we use a technique reminiscent of the Green ansatz for
parastatistics. Several results which appeared in various contexts are shown to
be particular cases of this generalized Poincar\'e lemma. We furthermore
identify the nontrivial part of the generalized cohomology. Many of the results
presented here were announced in [10].Comment: 47 page
Gravitation as a Plastic Distortion of the Lorentz Vacuum
In this paper we present a theory of the gravitational field where this field
(a kind of square root of g) is represented by a (1,1)-extensor field h
describing a plastic distortion of the Lorentz vacuum (a real substance that
lives in a Minkowski spacetime) due to the presence of matter. The field h
distorts the Minkowski metric extensor in an appropriate way (see below)
generating what may be interpreted as an effective Lorentzian metric extensor g
and also it permits the introduction of different kinds of parallelism rules on
the world manifold, which may be interpreted as distortions of the parallelism
structure of Minkowski spacetime and which may have non null curvature and/or
torsion and/or nonmetricity tensors. We thus have different possible effective
geometries which may be associated to the gravitational field and thus its
description by a Lorentzian geometry is only a possibility, not an imposition
from Nature. Moreover, we developed with enough details the theory of multiform
functions and multiform functionals that permitted us to successfully write a
Lagrangian for h and to obtain its equations of motion, that results equivalent
to Einstein field equations of General Relativity (for all those solutions
where the manifold M is diffeomorphic to R^4. However, in our theory,
differently from the case of General Relativity, trustful energy-momentum and
angular momentum conservation laws exist. We express also the results of our
theory in terms of the gravitational potential 1-form fields (living in
Minkowski spacetime) in order to have results which may be easily expressed
with the theory of differential forms. The Hamiltonian formalism for our theory
(formulated in terms of the potentials) is also discussed. The paper contains
also several important Appendices that complete the material in the main text.Comment: Misprints and typos have been corrected, Chapter 7 have been
improved. Appendix E has been reformulated and Appendix F contains new
remarks which resulted from a discussion with A. Lasenby. A somewhat modified
version has been published in the Springer Series: Fundamental Theories of
Physics vol. 168, 2010. http://www.ime.unicamp.br/~walrod/plastic2014.pd
Dynamics of Higher Spin Fields and Tensorial Space
The structure and the dynamics of massless higher spin fields in various
dimensions are reviewed with an emphasis on conformally invariant higher spin
fields. We show that in D=3,4,6 and 10 dimensional space-time the conformal
higher spin fields constitute the quantum spectrum of a twistor-like particle
propagating in tensorial spaces of corresponding dimensions. We give a detailed
analysis of the field equations of the model and establish their relation with
known formulations of free higher spin field theory.Comment: JHEP3 style, 40 pages; v2 typos corrected, comments and references
added; v3 published versio
Lagrangian multiforms on Lie groups and non-commuting flows
We describe a variational framework for non-commuting flows, extending the
theories of Lagrangian multiforms and pluri-Lagrangian systems, which have
gained prominence in recent years as a variational description of integrable
systems in the sense of multidimensional consistency. In the context of
non-commuting flows, the manifold of independent variables, often called
multi-time, is a Lie group whose bracket structure corresponds to the
commutation relations between the vector fields generating the flows. Natural
examples are provided by super-integrable systems for the case of Lagrangian
1-form structures, and integrable hierarchies on loop groups in the case of
Lagrangian 2-forms. As particular examples we discuss the Kepler problem, the
rational Calogero-Moser system, and a generalisation of the
Ablowitz-Kaup-Newell-Segur system with non-commuting flows. We view this
endeavour as a first step towards a purely variational approach to Lie group
actions on manifolds.Comment: 49 page
The metaphysics of the Time-Machine
The concept of time-travel is a modern idea which combines the imaginary signification of rational domination, the imaginary signification of technological omnipotence, the imaginary concept of eternity and the imaginary desire for immortality.
It is a synthesis of central conceptual schemata of techno-science, such as the linearity and homogeneity of time, the radical separation of subjectivity from the world, the radical separation of the individual from his/her social-historical environment. The emergence of this idea, its spread during the 20th century as a major theme of science fiction literature alongside its dissemination as a scientific hypothesis, its popularity with both the public and the scientific community, are indications of the religious role of techno-science.
It is my opinion, finally, that, as a chimera, time-travel is non-feasible and impossible.
In order to support my claims, I will briefly outline the origins of the time-travel concept and its epistemological and metaphysical/ontological conditions. If these conditions prove to be absurd, the logical impossibility of time-travel will have been demonstrated
Lagrangian multiforms on Lie groups and non-commuting flows
We describe a variational framework for non-commuting flows, extending the theories of Lagrangian multiforms and pluri-Lagrangian systems, which have gained prominence in recent years as a variational description of integrable systems in the sense of multidimensional consistency. In the context of non-commuting flows, the manifold of independent variables, often called multi-time, is a Lie group whose bracket structure corresponds to the commutation relations between the vector fields generating the flows. Natural examples are provided by superintegrable systems for the case of Lagrangian 1-form structures, and integrable hierarchies on loop groups in the case of Lagrangian 2-forms. As particular examples we discuss the Kepler problem, the rational Calogero-Moser system, and a generalisation of the Ablowitz-Kaup-Newell-Segur system with non-commuting flows. We view this endeavour as a first step towards a purely variational approach to Lie group actions on manifolds
Lagrangian Multiform Structures, Discrete Systems and Quantisation
Lagrangian multiforms are an important recent development in the study of integrable variational problems. In this thesis, we develop two simple examples of the discrete Lagrangian one-form and two-form structures. These linear models still display all the features of the discrete Lagrangian multiform; in particular, the property of Lagrangian closure. That is, the sum of Lagrangians around a closed loop or surface, on solutions, is zero. We study the behaviour of these Lagrangian multiform structures under path integral quantisation and uncover a quantum analogue to the Lagrangian closure property. For the
one-form, the quantum mechanical propagator in multiple times is found to be independent of the time-path, depending only on the endpoints. Similarly, for the two-form we define a propagator over a surface in discrete space-time and show that this is independent of the
surface geometry, depending only on the boundary.
It is not yet clear how to extend these quantised Lagrangian multiforms to non-linear or continuous time models, but by examining two such examples, the generalised McMillan maps and the Degasperis-Ruijsenaars model, we are able to make some steps towards that goal. For the generalised McMillan maps we find a novel formulation of the r-matrix for the dual Lax pair as a normally ordered fraction in elementary shift matrices, which offers a new perspective on the structure. The dual Lax pair may ultimately lead to commuting flows and a one-form structure. We establish the relation between the Degasperis-Ruijsenaars model and the integrable Ruijsenaars-Schneider model, leading to a Lax pair and two
particle Lagrangian, as well as finding the quantum mechanical propagator. The link between these results is still needed.
A quantum theory of Lagrangian multiforms offers a new paradigm for path integral quantisation of integrable systems; this thesis offers some first steps towards this theory
A weighted de Rham operator acting on arbitrary tensor fields and their local potentials
We introduce a weighted de Rham operator which acts on arbitrary tensor
fields by considering their structure as r-fold forms. We can thereby define
associated superpotentials for all tensor fields in all dimensions and, from
any of these superpotentials, we deduce in a straightforward and natural manner
the existence of 2r potentials for any tensor field, where r is its
form-structure number. By specialising this result to symmetric double forms,
we are able to obtain a pair of potentials for the Riemann tensor, and a single
(2,3)-form potential for the Weyl tensor due to its tracelessness. This latter
potential is the n-dimensional version of the double dual of the classical four
dimensional (2,1)-form Lanczos potential. We also introduce a new concept of
harmonic tensor fields, demonstrate that the new weighted de Rham operator has
many other desirable properties and, in particular, it is the natural operator
to use in the Laplace-like equation for the Riemann tensor.Comment: 33 pages: corrected typos and minor additions; reference [39] adde
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