Lorenzo Fatibene

Natural and Gauge Natural Formalism for Classical Field Theorie

I The Geometric Setting Introduction.- 1. Fiber Bundles.- 2. Jet Bundles.- 3. Principal Bundles and Connections.- 4. Natural Bundles.- 5. Gauge Natural Bundles.- II The Variational Structure of Field Theories Introduction.- 6. The Lagrangian Formalism.- 7. Natural Theories.- 8. Gauge Natural Theories.- III Spinor Fields Introduction.- 9. Spin Structures and Spin Frames.- 10. Spinor Theories.- Final Word.- References.- Analitic Index.

Gauge formalism for general relativity and fermionic matter

A new formulation for General Relativity is developed; it is a canonical, global and geometrically well posed formalism in which gravity is described using only variables related to spin structures. It does not require any background metric fixing and it applies to quite general manifolds, i.e. it does not need particular symmetries requirement or global frames. A global Lagrangian framework for Dirac spinors is also provided; conserved quantities and superpotentials are given. The interaction between gravity and spinors is described in a minimal coupling fashion with respect to the new variables and the Hilbert stress tensor of spinor fields is computed, providing the gravitational field generated by spinors. Finally differences and analogies between this formalism and gauge theories are discussed.

AUGMENTED VARIATIONAL PRINCIPLES AND RELATIVE CONSERVATION LAWS IN CLASSICAL FIELD THEORY

Augmented variational principles are introduced in order to provide a definition of relative conservation laws. As it is physically reasonable, relative conservation laws define in turn relative conserved quantities which measure, for example, how much energy is needed in a field theory to go from one configuration (called the reference or vacuum) to another configuration (the physical state of the system). The general prescription we describe solves in a covariant way the well known observer dependence of conserved quantities. The solution found is deeply related to the divergence ambiguity of the Lagrangian and to various formalisms recently appeared in literature to deal with the variation of conserved quantities (of which this is a formal integration). A number of examples relevant to fundamental Physics are considered in detail, starting from classical Mechanics.Augmented variational principles are introduced in order to provide a definition of relative conservation laws. As it is physically reasonable, relative conservation laws define in turn relative conserved quantities that measure, for example, how much energy is needed in a field theory to go from one configuration (called the reference or vacuum) to another configuration (the physical state of the system). The general prescription we describe solves in a covariant way the well-known observer dependence of conserved quantities. The solution found is deeply related to the divergence ambiguity of the Lagrangian and to various formalisms that have recently appeared in literature to deal with the variation of conserved quantities (of which this is a formal integration). A number of examples relevant to fundamental physics are considered in detail, starting from classical mechanics.

Generalized symmetries in mechanics and field theories

Generalized symmetries are introduced in a geometrical and global formalism. Such a framework applies naturally to field theories and specializes to mechanics. Generalized symmetries are characterized in a Lagrangian context by means of the transformation rules of the Poincare–Cartan form and the (generalized) Nother theorem is applied to obtain conserved quantities (first integrals in mechanics). In the particular case of mechanics it is shown how to use generalized symmetries to study the separation of variables of Hamilton–Jacobi equations recovering standard results by means of this new method. Supersymmetries (Wess–Zumino model) are considered as an intriguing example in field theory.

COVARIANT LAGRANGIAN FORMULATION OF CHERN–SIMONS AND BF THEORIES

We investigate the covariant formulation of Chern–Simons theories in a general odd dimension which can be obtained by introducing a vacuum connection field as a reference. Field equations, Nother currents and superpotentials are computed so that results are easily compared with the well-known results in dimension 3. Finally we use this covariant formulation of Chern–Simons theories to investigate their relation with topological BF theories.

Remarks on conserved quantities and entropy of BTZ black hole solutions. I. The general setting

The BTZ stationary black hole solution is considered and its mass and angular momentum are calculated by means of Noether theorem. In particular, relative conserved quantities with respect to a suitably fixed background are discussed. Entropy is then computed in a geometric and macroscopic framework, so that it satisfies the first principle of thermodynamics. In order to compare this more general framework to the prescription by Wald et al. we construct the maximal extension of the BTZ horizon by means of Kruskal-like coordinates. A discussion about the different features of the two methods for computing entropy is finally developed.

Nöther formalism for conserved quantities in classical gauge field theories

Gauge theories coupled with the gravitational field and external matter fields are considered and Notherian techniques are used to build the canonical conserved quantities. Their superpotentials are found and the relation between the canonical stress tensor and the Hilbert tensor is determined. The case of Yang–Mills theories coupled with gravity and scalar matter fields is considered in detail and it is shown that in this case the canonical stress tensor and the Hilbert tensor coincide.

Nöther charges, Brown–York quasilocal energy, and related topics

The Lagrangian proposed by York et al. and the covariant first-order Lagrangian for general relativity are reviewed. They both deal with the (vacuum) gravitational field on a reference background and were conjectured to be equivalent. The two corresponding actions are compared and we show that the first one can in fact be obtained from the latter under suitable hypotheses. A conditioned correspondence among Nother conserved quantities of covariant first-order Lagrangian, Brown–York quasilocal energy and the standard ADM Hamiltonian is also established.

Spacetime Lagrangian Formulation of Barbero-Immirzi Gravity

We shall discuss a new spacetime gauge-covariant Lagrangian formulation of general relativity by means of the Barbero?Immirzi SU(2)-connection on spacetime. To the best of our knowledge, the Lagrangian based on SU(2) spacetime fields seems to appear for the first time here.

Remarks on Nöther Charges and Black Holes Entropy

Abstract We criticize and generalize some properties of Nother charges presented in a paper by V. Iyer and R. M. Wald and their application to entropy of black holes. The first law of black holes thermodynamics is proven for any gauge–natural field theory. As an application charged Kerr–Newman solutions are considered. As a further example we consider the (1+2) BTZ black hole solution.