35 research outputs found
Parametrization and Stress-Energy-Momentum Tensors in Metric Field Theories
We give an exposition of the parametrization method of Kuchar [1973] in the
context of the multisymplectic approach to field theory, as presented in Gotay
and Marsden [2008a]. The purpose of the formalism developed herein is to make
any classical field theory, containing a metric as a sole background field,
generally covariant (that is, "parametrized," with the spacetime diffeomorphism
group as a symmetry group) as well as fully dynamic. This is accomplished by
introducing certain "covariance fields" as genuine dynamic fields. As we shall
see, the multimomenta conjugate to these new fields form the Piola-Kirchhoff
version of the stress-energy-momentum tensor field, and their Euler-Lagrange
equations are vacuously satisfied. Thus, these fields have no additional
physical content; they serve only to provide an efficient means of
parametrizing the theory. Our results are illustrated with two examples, namely
an electromagnetic field and a Klein-Gordon vector field, both on a background
spacetime.Comment: 13 pages, 1 figur
Routhian reduction for quasi-invariant Lagrangians
In this paper we describe Routhian reduction as a special case of standard
symplectic reduction, also called Marsden-Weinstein reduction. We use this
correspondence to present a generalization of Routhian reduction for
quasi-invariant Lagrangians, i.e. Lagrangians that are invariant up to a total
time derivative. We show how functional Routhian reduction can be seen as a
particular instance of reduction of a quasi-invariant Lagrangian, and we
exhibit a Routhian reduction procedure for the special case of Lagrangians with
quasi-cyclic coordinates. As an application we consider the dynamics of a
charged particle in a magnetic field.Comment: 24 pages, 3 figure
The Tulczyjew triple for classical fields
The geometrical structure known as the Tulczyjew triple has proved to be very
useful in describing mechanical systems, even those with singular Lagrangians
or subject to constraints. Starting from basic concepts of variational
calculus, we construct the Tulczyjew triple for first-order Field Theory. The
important feature of our approach is that we do not postulate {\it ad hoc} the
ingredients of the theory, but obtain them as unavoidable consequences of the
variational calculus. This picture of Field Theory is covariant and complete,
containing not only the Lagrangian formalism and Euler-Lagrange equations but
also the phase space, the phase dynamics and the Hamiltonian formalism. Since
the configuration space turns out to be an affine bundle, we have to use affine
geometry, in particular the notion of the affine duality. In our formulation,
the two maps and which constitute the Tulczyjew triple are
morphisms of double structures of affine-vector bundles. We discuss also the
Legendre transformation, i.e. the transition between the Lagrangian and the
Hamiltonian formulation of the first-order field theor
Geometric Generalisations of SHAKE and RATTLE
A geometric analysis of the Shake and Rattle methods for constrained
Hamiltonian problems is carried out. The study reveals the underlying
differential geometric foundation of the two methods, and the exact relation
between them. In addition, the geometric insight naturally generalises Shake
and Rattle to allow for a strictly larger class of constrained Hamiltonian
systems than in the classical setting.
In order for Shake and Rattle to be well defined, two basic assumptions are
needed. First, a nondegeneracy assumption, which is a condition on the
Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy
assumption, which is a condition on the geometry of the constrained phase
space. Non-trivial examples of systems fulfilling, and failing to fulfill,
these assumptions are given
Timeless path integral for relativistic quantum mechanics
Starting from the canonical formalism of relativistic (timeless) quantum
mechanics, the formulation of timeless path integral is rigorously derived. The
transition amplitude is reformulated as the sum, or functional integral, over
all possible paths in the constraint surface specified by the (relativistic)
Hamiltonian constraint, and each path contributes with a phase identical to the
classical action divided by . The timeless path integral manifests the
timeless feature as it is completely independent of the parametrization for
paths. For the special case that the Hamiltonian constraint is a quadratic
polynomial in momenta, the transition amplitude admits the timeless Feynman's
path integral over the (relativistic) configuration space. Meanwhile, the
difference between relativistic quantum mechanics and conventional
nonrelativistic (with time) quantum mechanics is elaborated on in light of
timeless path integral.Comment: 41 pages; more references and comments added; version to appear in
CQ
Classical field theory on Lie algebroids: Variational aspects
The variational formalism for classical field theories is extended to the
setting of Lie algebroids. Given a Lagrangian function we study the problem of
finding critical points of the action functional when we restrict the fields to
be morphisms of Lie algebroids. In addition to the standard case, our formalism
includes as particular examples the case of systems with symmetry (covariant
Euler-Poincare and Lagrange Poincare cases), Sigma models or Chern-Simons
theories.Comment: Talk deliverd at the 9th International Conference on Differential
Geometry and its Applications, Prague, September 2004. References adde
Nonlinear quantum gravity on the constant mean curvature foliation
A new approach to quantum gravity is presented based on a nonlinear
quantization scheme for canonical field theories with an implicitly defined
Hamiltonian. The constant mean curvature foliation is employed to eliminate the
momentum constraints in canonical general relativity. It is, however, argued
that the Hamiltonian constraint may be advantageously retained in the reduced
classical system to be quantized. This permits the Hamiltonian constraint
equation to be consistently turned into an expectation value equation on
quantization that describes the scale factor on each spatial hypersurface
characterized by a constant mean exterior curvature. This expectation value
equation augments the dynamical quantum evolution of the unconstrained
conformal three-geometry with a transverse traceless momentum tensor density.
The resulting quantum theory is inherently nonlinear. Nonetheless, it is
unitary and free from a nonlocal and implicit description of the Hamiltonian
operator. Finally, by imposing additional homogeneity symmetries, a broad class
of Bianchi cosmological models are analyzed as nonlinear quantum
minisuperspaces in the context of the proposed theory.Comment: 14 pages. Classical and Quantum Gravity (To appear
Vector coherent state representations, induced representations, and geometric quantization: I. Scalar coherent state representations
Coherent state theory is shown to reproduce three categories of
representations of the spectrum generating algebra for an algebraic model: (i)
classical realizations which are the starting point for geometric quantization;
(ii) induced unitary representations corresponding to prequantization; and
(iii) irreducible unitary representations obtained in geometric quantization by
choice of a polarization. These representations establish an intimate relation
between coherent state theory and geometric quantization in the context of
induced representations.Comment: 29 pages, part 1 of two papers, published versio
On the k-Symplectic, k-Cosymplectic and Multisymplectic Formalisms of Classical Field Theories
The objective of this work is twofold: First, we analyze the relation between
the k-cosymplectic and the k-symplectic Hamiltonian and Lagrangian formalisms
in classical field theories. In particular, we prove the equivalence between
k-symplectic field theories and the so-called autonomous k-cosymplectic field
theories, extending in this way the description of the symplectic formalism of
autonomous systems as a particular case of the cosymplectic formalism in
non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric
character of the solutions to the Hamilton-de Donder-Weyl and the
Euler-Lagrange equations in these formalisms. Second, we study the equivalence
between k-cosymplectic and a particular kind of multisymplectic Hamiltonian and
Lagrangian field theories (those where the configuration bundle of the theory
is trivial).Comment: 25 page