8 research outputs found
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
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
Symmetry aspects of nonholonomic field theories
The developments in this paper are concerned with nonholonomic field theories
in the presence of symmetries. Having previously treated the case of vertical
symmetries, we now deal with the case where the symmetry action can also have a
horizontal component. As a first step in this direction, we derive a new and
convenient form of the field equations of a nonholonomic field theory.
Nonholonomic symmetries are then introduced as symmetry generators whose
virtual work is zero along the constraint submanifold, and we show that for
every such symmetry, there exists a so-called momentum equation, describing the
evolution of the associated component of the momentum map. Keeping up with the
underlying geometric philosophy, a small modification of the derivation of the
momentum lemma allows us to treat also generalized nonholonomic symmetries,
which are vector fields along a projection. Such symmetries arise for example
in practical examples of nonholonomic field theories such as the Cosserat rod,
for which we recover both energy conservation (a previously known result), as
well as a modified conservation law associated with spatial translations.Comment: 18 page
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