3 research outputs found
Multivector Fields and Connections. Setting Lagrangian Equations in Field Theories
The integrability of multivector fields in a differentiable manifold is
studied. Then, given a jet bundle , it is shown that integrable
multivector fields in are equivalent to integrable connections in the
bundle (that is, integrable jet fields in ). This result is
applied to the particular case of multivector fields in the manifold and
connections in the bundle (that is, jet fields in the repeated jet
bundle ), in order to characterize integrable multivector fields and
connections whose integral manifolds are canonical lifting of sections. These
results allow us to set the Lagrangian evolution equations for first-order
classical field theories in three equivalent geometrical ways (in a form
similar to that in which the Lagrangian dynamical equations of non-autonomous
mechanical systems are usually given). Then, using multivector fields; we
discuss several aspects of these evolution equations (both for the regular and
singular cases); namely: the existence and non-uniqueness of solutions, the
integrability problem and Noether's theorem; giving insights into the
differences between mechanics and field theories.Comment: New sections on integrability of Multivector Fields and applications
to Field Theory (including some examples) are added. The title has been
slightly modified. To be published in J. Math. Phy
Geometric quantization of mechanical systems with time-dependent parameters
Quantum systems with adiabatic classical parameters are widely studied, e.g.,
in the modern holonomic quantum computation. We here provide complete geometric
quantization of a Hamiltonian system with time-dependent parameters, without
the adiabatic assumption. A Hamiltonian of such a system is affine in the
temporal derivative of parameter functions. This leads to the geometric Berry
factor phenomena.Comment: 20 page