Lagrangian and Hamiltonian descriptions of Yang-Mills particles

Abstract

A new Lagrangian L is proposed for the description of a particle with a non-Abelian charge in interaction with aYang-Mills field. The canonical quantization of L is discussed. At the quantum level L leads t o both irreducible and reducible multiplets of the particle depending upon which of the parameters in L are regarded as dynamical.The case which leads to the irreducible multiplet is the minimal non-Abelian generalization of the usual Lagrangian for a charged point particle in an electromagnetic field. Some of the Lagrangians proposed before for such systems are either special cases of ours or can be obtained from ours by simple modifications. Our formulation bears some resemblance t o Dirac's theory of magnetic monopoles in the following respects: (1) Quantization is possible only if the values of certain parameters in L are restricted to a certain discrete set, this is analogous to the Dirac quantization condition; (2) in certain cases, L depends on external (nondynamical) directions in the internal-symmetry space. This is analogous to the dependence of the magnetic-monopole Lagran'gian on the direction of the Dirac string.

Noncommutative Point Sources

“…The classical equations of motion would be analogous to the Wong equations in Yang-Mills theory. [12], [13] d) Generalizations to other gauge theories, including gravity, should be possible. For the case of gravity this should lead to yet another description of noncommutative black holes.…”

Non-Abelian fluid dynamics in Lagrangian formulation

“…Indeed the canonical (first) term of the Lagrangian (45) is like a Kirillov-Kostant 1-form, which has been previously used to give a Lagrangian for the point particle Wong's equation (31) [9]. Moreover, as we show in the Appendix B, the canonical brackets implied by the canonical 1-form ensure that the charge density algebra is represented canonically…”

Surface-invariants in 2D classical Yang-Mills theory

“…It can be seen that equations (15) are equivalent to the gauge-covariant conservation of the non-Abelian charge of each particle along its world line [16] (this conservation-law arises by taking the covariant derivative on both sides of equation (13))…”

“…To write down the equations of motion for the internal variables g i (τ ) we follow the procedure given in [16]. Take a parametrization of the group elements…”