Zero levels of momentum mappings for cotangent actions (original) (raw)

Dirac constraint analysis and symplectic structure of anti-self-dual Yang-Mills equations

Pramana-journal of Physics, 2006

We present the explicit form of the symplectic structure of anti-self-dual Yang-Mills (ASDYM) equations in Yang's J-and K-gauges in order to establish the bi-Hamiltonian structure of this completely integrable system. Dirac's theory of constraints is applied to the degenerate Lagrangians that yield the ASDYM equations. The constraints are second class as in the case of all completely integrable systems which stands in sharp contrast to the situation in full Yang-Mills theory. We construct the Dirac brackets and the symplectic 2-forms for both J-and K-gauges. The covariant symplectic structure of ASDYM equations is obtained using the Witten-Zuckerman formalism. We show that the appropriate component of the Witten-Zuckerman closed and conserved 2-form vector density reduces to the symplectic 2-form obtained from Dirac's theory. Finally, we present the Bäcklund transformation between the J-and K-gauges in order to apply Magri's theorem to the respective two Hamiltonian structures.

Generalized Symplectic Geometry for Classical Fields and Spinors

1994

The theory of symplectic geometry on the cotangent bundle T * M of a finite-dimensional manifold M has been extended by Norris [9] to the theory of n-symplectic geometry on the bundle of linear frames LM. The bundle LM is considered to be the momentum frame space for particle mechanics, and it is the arena for studying dynamics from the reference point of inertial observers who exist on a covariant configuration space. This dissertation investigates generalizations of n-symplectic geometry that model two classical physical theories, that of classical fields (sections 2-5) and of Dirac spinors (section 6). For the first model we reduce the linear frame bundle of a fiber bundle, and for the second we prolong the orthonormal frame bundle of spacetime [11]. On each, a pullback of the soldering form generates a generalized symplectic geometry. 2. Multisymplectic geometry. Let M be oriented and let π MY : Y → M be a fiber bundle with a k-dimensional fiber. (Note: In general, we shall denote a projection from A onto B as π BA .) A classical field is a section of the configuration bundle Y over the parameter space M. Let V Y ⊂ T Y be the subbundle of vertical vectors. The linear velocity bundle is Hom Y (T M, V Y), the vector bundle over Y with standard fiber Lin (T π MY (y) M, V y Y). The linear multiphase space J * Y is the vector bundle dual to the linear velocity bundle. We use two equivalent representations of J * Y , the Günther [5] and the Kijowski-Tulczyjew (KT) [6] representations.

Hamiltonian Formulation of the Yang–Mills field on the null–plane

Nuclear Physics B - Proceedings Supplements, 2010

We have studied the null-plane hamiltonian structure of the free Yang-Mills fields. Following the Dirac's procedure for constrained systems we have performed a detailed analysis of the constraint structure of the model and we give the generalized Dirac brackets for the physical variables. Using the correspondence principle in the Dirac's brackets we obtain the same commutators present in the literature and new ones.

On the configuration space of gauge theories

Nuclear Physics B, 1994

We investigate the structure of the configuration space of gauge theories and its description in terms of the set of absolute minima of certain Morse functions on the gauge orbits. The set of absolute minima that is obtained when the background connection is a pure gauge is shown to be isomorphic to the orbit space of the pointed gauge group. We also show that the stratum of irreducible orbits is geodesically convex, i.e. there are no geometrical obstructions to the classical motion within the main stratum. An explicit description of the singularities of the configuration space of SU(2) theories on a topologically simple space-time and on the lattice is obtained; in the continuum case we find that the singularities are coni-

B.L. Davis and A. Wade DIRAC STRUCTURES AND GAUGE SYMMETRIES OF PHASE SPACES

2007

We study the geometry of the phase space of a particle in a Yang -Mills-Higgs field in the context of the theory of Dirac structures. Several kno wn constructions are merged into the framework of coupling Dirac structures. Functorial pro perties of our constructions are discussed and examples are provided. Finally, application s t fibered symplectic groupoids are given.

Canonical structure of Yang-Mills theory

1998

I consider the problem of defining canonical coordinates and momenta in pure Yang-Mills theory, under the condition that Gauss' law is identically satisifed. This involves among other things particular boundary conditions for certain dependent variables. These boundary conditions are not postulated a priori, but arise as consistency conditions related to the equations of motion. It is shown that the theory indeed has a canonical structure, provided one uses a special gauge condition, which is a natural generalisation to Yang-Mills theory of the Coulomb gauge condition in electrodynamics. The canonical variables and Hamiltonian are explicitly constructed. Quantisation of the theory is briefly discussed.

2D Yang-Mills Theories, Gauge Orbit Space and The Path Integral Quantization

Eprint Arxiv Hep Th 9312160, 1993

The role of a physical phase space structure in a classical and quantum dynamics of gauge theories is emphasized. In particular, the gauge orbit space of Yang-Mills theories on a cylindrical spacetime (space is compactified to a circle) is shown to be the Weyl cell for a semisimple compact gauge group, while the physical phase space coincides with the quotient IR 2r /W A , r a rank of a gauge group, W A the affine Weyl group. The transition amplitude between two points of the gauge orbit space (between two Wilson loops) is represented via a Hamiltonian path integral over the physical phase space and explicitly calculated. The path integral formula appears to be modified by including trajectories reflected from the boundary of the physical configuration space (of the Weyl cell) into the sum over pathes. The Gribov problem of gauge fixing ambiguities is considered and its solution is proposed in the framework of the path integral modified. Artifacts of gauge fixing are qualitatively analyzed with a simple mechanical example. A relation between a gauge-invariant description and a gauge fixing procedure is established. * Works is supported by an MRT grant of the government of France

The Hamiltonian analysis for Yang–Mills theory on

Nuclear Physics B, 2009

Pure Yang-Mills theory on R × S 2 is analyzed in a gauge-invariant Hamiltonian formalism. Using a suitable coordinatization for the sphere and a gauge-invariant matrix parametrization for the gauge potentials, we develop the Hamiltonian formalism in a manner that closely parallels previous analysis on R 3. The volume measure on the physical configuration space of the gauge theory, the nonperturbative mass-gap and the leading term of the vacuum wave functional are discussed using a point-splitting regularization. All the results carry over smoothly to known results on R 3 in the limit in which the sphere is de-compactified to a plane.

Classical and quantum mechanics of non-abelian gauge fields

Nuclear Physics B, 1984

We i n v estigate the classical and quantum properties of a system of SU(N) non-Abelian Chern-Simons (NACS) particles. After a brief introduction to the subject of NACS particles, we rst discuss about the most general phase space of SU(N) i n ternal degrees of freedom or isospins which can be identied as one of the coadjoint orbits of SU(N) group by the method of symplectic reduction. A complete Dirac's constraint analysis is carried out on each orbit and the Dirac bracket relations among the isospin variables are calculated. Then, the spatial degrees of freedom and interaction with background gauge eld are introduced by considering the phase space of associated bundle which has one of the coadjoint orbit as the ber. Finally, the theory is quantized by using the coherent state method and various quantum mechanical properties are discussed in this approach. In particular, a coherent state representation of the Knizhnik-Zamolodchikov equation is given and possible solutions in this representation are discussed.

On the number of parameters of self-dual Yang-Mills configurations

Letters in Mathematical Physics, 1982

Using pure differential-geometric ideas (Lie groups as R-spaces and related properties) a new method of determining the number of parameters of a self-dual Yang-Mills configuration is proposed. Some connections with the Atiyah-Ward twistor approach are also revealed.