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

On the Hamiltonian formulation of Yang--Mills gauge theories

2005

The Hamiltonian formulation of the theory of J-bundles is given both in the Hamilton--De Donder and in the Multimomentum Hamiltonian geometrical approaches. (3+3) Yang-Mills gauge theories are dealt with explicitly in order to restate them in terms of Einstein-Cartan like field theories.

The geometrical setting of gauge theories of the Yang-Mills type

Reviews of Modern Physics, 1980

The pure Yang-Mills theory defined in the four-dimensional Euclidean space has a rich and interesting structure even at the classical level. The discovery of regular solutions to the Yang-Mills field equations, which correspond to absolute minima of the action (Belavin et al. , 1975},has led to an intensive study ot such a

Yang-Mills on Surfaces with Boundary: Quantum Theory and Symplectic Limit

Communications in Mathematical Physics, 1997

The quantum field measure for gauge fields over a compact surface with boundary, with holonomy around the boundary components specified, is constructed. Loop expectation values for general loop configurations are computed. For a compact oriented surface with one boundary component, let M(O) be the moduli space of fiat connectiwis with boundary holonomy lying in a conjugacy class 0 in the gauge group G. We prove that a certain natural closed 2-form on M{&X introduced in an earlier work by C. King and the author, is a symplectic structure on the generic stratum of M{0) for generic O. We then prove that the quantum Yang-Mills measure, with the boundary holonomy constrained to lie in @, converges in a natural sense to the corresponding symplectic volume measure in the classical limit. We conclude with a detailed treatment of tile case G = SU(2), and determine the symplectic volume of this moduli space. 664 A. Seagupta C = q(xoO)q(K 7 K 6 K 5)q(Ox 0) generate K\(E, o) subject to the relation CB~ABA = I (2.1.1) wherein/is the identity in 7ri(i7, 6). The presence of K5 andifv is not essential; however, when E is a general compact surface with more than one boundary component (as will be the case in section 5) then arcs like Ks and Kj must be included, and this is why we choose to keep them in our framework. 2.2 The G-bundle w : P->• E, space of connections A, holonomies /i(/c;u;), and curvature fi"'. We shall work with a principal G-bundle TT : P->-E, where G is a compact connected Lie group with Lie algebra g having an Ad-invariant inner-product (',-)g on **• Tn e set of all connections on P will be denoted A. The metrics on E and on g induce a metric on A in a standard way. Fix once and for all a basepoint u G n~l(p). If /c : [r, s]-> E is a path on 17 with /c(r) = o, and if a; G A, then we denote by T W (K)U the parallel translate of u along /c with respect to w. Thus if AC is a loop based at o then the holonomy of u; around K, with u as initial point, is h(K\u) G G given by T W (K)U = u/i(/c;a;). The curvature of UJ G .4 will be denoted QP; 2.3 The gauge transformation groups G,Go, acting on A. We shall use the set Q of automorphisms of P, i.e. diffeomorphisms : P->-P for which TT o = n and <^(p^) = <^(p)^ for all p G P, # G G. This is a group under composition, and the subgroup Go = { G G : <^(w) = «} will be of use. These groups act on .4 by 2.^ 7%^ Yang-Mills action. If a; G A then the Yang-Mills action 5 Y M(^) is given by da, (2.4.1) where da is the Riemannian surface area measure on E 9 and {Q^ |^ is the function on E given by where ar runs over E 9 and ei, e 2 G ^>P, for any p G TT" 1^) , are such that (7r*ei, 7r*e2) is an orthonormal basis of T X E. Since Of* vanishes when to applied vertical vectors and since it is a 2-form, it follows from the Ad-invariance of (•,-) 9 that [f?*'!^ is a well-defined function E. Furthermore, 5 Y M is invariant under the fir-action and therefore defines a function on A/G and on A/Go-2.5 The curvature function F w , and the parallel-transport equation. Let u G A. We shall use the map s w : D-> P given by s w (x) = r^ (q(Ox)ju 9 where Ox is the radial path from O to x. A convenient way to express the curvature is by means of the map F»:D->g:x^ F»(x) = f ^(e u e 2 \ (2.5.1) ', (8.1.4) a,-=*C4,)A =xCB,),c = x(C). (8.1.5) Conversely, given t/^ ,"., y^m, {^,6*}, c in G, satisfying there is an assignm^it e ^ x e9 with xj = x~l for every oriented 1-simplex e of 5, such that (8.1.4) and (8.1.5) hold. With this change of variables it follows, as for (4.4.5), J i)dy Aj , (8.1.6)

Generalized coordinates on the phase space of Yang-Mills theory

Classical and Quantum Gravity, 1995

We study the suitability of complex Wilson loop variables as (generalized) coordinates on the physical phase space of SU (2)-Yang-Mills theory. To this end, we construct a natural one-to-one map from the physical phase space of the Yang-Mills theory with compact gauge group G to a subspace of the physical configuration space of the complex G C-Yang-Mills theory. Together with a recent result by Ashtekar and Lewandowski this implies that the complex Wilson loop variables form a complete set of generalized coordinates on the physical phase space of SU (2)-Yang-Mills theory. They also form a generalized canonical loop algebra. Implications for both general relativity and gauge theory are discussed.

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.