Hamiltonian analysis for topological and Yang-Mills theories expressed as a constrained BF-like theory (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.

8 Four-Dimensional Yang–Mills Theory as a Deformation of Topological BF Theory

2016

The classical action for pure Yang-Mills gauge theory can be formulated as a deformation of the topological BF theory where, beside the two-form field B, one has to add one extra-field η given by a one-form which transforms as the difference of two connections. The ensuing action functional gives a theory that is both classically and quantistically equivalent to the original Yang-Mills theory. In order to prove such an equivalence, it is shown that the dependency on the field η can be gauged away completely. This gives rise to a field theory that, for this reason, can be considered as semi-topological or topological in some but not all the fields of the theory. The symmetry group involved in this theory is an affine extension of the tangent gauge group acting on the tangent bundle of the space of connections. A mathematical

Four-Dimensional Yang-Mills Theory as a Deformation of Topological BF Theory

Communications in Mathematical Physics, 1998

The classical action for pure Yang-Mills gauge theory can be formulated as a deformation of the topological BF theory where, beside the two-form field B, one has to add one extra-field η given by a one-form which transforms as the difference of two connections. The ensuing action functional gives a theory that is both classically and quantistically equivalent to the original Yang-Mills theory. In order to prove such an equivalence, it is shown that the dependency on the field η can be gauged away completely. This gives rise to a field theory that, for this reason, can be considered as semi-topological or topological in some but not all the fields of the theory. The symmetry group involved in this theory is an affine extension of the tangent gauge group acting on the tangent bundle of the space of connections. A mathematical

Covariant Canonical Method for Yang-Mills Theory Expressed as a Constrained BF-Like Theory

Advances in Mathematical Physics, 2012

The covariant canonical analysis for Yang-Mills theory expressed as a BF-like action is performed. We study a BF-like action, that in spite of being the coupling of two topological terms, yield, on shell to Yang-Mills action. In addition, by using the results obtained in the covariant canonical approach we study the symmetries of the action, in particular we calculate its energy-momentum tensor obtaining the same tensor found for Yang-Mills theory; then we confirm those results by using…

Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories

Journal of Geometric Mechanics

The multisymplectic formalism of field theories developed by many mathematicians over the last fifty years is extended in this work to deal with manifolds that have boundaries. In particular, we develop a multisymplectic framework for first order covariant Hamiltonian field theories on manifolds with boundaries. This work is a geometric fulfillment of Fock's characterization of field theories as it appears in recent work by Cattaneo, Mnev and Reshetikhin [Ca14]. This framework leads to a true geometric understanding of conventional choices for boundary conditions. For example, the boundary condition that the pull-back of the 1-form on the cotangent space of fields at the boundary vanish, i.e. Π * α = 0 , is shown to be a consequence of our finding that the boundary fields of the theory lie in the 0-level set of the moment map of the gauge group of the theory.

A pure Dirac's method for Yang-Mills expressed as a constrained BF-liketheory

A pure Dirac's method of Yang-Mills expressed as a constrained BF-like theory is performed. In this paper we study an action principle composed by the coupling of two topological BF-like theories, which at the Lagrangian level reproduces Yang-Mills equations. By a pure Dirac's method we mean that we consider all the variables that occur in the Lagrangian density as dynamical variables and not only those ones that involve temporal derivatives. The analysis in the complete phase space enable us to calculate the extended Hamiltonian, the extended action, the constraint algebra, the gauge transformations and then we carry out the counting of degrees of freedom. We show that the constrained BF-like theory correspond at classical level to Yang-Mills theory. From the results obtained, we discuss briefly the quantization of the theory. In addition we compare our results with alternatives models that have been reported in the literature

Hamiltonian study for Chern-Simons and Pontryagin theories

Annals of Physics, 2011

The Hamiltonian analysis for the Chern-Simons theory and Pontryagin invariant, which depends of a connection valued in the Lie algebra of SO(3, 1), is performed. By applying a pure Dirac's method we find for both theories the extended Hamiltonian, the extended action, the constraint algebra, the gauge transformations and we carry out the counting of degrees of freedom. From the results obtained in the present analysis, we will conclude that the theories under study have a closed relation among its constraints and defines a topological field theory. In addition, we extends the configuration space for the Pontryagin theory and we develop the Hamitonian analysis for this modified version, finding a best description of the results previously obtained.

Hamiltonian analysis of the BFCG theory for the Poincaré 2-group

Classical and Quantum Gravity, 2016

We perform the full Hamiltonian analysis of the topological BF CG action based on the Poincaré 2-group. The Hamiltonian of the theory is constructed, and the algebra of constraints is computed. The Dirac brackets are evaluated, and the second class constraints are then eliminated from the theory. The results are contrasted to those of the topological Poincaré gauge theory, which is equivalent to the BF CG model at the level of the classical action, but has a very different Hamiltonian structure.

Quantization ambiguities for Chern-Simons topological theories

Physics Letters B, 1991

We point out that the smgulantes of the gauge mvarlant phase space of the three-dlmensmnal topological Chern-S~mons theory with compact gauge group G give rise to amblgulnes of the canomcal quantlzatlon which are classified by the one-d~mensmnal representatmns of the Weyl group of G We show that the quantum Hflbert space assocmted w~th the non-tnwal representanon of the Weyl group has the unitary structure of the space of current blocks of two-dlmensmnal Wess-Zummo-Wltten theory on the group manifold G This provides a topological understanding of the "shift" of the central charge m the two-dimensional Sugawara construction In 1988 W~tten [ 1 ] discovered that the quantlzanon of the Chern-Slmons gauge theory in 2+ 1 dlmenstons wtth a compact gauge group G produces a quantum Hflbert space which is tsomorphtc to the hnear space of the current blocks of the two-dlmenstonal Wess-Zummo-Witten theory on the group manifold G. Since the original paper, the quanttzatlon of the Chern-S~mons theory has been discussed by a number of authors both from the standpomt of the canomcal formahsm [2-6] and in the perturbattve lagranglan framework [ 7-1 1 ]. What makes the theory exactly solvable ts the fact that the gauge mvariance ehmmates all but a finite number of phystcal degrees of freedom so that the usually untreatable field theoretical dlfficulnes become manageable. Rather curiously, however, the theory has been largely analyzed with the full machmery of mfinite-dlmensmnal quantum field theory. In the hamlltonlan formahsm th~s went under the name of"quannze first" approach [3], namely: one fixes a gauge, m particular the Ao = 0 gauge, quanhzes the resulting free-field theory, and eventually xmposes the Gauss-law constralnt as an operator equanon to select physical states. To make the contact with conformal field theory exphcit, one quannzes the gauge-fixed free theory m the "holomorph~c" quannzation scheme, first by choosing a complex structure on the two-dimensional "space-hke" manifold and then by takmg wavefunctlonals representing the quantum states which depend on a single holomorphlc component of the gauge connectton along the space d~rectlons. The holomorphtc wave functtonals are then tdenttfied with the generatmg funcnonals for the correlatton funcnons of the Wess-Zumino-Wttten conformal theory, and the Gauss-law constraint ts shown to be equtvalent to the current algebra Ward tdenmles.

COVARIANT LAGRANGIAN FORMULATION OF CHERN–SIMONS AND BF THEORIES

International Journal of Geometric Methods in Modern Physics, 2006

We investigate the covariant formulation of Chern-Simons theories in a general odd dimension which can be obtained by introducing a vacuum connection field as a reference. Field equations, Nöther currents and superpotentials are computed so that results are easily compared with the wellknown results in dimension 3. Finally we use this covariant formulation of Chern-Simons theories to investigate their relation with topological BF theories.