Abstract
The Poisson gauge algebra is a semi-classical limit of complete non- commutative gauge algebra. In the present work we formulate the Poisson gauge theory which is a dynamical field theoretical model having the Poisson gauge algebra as a corresponding algebra of gauge symmetries. The proposed model is designed to investigate the semi-classical features of the full non-commutative gauge theory with coordinate dependent non-commutativity Θab(x), especially whose with a non-constant rank. We derive the expression for the covariant derivative of matter field. The commutator relation for the covariant derivatives defines the Poisson field strength which is covariant under the Poisson gauge transformations and reproduces the standard U(1) field strength in the commutative limit. We derive the corresponding Bianchi identities. The field equations for the gauge and the matter fields are obtained from the gauge invariant action. We consider different examples of linear in coordinates Poisson structures Θab(x), as well as non-linear ones, and obtain explicit expressions for all proposed constructions. Our model is unique up to invertible field redefinitions and coordinate transformations.
Highlights
The associative non-commutativity of space-time is usually introduced in the theory by substituting the standard pointwise multiplication of fields f · g on some manifold M with the star multiplication, f g
The Poisson gauge algebra is a semi-classical limit of complete noncommutative gauge algebra
We derive the expression for the covariant derivative of matter field
Summary
We will summarize the necessary ingredients from the symplectic geometry that we will use throughout the paper. The problem of the construction of the symplectic embedding for the given Poisson structure (1.2) formulated in the introduction consists basically in finding the matrix γji(x, p) which defines the Poisson bracket {xi, pj} in such a way that the complete algebra of Poisson brackets (1.2) and (1.5) should satisfy the Jacobi identity. = 0, is satisfied automatically since Θij(x) is a Poisson bi-vector. The Jacobi identity with two original coordinates and one p-variable, {xi, {xj, pk}} + cycl. The matrix γji(x, p) is defined as a solution of the equation (2.1) with the condition, γji(x, p)|α=0 = δji , to guarantee that the complete algebra of the Poisson brackets (1.2) and (1.5) is a deformation in α of the canonical Poisson brackets, i.e., forms the symplectic algebra. The recurrence relations for the construction of the matrix γak(x, p) in any order in α are given in [30]
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