Geometric prequantization on the spin bundle based onN-symplectic geometry: The dirac equation

Abstract

We present preliminary results for a prequantization procedure that leads in a natural way to the Dirac equation. The starting point is the recently introduced n-symplectic geometry on the bundle of linear frames LM of an n-dimensional manifold M in which the Rn-valued soldering 1-form 0 on LM plays the role of the n-symplectic potential. On a 4-dimensional spacetime manifold we consider the tensorial R4@ •4-valued function ~ on LM determined by the spacetime metric tensor g as the Hamiltonian for free observers and determine the associated 0~4-valued Hamiltonian vector field )?~ = Xi~| "Integration" of the X~ yields the dynamics of free observers on spacetime, namely parallel transport of linear frames along spacetime geodesics. In order to obtain a vector field on the spin bundle SM which is a lift of .~ and which is induced by a vector field .~ for an appropriate mapping ~, it is useful to define a prolongation L~ of some bundle L~ of oriented frames of M. If GL +(4, R) denotes the identity component of GL(4~, ~), then GL +(4, R) is the structure group of L~ and its double cover GL+(4, R) is the structure group of L'~. We show that the lift ~ of 0 on L~ to L~ induces a natural 4-sympleetic potential on L~ If ~ is the lift of ~ to L~ then we find the R4-valued Hamiltonian vector field .~ on L~ determined by ~ and show that the vector fields X~ on L'~ are tangent to the subbundle SM. "Integration" of the restriction of the X~ to SM now yields parallel transport of spin frames and thus tetrads along spacetime ~odesics of g. We consider a naive prequantization operator assignment X~,~,=ih?~X~ acting on C4-spinors in the standard representation of SL(2, C). The eigenvalue equation for the system of new Hilbert space operators yields the Dirac equation.

Covariant field theory on frame bundles of fibered manifolds

“…We present here a sketch of the structure of the theory and refer the reader to the literature [5,6,12,13,14,15] for more details. See also the works of de León, Salgado et al [3] and Awane [1].…”

“…n-symplectic geometry on LE is the generalized symplectic geometry that emerges upon taking the soldering 1-form θ as the generalized symplectic potential. This geometry, including the notions of n-symplectic observables, the corresponding generalized Hamiltonian vector-valued vector fields, and generalized Poisson and graded Poisson brackets, has been developed in a series of papers [5,6,12,13,14,15]. A sketch of the basic structure of the theory can be found in section 2, but let us point out here that in [14] it was shown that the fundamentals of the canonical symplectic geometry on the cotangent bundle T * E can be constructed entirely in terms of the n-symplectic geometry on LE.…”

n -symplectic algebra of observables in covariant Lagrangian field theory

“…In this paper we construct an algebra of observables for covariant Lagrangian field theories using the n-symplectic theory [3,4,5,6,7,8,9] as the engine for the construction.…”

“…Remark: In references [3,4,5,6,7,8,9,15] the term n-symplectic structure refers to the two-form that is the exterior derivative of the R n -valued soldering 1-form on frame bundles or subbundles of frame bundles.…”

Background geometry in the gauge theory of gravity

“…To construct S, let us consider the two-fold universal covering group GL 4 of the group GL 4 and the corresponding principal bundle LX covering the frame bundle LX [2,[15][16][17]. Note that the group GL 4 admits only infinite-dimensional spinor representations [18].…”