GENERALIZED SYMPLECTIC GEOMETRY ON THE FRAME BUNDLE OF A MANIFOLD† by

In this paper we develope the fundamentals of the generalized symplectic geometry on the bundle of linear frames LM of an n-dimensional manifold M that follows upon taking the R-valued soldering 1-form θ on LM as a generalized symplectic potential. The development is centered around generalizations of the basic structure equation df = −Xf ω of standard symplectic geometry to LM when the symplectic 2-form ω is replaced by the closed and non-degenerate R-valued 2-form β = dθ = dθri. The fact that dθ is R-valued necessitates generalizing from R-valued observables to vector-valued observables on LM , and there is a corresponding increase in the number of Hamiltonian vector fields assigned to each observable. We show that the algebras of symmetric and anti-symmetric contravariant tensor fields on the base manifold have natural interpretations in terms of symplectic geometry on LM . For the analysis we consider in place of each rank p contravariant tensor field on the base manifold the uniquely related ⊗pR-valued tensorial function on LM . For symmetric contravariant tensor fields on M we show that the associated algebra (ST,⊗s), where ST = ∑∞ p=1 ST p is the vector space of all ⊗sR -valued tensorial functions on LM , becomes a Poisson algebra under a generalized Poisson bracket. In addition the associated set of locally defined ⊗p−1 s R -valued Hamiltonian vector fields X̂f̂ forms a Lie algebra under a generalized Lie bracket. In the case of anti-symmetric contravariant tensor fields onM we show that the corresponding vector space AT = ∑∞ p=1AT p of ⊗aR -valued functions on LM becomes a Poisson super algebra under a naturally defined bracket. The associated set of locally defined ⊗p−1 a R -valued Hamiltonian vector fields X̂f̂ forms a super algebra under a generalized super bracket. The naturally defined brackets of the tensorial functions on LM give the Schouten differential concomitants when reinterpreted on the base manifold. Generalized symplectic geometry on the frame bundle of a manifold thus unifies and clarifies the many different approaches to the differential concomitants of Schouten. Two applications of the geometry to physics are presented. First the dynamics of free inertial observers in spacetime is shown to follow upon taking the metric tensor as the Hamiltonian for free observers. We then show that the Dirac equation arises in a natural way as an eigenvalue equation for a naive prequantization operator assigned to the spacetime metric tensor Hamiltonian.