GEOMETRIC PREQUANTIZATION ON THE SPIN BUNDLE BASED ON N-SYMPLECTIC GEOMETRY - THE DIRAC-EQUATION

We present preliminary results for a prequantization procedure that le ads in a natural way to the Dirac equation. The starting point is the recently introduced n-symplectic geometry on the bundle of linear fram es LM of an n-dimensional manifold M in which the R(n)-valued solderin g 1-form theta on LM plays the role of the n-symplectic potential. On a 4-dimensional spacetime manifold we consider the tensorial R4 X R4-v alued function g on LM determined by the spacetime metric tensor g as the Hamiltonian for free observers and determine the associated R-valu ed Hamiltonian vector field X(g) = X(g)i X r(i). ''Integration'' of th e X(g)i yields the dynamics of free observers on spacetime, namely par allel transport of linear frames along spacetime geodesics. In order t o obtain a vector field on the spin bundle SM which is a lift of X, an d which is induced by a vector field X(g) for an appropriate mapping g , it is useful to define a prolongation L(o)M of some bundle L(o)M of oriented frames of M. If GL+(4, R) denotes the identity component of G L(4, R), then GL+(4, R) is the structure group of L(o)M and its double cover GL+(4, R) is the structure group of L(o)M. We show that the lif t theta of theta on L(o)M to L(o)M induces a natural 4-symplectic pote ntial on L(o)M. If g is the lift of g to L(o)M, then we find the R4-va lued Hamiltonian, vector field X(g) on L(o)M determined by g and show that the vector fields X(g)i on L(o)M are tangent to the subbundle SM. ''Integration'' of the restriction of the X(g)i to SM now yields para llel transport of spin frames and thus tetrads along spacetime geodesi cs of g. We consider a naive prequantization operator assignment X(g) bar arrow pointing right P(g) := ihgamma(i)X(g)iBAR acting on C4-spino rs in the standard representation of SL(2, C). The eigenvalue equation for the system of new Hilbert space operators yields the Dirac equati on.