Abstract
In a previous paper, we have shown how the classical and quantum relativistic dynamics of the Stueckelberg-Horwitz-Piron [SHP] theory can be embedded in general relativity (GR). We briefly review the SHP theory here and, in particular, the formulation of the theory of spin in the framework of relativistic quantum theory. We show here how the quantum theory of relativistic spin can be embedded, using a theorem of Abraham, Marsden and Ratiu, and also explicit derivation, into the framework of GR by constructing a local induced representation. The relation to the work of Fock and Ivanenko is also discussed. We show that in a gravitational field there is a highly complex structure for the spin distribution in the support of the wave function. We then discuss entanglement for the spins in a two body system.
Highlights
The relativistic canonical Hamiltonian dynamics of Stueckelberg, Horwitz and Piron (SHP)[1] with scalar potential and gauge field interactions for single and many body theory can, by local coordinate transformation, be embedded into the framework of general relativity (GR)[2][3]
In the following we briefly review the SHP theory and its imbedding into the curved space of GR [2][3](SHPGR) for a single spinless particle and turn to discuss the representations of a particle with spin in GR
The two body initial state is necessarily formed in the induced representation based on a common N μ, where we assume the atom is small compared to variations in the gravitational field; otherwise the two N μ vectors would be related by parallel transport
Summary
The relativistic canonical Hamiltonian dynamics of Stueckelberg, Horwitz and Piron (SHP)[1] with scalar potential and gauge field interactions for single and many body theory can, by local coordinate transformation, be embedded into the framework of general relativity (GR)[2][3] (to be called SHPGR). The invariance of the Poisson bracket* under local coordinate transformations provides a basis for the canonical quantization of the theory, for which the evolution under τ is determined by the Lorentz covariant form of the Stueckelberg-Schrodinger equation [1][4] This method was applied to the many body case [13], for which the SHP Hamiltonian is a sum of terms quadratic in four momentum with a many body potential term. The Lorentz group acts in the locally flat freely falling frame (tangent space) It is essential for the embedding of the special relativistic theory into GR that the set of local generators transform under the local embedding diffeomorphisms as covariant tensors. We treat the manifestation of long range correlations in GR resulting in spin entanglement
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
