Abstract
Consider a formally self-adjoint first order linear differential operator acting on pairs (two-columns) of complex-valued scalar fields over a four-manifold without boundary. We examine the geometric content of such an operator and show that it implicitly contains a Lorentzian metric, Pauli matrices, connection coefficients for spinor fields and an electromagnetic covector potential. This observation allows us to give a simple representation of the massive Dirac equation as a system of four scalar equations involving an arbitrary two-by-two matrix operator as above and its adjugate. The point of the paper is that in order to write down the Dirac equation in the physically meaningful four-dimensional hyperbolic setting one does not need any geometric constructs. All the geometry required is contained in a single analytic object—an abstract formally self-adjoint first order linear differential operator acting on pairs of complex-valued scalar fields.
Highlights
The paper is an attempt at developing a relativistic field theory based on the concepts from the analysis of partial differential equations as opposed to geometric concepts
The potential advantage of formulating a field theory in ‘analytic’ terms is that there might be a chance of describing the interaction of different physical fields in a more consistent, and, hopefully, non-perturbative manner
We equip our manifold M with a prescribed Lorentzian metric and a prescribed electromagnetic covector potential, and write the Dirac equation using the rules of spinor calculus, see appendix A
Summary
The paper is an attempt at developing a relativistic field theory based on the concepts from the analysis of partial differential equations as opposed to geometric concepts. The transformation (1.2) of the differential operator L induces the following transformations of its principal and subprincipal symbols: Lprin ↦ Q*L prin Q,. The differential operator L can be written down explicitly, in local coordinates, via the principal symbol Lprin and covariant subprincipal symbol Lcsub in accordance with formula (5.4), so formula (1.18) is shorthand for (5.4). Note that in the case when the principal symbol does not depend on the position variable x (this corresponds to Minkowski spacetime, which is the case most important for applications) the definition of the adjugate differential operator simplifies In this case the subprincipal symbol coincides with the covariant subprincipal symbol and one can treat the differential operator L as if it were a matrix: formula (1.20) becomes. This fact will be presented in the form of theorem 9.1, the main result of our paper
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