Analysis as a source of geometry: a non-geometric representation of the Dirac equation

Abstract

Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4manifold 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 4-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.

Spectral asymptotics for first order systems

“…In order to understand the geometric meaning of formula (6.13) we observe that the covariant subprincipal symbol can be uniquely represented in the form L csub (x) = L prin (x, A(x)) + IA 4 (x), (6.14) where A = (A 1 , A 2 , A 3 ) is some real-valued covector field, A 4 is some real-valued scalar field, x = (x 1 , x 2 , x 3 ) are local coordinates on M (we are working in the nonrelativistic setting) and I is the 2 × 2 identity matrix. Applying the results of [11] to the relativistic operator appearing in the LHS of (2.3) we conclude that A = (A 1 , A 2 , A 3 ) is the magnetic covector potential and A 4 is the electric potential. Note that Lemma 6.1 and formulae (6.8) and (6.14) tell us that the magnetic covector potential and electric potential are invariant under gauge transformations (6.7).…”

“…, x n and x n+1 are 'mixed up'. Such an approach was pursued in [11] and, to a certain extent, in [12].…”

“…The proof of Lemma 6.1 was given in [11]. Note that the analysis in [11] was performed in a more general, 4-dimensional Lorentzian setting.…”

Spectral analysis of the Dirac operator on a 3-sphere

“…Let us emphasise that the underlying reason why we can use three equivalent definitions presented in subsections B.1, B.2 and B.3 is that our manifold is 3-dimensional. One can define the Dirac operator via frames (subsection B.2) or the covariant symbol (subsection B.3) in dimension 3 (Riemannian signature) or in dimension 3 + 1 (Lorentzian signature), see [18] for details, but for higher dimensions these approaches do not seem to work.…”

A non-geometric representation of the Dirac equation in curved spacetime

“…Of course, the coefficients of the adjugate operator can be written down explicitly in local coordinates via the coefficients of the original operator (1), see Ref. 2 …”