Local U(2, 2) Symmetry in Relativistic Quantum Mechanics

Local gauge freedom in relativistic quantum mechanics is derived from a measurement principle for space and time. For the Dirac equation, one obtains local U(2, 2) gauge transformations acting on the spinor index of the wave functions. This local U(2, 2) symmetry allows a unified description of electrodynamics and general relativity as a classical gauge theory. 1 Connection between Local Gauge Freedom and Position Measurements In [3], it was suggested to link the physical gauge principle with quantum mechanical measurements of the position variable. In the present paper, we will extend this concept to relativistic quantum mechanics and apply it to the Dirac equation. For Dirac spinors, we obtain local U(2, 2) gauge freedom. Our main result is that this U(2, 2) symmetry allows a natural description of both electrodynamics and general relativity as a classical gauge theory. This is shown by deriving a U(2, 2) spin connection from the Dirac operator and analyzing the geometry of this connection. Although we develop the subject from a particular point of view, this paper can be used as an introduction to the Dirac theory in curved space-time. In contrast to [3], where the point of interest is the measure theoretic derivation of local gauge transformations, we will here concentrate on the differential geometry of the Dirac operator. In order to keep measure theory out of this paper, we will use a bra/ket notation in position space. This allows us to explain the basic ideas and results of [3] in a simple, non-technical way which will be sufficient for the purpose of this paper. Nevertheless one should keep in mind that the bra/ket symbols and the δ-normalizations are only a formal notation; the mathematical justification for this formalism is given in [3]. We begin with the example of a scalar particle in nonrelativistic quantum mechanics. The particle is described by a wave function Ψ(~x), which is a vector of the Hilbert space H = L2(IR). The physical observables correspond to self-adjoint operators on H. The position operators ~ X , for example, are given as multiplication operators with the coordinate functions, X : Ψ(~x) → x Ψ(~x), i = 1, . . . , 3. Our definition of the Hilbert space as a space of functions in the position variable was only a matter of convenience; e.g. we could just as well have introduced H as functions in momentum space. Therefore it seems reasonable to forget about the fact that H is a function space and consider it merely as ∗Supported by the Deutsche Forschungsgemeinschaft, Bonn.