Abstract
The Dirac equation, as a 4×4-hyperbolic system on ℝ3, possesses an invariant algebra of global pseudodifferential operators-in the sense that conjugation with the Dirac time propagator leaves the algebra invariatn (cf. [CX]. Chapter 10). In this paper we examine the relation between the two invariant algebras att=0 and att'=0 when (t,x) and (t',x') are coordinates of Minkowsky space related by a (proper) Lorentz transform. For vanishing electromagnetic potentials these algebras are transforms of each other by the implied change of dependent and independent variables. In the general case such a space-time transform will make the potentials time dependent, hence also the algebra dependent on the initial plane.
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References
[BLT] N. N. Bogoliubov, A. A. Logunov and I. T. Todorov,Introduction to Axiomatic Quantum Field Theory, Benjamin, Reading, Massachusetts, 1975.
[CD] H. O. Cordes, “A pseudo-algebra of observables for the Dirac equation,”Manuscripta Math. 45 (1983), 77–105.
[CE] H. O. Cordes, “A version of Egorov's theorem for systems of hyperbolic pseudodifferential equations,”J. of Functional Analysis 48 (1982), 285–300.
[CF] H. O. Cordes, “A pseudodifferential Foldy-Wouthuysen transform,”Comm. in Partial Differential Equations 8(13) (1983), 1475–1485.
H. O. Cordes,The Technique of Pseudodifferential Operators, London Math. Soc. Lecture Notes 202, Cambridge University Press, 1995.
[IZ] C. Itzykson and J. B. Zuber,Quantum Field Theory, McGraw Hill, New York, 1980.
[Th] B. Thaller,The Dirac Equation, Springer Verlag, Berlin Heidelberg New York, 1992.
A. Unterberger, “A calculus of observables on a Dirac particle,”Preprint 96.02,Dept. Math. Univ. de Reims, URA (1996), 1870. To appear in Annales Inst. Henri Poincaré (Phys. Théor.).
[Un2] A. Unterberger, “Quantization, symmetries and relativity,”Preprint 96.09,Dept. Math. Univ. de Reims, URA, 1870,Perspectives on Quantization, Contemporary Math. 214, AMS (1998), 169–187.