PhysRev.74.1439.pdf - VOLUM E 74 NUMBER PH YSI CAL R EVI EW Quantum 10 I A Covariant Electrodynamics NOVEM Formulation JULIAN SCHWINGER Harvard

PhysRev.74.1439.pdf - VOLUM E 74 NUMBER PH YSI CAL R EVI EW...

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PH YSI CAL R EVI EW VOLUM E 74, NUMBER 10 NOVEM BER 15, 1948 Quantum Electrodynamics. I. A Covariant Formulation JULIAN SCHWINGER Harvard University, Cambridge, Massachusetts (Received July 29, 1948) Attempts to avoid the divergence difhculties of quan- tum electrodynamics by mutilation of the theory have been uniformly unsuccessful. The lack of convergence does in- dicate that a revision of electrodynamic concepts at ultra- relativistic energies is indeed necessary, but no appreciable alteration of the theory for moderate relativistic energies can be tolerated. The elementary phenomena in which divergences occur, in consequence of virtual transitions involving particles with unlimited energy, are the po- larization of the vacuum and the self-energy of the elec- tron, e6ects which essentially express the interaction of the electromagnetic and matter fields with their own vacuum fluctuations. The basic result of these fluctuation inter- actions is to alter the constants characterizing the prop- erties of the individual fields, and their mutual coupling, albeit by infinite factors. The question is naturally posed whether all divergences can be isolated in such unob- servable renormalization factors; more specifically, we in- quire whether quantum electrodynamics can account unambiguously for the recently observed deviations from the Dirac electron theory, without the introduction of fundamentally new concepts. This paper, the first in a series devoted to the above question, is occupied with the formulation of a completely covariant electrodynamics. Manifest covariance with respect to Lorentz and gauge transformations is essential in a divergent theory since the use of a particular reference system or gauge in the course of calculation can result in a loss of covariance in view of the ambiguities that may be the concomitant of infinities. It is remarked, in the first section, that the customary canonical commutation relations, which fail to exhibit the desired covariance since they refer to field variables at equal times and different points of space, can be put in covariant form by replacing the four-dimensional surface t=const. by a space-like surface. The latter is such that light signals cannot be propagated between any two points on the surface. In this manner, a formulation of quantum electrodynamics is constructed in the Heisenberg repre- sentation, which is obviously covariant in all its aspects. It is not entirely suitable, however, as a practical means of treating electrodynamic questions, since commutators of field quantities at points separated by a time-like in- terval can be constructed only by solving the equations of motion. This situation is to be contrasted with that of the Schrodinger representation, in which all operators refer to the same time, thus providing a distinct separation between kinematical and dynamical aspects.
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