PhysRev.75.651.pdf - PHYSICAL REVIEW Electrodynamics Quantum F EB RUAR Y VOLUM E ZS II Vacuum 15 1949 Polarization and Self-Energy JULIAN SCH'INGER

PhysRev.75.651.pdf - PHYSICAL REVIEW Electrodynamics...

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PHYSICAL REVIEW VOLUM E ZS, F EB RUAR Y 15, 1949 Quantum Electrodynamics. II. Vacuum Polarization and Self-Energy JULIAN SCH%'INGER Department of Physics, Harvard University, Cambridge, Massachusetts (Received November 1, 1948) The covariant formulation of quantum electrodynamics, developed in a previous paper, is here applied to two elementary problems the polarization of the vacuum and the self-energies of the electron and photon. In the first section the vacuum of the non-interacting electromagnetic and matter fields is covariantly defined as that state for which the eigenvalue of an arbitrary time-like component of the energy-momentum four-vector is an absolute minimum. It is remarked that this definition must be compatible with the requirement t'hat the vacuum expec- tation values of a physical quantity in various coordinate systems should be, not only covariantly related, but identical, since the vacuum has a significance that is inde- pendent of the coordinate system. In order to construct a suitable characterization of the vacuum state vector, a covariant decomposition of the field operators into positive and negative frequency components is introduced, and the properties of these associated fields developed. It is shown that the state vector for the electromagnetic vacuum is annihilated by the positive frequency part of the trans- verse four-vector potential, while that for the matter vacuum is annihilated by the positive frequency part of the Dirac spinor and of its charge conjugate. These de- fining properties of the vacuum state vector are employed in the calculation of the vacuum expectation values of quadratic field quantities, specifically the energy-mo- mentum tensors of the independent electromagnetic and matter fields, and the current four-vector. It is inferred that the electromagnetic energy-momentum tensor, and the current vector must vanish in the vacuum, while the matter field energy-momentum tensor vanishes in the vacuum only by the addition of a suitable multiple of the unit tensor. The second section treats the induction of a current in the vacuum by an external electromagnetic field. It is supposed that the latter does not produce actual elec- tron-positron pairs; that is, we consider only the phe- nomenon of virtual pair creation. This restriction is introduced by requiring that the establishment and sub- sequent removal of the external field produce no net change in state for the matter field. It is demonstrated, in a general manner, that the induced current at a given space-time point involves the external current in the vicinity of that point, and not the electromagnetic potentials. This gauge invariant result shows that a light wave, propagating at remote distances from its source, induces no current in the vacuum and is therefore undisturbed in its passage through space.
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