Schwinger limit
In quantum electrodynamics (QED), the Schwinger limit is a scale above which the electromagnetic field is expected to become nonlinear. The limit was first derived in one of QED's earliest theoretical successes by Fritz Sauter in 1931 and discussed further by Werner Heisenberg and his student Hans Euler. The limit, however, is commonly named in the literature for Julian Schwinger, who derived the leading nonlinear corrections to the fields and calculated the production rate of electron–positron pairs in a strong electric field. The limit is typically reported as a maximum electric field before nonlinearity for the vacuum of
where me is the mass of the electron, c is the speed of light in vacuum, qe is the elementary charge, and ħ is the reduced Planck constant.
In a vacuum, the classical Maxwell's equations are perfectly linear differential equations. This implies – by the superposition principle – that the sum of any two solutions to Maxwell's equations is yet another solution to Maxwell's equations. For example, two intersecting beams of light should simply add together their electric fields and pass right through each other. Thus Maxwell's equations predict the impossibility of any but trivial elastic photon–photon scattering. In QED, however, non-elastic photon–photon scattering becomes possible when the combined energy is large enough to create virtual electron–positron pairs spontaneously, illustrated by the Feynman diagram in the adjacent figure.
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