Historical research guide · 0039 · 1951
On gauge invariance and vacuum polarization (Schwinger limit)
Julian Schwinger
In one minute
Julian Schwinger's 1951 paper is the one that says how hard you have to push before empty space pushes back. Its stated theme is modest: if a theory is gauge invariant in form, you keep that invariance in the answers by using only gauge-covariant quantities along the way. Schwinger works the point through the problem of vacuum polarisation — what a prescribed electromagnetic field does to the vacuum's sea of virtual electron and positron pairs. His device is proper time. He treats the calculation as the history of a particle whose space-time coordinates depend on a proper-time parameter, so that only field strengths ever appear. Renormalise the field strength and the charge, and out comes a finite, gauge-invariant result in which the vacuum has non-linear electromagnetic properties of its own. For a constant electric field the answer has an imaginary part, and an imaginary part means decay: past a critical field strength, the vacuum breaks down and makes real electron and positron pairs.
Connection
This is where the number in chapter 8 comes from. Whenever a proposal talks about polarising the vacuum with electromagnetic fields, Schwinger's critical field is the scale it has to be measured against — and his finding that a plane-wave background gives a vanishing effective action is the precision that keeps the argument honest.
From the source
The extraction of gauge-invariant results from a formally gauge-invariant theory is ensured if one employs methods of solution that involve only gauge-covariant quantities; Schwinger illustrates this in connection with the problem of vacuum polarisation by a prescribed electromagnetic field.Abstract; Section I
The vacuum current of a charged Dirac field, which can be expressed in terms of the Green's function of that field, implies an addition to the action integral of the electromagnetic field — the vacuum itself contributes to electrodynamics.Abstract
These quantities can be related to the dynamical properties of a particle whose space-time coordinates depend on a proper-time parameter; the proper-time equations of motion involve only electromagnetic field strengths and so give a gauge-invariant basis for calculation, with rigorous solutions for a constant field and for a plane-wave field.Abstract; proper-time formulation
A renormalisation of the field strength and charge, applied to the modified Lagrange function for constant fields, yields a finite, gauge-invariant result which implies non-linear properties for the electromagnetic field in the vacuum.Abstract; constant-field effective Lagrangian
For a constant electric field the effective Lagrangian has an imaginary part, which Schwinger extracts in full as a sum whose leading exponential goes as minus pi times the electron mass squared over the field: a strong enough electric field makes the vacuum unstable and it produces real electron and positron pairs. The field scale at which this sets in — the Schwinger critical field, the electron mass squared times c cubed divided by e times ħ, about 10¹⁶ volts per centimetre — marks the onset of non-linear QED.Constant electric field; imaginary part of the effective Lagrangian
For a plane-wave background field, for which the Dirac equation had been solved by Volkov, the effective action vanishes — a travelling light wave on its own does not polarise the vacuum this way, however intense it is.Plane-wave field case
The way in
https://doi.org/10.1103/PhysRev.82.664How to cite it
Julian Schwinger (1951) On gauge invariance and vacuum polarization (Schwinger limit) doi:10.1103/PhysRev.82.664