The Schwinger effect is a predicted physical phenomenon whereby matter is created by a strong electric field. It is also referred to as the Sauter–Schwinger effect, Schwinger mechanism, or Schwinger pair production. It is a prediction of quantum electrodynamics (QED) in which electron–positron pairs are spontaneously created in the presence of an electric field, thereby causing the decay of the electric field. The effect was originally proposed by Fritz Sauter in 1931 and further important work was carried out by Werner Heisenberg and Hans Heinrich Euler in 1936, though it was not until 1951 that Julian Schwinger gave a complete theoretical description. The Schwinger effect can be thought of as vacuum decay in the presence of an electric field. Although the notion of vacuum decay suggests that something is created out of nothing, physical conservation laws are nevertheless obeyed. To understand this, note that electrons and positrons are each other's antiparticles, with identical properties except opposite electric charge. To conserve energy, the electric field loses energy when an electron–positron pair is created, by an amount equal to 2 m e c 2 {\displaystyle 2m_{\text{e}}c^{2}} , where m e {\displaystyle m_{\text{e}}} is the electron rest mass and c {\displaystyle c} is the speed of light. Electric charge is conserved because an electron–positron pair is charge neutral. Linear and angular momentum are conserved because, in each pair, the electron and positron are created with opposite velocities and spins. In fact, the electron and positron are expected to be created at (close to) rest, and then subsequently accelerated away from each other by the electric field.
Mathematical description Schwinger pair production in a constant electric field takes place at a constant rate per unit volume, commonly referred to as Γ {\displaystyle \Gamma } . The rate was first calculated by Schwinger and at leading (one-loop) order is equal to
Γ = ( e E ) 2 4 π 3 ℏ 2 c ∑ n = 1 ∞ 1 n 2 exp { − π m 2 c 3 n ℏ e E } {\displaystyle \Gamma ={\frac {(eE)^{2}}{4\pi ^{3}\hbar ^{2}c}}\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}\exp \left\{-{\frac {\pi m^{2}c^{3}n}{\hbar eE}}\right\}}
where m {\displaystyle m} is the mass of an electron, e {\displaystyle e} is the elementary charge, E {\displaystyle E} is the electric field strength, and ℏ {\displaystyle \hbar } is the reduced Planck constant. This formula cannot be expanded in a Taylor series in e 2 {\displaystyle e^{2}} , showing the nonperturbative nature of this effect. In terms of Feynman diagrams, one can derive the rate of Schwinger pair production by summing the infinite set of diagrams shown below, containing one electron loop and any number of external photon legs, each with zero energy.
The infinite sum in the expression above can be written in terms of the dilogarithm, and then the equation becomes
Γ = ( e E ) 2 4 π 3 ℏ 2 c Li 2 ( exp { − π m 2 c 3 ℏ e E } ) {\displaystyle \Gamma ={\frac {(eE)^{2}}{4\pi ^{3}\hbar ^{2}c}}\operatorname {Li} _{2}\left(\exp \left\{-{\frac {\pi m^{2}c^{3}}{\hbar eE}}\right\}\right)} .
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