# Schwinger limit

A Feynman diagram (box diagram) for photon–photon scattering; one photon scatters from the transient vacuum charge fluctuations of the other

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[1] and discussed further by Werner Heisenberg and his student Hans Euler.[2] The limit, however, is commonly named in the literature[3] 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.[4] The limit is typically reported as a maximum electric field before nonlinearity for the vacuum of

$E_S = \frac{m_e^2 c^3}{q_e \hbar} \simeq 1.3 \times 10^{18} \, \mathrm{V} / \mathrm{m},$

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 beams of light pointed toward each other 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 figure on the right.

A single plane wave is insufficient to cause nonlinear effects, even in QED.[4] The basic reason for this is that a single plane wave of a given energy may always be viewed in a different reference frame, where it has less energy (the same is the case for a single photon). A single wave or photon does not have a center of momentum frame where its energy must be at minimal value. However, two waves or two photons not traveling in the same direction always have a minimum combined energy in their center of momentum frame, and it is this energy and the electric field strengths associated with it, which determine particle-antiparticle creation, and associated scattering phenomena.

Photon–photon scattering and other effects of nonlinear optics in vacuum is an active area of experimental research, with current or planned technology beginning to approach the Schwinger limit.[5] It has already been observed through inelastic channels in SLAC Experiment 144.[6][7] However, the direct effects in elastic scattering have not been observed. As of 2012, the best constraint on the elastic photon–photon scattering cross section belongs to PVLAS, which reports an upper limit far above the level predicted by the Standard Model.[8] Observation of a cross section larger than that predicted by the Standard Model could signify new physics such as axions, the search of which is the primary goal of PVLAS and several similar experiments. Even the planned, funded ELI–Ultra High Field Facility, which will study light at the intensity frontier, is likely to remain well below the Schwinger limit[9] although it may still be possible to observe some nonlinear optical effects.[10] Such an experiment, in which ultra-intense light causes pair production, has been described in the popular media as creating a "hernia" in spacetime.[11]

## References

1. ^ F. Sauter, "Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs", Zeitschrift für Physik, 82 (1931) pp. 742–764. doi:10.1103/PhysRev.82.664
2. ^ W. Heisenberg and H. Euler, "Folgerungen aus der Diracschen Theorie des Positrons", Zeitschrift für Physik, 98 (1936) pp. 714-732. doi:10.1007/BF01343663 English translation
3. ^ M. Buchanan, "Thesis: Past the Schwinger limit", Nature Physics, 2 (2006) pp. 721. doi:10.1038/nphys448
4. ^ a b J. Schwinger, "On Gauge Invariance and Vacuum Polarization", Phys. Rev.,82 (1951) pp. 664–679. doi:10.1103/PhysRev.82.664
5. ^ S. S. Bulanov et al., "On the Schwinger limit attainability with extreme power lasers", Phys. Rev. Lett., 105 (2010) 220407. doi:10.1103/PhysRevLett.105.220407
6. ^ C. Bula et al., "Observation of Nonlinear Effects in Compton Scattering", Phys. Rev. Lett., 76 (1996) pp. 3116–3119. doi:10.1103/PhysRevLett.76.3116
7. ^ C. Bamber et al., "Studies of nonlinear QED in collisions of 46.6 GeV electrons with intense laser pulses", Phys. Rev. D, 60 (1999) 092204. doi:10.1103/PhysRevD.60.092004
8. ^ G. Zavattini et al., "Measuring the magnetic birefringence of vacuum: the PVLAS experiment", Accepted for publication in the Proceedings of the QFEXT11 Benasque Conference, [1]
9. ^ T. Heinzl, "Strong-Field QED and High Power Lasers", Plenary talk QFEXT11 Benasque Conference, [2][3]
10. ^ G. Yu. Kryuchkyan and K. Z. Hatsagortsyan, "Bragg Scattering of Light in Vacuum Structured by Strong Periodic Fields", Phys. Rev. Lett., 107 (2011) 053604. doi:10.1103/PhysRevLett.107.053604
11. ^ I. O'Neill, "A Laser to Give the Universe a Hernia?", Discovery News, (2011). [4]