I found this paper, the most helpful paper I saw for trying to comprehend this. Barry Holstein seems to be good at explaining puzzling physics to the stupid (i.e. me) in mucho papers for the Amer.J.Physics, the journal put out by the Amer. Assoc of Physics Teachers which is big in trying for good writing and clarity. [At least one physics professor told me AJP is "not a real journal" and expressed contempt for it. But my experiences reading this journal have usually been good, in fact in the top class of physics journals, and I don't think his contempt was warranted. Just my opinion. I also like the J. Math'l Physics and Communications in Math'l Physics one hell of a lot more than most physicists I generally encounter; they ooze contempt for them too.] Barry R. Holstein: Strong field pair production, American Journal of Physics 67,6 (1999) 499-507. [Other AJP pieces by Holstein: Van der Waals interaction: AJP 69,4 (2001) 441-449; 69,12 (2001) 1283. Klein Paradox, AJP 66,6 (1998) 507- Second Born approximation and Coulomb scattering, AJP 75,6 (200-7) 537-539 Graviton physics, AJP 74,11 (2006) 1002-1011. How large is the "natural" magnetic moment? AJP 74,12 (2006) 1104-1111 Effective interactions and the hydrogen atom, AJP 72,3 (2004) 333-334. Neutrino physics AJP 68,1 (2000) 15-32: update, AJP 72,1 (2004) 18-24. Anomalies in quantum mechanics: The 1/r^2 potential, AJP 70,5 (2002) 513-. Gyroscope precession and general relativity, AJP 69,12 (2001) 1248-1256 Bound states of a uniform spherical charge distribution - revisited! AJP 68,7 (2000) 640- Blue skies and effective interactions, AJP 67,5 (1999) 422-427 Answer to question #76. Neutrino mass and helicity, AJP 66,12 (1998) 1045 The harmonic oscillator propagator, AJP 66,7 (1998) 583-589 Answer to question #4. Is there a physics application that is best analyzed in terms of continued fractions? AJP 65,12 (1997) 1133-1135. Diagonalization of the Dirac equation: An alternative approach, AJP 65,6 (1997) 519-522. The linear potential propagator, AJP 65,5 (1997) 414-418 Understanding alpha decay, AJP 64,8 (1996) 1061-1071. Quantum-Mechanics In Momentum-Space - The Coulomb System, AJP 63,8 (1995) 710-716. Quark Masses and Binding-Energy in a Proton - Answer to Question Number-2a, AJP 63,1 (1995) 14. Anomalies for Pedestrians, AJP 61,2 (1993) 142-147. Variations on the Aharonov-Bohm Effect, AJP 59,12 (1991) 1080-1085. Effective Lagrangians and Quantum-Mechanics - The Index of Refraction, AJP 57,2 (1989) 142-148. The Adiabatic Propagator, AJP 57,8 (1989) 714-720. The Adiabatic Theorem and Berry's Phase, AJP 57,12 (1989) 1079-1084 Gauge-Invariance and Quantization, AJP 56,5 (1988) 425-429 Temperature Measured by a Uniformly Accelerated Observer, AJP 52,8 (1984) 730. Bound-States, Virtual States, and Non-Exponential Decay via Path-Integrals, AJP 51,10 (1983) 897-901. Barrier Penetration via Path-Integrals, AJP 50,9 (1982) 833-839. Path-Integrals and the WKB Approximation, AJP 50,9 (1982) 829-832. Elementary Derivation Of The Radiation-Field From An Accelerated Charge, AJP 49,4 (1981) 346.] Holstein also follows: Fritz Sauter: U"ber das Verhalten eines Elektrons im Homogenen Elektrischen Feld Nach der Relativistischen Theorie Diracs, Zeitschrift Physik 69,11-12 (1931) 742-764 Zum Kleinschen Paradoxon, Zeitschrift Physik 73,7-8 (1932) 547-552 Sauter actually was probably the first person who understood the "Schwinger limit," and in my opinion better than Schwinger ever did 20 years later. The "Klein paradox" is the fact that, when you compute 1 dimensional plane wave transmission thru a "potential barrier" with Dirac's equation (the "barrier" actually might be a downstep, anyhow it is some localized fields) you can end up with reflection coefficient R and transmission coefficient T, with |R|>1. This historically was a big puzzle. Eventually interpreted as pair production. Sauter found exact solution of Dirac equation in an electrostatic potential of the form (V/2)*tanh(a*x/2) which steps V volts up, and avoids issues that might have worried somebody about non-smooth potentials with corners. The max electric field is a*V/4. Using this exact solution, we then can consider the "pre-filled Dirac sea" of electron states with negative energies in the field-free vacuum (all these states are exactly-known plane-wave states) and regard them as the input to Sauter. Sauter's formula for |R|^2+|T|^2-1, times an "attempt rate" aka "input current" for each state, can be summed over all Dirac-sea states, and regarded as a particle production rate. If the Dirac sea were restricted to a finite-volume box this actually effectively is a finite set of states to sum over (the ones with too-negative energies do not matter since they cannot be promoted to positive energies by the available step-size V). So it seems to me that, at least in principle, one can exactly obtain the pair production rate from the Sauter electric field, in this way. The details can get complicated. But that is, in principle, a way to do the calculation. Now. Suppose we try the same thing, but for a magnetic field. Incidentally: At least for *classical* point charges: if you have a magnetic field in the z-direction, which is a function of x, which is 0 when x<0, and positive when x>0, and bounded below by a positive constant for all large-enough x, then any such field is a "perfect mirror" in the sense that electrons flying into it from the x<0 region, get reflected back out, angle of incidence=angle of reflection. The same "perfect mirror" statement also would have classically been true for Sauter's pure-electric field provided the incident electron did not have enough energy to surmount the barrier. Anyhow, suppose we have magnetic field in the z-direction which is +B when x>0 and -B when x<0. In that case, a "Landau level" n in the right halfspace, with spin parameter -, automagically transitions to a Landau level n with spin parameter +, in the left halfspace, whenever cross the x=0 plane. I.e. the spin reverses with respect to the B-field, but aside from that the Landau states stay the same whenever cross the plane. This corresponds to an energy step up (or down) by 1 level. Nonrelativistically this step size is constant hbar*e*B / (m*c) but relativistically it is nonconstant i.e. depends upon n and decreases for n large, ultimately to 0. We regard the "Dirac sea" in this scenario as consisting entirely of Landau levels. In that case we have simply a potential step, from the view of the two Dirac seas. If because B is large, the size of this step exceeds 2*m*c^2, then it seems to me we ought to get pair production since a negative energy state in sea#1, can have positive energy in sea #2. Furthermore, the rate of particle production should be calculable since there are only a finite set of Landau levels for which the step size exceeds the threshold; we can sum over all of them therefore. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)