My replies: 1. (a) It matters not how the electron is moving in its lowest Landau level. What does matter is that the energy of the lowest level exceeds the magnetic moment alignment energy. The energy cost of creating a pair not only never gets below zero, it always exceeds 2 mc^2. (b) What do you find unacceptable about solving the Dirac equation for an electron in a constant magnetic field? So go solve the problem in your more complicated configuration, and tell us how it transpires. 2. I would like to use * to denote complex conjugation, so I will rewrite what you said using juxtaposition for multiplication. "If I understand your problem aright (which I might not), you want a device that changes the state a L + b R into the state (b L - a R) P where P is a complex phase factor, whose value you do not care about so long as |P|=1." For two states to be orthogonal in the quantum mechanical sense, we require their hermitian inner product to be zero. The output state should be (a' L + b' R) P, where a* a' + b* b' = 0. (3). The everyday world (excluding some exotic particle physics) is time reversal invariant. Time reversal requires reversing the current in the Faraday coil. But we don't reverse the current, and that's how the Faraday rotator works. -- Gene From: Warren D Smith <warren.wds@gmail.com> To: math-fun@mailman.xmission.com Sent: Wednesday, June 3, 2015 12:02 PM Subject: [math-fun] Replies to Salamin 1. About pair creation in magnetic field, (a) the lowest Landau level has no circular motion, (b) I'm not sure if I believe the usual physicist's claims. I actually was looking at that same paper you were. If I am to dispute this, it seems like the source of the dispute must be the unrealistic model that the magnetic field is constant everywhere. More realistic is both a localized field and a localized electron. Would that yield pair creation? I'm not presently sure, but suspect it is possible to answer this question in a model involving a cylindrical box with walls impermeable to electrons, full of constant magnetic field; compare the lowest electron energy in both the no-field and field cases. If the latter is lower, then there should be pair creation. (c) I agree with you that in the crossed field E=B case, there is no pair creation. 2.about your polarization problem. Let L and R denote -- maybe I really should call them |L> and |R> but I won't -- photon in circular anticlockwise and clockwise helicity states, for definiteness with electric field pointing in +x-direction at time t=0, and photon moving in +z direction. If I understand your problem aright (which I might not), you want a device that changes the state a*L + b*R into the state (b*L - a*R) * P where P is a complex phase factor, whose value you do not care about so long as |P|=1. [If this is not what you want, then I'd like you to say what the heck you do want, expressed in this formalism please. If you cannot express your desires this way, then answer is: it is impossible.] The X linear-polarization state is X = (L+R) / sqrt(2). The Y linear-polarization state is Y = i * (R-L) / sqrt(2). So anyhow, using the basis L and R, it seems you want to apply the 2x2 matrix (0 P) (-P 0) to the coefficient column vector a,b. If P=1 this is just a 90-degree rotation matrix. This matrix (indeed for any P) is certainly allowed as a Schrodinger time-evolution, since being a rotation, it is unitary. So this indicates this device is allowable by quantum mechanics... I then am disputing Mike Stay's conclusion to the contrary, so one of us made a mistake. OK, assuming optimistically that he's the one who made the mistake, then the question becomes, what physical device will implement this linear transformation? Simply a mirror implements (0 1) (1 0). Suitable mix of sugars ought to be able to delay L by 1/2 a period more than it delays R (with the latter delay being integer #periods) causing the matrix (0 1) (-1 0) which is what we wanted. So, I claim that either the mirror+sugar+Goucher's "move it aside" trick solves the problem, or I am not understanding the problem in which case there probably is no solution (which is what Salamin has claimed). 3. Faraday effect violates what I said earlier, that Maxwell's equations are time-reversal invariant. The resolution of that paradox is Faraday's magnetic field is generated by moving electrons; time reverse and they move backward, magnetic field also backward. So it still works if your view is wide enough. But anyhow this is a loophole allowing mirror+faraday to work, even without "move aside" trick. As Salamin recognized. But he claims this is not a solution to his problem, which presumably is because I do not understand what his problem is asking for. Probably related is: I have no idea what "poincare sphere" means.