[math-fun] Can a strong magnetic field create electron-positron pairs?
1. If magnetic moment mu_e of the electron works like it says in freshman physics textbooks, then YES, since the energy of stationary electron in a B-field is +-mu_e * B if aligned/antialigned, and this can have either sign and be made arbitrarily larger than m*c^2. 2. But the exact solution of Dirac equation in a constant magnetic field (EQ21 of http://arxiv.org/abs/0705.4275), yields the permitted "Landau energy levels" E[n] obeying E[n]^2 = m^2 * c^4 + c^2 * pz^2 + 2*n*hbar*c^2*e*B where n=0,1,2,3,... and pz is the momentum of electron in the z-direction (magnetic field also in that direction and B>=0) indicates NO, since energy always >=m*c^2 if nonnegative. Here "energy" is defined by E=h*f via the frequency f of the solution (which depends on time t only via an overall factor exp(2*pi*i*t*f)], and all these levels have 2 states each, except the n=0 level has only 1 state. This would indicate magnetic moment of electron does NOT behave like in freshman physics textbooks since it cannot result in a negative energy contribution on top of m*c^2, no matter what the field and alignment are. 3. The pair creation rate (per volume per time) is a Lorentz invariant and must be a function of the two fundamental invariants E^2-B^2 and E.B. So in the "equal crossed-field" case where both invariants 0, the creation rate must be 0. This strongly suggests that electron magnetic moment does NOT behave like in freshman physics textbooks. Those textbooks would have said the electron could get an arbitrarily negative energy contribution from the B and its magnetic moment. 4. It would be more realistic to solve Dirac equation in a situation with finite magnetic field energy, localized electrons, and total energy (of the whole system -- the electron plus all the other stuff that generates the magnetic field) all assessed. Might that yield a different conclusion? 5. One situation like (4) is the hydrogen atom's "hyperfine structure" https://en.wikipedia.org/wiki/Hyperfine_structure which is claimed to be an energy perturbation due to the magnetic moments of the proton and electron interacting, which can be OF EITHER SIGN according to their picture, plus is claimed to work just like the freshman physics textbooks say. Why is it of either sign? Why is it not always positive like in the Landau level calculation? Because either (a) this calculation is a lie, or (b) my worries in (4) really do matter. I'm guessing it is not a lie since supposedly hydrogen spectra have confirmed experiment-theory match to huge precision. For hydrogen ground state the hyperfine splitting is 5.9*10^(-6) eV versus the ionization energy 13.6 eV, a factor of 2 million smaller. I think they have more than enough precision to handle that. 6. Another situation like (4) is http://arxiv.org/abs/hep-th/9503051 where he solves Dirac equation in a 4-parameter potential, see EQs 100-106 where M=m*c^2 and N=integer of either sign. Anyhow, it looks like when the monopole g is increased the energy levels always rise, which if so confirms the notion from (2) that you cannot create pairs using a magnetic field. 7. Another situation like (4) is http://arxiv.org/pdf/1401.7144.pdf and it appears their energy levels usually monotonically rise as you ramp up magnetic field (e.g. pictures at end), but not always -- there are some dips, indeed the top curve in the last figure even appears to be monotone decreasing. 8. Abdulaziz D. Alhaidari, Hocine Bahlouli, Ahmed Jellal: Confined Dirac Particles in Constant and Tilted Magnetic Field, http://arxiv.org/abs/1202.5226 EQs 19 & 22 give energy levels where R- and R+ are two lengths and n=0,1,2,3,... and their energy levels are always >=m*c^2 if nonnegative. Their y-coordinate is cyclic, and B-field lies in the xz plane. 9. http://streaming.ictp.trieste.it/preprints/P/96/231.pdf claims to solve hydrogen atom in external magnetic field, positive energy levels on page 8 always increase as function of applied field. 10. RE Cutkovsky & KC Wang: Relativistic Landau levels in a hyperspherical finite cavity, Czech J Physics B 38,8 (1988) 825-836. Looks highly relevant for future reading. 11. Another situation like (4) would be a cylindrical box with walls impermeable to electrons axis-aligned with a constant B-field. What are the energy levels of electron in such a box? (This model still is not fully satisfactory since it does not assess the total energy of the whole system, unlike the hydrogen atom hyperfine calculation -- but it is better than the plain Landau calculation.) CONCLUSION: Sorry, I haven't got one. There are indications in both directions. There seem to be more indications in the mainstream direction that magnetic field cannot create pairs, but as far as I can see the question was never clearly settled. It was just falsely claimed it was clearly settled, which is different. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
AE Shabad & VV Usov contended in a number of papers that a large-enough magnetic field would create pairs because the positronium "atom" would be compressed by the magnetic field, and then have an arbitrarily negative Coulomb energy. http://arxiv.org/abs/hep-th/0605020 http://arxiv.org/abs/astro-ph/0601542 http://arxiv.org/abs/hep-th/0603070 http://arxiv.org/abs/hep-th/0512236 Their results were happily used in http://arxiv.org/pdf/astro-ph/0610441v2.pdf and at least 20 other papers continuing thru year 2014. Meanwhile CN Leung & S-Y Wang claimed to have refuted all that: http://arxiv.org/abs/0901.3413 (All papers refereed, published, etc.) Who is right? I do not find ANY of these convincing/understandable. The alleged exact solution of a hydrogenic atom in a magnetic field by CC Bernido, NS Lim, MV Carpio-Bernido (1996) http://streaming.ictp.trieste.it/preprints/P/96/231.pdf [energy levels on page 8; later apparently published as C.C. Bernido, M.V. Carpio-Bernido and N.S. Lim: "Sum-Over-Histories for the Dirac Electron of a Hydrogenic Atom in Superstrong Magnetic Fields," in Frontiers in Quantum Physics, eds. S.C. Lim, R. Abd-Shukor, K. H. Kwek (Springer-Verlag, Singapore, 1998) p. 251. ] seems however to flat out contradict the claims of Shabad+Usov about the positronium "atom" and was cited by none of those papers, despite all of them coming way after 1996. Apparently theoretical physics is a write-only field, and any claim is ok. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
Here is an argument, which seems absolutely indisputable, that an INHOMOGENOUS magnetic field can and will create pairs. Let the field be of form (Bx,By,Bz) = (0, 0, K*x) incidentally arising from vector potential (Ax,Ay,Az) = (0, K*x*x/2, 0). We know from the Stern-Gerlach effect that an electron (positron also) in such a field will be sucked in either the +x or -x direction depending on its spin. And the force sucking it will be F = mu_e * K. And this is experimentally confirmed, not mere theorizing. Therefore, if the distance L we can go in the x direction obeys L*F > m*c^2 then the total energy from all that sucking is enough to create an electron. A pair could be created at x=0, then suck distance L, and then have regained enough energy to pay for the pair creation. Therefore, pairs will be created. And if a distance L <= hbar/(m*c) suffices, then this creation process should be very fast. I don't see how anybody can possibly argue with this. Now any real-world magnetic field is NOT uniform, and it must go to 0 someplace. Therefore, any real magnetic field with enough strength, specifically if max strength B>m*c^2/mu_e, will generate pairs at a nonzero rate. Then the only question, it seems to me, is "what is that rate?" And how does the rate depend on the specific geometry of the field, for example on grad|B|? Optimally, we would express the pair generation rate (pairs/second) in terms of some triple integral of some function (or functional) of the magnetic field. -- Warren D. Smith
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Warren D Smith