June 2015 - mathfun@mailman.xmission.com

[math-fun] Reply to Salamin re Stern-Gerlach.
by Warren D Smith 04 Jun '15

04 Jun '15

WDS previously: Let the field be of form (Bx,By,Bz) = (0, 0, K*x) ES: Whoa there!? Your magnetic field fails to satisfy the Maxwell equation div B = 0. WDS: It *does* satisfy div B=0. [It is not however a vacuum static solution -- this B-field arises from a constant current density in the y-direction: curl B = (0,-K,0). That's fine.]

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[math-fun] A polarization puzzle - my solution
by Eugene Salamin 04 Jun '15

04 Jun '15

Here is a puzzle concerning the optics of polarized light. Every state of polarization has its opposite. For linear polarization, it's linear but rotated 90 degrees. For circular polarization, it's circular with opposite helicity. For general elliptic polarization, it's elliptic with the ellipse rotated 90 degrees, and the helicity reversed. On the Poincaré sphere, opposite states of polarization are represented by diametrically opposite points. The puzzle is to construct an optical device that reverses the polarization state. For any input, the output is the opposite polarization. Or, prove that it can't be done. I need to describe some of the machinery of polarization optics. The electric field of a plane wave propagating in the z direction is x component: Re(Ex exp(i(kz - omega t)), y component: Re(Ey exp(i(kz - omega t)). Ex and Ey are complex, encoding a magnitude and phase shift. It is crucial to the proof that these amplitudes are complex. The input and output beams may be in different directions. We pick coordinates in the transverse planes, (x,y) for the input beam, (x',y') for the output beam. The state of polarization is specified by the ratio P = Ey/Ex. If P is real, the x and y components are in phase, and the light is linearly polarized. If P = +-i, the components are equal in magnitude and 90 degrees out of phase, and the light is circularly polarized. Topologically, the values P form a sphere, the Poincaré sphere. When presenting my puzzle to optical science people, it's convenient to use the Poincaré sphere to make precise the meaning of "opposite", but I don't need to make use of it here. The output amplitudes are related to the input amplitudes by a linear transformation. [ Ex' ] = [ Jxx Jxy ] [ Ex ][ Ey' ] [ Jyx Jyy ] [ Ey ]. This matrix J is known as the Jones matrix. My Jones matrix must take every polarization state P to its opposite P'. I don't need to say exactly what "opposite" means. The only property I need is that no state is its own opposite. An eigenvector of J satisfies Ey'/Ex' = Ey/Ex. A polarization state P is preserved by J if and only if (Ex,Ey) with P = Ey/Ex is an eigenvector. But my J preserves no polarization state; it has no eigenvectors. This contradicts a theorem of linear algebra that every linear operator on a finite dimensional vector space over an algebraically closed field has at least one eigenvector. It follows that every optical system must preserve at least one polarization state. -- Gene

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[math-fun] A polarization puzzle - my solution (Eugene Salamin)
by Warren D Smith 04 Jun '15

04 Jun '15

Salamin's proof is stronger than mine in the sense that it works for any linear transformation as the effect of the optics; mine only works for "normal" linear transformations. Both are valid for quantum mechanics. Meeker's objection to our proofs is somewhat scary. Both my & Salamin's impossibility proofs seem valid provided the optical device is "passive" in the sense that it sort of acts like a potential in the Schrodinger equation, rather than being part of the wavefunction -- i.e. the state of the optical device itself is assumed unaltered. Meeker's objection pertains if that state is allowed to alter. I think Meeker is correct if that were allowed.

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[math-fun] Reply to Wechsler re polarization "demon"
by Warren D Smith 04 Jun '15

04 Jun '15

You cannot measure the polarization state of a photon. So your demon is impossible. More generally, see the "no cloning theorem" in quantum mechanics.

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[math-fun] Can a strong magnetic field create electron-positron pairs?
by Warren D Smith 04 Jun '15

04 Jun '15

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)

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[math-fun] Salamin polarization puzzle. Impossibility proof.
by Warren D Smith 04 Jun '15

04 Jun '15

OK, so Salamin wants a L + b R to be transformed to c L + d R where a' c + b' d = 0 where prime denotes complex conjugation. Because quantum mechanics is linear and unitary, (c,d) has to be got by multiplying (a,b) by a unitary 2x2 matrix U. So the question becomes: Does there exist a 2x2 unitary matrix U which converts any complex 2-vector to one complex-orthogonal to it? If a, b both real, then a solution is, as I'd said, c=-b, d=a and the matrix is 2x2 rotation by 90. If a,b both imaginary, then same solution also works. The trouble is if a,b general complex then the real-imag cross terms + ai br - ar bi + ai br - ar bi = 2 (ai br - ar bi) kill you (i.e. in general this is nonzero) hence the proposed solution fails because the imaginary part of the dot product, is in general nonzero. We can PROVE impossibility as follows. Wlog U is diagonal since we may take the eigendecomposition, because unitary matrices are normal hence this yields a diagonal because any normal matrix is unitarily similar to a diagonal matrix. (Note, the "complex dot product" is invariant under unitary transformations.) But obviously, no diagonal U can work. QED. On 6/4/15, math-fun-request(a)mailman.xmission.com <math-fun-request(a)mailman.xmission.com> wrote: > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. Re: A polarization puzzle (Allan Wechsler) > 2. Re: A polarization puzzle (Fred Lunnon) > 3. Re: A polarization puzzle (James Propp) > 4. Re: A polarization puzzle (Mike Stay) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Thu, 4 Jun 2015 09:38:02 -0400 > From: Allan Wechsler <acwacw(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] A polarization puzzle > Message-ID: > <CADy-sGHBVqwuETFjYJB84biNFuL5VzmE_tyVH2JJd-a3uuMzmQ(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > This whole discussion has exposed an area of deep ignorance in my physical > knowledge. I find myself now not sure whether polarization is an ensemble > property of a whole cohort of photons, or whether it is something one can > attribute to a single photon, and how much "state" there is at each level. > After about ten minutes I declared myself utterly incapable of thinking > about the problem, and if someone has an elementary reference to point me > at, that would be greatly appreciated. (Or, if somebody could hazard a > math-fun-level explanation of the basic concepts, that would be even > better!) > > > On Thu, Jun 4, 2015 at 12:00 AM, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > >> A monochromatic beam of light may be resolved as the sum >> of two linearly polarised beams in perpendicular planes, their >> phases displaced by some constant angle p . The device >> must reverse this displacement, delaying (say) one linear >> component by angle 2 p . >> >> Now consider an individual photon in such an input beam. >> The corresponding output phase must somehow be smeared >> between values at separation 2 p , in general impossibly. >> So sticking my neck out, I conclude that >> >> *** Salamin's demon (or genie?) is nonexistent. *** >> >> Caveat: my regrettable ignorance concerning elementary physics >> in general and optics in particular has been only barely perceptibly >> ameliorated by wrestling with this engaging problem. >> But even if I have (once again) blown it above, at least I now >> understand circular polarisation: an ultimately simple business which >> previously appeared ineffably mysterious. >> >> Fred Lunnon >> >> >> On 6/1/15, Eugene Salamin via math-fun <math-fun(a)mailman.xmission.com> >> wrote: >> > Here is a puzzle concerning the optics of polarized light. >> > Every state of polarization has its opposite. For linear polarization, >> it's >> > linear but rotated 90 degrees. For circular polarization, it's >> > circular >> > with opposite helicity. For general elliptic polarization, it's >> > elliptic >> > with the ellipse rotated 90 degrees, and the helicity reversed. On the >> > Poincar? sphere, opposite states of polarization are represented by >> > diametrically opposite points. >> > The puzzle is to construct an optical device that reverses the >> polarization >> > state. For any input, the output is the opposite polarization. Or, >> prove >> > that it can't be done. >> > -- Gene >> > >> > _______________________________________________ >> > math-fun mailing list >> > math-fun(a)mailman.xmission.com >> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > ------------------------------ > > Message: 2 > Date: Thu, 4 Jun 2015 15:48:02 +0100 > From: Fred Lunnon <fred.lunnon(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] A polarization puzzle > Message-ID: > <CAN57YqsE4xX5t3SyFuDRBniemVfzmmS7cEy5Gj5HoMkqvFW2Uw(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > Join the club! I found this link enlightening, but it's surely only a > start > --- particularly regarding quantum-theoretical aspects (which I haven't > even attempted to address yet). > > https://en.wikipedia.org/wiki/Polarization_%28waves%29 > > WFL > > > > On 6/4/15, Allan Wechsler <acwacw(a)gmail.com> wrote: >> This whole discussion has exposed an area of deep ignorance in my >> physical >> knowledge. I find myself now not sure whether polarization is an ensemble >> property of a whole cohort of photons, or whether it is something one can >> attribute to a single photon, and how much "state" there is at each >> level. >> After about ten minutes I declared myself utterly incapable of thinking >> about the problem, and if someone has an elementary reference to point me >> at, that would be greatly appreciated. (Or, if somebody could hazard a >> math-fun-level explanation of the basic concepts, that would be even >> better!) >> >> >> On Thu, Jun 4, 2015 at 12:00 AM, Fred Lunnon <fred.lunnon(a)gmail.com> >> wrote: >> >>> A monochromatic beam of light may be resolved as the sum >>> of two linearly polarised beams in perpendicular planes, their >>> phases displaced by some constant angle p . The device >>> must reverse this displacement, delaying (say) one linear >>> component by angle 2 p . >>> >>> Now consider an individual photon in such an input beam. >>> The corresponding output phase must somehow be smeared >>> between values at separation 2 p , in general impossibly. >>> So sticking my neck out, I conclude that >>> >>> *** Salamin's demon (or genie?) is nonexistent. *** >>> >>> Caveat: my regrettable ignorance concerning elementary physics >>> in general and optics in particular has been only barely perceptibly >>> ameliorated by wrestling with this engaging problem. >>> But even if I have (once again) blown it above, at least I now >>> understand circular polarisation: an ultimately simple business which >>> previously appeared ineffably mysterious. >>> >>> Fred Lunnon >>> >>> >>> On 6/1/15, Eugene Salamin via math-fun <math-fun(a)mailman.xmission.com> >>> wrote: >>> > Here is a puzzle concerning the optics of polarized light. >>> > Every state of polarization has its opposite. For linear >>> > polarization, >>> it's >>> > linear but rotated 90 degrees. For circular polarization, it's >>> > circular >>> > with opposite helicity. For general elliptic polarization, it's >>> > elliptic >>> > with the ellipse rotated 90 degrees, and the helicity reversed. On >>> > the >>> > Poincar? sphere, opposite states of polarization are represented by >>> > diametrically opposite points. >>> > The puzzle is to construct an optical device that reverses the >>> polarization >>> > state. For any input, the output is the opposite polarization. Or, >>> prove >>> > that it can't be done. >>> > -- Gene >>> > >>> > _______________________________________________ >>> > math-fun mailing list >>> > math-fun(a)mailman.xmission.com >>> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> > >>> >>> _______________________________________________ >>> math-fun mailing list >>> math-fun(a)mailman.xmission.com >>> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > > ------------------------------ > > Message: 3 > Date: Thu, 4 Jun 2015 10:53:06 -0400 > From: James Propp <jamespropp(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] A polarization puzzle > Message-ID: > <CA+G9J-cWv9XbZYFwjKKqKnY09BPZh83Bqr9BaQjJYP5Ke8c7OA(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > While we're at it, Feynman figured out something about polarization of > light and complex numbers that he seemed to view as the best illustration > of why complex numbers are "real". An attempt at a lay explanation appears > in Jim Ottaviani and Leland Myrick's 2011 graphic biography "Feynman". I > couldn't make sense of it. > > Jim Propp > > On Thu, Jun 4, 2015 at 10:48 AM, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > >> Join the club! I found this link enlightening, but it's surely only a >> start >> --- particularly regarding quantum-theoretical aspects (which I haven't >> even attempted to address yet). >> >> https://en.wikipedia.org/wiki/Polarization_%28waves%29 >> >> WFL >> >> >> >> On 6/4/15, Allan Wechsler <acwacw(a)gmail.com> wrote: >> > This whole discussion has exposed an area of deep ignorance in my >> physical >> > knowledge. I find myself now not sure whether polarization is an >> > ensemble >> > property of a whole cohort of photons, or whether it is something one >> > can >> > attribute to a single photon, and how much "state" there is at each >> level. >> > After about ten minutes I declared myself utterly incapable of thinking >> > about the problem, and if someone has an elementary reference to point >> > me >> > at, that would be greatly appreciated. (Or, if somebody could hazard a >> > math-fun-level explanation of the basic concepts, that would be even >> > better!) >> > >> > >> > On Thu, Jun 4, 2015 at 12:00 AM, Fred Lunnon <fred.lunnon(a)gmail.com> >> wrote: >> > >> >> A monochromatic beam of light may be resolved as the sum >> >> of two linearly polarised beams in perpendicular planes, their >> >> phases displaced by some constant angle p . The device >> >> must reverse this displacement, delaying (say) one linear >> >> component by angle 2 p . >> >> >> >> Now consider an individual photon in such an input beam. >> >> The corresponding output phase must somehow be smeared >> >> between values at separation 2 p , in general impossibly. >> >> So sticking my neck out, I conclude that >> >> >> >> *** Salamin's demon (or genie?) is nonexistent. *** >> >> >> >> Caveat: my regrettable ignorance concerning elementary physics >> >> in general and optics in particular has been only barely perceptibly >> >> ameliorated by wrestling with this engaging problem. >> >> But even if I have (once again) blown it above, at least I now >> >> understand circular polarisation: an ultimately simple business which >> >> previously appeared ineffably mysterious. >> >> >> >> Fred Lunnon >> >> >> >> >> >> On 6/1/15, Eugene Salamin via math-fun <math-fun(a)mailman.xmission.com> >> >> wrote: >> >> > Here is a puzzle concerning the optics of polarized light. >> >> > Every state of polarization has its opposite. For linear >> polarization, >> >> it's >> >> > linear but rotated 90 degrees. For circular polarization, it's >> circular >> >> > with opposite helicity. For general elliptic polarization, it's >> >> > elliptic >> >> > with the ellipse rotated 90 degrees, and the helicity reversed. On >> the >> >> > Poincar? sphere, opposite states of polarization are represented by >> >> > diametrically opposite points. >> >> > The puzzle is to construct an optical device that reverses the >> >> polarization >> >> > state. For any input, the output is the opposite polarization. Or, >> >> prove >> >> > that it can't be done. >> >> > -- Gene >> >> > >> >> > _______________________________________________ >> >> > math-fun mailing list >> >> > math-fun(a)mailman.xmission.com >> >> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >> > >> >> >> >> _______________________________________________ >> >> math-fun mailing list >> >> math-fun(a)mailman.xmission.com >> >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >> >> > _______________________________________________ >> > math-fun mailing list >> > math-fun(a)mailman.xmission.com >> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > ------------------------------ > > Message: 4 > Date: Thu, 4 Jun 2015 08:47:33 -0700 > From: Mike Stay <metaweta(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] A polarization puzzle > Message-ID: > <CAKQgqTZ9-Nt4StxNobOH+M36yK1jqWJhj4LeqOcB6PXyKv3-Lg(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > Spin is a property of individual quantum particles. It arises from > considering angular momentum in a quantized setting. There are spin-0 > particles like pions, spin-1/2 particles like electrons, protons, and > neutrons, spin-1 particles like photons, spin-3/2 particles like Delta > baryons, and spin-2 particles like gravitons. We can also treat > collections of particles like nuclei as having spin. > > A spin-0 particle has only one polarization. A spin-1/2 particle has > two polarizations, +1/2 = "up" and -1/2 = "down". The Stern-Gerlach > experiment shoots a beam of electrons through a magnetic field at a > detector; you get two bright dots. A spin-1 particle has three > polarizations, -1, 0, 1. In general, a spin s particle has 2s+1 > polarizations: -s, -s+1, ..., s-1, s. > > Pauli predicted the neutrino in part because a neutron can decay via > the beta process to a proton and an electron---but that meant the > total spin for the neutron should be 1, not 1/2 as had already been > observed. Pauli added the neutrino so that the spin of a neutron plus > a neutrino would equal the spin of a proton plus an electron. > > In quantum mechanics, we have a Hilbert space with one dimension for > each possible outcome. A photon is a spin-1 particle, but outside of > a waveguide, we never see the 0 polarization, so the state of an > individual photon lives in a 2-dimensional Hilbert space. We can > write the polarization state of a free photon as a complex-weighted > sum of the left and right circular polarizations: a|L> + b|R>. The > condition |a|^2 + |b|^2 = 1 reduces the four degrees of freedom (two > per complex coefficient) to three; this is what's known as the Riemann > sphere. > > The linear polarizations are |H> = (|L> + i|R>)/sqrt(2) and |V> = (|L> > - i|R>)/sqrt(2). The inner product of the two is denoted <H|V>, where > <H| = <L| - i*<R| denotes the conjugate transpose of |H>. The result > is (1*1 + (-i)*(-i))/2 = 0, so they're orthogonal, whereas <H|H> = > <V|V> = (1*1 + (-i)*i)/2 = 1. > > On Thu, Jun 4, 2015 at 6:38 AM, Allan Wechsler <acwacw(a)gmail.com> wrote: >> This whole discussion has exposed an area of deep ignorance in my >> physical >> knowledge. I find myself now not sure whether polarization is an ensemble >> property of a whole cohort of photons, or whether it is something one can >> attribute to a single photon, and how much "state" there is at each >> level. >> After about ten minutes I declared myself utterly incapable of thinking >> about the problem, and if someone has an elementary reference to point me >> at, that would be greatly appreciated. (Or, if somebody could hazard a >> math-fun-level explanation of the basic concepts, that would be even >> better!) >> >> >> On Thu, Jun 4, 2015 at 12:00 AM, Fred Lunnon <fred.lunnon(a)gmail.com> >> wrote: >> >>> A monochromatic beam of light may be resolved as the sum >>> of two linearly polarised beams in perpendicular planes, their >>> phases displaced by some constant angle p . The device >>> must reverse this displacement, delaying (say) one linear >>> component by angle 2 p . >>> >>> Now consider an individual photon in such an input beam. >>> The corresponding output phase must somehow be smeared >>> between values at separation 2 p , in general impossibly. >>> So sticking my neck out, I conclude that >>> >>> *** Salamin's demon (or genie?) is nonexistent. *** >>> >>> Caveat: my regrettable ignorance concerning elementary physics >>> in general and optics in particular has been only barely perceptibly >>> ameliorated by wrestling with this engaging problem. >>> But even if I have (once again) blown it above, at least I now >>> understand circular polarisation: an ultimately simple business which >>> previously appeared ineffably mysterious. >>> >>> Fred Lunnon >>> >>> >>> On 6/1/15, Eugene Salamin via math-fun <math-fun(a)mailman.xmission.com> >>> wrote: >>> > Here is a puzzle concerning the optics of polarized light. >>> > Every state of polarization has its opposite. For linear >>> > polarization, >>> it's >>> > linear but rotated 90 degrees. For circular polarization, it's >>> > circular >>> > with opposite helicity. For general elliptic polarization, it's >>> > elliptic >>> > with the ellipse rotated 90 degrees, and the helicity reversed. On >>> > the >>> > Poincar? sphere, opposite states of polarization are represented by >>> > diametrically opposite points. >>> > The puzzle is to construct an optical device that reverses the >>> polarization >>> > state. For any input, the output is the opposite polarization. Or, >>> prove >>> > that it can't be done. >>> > -- Gene >>> > >>> > _______________________________________________ >>> > math-fun mailing list >>> > math-fun(a)mailman.xmission.com >>> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> > >>> >>> _______________________________________________ >>> math-fun mailing list >>> math-fun(a)mailman.xmission.com >>> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > -- > Mike Stay - metaweta(a)gmail.com > http://www.cs.auckland.ac.nz/~mike > http://reperiendi.wordpress.com > > > > ------------------------------ > > Subject: Digest Footer > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > ------------------------------ > > End of math-fun Digest, Vol 148, Issue 10 > ***************************************** > -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] amazing computer-created black hole videos
by Warren D Smith 04 Jun '15

04 Jun '15

https://www.youtube.com/watch?v=Qg6PwRI2uS8 https://www.youtube.com/watch?v=ENd8Sz0AFOk searching youtube for "black hole SXS" finds more. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Replies to Salamin
by Warren D Smith 03 Jun '15

03 Jun '15

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.

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[math-fun] Are magnetars baloney? What is the maximum allowed magnetic field?
by Warren D Smith 03 Jun '15

03 Jun '15

If a volume of vacuum contains a magnetic field B greater than m_e c^2 / mu_e = 8.82x10^9 tesla (here mu_e is magnetic moment of electron) then it becomes energetically advantageous for the vacuum to break down spontaneously, creating electron-positron pairs, because the energy mu_e B is enough to pay for the energy m_e c^2. Note due to angular momentum and charge conservation the e+e- pair have opposite spins, and opposite charges, which means they have aligned magnetic moments. I conclude it should be impossible to sustain any magnetic field larger than this, in vacuum, for longer than about 10^(-16) seconds. OK, sounds straightforward enough. BUT, (1) I have not seen this "maximum magnetic field" discussed before. (Although the analogous "maximum electric field" is often discussed, it is sometimes called the "Schwinger limit" http://en.wikipedia.org/wiki/Schwinger_limit and has value of order 1.3*10^18 volts/meter. ) (2) http://www.physics.mcgill.ca/~pulsar/magnetar/main.html claims that the "magnetar" (very magnetic type of neutron star) "SGR 1806-20" has a surface magnetic field of about 23 times my upper bound, namely 2*10^11 tesla. There are also are two other magnetars listed with claimed surface magnetic fields each about 8 times my upper bound. Contradiction. I conclude that either there is some reason I do not understand for why magnetic fields much larger than my upper bound can exist for 10000 years -- or -- the magnetar astronomers are pressurized with excrement. Or both. I have looked into this and essentially all the claimed evidence over the last 25 years for the huge magnetic fields around magnetars is based on Maxwell's equations. Maxwell equations, are, however, simply wrong when the fields are this large. (The Euler-Kockel nonlinear equations are less wrong, but probably still very inadequate.) Therefore, everything, ever, that the magnetar astronomers said about these huge fields, is suspect. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Are magnetars baloney? What is the maximum allowed magnetic field?
by Warren D Smith 03 Jun '15

03 Jun '15

> 1.Naohiro Kanda > http://arxiv.org/pdf/1106.0592v1.pdf > claims the QED calculation of light by light scattering cross section > was done wrong by everybody until now (year 2011) when Kanda finally did > it right getting the right answer for the first time, apparently > nobody ever checked it during > 70 years, no matter how many times it appears in no matter how many > textbooks, > except once when Karplus+Kroll did it wrong out of a > too-strong desire to duplicate the previous wrong result. > > Unless Kanda is wrong. --apparently Kanda was wrong: refuted refute: http://arxiv.org/pdf/1111.6126v2.pdf --About my hypothesis a large magnetic field would create pairs in vacuum: the lowest energy of an electron in a magnetic field is supposedly m*c^2 independent of the field.... but I suspect that if we put, not only a constant magnetic field, but also a compact confining-box, then that solution becomes forbidden. The non-forbidden energies then would depend on the field roughly as I wanted... except the discrepancy from m*c^2 has the wrong sign?... -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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