by Yablon » Thu Dec 24, 2015 8:42 pm
Yablon wrote:I agree, there are two goals to be achieved in obtaining the Lorentz force from varying an action. First, is to get the structural relationship correct involving the acceleration, the field strength, and the velocity, on the physically fictitious supposition that e=m (which is a form of equivalence between the gravitational and electrical masses). Then, the other goal is to get the e and m in there in their real physical form. That may not be as hard as it seems, however. If there is a gauge field
in the action, one can always introduce that as
given that the covariant derivatives are
. Then, the mass needs to get in there somehow. The Klein-Gordon equation with a potential written as
may be a template for doing this, because that will implicitly contain an e/m (really,
) ratio in some of its terms. So too with the QED Lagrangian referred to by guest 1202 which can be manipulated to get out a ratio e/m. But on variation, that yields the Maxwell field equations, not the Lorentz equations of motion. I am liking the idea of using the field strength in the commutator form
, because that expressly represents the field strength as a curvature, and the Lorentz force should then emerge via the geodesic travels of electrons that are placed in this curvature. I will keep playing around with this.Jay
Amidst trying to complete a derivation of the Lorentz force law to mirror the geodesic derivation in
https://en.wikipedia.org/wiki/Geodesics ... _an_action, I went back to Einstein's original paper on General Relativity linked here
http://hermes.ffn.ub.es/luisnavarro/nue ... y_1916.pdf for convenience. First, I was reminded that this is the same path Einstein originally followed to obtain the geodesic motion, see section 9 where he starts with equation (20) and ends up with equation (22). Then, I went to section 20 where he deals with Maxwell's electrodynamics. There, Einstein shows both of Maxwell's equations and derives the Maxwell energy tensor. But of high interest, there is nothing mentioned about the Lorentz force law, much less deriving that by varying an action to obtain a geodesic equation for the Lorentz force. So that is very interesting. Also, keep in mind that Weyl developed gauge theory after GR in order to place EM theory on a similar geometric foundation. Weyl went so far as to identify the field strength
with a gauge space curvature akin to the Reimann tensor
for spacetime curvature based on non-commuting covariant derivatives. But I still to this day do not recall seeing a derivation which brings this program to its logical conclusion by showing some direct geodesic origin for the Lorentz force law just like that used for the geodesics in GR. If in fact such a derivation has still not been done (and again, maybe I just missed it), then it important that this aspect of Weyl's program finally be consummated. So, even though I started this to help prove that I am correctly using the single valuedness of electron wavefunctions to derive the half-integer charge fractions after guest1202 probed me on this point, I now think this derivation has quite independent merit. The most important merit is that it will give us an electrodynamic equivalent to the metric relationship
that is the starting point for the gravitational geodesic relationship. Especially given the e/m issues that Joy Christian raised, getting this done right could shed some new light on the nature of mass, because the inertial mass term in the Lorentz force law is is the one thing that does not seem to fit with everything else, because one way in which all the other interactions differ from gravitation is that only in gravitation is the "interaction mass" equal the "inertial mass." Keep in mind that Joy's
gambit is a simplification that fictionalizes the problem by removing this annoying but very important complication from the mix. So I am going to set my other work aside for a little while, until I find the way to do this. Indeed, now that I raise this, I realize that in the back of my mind I have always wondered how to expose the geodesic nature of the Lorentz force law, without resort to 5-D Kaluza-Klein where I have seen this done and did it myself some seven or eight years ago.
Jay
[quote="Yablon"]I agree, there are two goals to be achieved in obtaining the Lorentz force from varying an action. First, is to get the structural relationship correct involving the acceleration, the field strength, and the velocity, on the physically fictitious supposition that e=m (which is a form of equivalence between the gravitational and electrical masses). Then, the other goal is to get the e and m in there in their real physical form. That may not be as hard as it seems, however. If there is a gauge field [tex]A^\mu[/tex] in the action, one can always introduce that as [tex]eA^\mu[/tex] given that the covariant derivatives are [tex]D_\mu=\partial_\mu +ieA_\mu[/tex]. Then, the mass needs to get in there somehow. The Klein-Gordon equation with a potential written as [tex]-m^2\phi=g_{\mu\nu}D^\mu D^\nu \phi[/tex] may be a template for doing this, because that will implicitly contain an e/m (really, [tex]e^2/m^2[/tex]) ratio in some of its terms. So too with the QED Lagrangian referred to by guest 1202 which can be manipulated to get out a ratio e/m. But on variation, that yields the Maxwell field equations, not the Lorentz equations of motion. I am liking the idea of using the field strength in the commutator form [tex]F_{\mu\nu}\psi=i[D_\mu , D_\nu]\Psi[/tex], because that expressly represents the field strength as a curvature, and the Lorentz force should then emerge via the geodesic travels of electrons that are placed in this curvature. I will keep playing around with this.Jay[/quote]
Amidst trying to complete a derivation of the Lorentz force law to mirror the geodesic derivation in https://en.wikipedia.org/wiki/Geodesics_in_general_relativity#Deriving_the_geodesic_equation_via_an_action, I went back to Einstein's original paper on General Relativity linked here http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_GRelativity_1916.pdf for convenience. First, I was reminded that this is the same path Einstein originally followed to obtain the geodesic motion, see section 9 where he starts with equation (20) and ends up with equation (22). Then, I went to section 20 where he deals with Maxwell's electrodynamics. There, Einstein shows both of Maxwell's equations and derives the Maxwell energy tensor. But of high interest, there is nothing mentioned about the Lorentz force law, much less deriving that by varying an action to obtain a geodesic equation for the Lorentz force. So that is very interesting. Also, keep in mind that Weyl developed gauge theory after GR in order to place EM theory on a similar geometric foundation. Weyl went so far as to identify the field strength [tex]F^{\mu\nu}[/tex] with a gauge space curvature akin to the Reimann tensor [tex]R^\alpha_{\beta\mu\nu}[/tex] for spacetime curvature based on non-commuting covariant derivatives. But I still to this day do not recall seeing a derivation which brings this program to its logical conclusion by showing some direct geodesic origin for the Lorentz force law just like that used for the geodesics in GR. If in fact such a derivation has still not been done (and again, maybe I just missed it), then it important that this aspect of Weyl's program finally be consummated. So, even though I started this to help prove that I am correctly using the single valuedness of electron wavefunctions to derive the half-integer charge fractions after guest1202 probed me on this point, I now think this derivation has quite independent merit. The most important merit is that it will give us an electrodynamic equivalent to the metric relationship [tex]ds^2=g_{\mu\nu}dx^\mu dx^\nu[/tex] that is the starting point for the gravitational geodesic relationship. Especially given the e/m issues that Joy Christian raised, getting this done right could shed some new light on the nature of mass, because the inertial mass term in the Lorentz force law is is the one thing that does not seem to fit with everything else, because one way in which all the other interactions differ from gravitation is that only in gravitation is the "interaction mass" equal the "inertial mass." Keep in mind that Joy's [tex]e=m[/tex] gambit is a simplification that fictionalizes the problem by removing this annoying but very important complication from the mix. So I am going to set my other work aside for a little while, until I find the way to do this. Indeed, now that I raise this, I realize that in the back of my mind I have always wondered how to expose the geodesic nature of the Lorentz force law, without resort to 5-D Kaluza-Klein where I have seen this done and did it myself some seven or eight years ago.
Jay