guest1202 wrote:Most advanced electrodynamics texts discuss such a Lagrangian, for example, Jackson's book 2nd edition, section 12.2
This is not the latest edition, but is all that I have on hand. I seem to recall a discussion in Barut's book, but don't have that on hand.
Joy Christian wrote:Just to add to my note above: The reason the derivation works in the case of gravity is the weak equivalence principle, giving a "non-flat" connection field.
If you set e = m in the EM case, then the derivation may go through with a kind of trick I suggested above. But then the derivation will not be physical.
Joy Christian wrote:Just to add to my note above: The reason the derivation works in the case of gravity is the weak equivalence principle, giving a "non-flat" connection field.
If you set e = m in the EM case, then the derivation may go through with a kind of trick I suggested above. But then the derivation will not be physical.
FrediFizzx wrote:Joy Christian wrote:Just to add to my note above: The reason the derivation works in the case of gravity is the weak equivalence principle, giving a "non-flat" connection field.
If you set e = m in the EM case, then the derivation may go through with a kind of trick I suggested above. But then the derivation will not be physical.
Torsion? As you know from the research we did it is complicated though.
Joy Christian wrote:FrediFizzx wrote:Joy Christian wrote:Just to add to my note above: The reason the derivation works in the case of gravity is the weak equivalence principle, giving a "non-flat" connection field.
If you set e = m in the EM case, then the derivation may go through with a kind of trick I suggested above. But then the derivation will not be physical.
Torsion? As you know from the research we did it is complicated though.
I don't think torsion is necessary here. I think the above comments by Jay is on the right track. I am confident that by setting e = m the derivation is possible. I am less confident, however, that it can be modified or reinterpreted as physical. But one can find out only by trying. Let us hope Jay succeeds, at least with the derivation.
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
FrediFizzx wrote:The wikipedia page for Lorentz Force in the Lagrangian section says,
"The action is the relativistic arclength of the path of the particle in space time, minus the potential energy contribution, plus an extra contribution which quantum mechanically is an extra phase a charged particle gets when it is moving along a vector potential."
Does that help at all?
Yablon wrote:I would be interest in whether anybody has ever seen the expression (1) before?
Joy Christian wrote:Yablon wrote:I would be interest in whether anybody has ever seen the expression (1) before?
I haven't seen the expression, but it reminds me of 5D Kaluza–Klein action. You essentially have 4D + an extra term. You may want to look up Witten's 1980-81 paper on Kaluza–Klein.
And one finds that electric charge is identified with motion in the fifth dimension.
Yablon wrote:I got it!
Here is the full derivation, as well as some new insights into the geometric nature of electromagnetism. And I believe that this fairly qualifies as a unification of classical gravitational and electromagnetic theory in 4 dimensions (not the five dimensions needed for Kaluza-Klein).
https://jayryablon.files.wordpress.com/ ... cs-1-1.pdf
I plan to submit this as a stand-alone paper to a journal after a bit more tweaking. (I suppose at least one reference might be nice, even though this is purely a calculation. )
Two questions:
1) Did I get this right?
2) Has anybody else seen / done this before?
Joy Christian wrote:Very nice, Jay. Congratulations!
I haven't checked all the details, but on my first reading your argument seems to hold. It would naturally take considerable amount of time, effort, and reflection to check all the details. A formal (and honest) peer review would therefore be useful.
My only concern at the moment is that the paper is a bit too long. I am pretty sure that -- since now you have the basic argument in place -- you can streamline the paper to, say, 4 PRL pages. I think it may be worthwhile to squeeze it in 4 pages and send it to PRL (provided a similar derivation doesn't already exist somewhere).
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