## Nice derivation of Lorentz Force Law from an Action?

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

### Nice derivation of Lorentz Force Law from an Action?

Dear Friends:

There is a very nice derivation of the geodesic equation of general relativity from an action using variational calculus, shown at the link below:

https://en.wikipedia.org/wiki/Geodesics ... _an_action

I was hoping somebody might point out a parallel derivation for the Lorentz force law:

$m \frac{d^2x^\mu}{d\tau^2}=eF^\mu_\sigma \frac {dx^\sigma}{d\tau}$.

I am inclined to think, since $g_{\mu\nu}$ is the gauge field in gravitational theory and $A^\mu$ is the gauge field of electrodynamics, that the simplest derivation via variational principles should start with the one-form $A=A_\sigma dx^\sigma$, but there is only one $dx^\sigma$ in this one-form as opposed to two in $ds^2 = g_{\mu\nu}dx^\mu dx^\nu$, and so I do not see how we get out a necessary $\frac{d^2x^\mu}{d\tau^2}$ from this.

Any good pointers would be appreciated.

Thanks,

Jay
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### Re: Nice derivation of Lorentz Force Law from an Action?

Hi Jay,

I doubt that you will succeed, but here is something worth trying. Try to bring the force term on the left hand side first, and try to construct an effective connection field analogous to the connection field in general relativity.

Here is one of my very old papers in which I do something similar for Newton's theory of gravity. Take a look at equations (2.24) and (2.31). See how I absorb the force term into an effective connection field using a derivative term involving a gauge potential. Needless to say, this is just a hint and an analogy. It may not work for the Lorentz force law.

Good luck,

Joy
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### Re: Nice derivation of Lorentz Force Law from an Action?

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.
guest1202

### Re: Nice derivation of Lorentz Force Law from an Action?

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.

Yes, I have gone back and reviewed approaches using the QED Lagrangian, but what comes out of varying those actions are the Maxwell equations, just like one gets the Einstein equation from varying the Einstein-Hilbert action, see https://en.wikipedia.org/wiki/Einstein% ... ert_action. I want to get the Lorentz force law, and wish to do so using as my template, https://en.wikipedia.org/wiki/Geodesics ... _an_action from which the geodesic equation of motion general relativity emerges. But I have not seen the analog of this derivation, for the Lorentz force. Note that this link does also discuss the Lorentz force at https://en.wikipedia.org/wiki/Geodesics ... d_particle. But it just plunks it in without a derivation.

The reason I am doing this, by the way, is to follow up on our discussions a couple of weeks ago, and specifically to be able to show that my use of single-valuedness to obtain the half-integer FQHE charges is correct when analyzed using variational calculus, using geodesic motion in the gauge space (which should be described by the Lorentz force law) to parallel transport electron wavefunctions over closed loops about a magnetic monopole.

Jay
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### Re: Nice derivation of Lorentz Force Law from an Action?

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.
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### Re: Nice derivation of Lorentz Force Law from an Action?

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.

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 $A^\mu$ in the action, one can always introduce that as $eA^\mu$ given that the covariant derivatives are $D_\mu=\partial_\mu +ieA_\mu$. Then, the mass needs to get in there somehow. The Klein-Gordon equation with a potential written as $-m^2\phi=g_{\mu\nu}D^\mu D^\nu \phi$ may be a template for doing this, because that will implicitly contain an e/m (really, $e^2/m^2$) 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 $F_{\mu\nu}\psi=i[D_\mu , D_\nu]\Psi$, 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
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### Re: Nice derivation of Lorentz Force Law from an Action?

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.
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### Re: Nice derivation of Lorentz Force Law from an Action?

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.
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### Re: Nice derivation of Lorentz Force Law from an Action?

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.

I meant that the physicality might be recoverable via torsion when e = m. IOW, huge electronic energy + torsion energy = observable mass-energy (keeping in mind that torsion energy may be negative). So it may very well be worthwhile for Jay to explore e = m.
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### Re: Nice derivation of Lorentz Force Law from an Action?

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 $A^\mu$ in the action, one can always introduce that as $eA^\mu$ given that the covariant derivatives are $D_\mu=\partial_\mu +ieA_\mu$. Then, the mass needs to get in there somehow. The Klein-Gordon equation with a potential written as $-m^2\phi=g_{\mu\nu}D^\mu D^\nu \phi$ may be a template for doing this, because that will implicitly contain an e/m (really, $e^2/m^2$) 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 $F_{\mu\nu}\psi=i[D_\mu , D_\nu]\Psi$, 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 $F^{\mu\nu}$ with a gauge space curvature akin to the Reimann tensor $R^\alpha_{\beta\mu\nu}$ 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 $ds^2=g_{\mu\nu}dx^\mu dx^\nu$ 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 $e=m$ 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
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### Re: Nice derivation of Lorentz Force Law from an Action?

"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?
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### Re: Nice derivation of Lorentz Force Law from an Action?

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?

Hi Fred,

I see that you are looking at https://en.wikipedia.org/wiki/Lorentz_f ... _mechanics. At a certain level, yes, because the last two terms are the covariant form of the four potential. But no, because I want to see a tight and direct connection to the metric of spacetime.

I have thought about this quite a bit over the weekend while in Boston / Cambridge, where I am always inspired back to my roots at MIT where the most important thing I learned way beyond any specific curriculum is that I can figure out anything if I really put my mind to it. MIT is the place I really learned how to think.

I am now pretty well convinced that the term to be varied must have something close to the form:

$ds'=\sqrt{g_{\mu\nu}dx^\mu dx^\nu}+\frac{e}{m}A_\mu dx^\mu$ (1)

Specifically, if we put the metric interval $ds=\sqrt{g_{\mu\nu}dx^\mu dx^\nu}$ together with a supplemental term $\frac{e}{m}A_\mu dx^\mu=\frac{e}{m}A$ for the potential one-form, then by dimensional analysis we are comparing apples to apples in $\hbar=c=1$ units because the mass dimension of both terms is -1 and because each of g and A are gauge potentials. Also, by putting the derivation of gravitational geodesics together with the derivation of the Lorentz force geodesics the former can lend the needed acceleration term $\frac{d^2x^\mu}{d\tau^2}$ to the latter and the latter can lend a resulting term with $F^\mu_\sigma \frac {dx^\sigma}{d\tau}$ to the former that contains a velocity and a field strength which will appear when the variation operates on the potential.

This week I will do the calculation of https://en.wikipedia.org/wiki/Geodesics ... _an_action but using (1) above rather than just $ds=\sqrt{g_{\mu\nu}dx^\mu dx^\nu}$ alone, and see if it in fact, as I suspect, it will lead to the fourth equation in https://en.wikipedia.org/wiki/Geodesics ... d_particle which then contains both forms of motion, without plunking in the Lorentz force as is done in the Wiki article.

I would be interest in whether anybody has ever seen the expression (1) before?

Jay
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### Re: Nice derivation of Lorentz Force Law from an Action?

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.
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### Re: Nice derivation of Lorentz Force Law from an Action?

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.

Go to https://en.wikipedia.org/wiki/Kaluza%E2 ... Hypothesis, fourth equation down in this section, set $dx^5=0$, and aside from the scalar coefficient you get my length element:

$ds'=\sqrt{g_{\mu\nu}dx^\mu dx^\nu}+\frac{e}{m}A_\mu dx^\mu$ (1)

So this confirms that I am on the right path, and can quite likely get the Lorentz force law out of a variation carried out on this exclusively in four spacetime dimensions.

Jay
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### Re: Nice derivation of Lorentz Force Law from an Action?

And one finds that electric charge is identified with motion in the fifth dimension.

Well, that makes me pleased as it fits with my preon model where electric charge is (screwing) motion into the colour dimension(s).
See https://ben6993.wordpress.com/2015/05/18/electric-charge-and-coloured-socks/
for some more details.

I think that mass is not a fundamental enough term to go into an equation as m1m2 instead of (the more fundamental) e1e2 [in a situation where one is trying to find the gravitational force between elementary particles].
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### Re: Nice derivation of Lorentz Force Law from an Action?

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?

Thanks,

Jay
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### Re: Nice derivation of Lorentz Force Law from an Action?

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?

Congrats. Hmm... I am not totally sure but I think you also validated a hypothesis from Joy's work that space has unique spinor properties. Or did that get plugged in manually for the Dirac equation?
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### Re: Nice derivation of Lorentz Force Law from an Action?

***
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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### Re: Nice derivation of Lorentz Force Law from an Action?

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).

Joy, good advice. I accepted that advice, and submitted a streamlined paper to PRL. I will post a copy shortly. Thanks, Jay
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### Re: Nice derivation of Lorentz Force Law from an Action?

Here is the condensed paper I prepared and then submitted to PRL, following Joy's earlier advice.

http://vixra.org/pdf/1512.0489v1.pdf

I appreciate any feedback.

Jay
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