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Feynman's Derivation of Maxwell's Equations from Commutator Relations 
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#1
May3111, 04:27 AM

P: 211

According to Dyson, Feynman in 1948 related to him a derivation, which, from
1) Newton's: [tex]m\ddot{x}_i=F_i(x,\dot{x},t)[/tex] 2) the commutator relations: [tex][x_i,x_j]=0[/tex][tex]m[x_i,\dot{x}_j]=i\hbar\delta_{ij}[/tex] deduces: 1) the 'Lorentz force': [tex]F_i(x,\dot{x},t)=E_i(x,t)+\epsilon_{ijk}\dot{x}_j B_k(x,t)[/tex] 2) and the homogenous 'Maxwell equations': [tex]\nabla\cdot\mathbf{B}=0[/tex] [tex]\nabla\times\mathbf{E}+\frac{\partial B}{\partial t}=0[/tex] Now the derivation is straightforward enough. (And apparently, the inhomogenous equations left underived just provide 'a definition of matter'.) But the question is: what does this mean? I can come up with several interpretations of various strengths: 1) It means nothing; it's just a mathematical oddity. 2) Dyson's view, apparently, is that the proof shows that the only possible ﬁelds that can consistently act on a quantum mechanical particle are gauge ﬁelds  I'm not sure I exactly understand what's meant by that (well, rather, I understand what it means, but I'm not sure I get why the proof implies it). 3) Electrodynamics is somehow 'built in' to quantum mechanics. Which, if any, of these is right? It just seems odd that all that information ought to be contained in the simple commutation relations  how are they supposed to 'know' about electromagnetic fields? Moreover, I understand there are various generalizations of the derivation, incorporating explicitly special or general relativity, nonAbelian gauge fields, or even higher dimensions. So... what's it all mean? 


#2
May3111, 05:06 AM

Sci Advisor
P: 4,603

It means
4) The derivation above works because the "quantum" commutators involved are essentially the same as the corresponding classical Poisson brackets. So it is really a classical derivation disguised into a quantum one. You can repeat the whole derivation by replacing the commutators with the appropriate Poisson brackets. One additional comment. The derivation above (either with commutators or Poisson brackets) suggests that electromagnetic forces are the only possible forces on particles. But then, what about gravitational forces? Are they impossible? Of course not. It can be shown that the equations of Einstein gravity in a nonrelativistic limit can be approximated by the Maxwell equations above. (Note also that these are only 2 Maxwell equations, not all 4 of them. This truncated set of equations alone does not imply Lorentz invariance.) For more details see R. J. Hughes, Am. J. Phys. 60, 301 (1992) 


#3
May3111, 06:32 AM

Sci Advisor
P: 4,603

To see this, the most elegant language is the language of differential forms. In this language, the homogeneous Maxwell equations can be written in a compact form dF=0 which do not imply F=dA 


#4
May3111, 10:09 AM

Sci Advisor
P: 905

Feynman's Derivation of Maxwell's Equations from Commutator Relations
You mean one has to say on top of that something about the topology of spacetime (that it should be simply connected)?



#5
May3111, 10:44 AM

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P: 4,603



#6
May3111, 12:43 PM

P: 211




#7
May3111, 02:09 PM

Sci Advisor
Thanks
P: 4,160

It means
1.5) There's another assumption being made, namely minimal coupling. A vector gauge field A is the only field that can couple to fermions via P = p + e/c A, and this assumption quickly leads to the homogeneous field equations for the field F that corresponds to A. 


#8
May3111, 04:28 PM

P: 47

Here is the article presenting the derivation, further down in the PDF are included the commentaries on the derivation. It's interesting that this thread was started today because I have been working through this derivation during the last couple of days.



#9
Jun111, 03:56 AM

Sci Advisor
P: 4,603

Anyway, I highly recommend to read the paper I cited in post #2, which is much better than the original Dyson's paper (to which Sybren gave the link). 


#10
Jun211, 10:00 AM

P: 1

great source! thanks! daryl



#11
Jun411, 03:38 PM

P: 1,235

Hello,
I would be interrested to read the same proof based on Poisson brackets instead of commutators. Would you know a reference for this? Thanks, Michel 


#13
Jun611, 03:49 AM

P: 1,235

Thanks a lot Demystifier.
I found a copy of this paper. 


#14
Jun611, 03:21 PM

P: 47

I also got a copy of the paper now, and in it they mention, just as the first post of this thread does, that the 'Lorentz force' follows from the commutator relations and newton's second law of motion. However, I see from the Dyson paper that the 'Lorentz force' (which I put between quotes because the charge [itex]q[/itex] is missing) is used as a definition. So it is not a result. Only the two maxwell equations are results of the derivation.



#15
Jun711, 10:47 AM

P: 661




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