- #1
S.Daedalus
- 221
- 7
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, non-Abelian gauge fields, or even higher dimensions. So... what's it all mean?
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, non-Abelian gauge fields, or even higher dimensions. So... what's it all mean?