Feynman's formulation is manifestly non-relativistic

[tavi_boada]

In reading about Feynmans functional approach to QM it is obvious that in summing over all paths we also consider paths with v>c. This troubles me because it is possible to set a Lagrangian and a time interval for which the non-relativistic classical path has the particle traveling at v>c. Take for instance the lagrangian of the free fall. Taking sufficiently far apart endpoints it is clear that we can have the particle arriving at v>c.

I have tried bypassing this problem by using the relativistic action and some how shortening the spatial integration limits in each time slice so as to not sum over supraluminal paths, but have been unsuccessful. Any solution to this problem?

I am aware of possible answers like " oh, don't worry about those paths, they cancell out". Please if you are tempted to say this, add a non-heuristic proof to your assertion.

Happy New Year!

 

[ohwilleke]

I think the reconcillation between SR and QM in a Feynman approach would be to state that the speed of light speed limit in SR is a classical theory which is true, on average, but which may be violated at that quantum level. Many QM phenomena, like quantum tunnelling have this character. Using the simplified "average" rule that applies in classical settings, they should be impossible, but at a QM level, for a very small number of particles they are possible. I've never seen any writing by Feynman that flatly reject the possibility of quantum moving at more than c and have seen coy allusions to the possiblity in his writings.

Also, I am not aware of any QM level experiments that have actually shown QM behavior triumphing over the c speed limit, although some have appeared to do so until carefully analyzed.

 

[tavi_boada]

Ok I understand what you say and recognize that it's very Feynmanee. But what I ask is how do we formalize this idea.

I've tried working this out in vain because, for instance, in the case of a freely moving particle we are obliged to sum over supraluminal paths and hence get a nonsensical (in the SR sense) propagator.

It's obvously meaningless to use a relativistic lagrangian because you get divergent arguments for many paths (the gammas explode for v=c and become imaginary for v>c). Any ideas to solve this?

 

[caribou]

I vaguely remember something about faster/slower photons actually being part of the explanation for why the speed of light is the speed of light or something about spacetime being how it is or something.

I've forgotten what it was and where I read it but all I really remember is that it's not a problem but actually an explanation or reinforces something fundamental.