Understanding the Concept of Instantaneous Potentials in Quantum Mechanics

[jostpuur]

If the private messages supported latex, I could ask this directly from meopemuk, but I'll put the question here now. (Edit: very logical, I'm not even using latex here... well maybe the replies use latex)

meopemuk, these instantaneous potentials you have been talking about seem very confusing. How precisely are they not in contradiction with everything in the special relativity?

Since I don't know the answer myself, I don't know if this should be in relativity forum instead of QM forum, but the discussion has usually been about QM when these instantaneous potentials have entered discussion, so I'll start here.

 

[meopemuk]

meopemuk, these instantaneous potentials you have been talking about seem very confusing. How precisely are they not in contradiction with everything in the special relativity?

jostpuur,
You are right that instantaneous potentials contradict some aspects of special relativity, in particular, the aspects related to interacting systems. The SR proof of impossibility of action-at-a-distance goes, basically, like this: If the effect and the cause were connected by a superluminal interaction, then one could find a moving reference frame in which (by applying Lorentz transformations) these two events would change their time order - the effect would occur before the cause. This is clearly an absurd.

However, this proof has a weak point. Can we be sure that usual Lorentz transformations for the time and position of events is applicable to events in interacting systems? A full discussion of this loophole can be found in http://www.arxiv.org/abs/physics/0504062. Here I'll mention just two arguments which (I hope) will force you to think twice before answering this question.

1. All existing proofs of Lorentz transformations assume that we are dealing with events associated with non-interacting particles. For example, photons in light rays. Special relativity assumes that these transformations can be generalized to all kinds of events, and, moreover, that these are just fundamental properties of space and time, which are totally independent on the kind of physical system we are observing. As far as I know, there is no rigorous theoretical or experimental justification for this generalization.

2. Any relativistic quantum desription of an isolated interacting system requires construction of an interacting representation of the Poincare group in the Hilbert space. The generators of this representation can be kinematical (interaction-independent) and dynamical (interaction-dependent). There are sufficient reasons to believe that generators of space translations and rotations are interaction-independent and that generators of time translations (the Hamiltonian) and boosts are dynamical, i.e., interaction-dependent.

P. A. M. Diracs, "Forms of relativistic dynamics", Rev. Mod. Phys. 21 (1949), 392

S. Weinberg, "The quantum theory of fields", 1995, vol.1

Since the generator of boost transformations in the Hilbert space is interaction-dependent, one can also expect that boost transformations of some observables should depend on the interaction acting in the system, i.e., they may not be given by standard Lorentz formulas.

Eugene.

 

[jostpuur]

The superluminal propagation of some wave functions probably acts as some motivation for this?

 

[meopemuk]

The superluminal propagation of some wave functions probably acts as some motivation for this?

No, there is no direct connection between superluminality in propagation of a single particle wave function and superluminality in propagation of interactions in multiparticle systems. I consider them as two different cases.

Eugene.