A Critique of Non-Interaction Theorem: Meaning & Validity

[andresB]

TL;DR Summary

What exactly does the non interaction theorem asserts?

At the end of appendix C (concerning the non-interaction theorem of classical relativistic hamiltonian systems) of the book "Classical Relativistic Many-Body Dynamics" by Trump and Schieve it is stated that"It follows that Currie's equation (C.21), and subsequently the assertion of vanishing acceleration in eq. (C.43), is valid only over an infinitesimal duration of time. Likewise, the conclusion that the world line is straight is valid only over an infinitesimal length of the particle trajectory in spacetime. This conclusion, however, is precisely what had been assumed in both eq. (C.36) and (C.50). It is simply the statement that any particle moves locally as a free particle, which is the foundation of all of kinematics. Likewise, any one-dimensional curve in a metric space is, to lowest order, straight. 8 The no interaction theorem makes no statement at all about the acceleration over finite lengths of the world line"

If this correct? if so, what's the meaning of the non-interaction theorem?

 

[meopemuk]

You should keep in mind that Trump and Schieve are not talking about straightforward relativistic Hamiltonian dynamics as presented, for example, in the original paper of Currie, Jordan and Sudarshan. I am not familiar with the details of the Trump-Schieve approach, but I've noticed that in addition to the "normal" time coordinate they introduce some "invariant" evolution time parameter. In addition to the "normal" Hamiltonian H, they consider an alternative generator of time evolution T. I suspect that one of the reasons for these modifications is exactly the desire to bypass the restrictions of the no-interaction theorem.

This is not the only attempt to reformulate relativistic particle mechanics to make it immune to the no-interaction theorem. For example, there are various kinds of "constraint dynamics" and other non-Hamiltonian approaches. You can find a short review in

Keister, B. D., Forms of relativistic dynamics: What are the possibilities? nucl-th/9406032

Eugene.

 

[andresB]

You should keep in mind that Trump and Schieve are not talking about straightforward relativistic Hamiltonian dynamics as presented, for example, in the original paper of Currie, Jordan and Sudarshan. I am not familiar with the details of the Trump-Schieve approach, but I've noticed that in addition to the "normal" time coordinate they introduce some "invariant" evolution time parameter. In addition to the "normal" Hamiltonian H, they consider an alternative generator of time evolution T. I suspect that one of the reasons for these modifications is exactly the desire to bypass the restrictions of the no-interaction theorem.

This is not the only attempt to reformulate relativistic particle mechanics to make it immune to the no-interaction theorem. For example, there are various kinds of "constraint dynamics" and other non-Hamiltonian approaches. You can find a short review in

They say

" In articles concerning on-mass-shell relativistic mechanics, 18 it is often mentioned that a particular theory "avoids the consequences of the no interaction theorem" because of certain postulates regarding the use
of canonical coordinates. Even though this statement has been applied as well to the dynamical theory in this work,l 9 it is possible to provide a stronger and more general critique of the theorem, as discussed in Appendix C."

By the way they talk about the theorem I think they are disagreeing with it completely.

 

[meopemuk]

I don't think there is any way around the no-interaction theorem, if you stick to the standard relativistic Hamiltonian theory (without constraints, special time parameters, and things like that). This theorem basically says that in an interacting relativistic theory, boost transformations of particle observables (coordinates, momenta, etc.) must be different from Lorentz formulas from special relativity. This statement is kind of obvious, because in order to keep Poincare commutation relations, interaction should be present in both the Hamiltonian and the boost generator, which means that boost transformations of observables must be interaction-dependent and cannot be given by universal Lorentz formulas.

There are hundreds of papers discussing the no-interaction theorem, but this is the first time I see somebody disagreeing with its basic conclusions.

Eugene.