No Interaction Theorem: Root Cause and Resolution
Actually, to follow up on my earlier reply (a few years back), a closer examination of the theorem (both the 1963 Currie et al. version and the later Leutwyler generalization of it) do not
assume additivity for angular momentum and momentum, they infer it from the transformation propeties posed for the worldline coordinates. Then, follows the slow descent into triviality.
The following also applies to a later formulation of the 2-body no interaction proof published by Jordan (1968, if I recall).
To be more precise, additivity for momentum and angular momentum only up to an adjustment of the momentum (i.e. a canonical transformation). The adjustment made by this transformation also yields certain transformation properties, as a result, for the momentum which, if assumed at the outset, would equivalently characterize the transformation made by the adjustment.
Denoting the infinitesimal forms of rotation, boost and translation respectively by
ω, <b>\upsilon</b> and
ε and infinitesimal time translation by \tau, the transformation assumed for a worldline coordinate
q is
\deltaq = ω\timesq + (α<b>\upsilon</b> - \tau)v + ε
where
v = {
q, E} is the velocity, with E being the time translation generator.
If you apply this to the bracket relation {
q,
q'} for two different particle coordinates (noting that the bracket is the 0 dyad), then the transform should be 0. Plugging in the time translation and boost you immediately get that the inverse of the "coefficients of inertia" matrix (i.e. the matrix of all the {
q,
v'} brackets has zero cross terms -- or more precisely, cross terms that admit only contact terms (ones with δ(
q -
q') factors in it). The contact terms are not included in the analyses posed by the no interaction theorems and probably won't change the situation much at all.
This entails that E separates into a sum of kinetic terms, i.e. terms dependent on only
one of the particle momenta; plus a potential term (i.e. a term independent of all the momenta).
None of this holds in the non-relativistic case. That extra factor α is 1/c
2 for relativity and 0 for non-relativistic theory. That factor controls the separability.
The root of the problem is clearly seen here: the correspondence limit employed by this theory has a discontinuity at α = 0. Empirically, this is a big no no. Qualitative differences should
never result from a continuous parameter adopting a specific value, because that would in principle provide a way of measuring a value with infinite precision.
In other words, we're using the wrong correspondence limit.
If you go back and take a closer -- more modern -- view of non-relativistic theory, you will see that its symmetry group is now understood to
not be the 10-dimensional Galilei group, but the 11-dimensional Bargmann group. In order to have a more cohesive correspondence limit, this then requires that whatever symmetry group we adopt for relativity should have the Bargmann group as its limit. This leave no alternative but for this group to have 11 dimensions, not 10.
The modification takes place in the brackets between the boost and trnslation generators and entails a slightly different notion of time translation as a result. Let H be the new time translation generator. It will be defined so that it has the kinetic energy as its non-relativistic limit. The corresponding to the central charge m in the Bargmann group will be a trivial central charge, which we'll call \mu. It is understood that it will have m as its non-relativistic limit, and that the original time translation generator E will continue to be regarded as the "total" energy, with the decomposition E = H + \mu/\alpha. But to hold true to the requirement for a consistent correspondence limit, it will be necessary to replace E by the "relativistic mass" M = \alphaE. Then the decomposition relation will take the form M = \mu + \alphaH. In the non-relativistic limit M also goes to m. On the other hand, E has no limit. It's divergent (as seen by the fact that the parameter \alpha was sitting in the denominator). So, it drops out.
Then the Lie brackets continue to have the same form as before, but with E rewritten as M. In particular [K, P] = M I, where I denotes the identity dyad. For \mu, the brackets are all 0.
The original time translation was {_, E}. Now it becomes {_, H}. The Leutwyler theorem
still applies to the rotation generator
J, boost generator
K, translation generator
P and E (which is now M). But H is exempt. Thus, {_, E} now represents "time translation that would occur if the bodies involve were free and non-interacting", while the difference {_, H-E} represents the interacting part of the time translation generator.
For Leutwyler, the boost transform for
q is now ambiguous. There are now two forms one can state this property in:
(a) { q, \upsilon\cdotK} = \upsilon\cdotq {q, αH}
or
(b) { q, \upsilon\cdotK} = \upsilon\cdotq {q, M}
In the absence of the 11th parameter, both forms are equivalent with H = E = M/\alpha. For the extended Poincare' group, they are no longer equivalent.
Version (a) yields a Leutwyler result for the full 11-dimensional symmetry group. Version (b) breaks the no interaction theorem and allows in non-trivial interactions.
The energy difference H - E = -\mu/α plays the role of the potential U. The only requirement imposed on it is that all the Poisson brackets formed by the 10 other functions -- the 3 vectors
J,
K,
P and the scalar M -- should be 0. This can be done with non-trivial results.
As a consequence, H decomposes into a sum of purely kinetic contributions from each body -- as in the non-relativistic case -- plus the potential U.
The only drawback to this revision is that this adjustment tot he correspondence limit is
still not quite enough! This is best seen by considering the general solution for interacting 2-body systems. When setting α = 0, one gets for U a function of the 3 scalar 2-body invariants formed from their relative speed and relative displacement, as expected. As soon as you turn on α, allowing it to be non-zero, it drops down to only 2 invariants: namely the ones whose non-relativistic limits are relative speed and "areal speed" of the radius vector. The component of velocity collinear to the displacement is lost in the translation. So, no Kepler laws.
So, there's even more subtlety that still lies hidden with the correspondence limit than what I've brought up here. That extra term went somewhere, after turning on α to allow it to be non-zero, and I'm hunting it down to see where it got lost. But it's there somewhere.
What makes the correspondence limit so non-trivial is (at least in part) that you have a complete topological change that takes place at α = 0. You can best see this by combining all groups for all α (both positive
and negative -- i.e. the 4-D Euclidean group -- as well as 0) into a single Poisson manifold. This manifold has the 11 coordinates of the duals of the Lie algebras above (which may simply be denoted
J,
K,
P, H and \mu as before), plus \alpha as the 12 coordinate; and it has the Poisson bracket formed in the natural way from the Lie brackets, with it understood that {\alpha, _} = 0.
The Galilei group is sitting in there as the Lie group that has the submanifold for (\alpha, \mu) = (0, 0). The oridinal Poincare' group produces each of the submanifolds (\alpha, \mu) = (1/V
2, 0), for all different values of V. They are immediately adjacent to one another, but the Galilei group is not simply connected, while the Poincare' group is.
This means that contraction can expose not just one order of \alpha, but an infinite tower of \alpha's -- much like the Einstein-Infeld analysis of the gravity field. The analysis above only pushes the correction to the first order. The 3rd scalar 2-body invariant is hidden somewhere in whatever extra infrastructure is required to recover the 2nd order or higher.
A similar problem occurs when attempting to unify Einstein and Newtonian gravity as a one-parameter family of infrastructures, geometries and Lagrangian theories. Since the Bargmann and extended Poincare' groups are all little groups of the 4+1 de Sitter group, then a unified model for gravity can be formulated in 4+1 dimensions by requiring that there be an invariant covariantly constant field. The norm of that field, with respect to the 4+1 metric, is just \alpha. The resulting field equations include Newton's equation at \alpha = 0, and Einstein's equations (for the most part) for \alpha > 0. But there are important terms that get lost as soon as you get to \alpha = 0 and the resulting field equations for the non-relativistic case become somewhat handicapped. The root of the problem is that the coupling coefficient in general relativity (when you do the dimensional analysis right) is actually 2nd order, not 1st -- it's proportional to 1/\alpha<sup>2</sup>. The conversion to a constrained 5-D geometry allows one to reduce this only by one order. But this isn't quite enough to remove the discontinuity in the correspondence limit at \alpha = 0.
Although: it
is enough to write non-relativistic gravity in Lagrangian form in a way that's sufficient to include Newton's equation.
