The Classical and/or Non-Relativistic Form of the Haag & Leutweiler

The Haag theorem is established within the setting of axiomatic
quantum field theory. Historically, it arose from the consideration of
the relation between vacuum polarization and interacting fields and
the earliest proofs were cast firmly in the language of quantum theory
-- which naturally gives one the impression that this is a result that
pertains specifically to quantum theory or that it has to do
specifically with field theory (as opposed to N-body dynamics).

However, there is another no go result

The No-Interaction Theorem for Relativistic Dynamics
http://federation.g3z.com/Physics.htm#NoInteraction

which takes place in CLASSICAL relativistic dynamics.

Despite the emphasis on vacuum polarization, I'm fairly sure that
Haag's Theorem is directly related to the classical no go theorem,
known as the Leutweiler or "No Interaction" Theorem, and that it is
simply the quantized version of the No Interaction Theorem -- and
there should be a way of setting both as special cases within a common
framework.

The parallels are:
Classical -- N-body dynamics
Quantum -- Fock space (quantized form of N-body dynamics, with
variable N allowed)

Classical -- particles treated as free objects, subject to mutual
interaction
Quantum -- dynamics treated as free interaction, subject to an
interaction Hamiltonian (the interaction picture)

The Leutweiler theorem, in particular, is cast in the following form.
Suppose each particle has position and velocity (r,v) subject to the
following infinitesimal transformations:

delta(r) = omega x r + alpha upsilon.r v + epsilon - tau v
delta(v) = omega x v - upsilon + alpha upsilon.v v + alpha upsilon.r A
- tau A

under an infinitesimal boost upsilon and infinitesimal time
translation tau.

The term A (acceleration) may be a function of all the body's (r,v)'s.
Then, the resulting conclusion is that A is trivial.

(Here, we can begin to see part of the REAL problem. To properly
account for time translation, one needs to first solve the equations
of motion and substitute the SOLUTION into the expression for time
translation).

The Leutweiler theorem thus forbids a large family of classical
relativistic N-body interactions, that includes all those that one
would naively pose as the relativistic versions of non-relativistic N-
body dynamics. The arguments run very close to those used in the Haag
theorem; particularly, that of exploiting the misfits brought about by
trying to mesh two or more irreducible representations together.

The root of the problem in the classical case is that the boost
generator has problems dealing with acceleration. If two or more
bodies are engaged in mutual interaction, then the time translation
generator has to also include the acceleration of the bodies. Non-
relativistically, the spatial translation generators are kept free and
disentangled from one another, so if (r,p) are the position and
momentum 3-vectors of a single body the spatial translation P(a) would
simply produce (r+a,p). With the boost generator K(v), one would get (r
+(???), p+Mv) where M is the relativistic mass and (???) is the part
that creates problems. The hammer come down when applying the Lie
bracket [K(v),P(a)] = M <v,a>; particularly if we adopt the expression
M = H/c^2, where H is the time translation generator. Since H is to
include the non-linear part of the motion and not just the motion of
the free body, then either K or P is stuck also having to include it.
P, by assumption, has no interaction part, so the hot potato is left
landing in K's corner. But then, when it comes time to apply the other
Lie bracket [K(v), H] = P(v), the fatal blow is struck.

In a way, this comes full circle and you can see where vacuum
polarization enters the picture, after the fact: the expression M = H/
c^2 is the mass-energy conversion formula and the sole mechanism by
which mass-energy conversion takes place (at least for fermions) is
pair production -- i.e. vacuum polarization. Thus, vacuum polarization
is seen to be a symptom ... and one which is also shared by the
Leutweiler theorem ... and not the problem.

A parallel situation exists with the Haag Theorem. In the interaction
picture, what you're doing is taking Fock space as the basis. This
sets the stage for the quantized version of N-body dynamics. Then
you're trying to inject an interaction term into H. The result is that
the very same problem which occurs with the Leutweiler Theorem crops
up here.

So, for the following, assume my observation does indeed pan out; and
that the Haag theorem is just the quantized form of the Leutweiler
Theorem; and that vacuum polarization (as in the case of the
Leutweiler theorem) is actually just a symptom of the problem grounded
deeper in the way M and H are related and the way M and H combine with
K and P.

Then what it means is that Quantum Theory, itself, is actually a red
herring; and that the problem raised by the Haag Theorem actually has
nothing to do with quantum theory at all! It cuts across paradigms and
applies to classical theory in the guise of the No Interaction
Theorem.

This, by itself, immediately excludes any and all explanations,
accounts or attempted solutions which focus specifically on quantum
theory or which try to tie the issue specifically to quantum theory or
which treat it as a "quantum issue" -- all are excluded as "treatments
of the symptoms" rather than as "cures of the underlying sickness".
The problem is deeply rooted in the very manner in which
representations for N-body dynamics are constructed out of the space-
time symmetry group -- and this problem is blind to any distinction
between classical vs. quantum theory.

It also means, therefore, that the solution must likewise run deeper
than quantum theory and must be classically rooted; solving also the
problem raised by the No Interaction Theorem.

But here is where the punchline really occurs: in closely examining
the proof of the No Interaction Theorem, the following question
naturally emerges -- just where does relativity actually enter the
picture? Why doesn't the no go result also apply non-relativistically?

Or does it?

In trying to construct specific examples of the Galilei group for N-
body dynamics -- like the 2-body Kepler problem -- the first thing one
notices is that there is, in fact, a problem with the [K, P] bracket.
The problem is this: the naive expressions K(v), P(a) and H do,
indeed, produce consistent results when applied to the 2-body problem.
But one gets the bracket [K(a), P(b)] = 0.

This yields the representation for the HOMOGENEOUS (10-parameter)
Galilei group; but not the representation for the MASSIVE (11-
parameter) central extension of the Galilei group.

This may actually get to the root of the problem. The problems
encountered in the relativistic version of the no interaction theorem
persist when taking the non-relativistic limit, provided the limit is
such that M -> m, rather than H/c^2 -> 0. And it is here is where the
misfit between H and M comes fully to bear.

In that case, the non-relativistic version of the Leutweiler Theorem
would assert that there exists no non-trivial representation for N-
body dynamics in the centrally extended Galilei group.

If this conjecture pans out, then what it will show is that it's not
only quantum theory that is a red herring to the problem underlying
the Haag Theorem, but relativity is a red herring too. That is, it
would show that the no go results expressed by the Leutweiler and Haag
Theorems run VERY deep, not only cutting across paradigm boundaries
between classical vs. quantum physics; but also cutting across the
paradigm boundary that separates non-relativistic from relativistic
physics.

In other words, it would should that the Haag Theorem is rooted in
CLASSICAL NON-RELATIVISTIC dynamics, and actually has nothing, per se,
to do with either quantum theory or even relativity.

This would go a long way toward explaining the difficulty in
interpreting and resolving the issue: the explanation being that
people are simply looking in the wrong place. They should be looking
to repairing classical non-relativistic dynamics, not quantum field
theory; and doing so in such a way that upon "relativization" of this
fix we obtain a resolution to the Leutweiler Theorem; and upon
"quantization" of the relativized fix, we obtain a resolution to the
Haag Theorem.
.

Relevant Pages

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