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 a
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.

On Jun 13, 7:27*pm, Rock Brentwood wrote:
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 Dynamicshttp://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 a
* *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.

This goes to the heart of the problem of trying to replace an
asynchronous (e.g. delayed) interaction particle model with a local
field theory. When the focus is the existence of two particles
separated in space then there is no SINGLE time that characterizes the
total system. Field theory (and classical potential theory: either
Hamiltonian or Lagrangian) replace the effect of INSTANTANEOUS
interactions from other particles (even one!) with a spatially
sensitive potential function. This reduces the problem to an
equivalent ONE time problem (relative to an arbitrary inertial
frame). Instantaneous (mathematical) velocity 'boosts' switch to
another equivalent frame. This is OK for the one time models but not
when reality requires two times - one for each of the interacting
particles.
Gentlemen, Newton knew what he was doing with his point collisions &
instantaneous gravity and so did Clerk-Maxwell, when he restricted his
EM aether to a fixed, global medium. "Those who ignore history are
condemned to go round & round in circles".

Interesting remarks and thoughts!

I am convinced that the QFT problems arise from the self-action ansatz
of interaction. It renders the physical theory non physical. This
ansatz was proposed by H. Lorentz to preserve the energy-momentum
conservation law but it failed. It was a wrong idea. Using the same
form j*A_rad for electron-classical or quantized field coupling, by
analogy with the external force action j*A_ext, brought unexpected
equation rebuilding: the solutions became non-physical. There is
another way of the energy-momentum conservation based in "interaction"
ansatz that essentially "linearises" the equations and is completely
physical.

Indeed, let us consider a non-relativistic electron (Hamiltonian H_e)
permanently coupled with the quantised EMF (Hamiltonian H_f). Without
coupling jA_rad the dynamics od the system is trivial. Now, how to
introduce their "interaction"? If one considers H_e and H_f as
Hamiltonians of independent systems, that can exist without each
other, it looks OK. The self-interaction terms "spoils" everything.
Renormalizations serve to remove the bad things introduced with this
term.

We can avoid this conceptual and mathematical complications if we look
at the sum H_e + H_f as at the sum of Hamiltonians of independent
_subsystems_ representing separated variables of a compound system:
the "electron" Hamiltonian describes the center of inertia motion and
the "photon" Hamiltonian represents internal or relative motions of
the compound system (that I call an electronium). So _no_ interaction
term is necessary to couple them. An external (potential) field Vext
(r_e) acting on the bound electron excites naturally the photon
oscillators. The charge-charge interaction can be described as a
potential one of compound systems. No problems with getting out of
Fock spaces arises. So the main problem is in correct understanding of
what is what and in correct interaction terms between charges. This
eliminates Haag's objections and permits to build a QFT ans an N-body
QM.

Details can be found in my articles "Reformulation instead of
Renormalizations" and "Atom as a "dressed" nucleus" by Vladimir
Kalitvianski available on arXiv. I believe that this approach is the
real exit from the dead-end caused with the self-action ansatz. It
should be studied and developed further. It may resolve the problem of
quantum gravity in a plane space-time as well as many other "gauge"
theories.

Regards,

Bob.

On Jun 13, 9:27 pm, Rock Brentwood wrote:
The Haag theorem is established within the setting of axiomatic
quantum field theory... there is another no go result [Leutweiler's Theorem]
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...
and there should be a way of setting both as special cases within a
common framework.

and the punchline:
In trying to construct specific examples of the Galilei group for N-
body dynamics -- like the 2-body Kepler problem...

one runs into a similar problem with the [K, P] bracket; hence,

a 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.

(Emphasis mine)

The general issue can be best seen this way. If one starts out by
taking a system composed of N elementary constituents, each separately
having its own representations of the Galilei/Poincare' group (I'll be
using the one-parameter family "Unified Group" that contains both
Galilei and Poincare in the following, for comprehensiveness), then
one has the following:
body a: (J_a, K_a, P_a, H_a, M_a).
The total system has the following
system (J, K, P, H, M)
with the brackets
{J(a), J(b) + K(c) + P(d)} = J(a x b) + K(a x c) + P(a x d)
{K(a), K(b)} = -alpha J(a x B)
{K(a), P(b)} = M (a.b)
{K, H} = P; {K, M} = alpha P
{J, H} = 0; {J, M} = 0
{P, H} = 0; {P, M} = 0; {H, M} = 0
where 3-vector notation is used, with J(a) = J_1 a^1 + J_2 a^2 + J_3
a^3, etc.

(alpha = 0 for the centrally extended Galilei group; alpha = (1/c)^2
0 for the 11-parameter extension of Poincare', alpha 0 for the 11-
parameter extension of the 4-D Euclidean group).

The basis of both theorems is to take the total system to be additive
with respect to J and P
J = sum_a J_a; P = sum_a P_a.
The place where we start to see a discrepancy starts to become clear
the moment we consider a specific problem, like the Kepler problem.
Here, we have
J = J_1 + J_2; P = P_1 + P_2; M = M_1 + M_2; K = K_1 + K_2
but
H = H_1 + H_2 + U.
For the equations of motion, J, P, M and K are conserved, but not (H_1
+ H_2).

The corresponding Hamiltonian flows, given by
X_J = -(P_1 x d/d(P_1) + r_1 x d/d(r_1) + P_2 x d/d(P_2) + r_2 x d/d
(r_2))
X_P = -(d/d(r_1) + d/d(r_2))
under the decomposition
J_i = r_i x p_i; K_i = m_i r_i - p_i t; P_i = p_i
H_i = p_i^2/(2m_i); M_i = m_i
and
X_H = -(v_1.d/d(r_1) + f_1.d/d(p_1) + v_2.d/d(r_2) + f_2.d/d(p_2))
where
v_i = p_i/m_i; f_1 = Gm_1m_2(r_2 - r_1)/|r_2 - r_1|^3 = -f_2
however, yield ONLY the homogeneous representation (where M = 0, X_K
is not written out but is implied by the other fields).

The simplest resolution is to pose an intermediary
(J_3, K_3, P_3, H_3, M_3)
such that
J_1 + J_2 + J_3, K_1 + K_2 + K_3, P_1 + P_2 + P_3
H_1 + H_2 + H_3, M_1 + M_2 + M_3
are all preserved. This requires that
d(J_3)/dt = 0, d(K_3)/dt = 0, d(P_3)/dt = 0, d(M_3)/dt = 0
but
d(H_3)/dt not 0.

In the Unified Group, the 3 invariants are
M - alpha H, P^2 - 2MH + alpha H^2
and
W^2 - alpha W_0^2
where
W = MJ + PxK, W_0 = P.J.

If we assume the interaction involves N systems each of which
preserves their respective invariants, then when applying this general
assumption here, we find that for the 3rd system, we have
dM_3/dt = alpha dH_3/dt
P_3 . dP_3/dt = M_3 dH_3/dt.
Given that dM_3/dt = 0 and dP_3/dt = 0, but not dH_3/dt, the only
solution is
alpha = 0; M_3 = 0.

These are the systems for which I've termed the name SYNCHRON --
systems that accord with the homogeneous Galilei group.

Thus, the root of the matter is that by making the additivity
assumption for J and P, we end up with an interaction that can only be
described consistently (as entirely additrive) only be making the
extra constituent a synchron.

To generalize this requires generalizing beyond additivity to "contact
interactions" where the intermediaries are NOT necessarily synchronic;
e.g. a "tachyonic" intermediary for the relativistic generalization of
Kepler.

Rock Brentwood wrote:

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

Hi Rocky! (Or whatever your real name is - this little girl
is getting a bit confused! :-)

The url you gave above doesn't work for me. Did you mean:

http://federation.g3z.com/Physics/in...#NoInteraction (?)

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, [...]

OK... so... the paper on your website is actually
your re- editing of

Marmo, Mukunda, Sudarshan,
"Relativistic particle dynamics -- Lagrangian proof of the
no-interaction theorem",
Phys Rev D, vol 30, no 10, (1984), p2110.

(Right?)

which is an improvement (on the older Currie-Jordan-Sudarshan
theorem) to cover (a subset of) singular Lagrangians?

I still don't get the exact connection to the proof
of Haag's thm, though. (Or are you still working on that? :-)

---
LoL from the Princess!

On Jun 16, 2:45 pm, Bob_for_short
wrote:
Interesting remarks and thoughts!

I am convinced that the QFT problems arise from the self-action ansatz
of interaction. It renders the physical theory non physical. This
ansatz was proposed by H. Lorentz to preserve the energy-momentum
conservation law but it failed. It was a wrong idea. Using the same
form j*A_rad for electron-classical or quantized field coupling, by
analogy with the external force action j*A_ext, brought unexpected
equation rebuilding: the solutions became non-physical. There is
another way of the energy-momentum conservation based in "interaction"
ansatz that essentially "linearises" the equations and is completely
physical.

Indeed, let us consider a non-relativistic electron (Hamiltonian H_e)
permanently coupled with the quantised EMF (Hamiltonian H_f). Without
coupling jA_rad the dynamics od the system is trivial. Now, how to
introduce their "interaction"? If one considers H_e and H_f as
Hamiltonians of independent systems, that can exist without each
other, it looks OK. The self-interaction terms "spoils" everything.
Renormalizations serve to remove the bad things introduced with this
term.

We can avoid this conceptual and mathematical complications if we look
at the sum H_e + H_f as at the sum of Hamiltonians of independent
_subsystems_ representing separated variables of a compound system:
the "electron" Hamiltonian describes the center of inertia motion and
the "photon" Hamiltonian represents internal or relative motions of
the compound system (that I call an electronium). So _no_ interaction
term is necessary to couple them. An external (potential) field Vext
(r_e) acting on the bound electron excites naturally the photon
oscillators. The charge-charge interaction can be described as a
potential one of compound systems. No problems with getting out of
Fock spaces arises. So the main problem is in correct understanding of
what is what and in correct interaction terms between charges. This
eliminates Haag's objections and permits to build a QFT ans an N-body
QM.

Details can be found in my articles "Reformulation instead of
Renormalizations" and "Atom as a "dressed" nucleus" by Vladimir
Kalitvianski available on arXiv. I believe that this approach is the
real exit from the dead-end caused with the self-action ansatz. It
should be studied and developed further. It may resolve the problem of
quantum gravity in a plane space-time as well as many other "gauge"
theories.

Regards,

Bob.

Feynman & Dirac both recognized the problem of using Hamiltonians in
QFT - that's why they always preferred Lagangians. Dyson was so
discouraged by the infinities that he too gave up on this bogus
approach to EM interactions. Hamiltonian densities are just a
classical analog (guess??) for reducing everything to a single point -
the exact opposite of what remote particle interactions are all
about. Even Lagrangian theory only "works" after one knows the answer
& can step backwards to create the Lagrangian function - not a very
fruitful way to make progress, as the last 50 years of theoretical
physics has demonstrated.
Any attempt to substitute a global concept (field) for a point concept
(particle) is bound to have extreme difficulties - real inertial mass
for a start. Gentlemen (& madam), the world consists of point
electrons - get used to it.