# QED Lagrangian lead to self-interaction?

## Main Question or Discussion Point

Hi, I'm not very good at QFT, but starting from Dirac's QED Lagrangian (look for example at: http://en.wikipedia.org/wiki/Quantum_electrodynamics" [Broken] )

$$L=\bar{\psi}\left(\gamma^{\mu}\left(i\hbar\frac{\partial}{\partial\mu}-q(A^{ext}_{\mu}+A_{\mu})\right)+ mc\gamma^0 \right)\psi-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}$$

From here we derive Dirac's equation and Maxwell's equations. Now omitting all derivation steps, every thing could be summarized into the non-relativistic limit as:

$$\left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}- \frac{q^2}{4\pi\epsilon}\int\frac{\mid\psi(\vec{r})\mid^2}{\mid \vec{r}-\vec{r}^{\prime} \mid}d^3\vec{r}^{\prime}\right)\psi=E\psi$$

This is in fact what the QED Lagrangian result in (ignoring the very small contribution from the magnetic vector potential A for simplicity), but the effective Schrödinger equ looks more like Kohn-Sham equation for a single particle. But is this correct?

The Coulomb integral suggest that the electron is self-interacting with itself! (compare coulomb blocking) I thought that the correct limit has to be Schrödinger equation, but with some funny coupling term (but not so strong as coulomb self-interaction). Is it the field operators that are needed in some way to vanish this term, when it is not "appropriate"?

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Hi, I'm not very good at QFT, but starting from Dirac's QED Lagrangian (look for example at: http://en.wikipedia.org/wiki/Quantum_electrodynamics" [Broken] )

$$L=\bar{\psi}\left(\gamma^{\mu}\left(i\hbar\frac{\partial}{\partial\mu}-q(A^{ext}_{\mu}+A_{\mu})\right)+ mc\gamma^0 \right)\psi-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}$$

From here we derive Dirac's equation and Maxwell's equations. Now omitting all derivation steps, every thing could be summarized into the non-relativistic limit as:

$$\left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}- \frac{q^2}{4\pi\epsilon}\int\frac{\mid\psi(\vec{r})\mid^2}{\mid \vec{r}-\vec{r}^{\prime} \mid}d^3\vec{r}^{\prime}\right)\psi=E\psi$$

This is in fact what the QED Lagrangian result in (ignoring the very small contribution from the magnetic vector potential A for simplicity), but the effective Schrödinger equ looks more like Kohn-Sham equation for a single particle. But is this correct?

The Coulomb integral suggest that the electron is self-interacting with itself! (compare coulomb blocking) I thought that the correct limit has to be Schrödinger equation, but with some funny coupling term (but not so strong as coulomb self-interaction). Is it the field operators that are needed in some way to vanish this term, when it is not "appropriate"?
I hoped somebody else would answer this question:-) I am afraid I have neither a clear answer nor time to sort this out, but I suspect this question is considered in Barut's works on self-field electrodynamics and in nightlight's posts in this forum.

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I hoped somebody else would answer this question:-) I am afraid I have neither a clear answer nor time to sort this out, but I suspect this question is considered in Barut's works on self-field electrodynamics and in nightlight's posts in this forum.
YES!!! you guys are correct! That is the correct form of the equation. Indeed this is Barut's SFQED.

Now you just have to put in the self-interaction from the vector potential, and you can do all of QED with a first-quantized formalism!

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Hi, I'm not very good at QFT, but starting from Dirac's QED Lagrangian (look for example at: http://en.wikipedia.org/wiki/Quantum_electrodynamics" [Broken] )

$$L=\bar{\psi}\left(\gamma^{\mu}\left(i\hbar\frac{\partial}{\partial\mu}-q(A^{ext}_{\mu}+A_{\mu})\right)+ mc\gamma^0 \right)\psi-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}$$

From here we derive Dirac's equation and Maxwell's equations. Now omitting all derivation steps, every thing could be summarized into the non-relativistic limit as:

$$\left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}- \frac{q^2}{4\pi\epsilon}\int\frac{\mid\psi(\vec{r})\mid^2}{\mid \vec{r}-\vec{r}^{\prime} \mid}d^3\vec{r}^{\prime}\right)\psi=E\psi$$

This is in fact what the QED Lagrangian result in (ignoring the very small contribution from the magnetic vector potential A for simplicity), but the effective Schrödinger equ looks more like Kohn-Sham equation for a single particle. But is this correct?

The Coulomb integral suggest that the electron is self-interacting with itself! (compare coulomb blocking) I thought that the correct limit has to be Schrödinger equation, but with some funny coupling term (but not so strong as coulomb self-interaction). Is it the field operators that are needed in some way to vanish this term, when it is not "appropriate"?
Also, you'll notice that on this view, the physically correct Schroedinger equation for a charged particle always has those self-interaction terms, and therefore the physically correct Schroedinger (or Dirac) equation is not actually a linear equation, but a nonlinear integro-differential equation. This has profound implications for the interpretations of QM. Most notably, the Everett MWI interpretation is not consistent with a Schroedinger equation like this with self-interaction terms.

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That is the correct form of the equation. Indeed this is Barut's SFQED.

Now you just have to put in the self-interaction from the vector potential, and you can do all of QED with a first-quantized formalism!
If you mean the standard second-quantized QED in its entirety, I suspect this statement requires a few caveats. First, Barut introduced the Feynman propagator for the electromagnetic field (which corresponds to a complex Lagrangian) - that was an additional assumption. Second, to implement Pauli's exclusion principle in his theory, Barut had to modify the Lagrangian. Furthermore, Barut did not claim his theory exactly reproduced all the results of the standard QED, rather he hoped experiments would favor his theory if and where it differed from QED.

Also, you'll notice that on this view, the physically correct Schroedinger equation for a charged particle always has those self-interaction terms, and therefore the physically correct Schroedinger (or Dirac) equation is not actually a linear equation, but a nonlinear integro-differential equation.
I have always suspected that this is the case, but have never seen the equations interpreted that way in a mainstream context.

If such a result was more widely accepted, it could have profound implications for quantum computing. Mark Oskin, from the University of Washington CS department, explains the time evolution of the quantum state in terms of a unitary operator:

ψ' = Uψ

"The fact that U cannot depend ψ and only on t1 and t2 is a subtle and disappointing fact. We will see later that if U could depend on ψ then quantum computers could easily solve NP complete problems!"

Amazing!

If you mean the standard second-quantized QED in its entirety, I suspect this statement requires a few caveats. First, Barut introduced the Feynman propagator for the electromagnetic field (which corresponds to a complex Lagrangian) - that was an additional assumption. Second, to implement Pauli's exclusion principle in his theory, Barut had to modify the Lagrangian. Furthermore, Barut did not claim his theory exactly reproduced all the results of the standard QED, rather he hoped experiments would favor his theory if and where it differed from QED.
I mean you can calculate all radiative effects that are calculate with 2nd quantized QED, but with Barut's 1st quantized formalism.

The Feynman propagator he uses, yes, was an additional assumption, but can't really be called "2nd quantization", and is not necessarily problematic.

The exclusion principle part I'm not sure about.

Even though Barut hoped his theory could be differentiated from standard QED, that was only a hope and he had no real physics-based reason to believe this would be the case (in fact there is reason to think the theories are empirically equivalent). And to the extent that SFQED was applied to all radiative processes in QED, it made the same predictions. That's why I said you could do all of QED with his theory.

I have always suspected that this is the case, but have never seen the equations interpreted that way in a mainstream context.

If such a result was more widely accepted, it could have profound implications for quantum computing. Mark Oskin, from the University of Washington CS department, explains the time evolution of the quantum state in terms of a unitary operator:

ψ' = Uψ

"The fact that U cannot depend ψ and only on t1 and t2 is a subtle and disappointing fact. We will see later that if U could depend on ψ then quantum computers could easily solve NP complete problems!"

Amazing!

That would be interesting if this was an implication for quantum computing, but I don't know enough about this.

That would be interesting if this was an implication for quantum computing, but I don't know enough about this.
I found the link I was looking for to the paper containing the statement:

"www.cs.washington.edu/homes/oskin/quantum-notes.pdf"[/URL]

I don't think that the quantum computing people will take notice until someone comes up with a method that could in theory exploit the nonlinear self-interaction term to create a general logic gate that transforms qubits with regard for their current state.

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I found the link I was looking for to the paper containing the statement:

"www.cs.washington.edu/homes/oskin/quantum-notes.pdf"[/URL]

I don't think that the quantum computing people will take notice until someone comes up with a method that could in theory exploit the nonlinear self-interaction term to create a general logic gate that transforms qubits with regard for their current state.[/QUOTE]

Sounds like it's worth looking into from a self-field approach.

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YES!!! you guys are correct! That is the correct form of the equation. Indeed this is Barut's SFQED.

Now you just have to put in the self-interaction from the vector potential, and you can do all of QED with a first-quantized formalism!
Thanks. I have to check this guy Barut I think.

I solved the self-consistent problem today for a free electron in vacuum. You then got a bound state and at the same time, the electric field that wave function produces also contain that energy, so energy is conserved. The electron is then a blob on its own and not a wave. But isn't this unphysical?

Thanks. I have to check this guy Barut I think.

I solved the self-consistent problem today for a free electron in vacuum. You then got a bound state and at the same time, the electric field that wave function produces also contain that energy, so energy is conserved. The electron is then a blob on its own and not a wave. But isn't this unphysical?
What do you mean it is a blob on its own? The wavefunction solution to the nonlinear S.E. is still a Fourier expansion of waves.

By the way, Barut et al. have treated the case of the nonrelativisic free particle of their theory:

Quantum electrodynamics based on self-fields, without second quantization: A nonrelativistic calculation of g-2
A. O. Barut, Jonathan P. Dowling, J. F. van Huele
http://prola.aps.org/abstract/PRA/v38/i9/p4405_1

I mean you can calculate all radiative effects that are calculate with 2nd quantized QED, but with Barut's 1st quantized formalism.

The Feynman propagator he uses, yes, was an additional assumption, but can't really be called "2nd quantization", and is not necessarily problematic.
"not necessarily problematic" - maybe, but some knowledgeable people (such as Bialynicki-Birula) contend the use of the Feynman propagator is equivalent to 2nd quantization of the electromagnetic field in the absence of external photon lines.

Maaneli said:
The exclusion principle part I'm not sure about.
I'll try to find a reference if and when I have time. Same for the reference to Bialynicki-Birula.

Maaneli said:
Even though Barut hoped his theory could be differentiated from standard QED, that was only a hope and he had no real physics-based reason to believe this would be the case (in fact there is reason to think the theories are empirically equivalent). And to the extent that SFQED was applied to all radiative processes in QED, it made the same predictions. That's why I said you could do all of QED with his theory.
Actually, it was the word "just" in your phrase "Now you just have to put in the self-interaction from the vector potential, and you can do all of QED with a first-quantized formalism!" that triggered my previous post.

Maybe I missed something, but I'm not sure he "hoped his theory could be differentiated from standard QED", my impression was he would have preferred if the results had been the same. Maybe I'm wrong though.

"not necessarily problematic" - maybe, but some knowledgeable people (such as Bialynicki-Birula) contend the use of the Feynman propagator is equivalent to 2nd quantization of the electromagnetic field in the absence of external photon lines.
I have already read the Birula paper. I meant two things: that the matter field in Barut is not 2nd quantized (that's obvious), and that although Feynman and Birula say the use of the Feynman propagator is equivalent to 2nd quantization of the EM field in the absence of external photon lines, this is not the same as saying that the self-field is 2nd quantized, in my opinion. Maybe it's just semantics, but it's not so clear to me what "2nd quantized" means with respect to the self-field. If they just mean use of the complex-valued Feynman propagator instead of the real-valued classical Green's propagator, well, OK, but then both parts of the term "2nd quantized" seem to me a misnomer. I mean, the self-field is not an operator-valued field, nor is it decomposable into quantized harmonic oscillators. One could certainly however say that it is a 1st quantized self-field because after all the electron charge is coupled to the 1st quantized matter current density, but that's about it as far as I can see.

I'll try to find a reference if and when I have time.
OK, for the Pauli exclusion principle, thanks.

Maybe I missed something, but I'm not sure he "hoped his theory could be differentiated from standard QED", my impression was he would have preferred if the results had been the same. Maybe I'm wrong though.
I have asked Dowling about this and read some papers where they pretty much say they did hope it would be a different theory than standard perturbative QED. The only reasons to think this is that their method of solution is different, being a nonperturbative iteration procedure with Mellin-Barnes transforms, as opposed to asymptotic expansions with renormalization. But all the QED phenomena they did treat in their theory gave the same results to lowest orders in Z*alpha. Moreover, the equivalence of eliminating the 2nd quantized free field with the self-field as Feynman and Birula mention would also suggest to me an empirical equivalence for QED phenomena, even if the methods of solution are different.

There is however one place where a difference of predictions does seem to exist between the two theories (and I think it suggests that perhaps perturbative QED is after all an approximation to the Barut theory), namely, the old cosmological constant problem. Perturbative QED predicts an infinite vacuum energy density (even in the absence of matter) whose absolute value induces infinite spacetime curvature according to the Einstein field equation. But the Barut theory does not predict any such infinite vacuum energy density, with or without the presence of matter. So it easily solves the old cosmological constant problem. That to me seems like a significant difference, but one based on an intertheoretic consideration. Actually, this also ties into the fact that SFQED gives finite answers whereas perturbative QED gives infinite bare values. So maybe you could indeed have good reason to say that perturbative QED is an approximation to SFQED.

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"not necessarily problematic" - maybe, but some knowledgeable people (such as Bialynicki-Birula) contend the use of the Feynman propagator is equivalent to 2nd quantization of the electromagnetic field in the absence of external photon lines.

I'll try to find a reference if and when I have time. Same for the reference to Bialynicki-Birula.

Actually, it was the word "just" in your phrase "Now you just have to put in the self-interaction from the vector potential, and you can do all of QED with a first-quantized formalism!" that triggered my previous post.

Maybe I missed something, but I'm not sure he "hoped his theory could be differentiated from standard QED", my impression was he would have preferred if the results had been the same. Maybe I'm wrong though.

I should also add that Barut and Dowling did in fact extend their approach to 2nd quantized matter fields and got the same answers.

reilly
The idea of self energy makes great sense, and is, in fact, forced on us by the basic structure of the QED interaction, for example. It is evident in Poynting's thrm, and in the old adiabatic assembling of a charge. That self energy shows up is no surprise, so the issue is what do you do with it? And, the jury is still out.

Fortunately, our inability to deal with this concept has not precluded great advances in QED, the Standard Model and on.... What we've learned from QED is that the corrections due to self energy and polarization of the vacuum and charge screening (Corrections due to vertex diagrams) are very small, and have virtually no effect on physics at an atomic or molecular or nuclear scale. Non-corrected theory works just fine in those regions of physics. So, typically we throw out the self energy terms, with a nod to empirical justification. In the relativistic case, we throw away the infinities that plague us, but in a way that is astonishingly accurate.

It ain't pretty, but it's the best we have. Great opportunity indeed.

Regards,
Reilly Atkinson

What we've learned from QED is that the corrections due to self energy and polarization of the vacuum and charge screening (Corrections due to vertex diagrams) are very small, and have virtually no effect on physics at an atomic or molecular or nuclear scale.
I strongly disagree with this characterization of corrections due to self energy as having "virtually no effect on physics at an atomic or molecular or nuclear scale". The Lamb shift, spontaneous emission, corrections to g-2, and cavity QED effects, are all examples of highly nontrivial physical phenomena in various parts of AMO and nuclear physics. Moreover, Barut's self-field approach is the most explicit example of how self-energy is indispensable to said QED phenomena.

OK, for the Pauli exclusion principle, thanks.
So here's the reference.

A.O. Barut, "Foundations of Self-Field Quantumelectrodynamics", in: "New Frontiers in Quantum Electrodynamics and Quantum Optics", Ed. by A.O. Barut, NATO ASI Series V.232, 1991, p. 358:

"For two identical particles we use the postulate of the first quantized quantum theory that the field is symmetric or antisymmetric under the interchange of all dynamical variables of identical particles. In our formulation we go back to the original action principle and assume that the current $j_\mu$ is antisymmetric in the two fields
$j_\mu=\frac{1}{2}e(\bar{\psi_1}\gamma_\mu\psi_2-\bar{\psi_2}\gamma_\mu\psi_1)$,
$e_1=e_2=e$ (52)

This implies in the interaction action

$W_{\textrm{int}}=\frac{1}{4}e^2\left[\int dx dy \bar{\psi_1}(x)\gamma_\mu\psi_2(x)D(x-y) \bar{\psi_1}(y)\gamma_\mu\psi_2(y)-\int dx dy \bar{\psi_1}\gamma_\mu\psi_2D(x-y) \bar{\psi_2}\gamma_\mu\psi_1+(1\leftrightarrow2)\right]$ (53)"

So here's the reference.

A.O. Barut, "Foundations of Self-Field Quantumelectrodynamics", in: "New Frontiers in Quantum Electrodynamics and Quantum Optics", Ed. by A.O. Barut, NATO ASI Series V.232, 1991, p. 358:

"For two identical particles we use the postulate of the first quantized quantum theory that the field is symmetric or antisymmetric under the interchange of all dynamical variables of identical particles. In our formulation we go back to the original action principle and assume that the current $j_\mu$ is antisymmetric in the two fields
$j_\mu=\frac{1}{2}e(\bar{\psi_1}\gamma_\mu\psi_2-\bar{\psi_2}\gamma_\mu\psi_1)$,
$e_1=e_2=e$ (52)

This implies in the interaction action

$W_{\textrm{int}}=\frac{1}{4}e^2\left[\int dx dy \bar{\psi_1}(x)\gamma_\mu\psi_2(x)D(x-y) \bar{\psi_1}(y)\gamma_\mu\psi_2(y)-\int dx dy \bar{\psi_1}\gamma_\mu\psi_2D(x-y) \bar{\psi_2}\gamma_\mu\psi_1+(1\leftrightarrow2)\right]$ (53)"

Thanks for this reference! I have seen this paper before, and only vaguely recall this part. I don't really consider this an "additional assumption". It is simply a consequence of staying in the first quantized matter formalism. In any case, thanks.

Thanks for this reference! I have seen this paper before, and only vaguely recall this part. I don't really consider this an "additional assumption". It is simply a consequence of staying in the first quantized matter formalism. In any case, thanks.
In my book, if one changes the Lagrangian, he does introduce an additional assumption. So I guess we disagree on this point.

Another thing. You see, to "stay in the first quantized matter formalism", Barut "smuggles" second quantization for the electromagnetic field by using the Feynman propagator, and "smuggles" something similar for the electron field by changing the Lagrangian. I am not trying to criticize Barut, I am just trying to say that his self-field electrodynamics is an unfinished business. It is not easy to determine its exact status. It took me quite some time to sort it out, and I am not sure I have a clear picture now. So I try to tread carefully. It is too easy to say something that is not quite accurate. Sometimes I had to admit in this forum that I had made a mistake. Things are just too often not quite what they look.

In my book, if one changes the Lagrangian, he does introduce an additional assumption. So I guess we disagree on this point.

Another thing. You see, to "stay in the first quantized matter formalism", Barut "smuggles" second quantization for the electromagnetic field by using the Feynman propagator, and "smuggles" something similar for the electron field by changing the Lagrangian. I am not trying to criticize Barut, I am just trying to say that his self-field electrodynamics is an unfinished business. It is not easy to determine its exact status. It took me quite some time to sort it out, and I am not sure I have a clear picture now. So I try to tread carefully. It is too easy to say something that is not quite accurate. Sometimes I had to admit in this forum that I had made a mistake. Things are just too often not quite what they look.

Hey dude,

I would just say that extra assumptions aren't necessarily bad if there is some reasonable way to interpret them and some reliable criterion for evaluating them. Also, I still disagree that Barut smuggles in second quantization of the EM field. I explained earlier why I think the term second quantization is an ambiguous misnomer in the context of Barut's theory. I agree with you that Barut's theory has alot of unfinished business. In fact, I am trying to work some of them out. In particular, how does one treat "photon" (of which there are really none in SFQED) entanglement behavior in Barut's formalism, and for that matter, all of quantum optics? Also, is the theory really finite, or was something overlooked? After all, this claim is in contradiction to the entire effective field theory and SUSY research program that dominates theoretical high energy physics; so it's really important to get this straight. Also, does the self-field introduce extraneous nonlocal effects which are experimentally testable? Also, is there any systematic way to calculate higher order corrections in the theory? Also, how does one treat the pair creation/annihilation phenomena in the Barut theory, using only the self-field and Dirac sea? Also, can the Barut theory be combined with a reasonable theory of measurement like pilot wave theory or stochastic mechanics or GRW collapse theory? So I agree with you, there are lots of important open questions.

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I would just say that extra assumptions aren't necessarily bad if there is some reasonable way to interpret them and some reliable criterion for evaluating them.
I did not say that extra assumptions are bad, I'm not saying the Barut's theory is bad, I'm just saying this is an extra assumption, and you seemed to disagree with that.

Maaneli said:
Also, I still disagree that Barut smuggles in second quantization of the EM field.
I guess we do disagree. I am afraid your reasoning did not convince me. Quantization can be implemented in several ways. As they say, "if it looks like a duck, swims like a duck and quacks like a duck, then it probably is a duck."

I did not say that extra assumptions are bad, I'm not saying the Barut's theory is bad, I'm just saying this is an extra assumption, and you seemed to disagree with that.

I guess we do disagree. I am afraid your reasoning did not convince me. Quantization can be implemented in several ways. As they say, "if it looks like a duck, swims like a duck and quacks like a duck, then it probably is a duck."
I was just talking generally about assumptions. Also, I don't think this is an extra assumption if it is already part of 1st quantized Dirac theory (whether you include the self-field or not), which is what Barut seemed to be saying.

Regarding 2nd quantization, it doesn't "look like a duck, swims like a duck, or and quack like a duck". Please tell me what specifically about my objection to second quantization you don't find reasonable. Again, I think it is probably more accurate to say the self-field is 1st quantized than 2nd quantized.

I was just talking generally about assumptions. Also, I don't think this is an extra assumption if it is already part of 1st quantized Dirac theory (whether you include the self-field or not), which is what Barut seemed to be saying.
This is certainly an extra assumption as far as your phrase "Now you just have to put in the self-interaction from the vector potential, and you can do all of QED with a first-quantized formalism!" is concerned. Furthermore, how come the Pauli exclusion principle is a part of 1st quantized Dirac theory?

Maaneli said:
Regarding 2nd quantization, it doesn't "look like a duck, swims like a duck, or and quack like a duck".
As you don't dispute that "the use of the Feynman propagator is equivalent to 2nd quantization of the EM field in the absence of external photon lines", it does "quack like a duck". If you do dispute it, I'll have to disagree (together with Feynman and Birula:-) ).

Maaneli said:
Please tell me what specifically about my objection to second quantization you don't find reasonable. Again, I think it is probably more accurate to say the self-field is 1st quantized than 2nd quantized.
I did not say you said something unreasonable, I said your reasoning did not convince me. And the reason is your arguments seem too strong to me. If I take them at face value I'll have to admit that the expressions for S-matrix elements written using the Feynman rules are not 2nd quantized: there are no operator-valued fields, no quantized harmonic oscillators. Along the same lines I would have to say there is no 2nd quantization in the path formulation of QFT. You insist that the self-field is not 2nd quantized. I am not sure it is technically correct (I suspect that the Pauli exclusion principle is a part of 2nd quantization). But I do believe that 2nd quantization has been introduced in the Barut's theory, and I think that is what really matters. I did not say the self-field is second-quantized, but I am sure it is not just 1st-quantized. Whether it is "more 1st" or "more 2nd", I don't know.

This is certainly an extra assumption as far as your phrase "Now you just have to put in the self-interaction from the vector potential, and you can do all of QED with a first-quantized formalism!" is concerned. Furthermore, how come the Pauli exclusion principle is a part of 1st quantized Dirac theory?
Why isn't the Pauli exclusion principle part of 1st quantized Dirac theory? It is after all a property of the Dirac sea.

As you don't dispute that "the use of the Feynman propagator is equivalent to 2nd quantization of the EM field in the absence of external photon lines", it does "quack like a duck". If you do dispute it, I'll have to disagree (together with Feynman and Birula:-) ).
With all do respect to them, I think they're being facile by just calling it "2nd quantization".

You insist that the self-field is not 2nd quantized. I am not sure it is technically correct (I suspect that the Pauli exclusion principle is a part of 2nd quantization). But I do believe that 2nd quantization has been introduced in the Barut's theory, and I think that is what really matters. I did not say the self-field is second-quantized, but I am sure it is not just 1st-quantized. Whether it is "more 1st" or "more 2nd", I don't know.
OK, fair enough. I think this is reasonable. But I still think his self-field is more 1st quantized. At least, it is clearer to me what is meant by 1st quantized. Moreover, there already is a full-fledged second quantizated formulation of radiation reaction effects (the self-field is actually an operator, and the matter field is also second quantized) developed by Eberly, Jaynes, Milonni, and others. Also, yes, I don't think 2nd quantization is an approrpiate term for the S-matrix approach either.

By the way, how did you get interested in the Barut theory, and what are your hopes for it? It's quite rare to find someone interested in this.