QED Lagrangian lead to self-interaction?

[per.sundqvist]

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" )

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

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:

<br /> \left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}-<br /> \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<br />

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"?

 

[akhmeteli]

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" )

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

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:

<br /> \left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}-<br /> \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<br />

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.

 

[Maaneli]

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!

 

[Maaneli]

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" )

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

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:

<br /> \left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}-<br /> \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<br />

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.

 

[akhmeteli]

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.

 

[Crosson]

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

And then he comments:

"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!

 

[Maaneli]

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.

 

[Maaneli]

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

And then he comments:

"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.

 

[Crosson]

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.

 

[Maaneli]

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.

 

[per.sundqvist]

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?

 

[Maaneli]

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

 

[akhmeteli]

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.

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.

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.

 

[Maaneli]

"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.

 

[Maaneli]

"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

 

[Maaneli]

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.

 

[akhmeteli]

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)"

 

[Maaneli]

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.

 

[akhmeteli]

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.

 

[Maaneli]

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 a lot 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.

 

[akhmeteli]

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.

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."

 

[Maaneli]

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.

 

[akhmeteli]

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?

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:-) ).

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.

 

[Maaneli]

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.

 

[reilly]

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.

My thesis involved lots of QED, including second order self-energy corrections and so on. So, naturally, I figure I know something about QED. Oh, and by the way, I've also taught QED. So I do know about the importance of self energy; it is very evident that this is so from the structure of the QED interaction.The Lamb shift, g-2 corrections play almost no role in atomic physics because they are very small, and usually can be ignored. The overwhelming majority of atomic physics, and then chemical bonding and considerable amount of nuclear theory are done very nicely without the 2nd order QED corrections(See Condon and Shortley's Theory of Atomic Spectra;Pauling's theory of Chemical Bonding; Blatt and Weisskopf's Theoretical Nuclear Physics all bibles in their time.


And, people apparently forget that 2nd quantization is nothing more than a unitary transformation away from configuration and momentum space representations. That is, the only magic of 2nd quantization is the ease it brings to many field theory issues. I'll admit that this is not stressed in many texts, although it is apparent from the usual discussions of Fock space. One can do all of QED and field theory, many-body theory without 2nd quantization, but the bookkeeping gets to be seriously difficult. Bookkeeping in Fock space is quite suitable for the problems of QED,FT, and many body problems like the electron gas and nuclear matter. If you want an example or two, I suggest Chapter 10 of a current bible, Optical Coherence and Quantum Optics by Mandel and Wolf -- in which they deal directly, in some detail, with the unitary transformations connecting Fock Space with regular p,q space.

Regards,
Reilly Atkinson

 

[Maaneli]

My thesis involved lots of QED, including second order self-energy corrections and so on. So, naturally, I figure I know something about QED. Oh, and by the way, I've also taught QED. So I do know about the importance of self energy; it is very evident that this is so from the structure of the QED interaction.The Lamb shift, g-2 corrections play almost no role in atomic physics because they are very small, and usually can be ignored. The overwhelming majority of atomic physics, and then chemical bonding and considerable amount of nuclear theory are done very nicely without the 2nd order QED corrections(See Condon and Shortley's Theory of Atomic Spectra;Pauling's theory of Chemical Bonding; Blatt and Weisskopf's Theoretical Nuclear Physics all bibles in their time.


And, people apparently forget that 2nd quantization is nothing more than a unitary transformation away from configuration and momentum space representations. That is, the only magic of 2nd quantization is the ease it brings to many field theory issues. I'll admit that this is not stressed in many texts, although it is apparent from the usual discussions of Fock space. One can do all of QED and field theory, many-body theory without 2nd quantization, but the bookkeeping gets to be seriously difficult. Bookkeeping in Fock space is quite suitable for the problems of QED,FT, and many body problems like the electron gas and nuclear matter. If you want an example or two, I suggest Chapter 10 of a current bible, Optical Coherence and Quantum Optics by Mandel and Wolf -- in which they deal directly, in some detail, with the unitary transformations connecting Fock Space with regular p,q space.

Regards,
Reilly Atkinson

That's great to hear that you have lots of experience with QED, and thanks much for the reference with regard to 2nd quantization. I'll have a look at them.


But I would still disagree that those radiative corrections are insignificant for AMO phenomena. For example, in the laser cooling work of Metcalf, Phillips, and Chu, spontaneous emission is absolutely essential. Laser cooling cannot happen without spontaneous emission. Also, vacuum polarization, spontaneous emission, and the Lamb shift are essential parts of the work of experimental cavity QED people like Jeff Kimble at Caltech or Umar Mohideen at UCR.

 

[akhmeteli]

Why isn't the Pauli exclusion principle part of 1st quantized Dirac theory? It is after all a property of the Dirac sea.

I don't know, maybe our dispute is just about terminology. In my book, the 1st quantized Dirac theory is just the Dirac equation. The Dirac sea and the Pauli principle (and any quantum statistics, such as Fermi or Bose) go a step further, towards 2nd quantization. If the 2nd quantization means just the specific form for you, that is your choice. For me, however, it means the specific content. If this is so, maybe there is no point in further dispute.

With all do respect to them, I think they're being facile by just calling it "2nd quantization".

Maybe they also cared more about contents, not the form.

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.

I see. So again, 2nd quantization is a matter of form for you. Nothing to dispute.

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.

It's a long story:-) Actually, in my work, I considered the Klein-Gordon-Maxwell (KGM) system in the unitary gauge, where the Klein-Gordon particle wavefunction is real (Schroedinger used this example (see the reference in my post https://www.physicsforums.com/showpost.php?p=1147276&postcount=9 ) to argue that, contrary to the widely accepted opinion, a charged particle can be described by one real field). I found out that the wavefunction can be eliminated in a natural way, and the resulting equations describe independent evolution of the electromagnetic field. I tried to apply this result to a hydrogen atom, but found out that the equations of the KGM system (with an external current to account for the nucleus) do not reduce to the standard Schroedinger equation, at least not directly, as there is no self-field in the Schroedinger equation, while such self-field is a direct consequence of KGM. I tried to resolve this contradiction, and immediately found references to the Barut's theory. So the reason I started to look for something like Barut's theory was I could not understand the same thing as per.sundqvist, the original poster to this thread. By the way, I much appreciated posts in this forum by nightlight, both on the Barut's theory and other topics. I think his numerous posts may be most interesting for you. Unfortunately, he does not post here anymore.

As for my hopes for the Barut's theory, I don't know. I'd say I was more enthusiastic about it three years ago, when I first found out about it. Now that I've read more about it, I have an impression Barut cut a lot of corners, some of which I mentioned in this thread. I suspect we are still missing something important here.

Maybe I'll send a PM to you.

 

[Maaneli]

I see. So again, 2nd quantization is a matter of form for you. Nothing to dispute.

If by content you mean the predicted results of experiments, and form the physical interpretation (at least that's what it seems to mean to me), then yes.

It's a long story:-) Actually, in my work, I considered the Klein-Gordon-Maxwell (KGM) system in the unitary gauge, where the Klein-Gordon particle wavefunction is real (Schroedinger used this example (see the reference in my post https://www.physicsforums.com/showpost.php?p=1147276&postcount=9 ) to argue that, contrary to the widely accepted opinion, a charged particle can be described by one real field). I found out that the wavefunction can be eliminated in a natural way, and the resulting equations describe independent evolution of the electromagnetic field. I tried to apply this result to a hydrogen atom, but found out that the equations of the KGM system (with an external current to account for the nucleus) do not reduce to the standard Schroedinger equation, at least not directly, as there is no self-field in the Schroedinger equation, while such self-field is a direct consequence of KGM. I tried to resolve this contradiction, and immediately found references to the Barut's theory. So the reason I started to look for something like Barut's theory was I could not understand the same thing as per.sundqvist, the original poster to this thread. By the way, I much appreciated posts in this forum by nightlight, both on the Barut's theory and other topics. I think his numerous posts may be most interesting for you. Unfortunately, he does not post here anymore.

As for my hopes for the Barut's theory, I don't know. I'd say I was more enthusiastic about it three years ago, when I first found out about it. Now that I've read more about it, I have an impression Barut cut a lot of corners, some of which I mentioned in this thread. I suspect we are still missing something important here.

Maybe I'll send a PM to you.

Very interesting story and very interesting paper! I can't wait to have a look at it. What do you think about the claim that Barut's theory is a finite theory? That to me is the most interesting of all.

 

[akhmeteli]

Very interesting story and very interesting paper! I can't wait to have a look at it.

If you mean the Shroedinger's paper, it is very short, and you can find its summary in my online paper (I sent a reference to you in a PM).

What do you think about the claim that Barut's theory is a finite theory? That to me is the most interesting of all.

I did not check that claim myself - it looks like the calculations are quite cumbersome. I think the claim can be true. However, I am not sure this is the most interesting question. I'd say it would be more interesting to know if the predictions of the Barut's theory and the standard QED are identical (I would expect that they are not), and if not, which one describes experimental results better (again, if I had to bet, my choice would be QED). If, however, the predictions are identical, finiteness might acquire great significance. My general feeling is the Barut's theory is not fundamental enough (it only acquires its final form after elimination of the electromagnetic field, and this final form does not look very attractive). As I said, something may be missing.

 

[Maaneli]

If you mean the Shroedinger's paper, it is very short, and you can find its summary in my online paper (I sent a reference to you in a PM).



I did not check that claim myself - it looks like the calculations are quite cumbersome. I think the claim can be true. However, I am not sure this is the most interesting question. I'd say it would be more interesting to know if the predictions of the Barut's theory and the standard QED are identical (I would expect that they are not), and if not, which one describes experimental results better (again, if I had to bet, my choice would be QED). If, however, the predictions are identical, finiteness might acquire great significance. My general feeling is the Barut's theory is not fundamental enough (it only acquires its final form after elimination of the electromagnetic field, and this final form does not look very attractive). As I said, something may be missing.

Why do you expect that the two theories are not empirically equivalent? I agree with you that the results are even more significant if the theories are empirically equivalent. Also, why does the attractiveness of the Barut formalism have anything to do with it being fudamental enough? Certainly QED is not an attractive formalism either. Oppenheimer called it a "stop gap" theory. In fact, I think QED is even less attractive than the Barut theory.

 

[akhmeteli]

Why do you expect that the two theories are not empirically equivalent?

Because I am not sure violations of the Bell inequalities are possible in the Barut's theory.

Also, why does the attractiveness of the Barut formalism have anything to do with it being fudamental enough?

I just tend to think that fundamental theories are typically simple and beautiful, but you may disagree. As Dirac said, "Physical laws should have mathematical beauty", or something like that.

Certainly QED is not an attractive formalism either. Oppenheimer called it a "stop gap" theory. In fact, I think QED is even less attractive than the Barut theory.

I don't now. Actually, QED looks more aesthetically appealing to me than the Barut's theory, maybe because the former is better developed.

 

[per.sundqvist]

I don't know, maybe our dispute is just about terminology. In my book, the 1st quantized Dirac theory is just the Dirac equation. The Dirac sea and the Pauli principle (and any quantum statistics, such as Fermi or Bose) go a step further, towards 2nd quantization. If the 2nd quantization means just the specific form for you, that is your choice. For me, however, it means the specific content. If this is so, maybe there is no point in further dispute.



Maybe they also cared more about contents, not the form.



I see. So again, 2nd quantization is a matter of form for you. Nothing to dispute.



It's a long story:-) Actually, in my work, I considered the Klein-Gordon-Maxwell (KGM) system in the unitary gauge, where the Klein-Gordon particle wavefunction is real (Schroedinger used this example (see the reference in my post https://www.physicsforums.com/showpost.php?p=1147276&postcount=9 ) to argue that, contrary to the widely accepted opinion, a charged particle can be described by one real field). I found out that the wavefunction can be eliminated in a natural way, and the resulting equations describe independent evolution of the electromagnetic field. I tried to apply this result to a hydrogen atom, but found out that the equations of the KGM system (with an external current to account for the nucleus) do not reduce to the standard Schroedinger equation, at least not directly, as there is no self-field in the Schroedinger equation, while such self-field is a direct consequence of KGM. I tried to resolve this contradiction, and immediately found references to the Barut's theory. So the reason I started to look for something like Barut's theory was I could not understand the same thing as per.sundqvist, the original poster to this thread. By the way, I much appreciated posts in this forum by nightlight, both on the Barut's theory and other topics. I think his numerous posts may be most interesting for you. Unfortunately, he does not post here anymore.

As for my hopes for the Barut's theory, I don't know. I'd say I was more enthusiastic about it three years ago, when I first found out about it. Now that I've read more about it, I have an impression Barut cut a lot of corners, some of which I mentioned in this thread. I suspect we are still missing something important here.

Maybe I'll send a PM to you.

Hi, I have been busy with other things for a while. This is an interesting discussion you have here, but I don't think I have got any real answers how to deal with the "unphysical" self-interaction yet, only disputes over which formalism is the best...

Well I know that a localized state, like the 1s state (the "blob"), which you get from the equation I presented in the original post, could be Fourier expanded into waves, but it does not imply its not physical. The non-physical nature must be a consequense of that the particle cannot feel its own electric field.

For the magnetic vector potential, this problem is solved, since its a time-dependent wave equation. you integrate over \vec{j}(\vec{r}-ct\hat{n}), like:

<br /> \vec{A}(\vec{r},t)=\frac{1}{\mu}\int\frac{\vec{j}(\vec{r}-ct\hat{n})}<br /> {\mid\vec{r}-\vec{r}^{\prime}\mid}d^3r^{\prime}<br />

I might missed some details here, because I don't have my books at hand right now. The result is however that the field is moving away from its source (by speed of light) and can not really interact with its source particle at the "same place". But for the scalar field A^0=\Phi/c, you obtain Poisson equation from the QED-Lagrangian, i.e., there is no second time-derivative in Poissons equation. This means that the field is instantanious, and therefore the unphysical self-interaction could take place. Maby there must be some special case with the scalar field? Because you get anyway an electric field in photon creation since \vec{E}=-\nabla\Phi-\partial\vec{A}/\partial t. Should you write \sqrt{i}qA^0 in the Lagrangian or should you postulate \Phi(\vec{r}-ct) in some way?

 

[Maaneli]

Hi, I have been busy with other things for a while. This is an interesting discussion you have here, but I don't think I have got any real answers how to deal with the "unphysical" self-interaction yet, only disputes over which formalism is the best...

Well I know that a localized state, like the 1s state (the "blob"), which you get from the equation I presented in the original post, could be Fourier expanded into waves, but it does not imply its not physical. The non-physical nature must be a consequense of that the particle cannot feel its own electric field.

For the magnetic vector potential, this problem is solved, since its a time-dependent wave equation. you integrate over \vec{j}(\vec{r}-ct\hat{n}), like:

<br /> \vec{A}(\vec{r},t)=\frac{1}{\mu}\int\frac{\vec{j}(\vec{r}-ct\hat{n})}<br /> {\mid\vec{r}-\vec{r}^{\prime}\mid}d^3r^{\prime}<br />

I might missed some details here, because I don't have my books at hand right now. The result is however that the field is moving away from its source (by speed of light) and can not really interact with its source particle at the "same place". But for the scalar field A^0=\Phi/c, you obtain Poisson equation from the QED-Lagrangian, i.e., there is no second time-derivative in Poissons equation. This means that the field is instantanious, and therefore the unphysical self-interaction could take place. Maby there must be some special case with the scalar field? Because you get anyway an electric field in photon creation since \vec{E}=-\nabla\Phi-\partial\vec{A}/\partial t. Should you write \sqrt{i}qA^0 in the Lagrangian or should you postulate \Phi(\vec{r}-ct) in some way?

Hi,


I believe your choices for A and V must be co-dependent. Barut always uses the Lorentz gauge for his choices of A and V, which means both depend on retarded time. I believe initially you wrote your V in the Coulomb gauge.

 

[Maaneli]

Because I am not sure violations of the Bell inequalities are possible in the Barut's theory.

I think there is good reason to expect that Barut's theory does violate the Bell inequalities. The self-field is defined in terms of the probability current j_mu, which is a current in configuration space. Therefore, for two electrons in an atom entangled in configuration space at the level of their probability currents, let's say, then their radiated source fields will also be entangled in terms of their polarizations and wavevectors.

I just tend to think that fundamental theories are typically simple and beautiful, but you may disagree. As Dirac said, "Physical laws should have mathematical beauty", or something like that.

I understand that POV, and am sympathetic to it. But there's nothing that logically implies physical laws <=> beauty.

I don't now. Actually, QED looks more aesthetically appealing to me than the Barut's theory, maybe because the former is better developed.

Interesting.

 

[akhmeteli]

I think there is good reason to expect that Barut's theory does violate the Bell inequalities. The self-field is defined in terms of the probability current j_mu, which is a current in configuration space. Therefore, for two electrons in an atom entangled in configuration space at the level of their probability currents, let's say, then their radiated source fields will also be entangled in terms of their polarizations and wavevectors.

I am not sure j_mu is a current in configuration space in the Barut's theory (which I'll call SFED hereafter - self-field electrodynamics) - just look how Barut defines the current for two particles (in the quote in one of my posts related to the Pauli principle) - the wavefunctions for the two particles, \psi_1 and \psi_2, are in 3D, and the current depends locally on them. nightlight, for example, does not believe there are any VBI in SFED, as far as I understand.

Furthermore, entanglement is not enough for VBI, as far as I understand, you need the projection postulate as well, or something like it, to obtain VBI.

I understand that POV, and am sympathetic to it. But there's nothing that logically implies physical laws <=> beauty.

Certainly, but I don't believe my wording ("I just tend to think that fundamental theories are typically simple and beautiful") was categorical.

 

[per.sundqvist]

Hi,


I believe your choices for A and V must be co-dependent. Barut always uses the Lorentz gauge for his choices of A and V, which means both depend on retarded time. I believe initially you wrote your V in the Coulomb gauge.

Hi, thanks, yes the Lorentz gauge makes sense! I had a look in Jackson so now its clear, so my equation in the original poster should have been:
<br /> <br /> \left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}-<br /> \frac{q^2}{4\pi\epsilon}<br /> \int\frac{\mid\psi(\vec{r}^{\prime},t-\frac{\mid \vec{r}-\vec{r}^{\prime}\mid}{c})\mid^2}<br /> {\mid\vec{r}-\vec{r}^{\prime}\mid}d^3r^{\prime}\right)\psi=<br /> -i\hbar\frac{\partial\psi}{\partial t}<br /> <br />

This makes sense, since self-interaction is restricted to a very short time. So now there is no problem, and you would not need second quantization (unless you like it)?

 

[Maaneli]

I am not sure j_mu is a current in configuration space in the Barut's theory (which I'll call SFED hereafter - self-field electrodynamics) - just look how Barut defines the current for two particles (in the quote in one of my posts related to the Pauli principle) - the wavefunctions for the two particles, \psi_1 and \psi_2, are in 3D, and the current depends locally on them. nightlight, for example, does not believe there are any VBI in SFED, as far as I understand.

Furthermore, entanglement is not enough for VBI, as far as I understand, you need the projection postulate as well, or something like it, to obtain VBI.



Certainly, but I don't believe my wording ("I just tend to think that fundamental theories are typically simple and beautiful") was categorical.

Sorry, I should have read your words more carefully. But another consideration is how do you judge what theory is more "beautiful" than another? Certainly there is no objective criterion, as you can see by the fact that we disagree about which is more "beautiful" a theory QED or SFED. And of course neither of us is more "correct". I agree however that simplicity is a more objective property of a theory, and there I would also suggest the Barut theory is superior (at least in form).

Actually, I have communicated with Jonathan Dowling (Barut's former graduate student who worked on SFED) in the past about some of these issues, and he did say to me that Asim and he hoped entanglement could be accounted for by SFED. However, while it is true that in the 2-particle examples that Barut uses the currents are in 3D, this is because he never considers an entanglement case to my knowledge. The wavefunctions in the cases you mention are factorizable, meaning that

psi(x1, x2) = psi(x1)\otimespsi(x2),

and so his currents will be given by

j(x1) + j(x2) = rho(x1)*v1 + rho(x2)*v2,

where

rho(x1) = |psi(x1)|^2 and rho(x2) = |psi(x2)|^2.

If he were to consider the basic singlet state for two electrons, and include the self-fields, then it seems obvious to me that not only would the wavefunctions not be factorizable, but neither would the currents. In other words, he would only have

rho(x1, x2) = |psi(x1, x2)|^2

and

j_n(x1, x2) = rho(x1, x2)*v_n = rho(x1, x2)*(v1 + v2).

So the probability currents would necessarily be in configuration space. This also means that the self-fields would not be distinctly separable as being sourced by two separate electrons.

As for the projection postulate, no you don't necessarily need it to get VBI. In the pilot wave theory, or stochastic mechanics, you can easily account for VBI due to the branching of wavefunctions after a measurement interaction, from the initial superposition state, and the observed point particle goes into only one of those branches. No postulates are needed. This is why I want to combine SFED with pilot wave theory and stochastic mechanics. It is the easiest and most rigorous way to account for measurement interactions, which Barut didn't really focus on with his theory. Of course, if you want a wavefunction collapse mechanism, you can certainly obtain it in a mathematically rigorous way, also without postulates, using the GRW stochastic collapse mechanism. By the way, all these mechanisms can be made relativistically covariant, so that is no problem either. And I see no fundamental obstacle to combining these measurement theories with SFED.

 

[Maaneli]

Hi, thanks, yes the Lorentz gauge makes sense! I had a look in Jackson so now its clear, so my equation in the original poster should have been:
<br /> <br /> \left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}-<br /> \frac{q^2}{4\pi\epsilon}<br /> \int\frac{\mid\psi(\vec{r}^{\prime},t-\frac{\mid \vec{r}-\vec{r}^{\prime}\mid}{c})\mid^2}<br /> {\mid\vec{r}-\vec{r}^{\prime}\mid}d^3r^{\prime}\right)\psi=<br /> -i\hbar\frac{\partial\psi}{\partial t}<br /> <br />

This makes sense, since self-interaction is restricted to a very short time. So now there is no problem, and you would not need second quantization (unless you like it)?

Hi Per,

Yes, you got it right. Now there is no problem. And you don't need second quantization for anything other than practical convenience in dealing with relativistic N-body systems, as Barut also mentions in his papers.

 

[Maaneli]

Hi, thanks, yes the Lorentz gauge makes sense! I had a look in Jackson so now its clear, so my equation in the original poster should have been:
<br /> <br /> \left(\frac{\hat{p}^2}{2m}+q\Phi_{ext}-<br /> \frac{q^2}{4\pi\epsilon}<br /> \int\frac{\mid\psi(\vec{r}^{\prime},t-\frac{\mid \vec{r}-\vec{r}^{\prime}\mid}{c})\mid^2}<br /> {\mid\vec{r}-\vec{r}^{\prime}\mid}d^3r^{\prime}\right)\psi=<br /> -i\hbar\frac{\partial\psi}{\partial t}<br /> <br />

This makes sense, since self-interaction is restricted to a very short time. So now there is no problem, and you would not need second quantization (unless you like it)?

And you know how to include the vector potential in this equation?

 

[per.sundqvist]

And you know how to include the vector potential in this equation?

Thanks Maaneli,

Oh yes, I only included the dominating part because of simplicity. I read part of Baruts paper also, interesting. In the non-relativistic limit it should be:

<br /> \frac{\hat{p}^2}{2m}\rightarrow \frac{\vec{(\sigma\cdot (\hat{p}+q\vec{A}))^2}}{2m}<br />

and A is similar given by the free Green function using the quantum current density as a source. I know its a dirty trick to replace E with -ihd/dt in Diracs equation to get the time-dependent Schrödinger, but I believe that the correction is very small any way.

 

[Maaneli]

Thanks Maaneli,

Oh yes, I only included the dominating part because of simplicity. I read part of Baruts paper also, interesting. In the non-relativistic limit it should be:

<br /> \frac{\hat{p}^2}{2m}\rightarrow \frac{\vec{(\sigma\cdot (\hat{p}+q\vec{A}))^2}}{2m}<br />

and A is similar given by the free Green function using the quantum current density as a source. I know its a dirty trick to replace E with -ihd/dt in Diracs equation to get the time-dependent Schrödinger, but I believe that the correction is very small any way.

OK. Yes, the corrections are small, and of course we aren't trying to do anything too precise here.

So what do you think of these results? I like to imagine how much easier QED might have been if it had started from this route.

 

[Maaneli]

Hi Per,

Yes, you got it right. Now there is no problem. And you don't need second quantization for anything other than practical convenience in dealing with relativistic N-body systems, as Barut also mentions in his papers.

Hey, woops I made a big error. This isn't quite right from the Barut POV. In particular, it is problematic to find a normalization based on rho = |psi(x, t - r/c)|^2. In Barut SFED you would have to replace the Green's function you're using with the Feynman propagator in order to satisfy the Lorentz gauge. The wavefunction does not depend on retarded time in that case. Then the self-interaction is self-consistent.

For more details on this, see this paper:

http://phys.lsu.edu/~jdowling/publications/Barut89b.pdf

 

[Maaneli]

Hi, I have been busy with other things for a while. This is an interesting discussion you have here, but I don't think I have got any real answers how to deal with the "unphysical" self-interaction yet, only disputes over which formalism is the best...

Well I know that a localized state, like the 1s state (the "blob"), which you get from the equation I presented in the original post, could be Fourier expanded into waves, but it does not imply its not physical. The non-physical nature must be a consequense of that the particle cannot feel its own electric field.

For the magnetic vector potential, this problem is solved, since its a time-dependent wave equation. you integrate over \vec{j}(\vec{r}-ct\hat{n}), like:

<br /> \vec{A}(\vec{r},t)=\frac{1}{\mu}\int\frac{\vec{j}(\vec{r}-ct\hat{n})}<br /> {\mid\vec{r}-\vec{r}^{\prime}\mid}d^3r^{\prime}<br />

I might missed some details here, because I don't have my books at hand right now. The result is however that the field is moving away from its source (by speed of light) and can not really interact with its source particle at the "same place". But for the scalar field A^0=\Phi/c, you obtain Poisson equation from the QED-Lagrangian, i.e., there is no second time-derivative in Poissons equation. This means that the field is instantanious, and therefore the unphysical self-interaction could take place. Maby there must be some special case with the scalar field? Because you get anyway an electric field in photon creation since \vec{E}=-\nabla\Phi-\partial\vec{A}/\partial t. Should you write \sqrt{i}qA^0 in the Lagrangian or should you postulate \Phi(\vec{r}-ct) in some way?

One other thing I should have added. Even if you use the Green's function, 1/|r - r'|, I think you can still use the Coulomb gauge here just as well as the Lorentz gauge and get the same empirical predictions. Recall that the gauge you choose does not affect the empirical predictions. In that sense, the instantaneous nonlocal interaction is not "unphysical".

 

[akhmeteli]

Sorry, I should have read your words more carefully. But another consideration is how do you judge what theory is more "beautiful" than another? Certainly there is no objective criterion, as you can see by the fact that we disagree about which is more "beautiful" a theory QED or SFED. And of course neither of us is more "correct". I agree however that simplicity is a more objective property of a theory, and there I would also suggest the Barut theory is superior (at least in form).

Again, I just described my "expectations" and "feelings" in reply to your "what do you think" question. I readily admit that I don't have much to support them.

Actually, I have communicated with Jonathan Dowling (Barut's former graduate student who worked on SFED) in the past about some of these issues, and he did say to me that Asim and he hoped entanglement could be accounted for by SFED. However, while it is true that in the 2-particle examples that Barut uses the currents are in 3D, this is because he never considers an entanglement case to my knowledge. The wavefunctions in the cases you mention are factorizable, meaning that

psi(x1, x2) = psi(x1)\otimespsi(x2),

and so his currents will be given by

j(x1) + j(x2) = rho(x1)*v1 + rho(x2)*v2,

where

rho(x1) = |psi(x1)|^2 and rho(x2) = |psi(x2)|^2.

If he were to consider the basic singlet state for two electrons, and include the self-fields, then it seems obvious to me that not only would the wavefunctions not be factorizable, but neither would the currents. In other words, he would only have

rho(x1, x2) = |psi(x1, x2)|^2

and

j_n(x1, x2) = rho(x1, x2)*v_n = rho(x1, x2)*(v1 + v2).

So the probability currents would necessarily be in configuration space. This also means that the self-fields would not be distinctly separable as being sourced by two separate electrons.

It does not seem obvious to me that the wavefunctions would not be factorizable. Furthermore, in the current form of SFED they are factorizable, to the best of my knowledge. If and when Dowling or somebody else proposes a new version of the theory, this issue may be discussed again.

As for the projection postulate, no you don't necessarily need it to get VBI. In the pilot wave theory, or stochastic mechanics, you can easily account for VBI due to the branching of wavefunctions after a measurement interaction, from the initial superposition state, and the observed point particle goes into only one of those branches. No postulates are needed. This is why I want to combine SFED with pilot wave theory and stochastic mechanics. It is the easiest and most rigorous way to account for measurement interactions, which Barut didn't really focus on with his theory. Of course, if you want a wavefunction collapse mechanism, you can certainly obtain it in a mathematically rigorous way, also without postulates, using the GRW stochastic collapse mechanism. By the way, all these mechanisms can be made relativistically covariant, so that is no problem either. And I see no fundamental obstacle to combining these measurement theories with SFED.

I just wanted to say that one needs the projection postulate or something like that to prove that there can be VBI in the standard quantum theory.

I'd say you have an ambitious program. It is not quite clear though why you need SFED at all, if you are going to make it nonlocal and stochastic anyway:-)

 

[per.sundqvist]

Hey, woops I made a big error. This isn't quite right from the Barut POV. In particular, it is problematic to find a normalization based on rho = |psi(x, t - r/c)|^2. In Barut SFED you would have to replace the Green's function you're using with the Feynman propagator in order to satisfy the Lorentz gauge. The wave function does not depend on retarded time in that case. Then the self-interaction is self-consistent.

For more details on this, see this paper:

http://phys.lsu.edu/~jdowling/publications/Barut89b.pdf

Ok I read it briefly, and what I understood was that you include heavyside functions also in the propagator. Is that right?

Now I was actually going to try to solve this numerically for some simple system using 3D FEM-numerics. So I want to solve Maxwell's equations and Schrödinger coupled (self-consistently) in time. But do I have to change Maxwell's equ to get the Feynman propagator? Normalizing time-Schrödinger is not a problem numerically these days. My little project is to make a movie showing how the EM-wave is created and the wave function is changed. But it all relies on that the equations are ok.

 

[Maaneli]

Again, I just described my "expectations" and "feelings" in reply to your "what do you think" question. I readily admit that I don't have much to support them.



It does not seem obvious to me that the wavefunctions would not be factorizable. Furthermore, in the current form of SFED they are factorizable, to the best of my knowledge. If and when Dowling or somebody else proposes a new version of the theory, this issue may be discussed again.



I just wanted to say that one needs the projection postulate or something like that to prove that there can be VBI in the standard quantum theory.

I'd say you have an ambitious program. It is not quite clear though why you need SFED at all, if you are going to make it nonlocal and stochastic anyway:-)

<< Furthermore, in the current form of SFED they are factorizable, to the best of my knowledge. >>

But that's my point, I don't think they are factorizable even in the current form of SFED. It's just that Barut never bothered to analyze the singlet state according to Dowling.

<< If and when Dowling or somebody else proposes a new version of the theory, this issue may be discussed again. >>

I think we can discuss it now.

<< I just wanted to say that one needs the projection postulate or something like that to prove that there can be VBI in the standard quantum theory.>>

That's true.

<< I'd say you have an ambitious program. It is not quite clear though why you need SFED at all, if you are going to make it nonlocal and stochastic anyway:-) >>

I don't think it's that ambitious in the sense that it is not that hard to combine SFED with pilot wave theory or stochastic mechanics. I have unpublished notes in which I have already done this. Of course, I already think the SFED wavefunction is generally nonlocal, until I see an argument otherwise. BTW, the reasons why I want to combine SFED with pilot wave theory and stochastic mechanics is first just because SFED as it is needs a measurement theory that solves the measurement problem, and second because I do not believe the wavefunction is a fundamental field (in either an ontological or nomological sense). A stochastic mechanical theory allows one to derive the wavefunction as a phenomenological approximation (much like the transition probability solution to a diffusion equation) to a more fundamental, causally symmetric, stochastic particle dynamics that, I believe, can show that nonlocality is only an approximation. Currently, it is believed that a stochastic mechanical derivation of the wavefuncton in QM is unsuccessful because of Timothy Wallstrom's criticism that such derivations cannot satisfy the Bohr-Sommerfeld quantization condition for the gradient of the phase of a wavefunction around a closed loop. However, I believe a stochastic mechanical derivation of the SFED wavefunction solves that problem simply because its phase doesn't have to satisfy the Bohr-Sommerfeld quantization condition.

And of course I do desire a pilot wave or stochastic theory of electrodynamics that is nonperturbative and finite; and combining SFED with pilot wave theory and stochastic mechanics is the only way to do that thus far.

Hope that helps clarify my view.

 

[Maaneli]

Ok I read it briefly, and what I understood was that you include heavyside functions also in the propagator. Is that right?

Now I was actually going to try to solve this numerically for some simple system using 3D FEM-numerics. So I want to solve Maxwell's equations and Schrödinger coupled (self-consistently) in time. But do I have to change Maxwell's equ to get the Feynman propagator? Normalizing time-Schrödinger is not a problem numerically these days. My little project is to make a movie showing how the EM-wave is created and the wave function is changed. But it all relies on that the equations are ok.

<< Ok I read it briefly, and what I understood was that you include heavyside functions also in the propagator. Is that right? >>

Unfortunately I can't look at the paper again right now, but that sounds OK. That paper should have defined everything you need.

<< Normalizing time-Schrödinger is not a problem numerically these days. >>

I meant two things. First, if I recall correctly, for some reason there is no way to interpret

rho = |psi(x, t - r/c)|^2

as a probability measure on configuration space, as Squires, Duerr, Goldstein, and Berndl have shown. Besides that, if the wavefunction did depend on retarded time, then it would be a locally propagating signal, just like a classical EM wave, and then this theory could never be nonlocal or violate the Bell inequalities, which means it would be empirically inadequate.

Although, such a physically incorrect theory could still be mathematically well-defined enough for you to do your simulations.

 

[akhmeteli]

But that's my point, I don't think they are factorizable even in the current form of SFED. It's just that Barut never bothered to analyze the singlet state according to Dowling.

What I'm saying is based on the quote from Barut on the current for two identical particles.

I think we can discuss it now.

So what version are we supposed to discuss?

BTW, the reasons why I want to combine SFED with pilot wave theory and stochastic mechanics is first just because SFED as it is needs a measurement theory that solves the measurement problem, and second because I do not believe the wavefunction is a fundamental field (in either an ontological or nomological sense). A stochastic mechanical theory allows one to derive the wavefunction as a phenomenological approximation (much like the transition probability solution to a diffusion equation) to a more fundamental, causally symmetric, stochastic particle dynamics that, I believe, can show that nonlocality is only an approximation. Currently, it is believed that a stochastic mechanical derivation of the wavefuncton in QM is unsuccessful because of Timothy Wallstrom's criticism that such derivations cannot satisfy the Bohr-Sommerfeld quantization condition for the gradient of the phase of a wavefunction around a closed loop. However, I believe a stochastic mechanical derivation of the SFED wavefunction solves that problem simply because its phase doesn't have to satisfy the Bohr-Sommerfeld quantization condition.

And of course I do desire a pilot wave or stochastic theory of electrodynamics that is nonperturbative and finite; and combining SFED with pilot wave theory and stochastic mechanics is the only way to do that thus far.

Hope that helps clarify my view.

Yes, it does. Although it is difficult for me to judge how promising is the direction you chose.

 

[Maaneli]

What I'm saying is based on the quote from Barut on the current for two identical particles.

But I don't recall he was explicitly talking about two entangled particles. One can still write down the two-body theory without entanglement even in standard QM. Can you refer me to the exact paper again?

So what version are we supposed to discuss?

As I had discussed, we should consider the singlet state for spinor wavefunctions. Then put in the self-fields. I see no reason why putting in the self-fields will make the singlet state wavefunctions factorziable, since the self-fields are already defined in terms of the same entangled probability currents of the singlet state in QM without the self-fields.

Yes, it does. Although it is difficult for me to judge how promising is the direction you chose.

Understandable that it's difficult to judge the fruitfulness of my approach. I would have to go into much more detail which I don't know if we can do here.