Hi, I'm not very good at QFT, but starting from Dirac's QED Lagrangian (look for example at: wikipedia/Quantum_electrodynamics ) [tex] 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} [/tex] 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: [tex] \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 [/tex] 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.
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!
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.
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 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!
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.
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: Quantum Computing Lecture Notes 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.
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
"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 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. OK, for the Pauli exclusion principle, thanks. 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.
I should also add that Barut and Dowling did in fact extend their approach to 2nd quantized matter fields and got the same answers.
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
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.
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.
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.