Form factors, self-energy, and dynamics of a single spin 1/2 particle

Question

The textbook Dirac particle is an idealized particle (often called a ''point particle''). Its scattering behavior from external electromagnetic fields is simple and explicitly computable, but not realistic for any particle observed in nature.

How does a real (as opposed to idealized) relativistic spin 1/2 particle respond to a classical electromagnetic field that is weak enough that pair creation can be neglected?

Comments

What is the difference between a real and an idealised relativistic spin-1/2 particle?

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The explanation was added to the question.

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Yes, I have no disagreement with you, but the answer is defining and using the form factors, and this is what people will google if they are interested in this stuff.

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Do you think Google ranks pages where form factor is in the title higher than if it is the body?

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@ArnoldNeumaier: I don't know if algorithms do this, but certainly a human reader navigating titles would. Thanks for the title.

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You mean form factors! I was curious too, I had a hard time interpreting. Can you add "Form factors for ..." at the beginning of the title to make it clear for casual visitors?

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I meant dynamics. To formulate the dynamics you need form factors and self energy. These can be (and are in practice) treated either as phenomenological functions to be fitted to experiments, or as objects computed in some approximation from an underlying quantum field theory.

Answer 1

The scattering behavior of a textbook Dirac particle (or ''point particle'') from an external electromagnetic field is simple and explicitly computable, but (at high accuracies) not realistic for any particle observed in Nature. Thus the Dirac particle is an idealized particle. The equation of motion of a  real (as opposed to idealized) relativistic spin 1/2 particle is given by a modified Dirac equation that correctly describes the response to a classical electromagnetic field that is weak enough that pair creation and higher order terms in the field can be neglected. (This still allows motion in the central Coulomb field generated by a nucleus of charge $Z$ up to appoximately 137, and in particular the motion of an electron in the hydrogen atom.)

In the following, I derive the most general form such a modified Dirac equation may have. Special cases of this equation is actually used (and were always used) to compute the hyperfine splitting. There is no other known way to derive the hyperfine splitting (the Lamb shift) or the anomalous magnetic moment of a spin 1/2 particle, except by using these equations. The functions entering this modified Dirac equations (the form factors and self-energies) are either fitted to experiment (for hadrons) or derived from QED (for electrons) or QCD (for hadrons). 

Since the equation must be covariant, we look for Poincare invariant field equations of Dirac type. The most general Hermitian Poincare invariant (pseudo-differential) operator on the space of a relativistic spin 1/2 particle is given in terms of $p:=i\hbar\partial$ by
\[
D_0(p):= p\!\!\!/\lambda(p^2)-m(p^2)
\]
with 
\[
p\!\!\!/:=\gamma\cdot p,
\]
where $\gamma$ is the Dirac 4-vector and $\lambda(p^2)$ and $m(p^2)$ are appropriate real-valued functions.

If we allow for terms linear in an external (classical) electromagnetic field - the mean field generated by all other charged matter in the universe - with 4-vector potential $A(x)$, we find the most general first order interaction terms possible, leading to an operator of the form
\[
D(p):=D_0(p)-(j(p)\cdot A(x))
\]
with a vector
\[
j(p):=e(p^2)\gamma+i\mu(p^2)\Sigma p,
\]
where $\Sigma$ is the antisymmetric matrix with components $\Sigma_{\mu\nu}:=\tfrac12[\gamma_\mu,\gamma_\nu]$ and $e(p^2)$ and $\mu(p^2)$ are appropriate real-valued functions. (Ensuring gauge invariance would in general require additional terms nonlinear in $A$ not treated here.)

Thus Poincare invariance dictates that the single-particle dynamics of a 1-particle wave function is (to first order in the field) necessarily of the form of a modified Dirac equation
\[
D(p)\psi=0. 
\]
Which functions appear must be determined for any kind of real particle by comparing with the results of spectral experiments (where the particle is confined by the field, leading to a discrete bound state spectrum)  and of scattering experiments (where the particle is not confined by the field). 

In the special case of a time-independent electric or magnetic field we may take a Fourier-Laplace transform, where $p$ becomes a momentum vector $p=Eu/c$, where $u$ is the 4-velocity of the mode and $E$ is a complex energy. The spectrum and the scattering matrix are then completely determined by the analytic behavior of the Greens function 
\[
G(E):=D(p)^{-1}.
\]
The $u$-dependence can be removed by an appropriate Lorentz transformation. The real singularities of the $E$-dependence define the spectrum, complex singularities on the second sheet determine the resonances, and the discontinuity along branch cuts determines the scattering angles.

An external electric Coulomb field defines a hydrogen-like atom. (The special case where the particle is an idealized electron and the Coulomb field is produced by an idealized proton gives more or less idealized hydrogen atoms. One may replace the electron and/or proton by other particles to get a variety of variants.) Adding a constant electric or magnetic field induces spectral shifts. These allow to probe experimentally the structure of the functions occuring in the definition of $D(p)$.

A point particle of mass $m_0$ and charge $-e_0$ corresponds to the ordinary minimally coupled Dirac equation, which is the special case of constant functions with values
\[
\lambda(p^2)=1,~~~m(p^2)=m_0,~~~e(p^2)=e_0,~~~\mu(p^2)=0.
\]
In the absence of an external field, its spectrum consists of a single energy $E=m_0c^2$. The Coulomb field of a nucleus produces a mixed spectrum, with highly degenerate bound states due to an $SO(4)$ symmetry (from the conserved Lenz-Runge vector) and a continuous spectrum for scattering states obtained by ionization. Adding electric or magnetic fields produces a fine-splitting of degenerate levels, but there is a residual symmetry leaving some spectral degeneracy.

Real hydrogen-like atoms do not have this degeneracy but produce a hyperfine splitting defining the Lamb shift. This proves that real electrons, muons, tauons are not point particles. Indeed, all functions mentioned are now momentum-dependent and can be obtained by fitting appropriate ansatz functions to information obtained from spectral and scattering experiments. For obvious reasons one calls $m(p^2)$ the running mass and $e(p^2)$ the running charge.

Deviations from the point particle structure are traditionally characterized by an electric form factor 
\[
F_1(p^2):=e(p^2)/e_0, 
\]
a magnetic form factor 
\[
F_2(p^2):=2m_0\mu(p^2)/e_0,
\]
and the so-called self-energy 
\[
\Sigma(p\!\!\!/):=p\!\!\!/-m_0-D_0(p)
\]
involving $\lambda(p^2)$ and $m(p^2)$. The latter definition allows one to write the free Greens function in the conventional form
\[
D_0(p)^{-1}=(p\!\!\!/-m_0-\Sigma(p\!\!\!/))^{-1}
\]
suitable for simple perturbation theory. 

For particles described by a quantum field theory such as the electron $e$, one can also compare with the results of renormalized perturbation theory for the scattering process $e+\gamma\to e$ at a virtual photon $\gamma$ to first order in the electromagnetic field but to arbitrary order in the number of loops. This defines the form factors and the self-energy perturbatively. For the electron, QED provides a very high accuracy description of these functions. 

Since the photon is massless, QED has a nontrivial infrared behavior nottypically treated well in the standard QFT textbooks. In particular, thecontinuous spectrum starts directly at the lowest possible energy of a physical electron. This implies that the Green's function of the electron has no pole at the physical electron mass but an essential power singularity with an exponent defined within QED by an anomalous dimension. Thus the physical electron (i.e., the asymptotic QED state with electron number 1) does not have a well-defined electron mass, but its mass has a continuous spectrum, even in the absence of an external field. (Intuitively, this happens because the physical electron is dressed by a coherent state of photons that slightly change the electron momentum and hence mass; in perturbative language by a cloud consisting of an arbitrary number of soft virtual photons.) This continuous spectrum is therefore encoded in $D_0(p)$. 

Thus QED tells us a lot about the nontrivial structure of the physical electron, encoded in the form factors and the self-energy, and proves that a single, nearly free electron is already a highly complex object. It is astonishing that it can be described to an exceedingly high accuracy (relative error of $10^{-12}$ in terms of two parameters only, namely the rest mass and rest charge of the electron. 

References:

Modified Dirac equations: L.L. Foldy, The Electromagnetic Properties of Dirac Particles, Phys. Rev. 87 (1952), 688--693.

Relativistic form factors: D.R. Yennie, M.M. Levy and D.G. Ravenhall, Electromagnetic Structure of Nucleons, Rev. Mod. Phys. 29 (1957), 144--157.

Popular article on particle form factors: R. Wilson, Form factors of elementary particles, Physics Today 22 (1969), 47-53. 

1-loop calculations of electron form factors: M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley, New York, 1996. (In particular pp. 186 and 220)

Electron branch cut and anomalous exponents: T. Appelquist and J. Carazzone, Infrared singularities and massive fields, Phys. Rev. D11 (1975), 2856--2861.

Comments

Does anyone know how to encode $\slashp$ so that the system can write it as a slashed $p$?

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Done. I used p\!\!\!/

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At first glance it looks that you derived the real electron equation solely from the Poincaré invariance ;-)

Apar from Poincaré invariance, what physics is described with those form-factors? Due to what they are different from 1 and what is physical origin of the "self-energy" $\Sigma$?

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I only derived the form of the equations, not the equations themselves. I.e., I know that there can be self-energy and form factors but know nothing about their values. A true derivation finds these values in terms of a few parameters, as for the electron in QED. For protons and neutrons, the form factors are accurately known from experiment but only through fitting, not through derivation - they should come from the standard model.

Informally, form factors describe the ''form'' of a particle in some general sense. http://en.wikipedia.org/wiki/Form_factor says:

Electric form factor, the Fourier transform of electric charge distribution in space
Magnetic form factor, the Fourier transform of an electric current distribution in space

But this is not fully accurate, as the charge and current distributions of a quantum particle depend on the state, while the form factors are state independent - essentially interpreted in something like the the ground state of the free particle. 

For example, one usually defines the charge radius by a semiclassical comparison with the scattering behavior of a classical charged ball. It is always positive, very tiny for the electron, whereas the electron charge density in a hydrogen atom essentially fills the whole atom. 

The physical origin of all these is the renormalization procedure. In semiclassical terms that you prefer, the physical origin of the self-energy is probably what you hate and call ''wrong'' self-induction. 

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Self-induction effect is subtracted, but permanent interaction with the radiated field remains. The $\Sigma$ describes the latter.

Speaking of the real electron, you forgot to write down an equation for the electromagnetic degrees of freedom whose coupling determines the form-factors and the "self-energy" (what a misleading label!)

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I didn't create the label self-energy. But the term isn't too bad; it describes the part of the energy of the free particle that isn't due to the Dirac equation itself (anonymous spin 1/2) but to what is special to the electron , i.e., its self.

There are no electromagnetic degrees of freedom in an equation for a single particle in an external field. There is, however, a similar equation for a single real photon interacting with a classical current. This has nothing to do with the equation for a real electron; it would be a different question, which you might want to ask in Q/A. Both equations are of course simplifications of the real thing, which is QED with its full asymptotic multiparticle structure.

I said already that I didn't determine the form-factors and the self-energy, only gave the general form and demonstrated (by reference to computations and experiment) that they are nontrivial. But for the electron they are determined by the field equations for QED, and the 1-loop approximations can be found in the textbooks. If you are really interested in how the field equations for QED look like in renormalized form, pose it as a question in Q/A, and I'll try to dig out what I can find about it.

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The term $\Sigma$ has a dimension of energy, but it is not really a self-energy. Before mass renormalization $\Sigma$ contains indeed a term $\delta m c^2\cdot \mathbf{I}$, but after renormalization it is gone and $\Sigma$ does not have diagonal matrix elements (the part proportional to the unit matrix $\mathbf{I}$ is absent). It is an interaction matrix, to be precise.

The real electron interacts permanently with the radiation field degrees of freedom. Their impact is seen in those additional terms that differ an idealized electron from the real one. When the real electron is scattered from a static potential, it radiates obligatorily, so the radiated field equations belong to the real electron.

Your explanation of an uncertain electron mass matches well with my electronium idea, I think.

I asked about form-factors not because I did not know them, but because I was sure we understand them differently.

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The terminology (form factors and self-energy) is standard; one cannot change it. The identification of the self-energy comes from matching the expressions for the Green's function in perturbation theory. 

The question was about a single quantum particle in a classical external field. There are no counterterms and there is no renormalization. The perturbation theory and the scattering theory are canonical, without any difficulties. 

There is no other particle around, and there are no radiation degrees of freedom (as one would have in QED). The self-field of the particle is the classical mean field generated by the current $j(x):=\psi(x)^*j(p)\psi(x)$ according to $p^2A(x)=j(x)$

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@dimension10: This ''answer'' by Vladimir is not an answer at all, but a comment to my answer. it should be treated accordingly.

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A single electron is not really real, sorry for pun.

OK, let us consider your real electron as a target and a heavy charged projectile as an external field or so. Does your electron radiate while being hit? What do you think the point-like projectile will see instead of a point-like electron? What will be the "smearing size" for your electron?

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If a single electron is not real then nothing in physics is real, as we always ignore the part deemed irrelevant for a particular analysis.

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There are two ways of "ignoring" what is deemed "irrelevant", but let us first concentrate on scattering questions above, please.

In addition to them, please tell me whether adding a Coulomb external field in your Dirac-with-form-factors equation can produce the Lamb shift?

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When the real electron is scattered from a static potential, it radiates obligatorily, so the radiated field equations belong to the real electron.

In a single particle equation, radiation is accounted for through a dissipative loss of energy, modeled by an imaginary part in $D_0(p)$ and hence the self-energy. When computing the self-energy from QED, the imaginary part comes from the fact that the $i\eps$ prescription needed for the branch cut in the Green's function of an electron-photon subsystem induces complex terms into the effective system where the photon is projected out. This leads to complex eigenvalues of the modified Dirac operator, and the corresponding line width is the contribution due to radiative scattering.

For example, it is well-known that the hyperfine splitting in a hydrogen electron is complex, and the Lamb shift is only its real part.

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Arnold, please answer directly, does your electron radiate? Yes or No question. I know how it is done in QED, but you speak of a single "real" electron in an external field.

And do you think in a Hydrogen atom your "real" electron gives the right shifts of energy levels? I mean, can we really deal with your "real" electron in an external field and forget about electromagnetic field degrees of freedom? Does your "real" electron without electromagnetic field degrees of freedom make any sense?

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 does your electron radiate?

The electron loses energy to the environment due to an optical potential part in $D_0(p)$, which shows in a spectral line width. In a more comprehensive QED treatment this would be seen as radiation. This is not so different from calculating the spectrum of a textbook hydrogen atom, which shows sharp lines since there is no dissipation. In a more comprehensive treatment modeling interaction with a family of harmonic oscillators it would show resonance at the spectral energies, which means that one can ''see'' the spectrum.

But as customary in physics one prefers the less comprehensive treatment if it already gives all the information. As QM models usually do not include the oscillators that allow us to see a spectrum (unless this seeing process is to be studied), so my model does not include the oscillators that allow us to see the radiation. To include them, one must in both cases extend the model, and can do so without difficulty. But then it is no longer a single-particle equation; hence this part is off-topic here.

do you think in a Hydrogen atom your "real" electron gives the right shifts of energy levels? I mean, can we really deal with your "real" electron in an external field and forget about electromagnetic field degrees of freedom?

Of course; that's the whole point. With the correct form factors and self-energy (obtainable from QED) the spectrum of my model equation is (including the line width due to radiation) just that predicted by QED (apart from corrections coming from adding higher order in $A$ terms to my equations). In fact, this is one way these spectra are actually calculated  (e.g., in Itzykson-Zuber)!

Everywhere in physics one uses simpler models than the quantum theory of the whole universe to reduce a phenomenon to its essence. Which usually means cutting off the part that is only responsible for observablity. Otherwise we couldn't do classical mechanics of rigid bodies, as we neglect both light (seeing the objects) and sound (hearing objects slide, etc.). Only microscopic light or sound studies add these degrees of freedom. On the macroscopic level the sliding is just a dissipative coefficient in the equations of motions, which lead to friction. 

In my model, the line broadening is the analogue of friction - it is the sign that some interaction with the environment is taken into account only in one direction (as it effects the particle) but not in the other (the radiation of sound waves or electromagnetic waves).

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Thank you, Arnold, for your detailed explanation. +1!

It is exactly what I wanted to learn about your model. I will give my remarks later.

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You can even accommodate in the above general setting fictitious particles like that of your own toy model or your electronium model if you project out the oscillators and work out the reduced particle Green's function and asymptotic scattering, and make everything relativistic. This just corresponds to different choices of form factors and self-energy.

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No, I do not think I can accommodate your equation to the relativistic electronium. It is true, my model needs further development badly, but I think, in another direction. The electron "form-factors" in QED are calculable namely due to taking into account all the interactions experienced by the (constituent, let me call it so) electron. They are not constant and do not characterize a "real" electron in your understanding. The constituent electron never comes alone.

I asked you about the QM smear of your real electron "seen" by a fast (even non relativistic) projectile (what your electric form-factor describes), and you did not answer. This QM smearing is essential for understanding what is going on and even in atom the additional smearing depends on the orbit (on the external field). It was described in my paper "Atom as a "Dressed" Nucleus" and mentioned in my comments above. Too bad it has passed unnoticed.

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''I do not think I can accommodate your equation to the relativistic electronium.''

I meant it the other way around: Given a Poincare invariant electronium model, you can compute perturbatively the self-energy and the form factors, and get a particular instance of my equations.

You can view the electron current in the rest frame of the projectile to find out which electromagnetic field it sees - apart from the amount of radiation, which is not determined by the current. This would have to be extracted from the spectral information, which tells about frequencies and energy loss. 

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Thank you Arnold, for your kind answer. I will reply later.

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I do not see any logic in this motivation: "

The scattering behavior of a textbook Dirac particle (or ''point particle'') from an external electromagnetic field is simple and explicitly computable, but (at high accuracies) not realistic for any particle observed in Nature. ... Since the equation must be covariant, we look for Poincare invariant field equations of Dirac type."

If Dirac Poincare invariant equation is wrong (i.e., we do not have the right description), then what is the significance of "Poincare invariance"?

Next, "a single spin 1/2 particle" means evidently not creating pairs while scattering, but creating photons is allowed. For creating photons the "single particle" must be permanently coupled with EMF oscillators, so it is not really "single". Then the form-factors make sense. And they exist indeed, but different from close to unity elastic $e(q)$ and $\mu(q)$ (where $q$ is the transferred momentum). For qualitative explanation, see my paper "On integrating out short-distance physics", which could be called by analogy with this post title "Form factors, self-energy, and dynamics of a "single" spin 0 heavy charged particle".

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What my long discussion is telling is that the textbook Dirac equation is only the simplest of a large number of Poincare covariant Dirac-like equation, and that a priori there is no reason to restrict discussion to the simplest Dirac equation. 

''what is the significance of "Poincare invariance"?'' It defines the constraints on a general free particle, i.e., in the absence of an external field. The interaction with the electromagnetic field is later added in the standard, gauge -invariant way. 

Experiment proves the above. Indeed, the simple Dirac equation gives wrong predictions of the hyperfine splitting when tested in a Coulomb field. Whereas the general form is consistent with experiment when the correct self-energy expression is used.

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@ArnoldNeumaier

What does the generalized (relativistic?) Dirac question that correctly predicts the hyperfine structure look like?

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As I already told you, a "free equation" means nothing but the case of a sufficiently weak external force (an inequality is implied), so for practical purposes we neglect its small contribution. In other words, we (and Dirac) write a free Dirac equation for a physical particle. That is why we require for it $E^2={\bf{p}}^2+m^2c^4$, that is why we require spin 1/2 and Poincare invariance. These are properties of an observable "free particle".

Hyperfine effects in a bound state is not a reaction on a weak external force. Weak external force means a continuous spectrum of the particle energy, i.e., the case of a fast particle scattering.

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Nowhere in my answer or my last comment make the assumption of a weak external field. Correct equations must also work for strong fields! 

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@Dilaton: $D(p)\psi=0$ in my answer. The standard Dirac equation is the special case where $\lambda,m,e,\mu$ are independent of $p^2$ and $\mu=0$.

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in the OP question you write; "How does a real (as opposed to idealized) relativistic spin 1/2 particle respond to a classical electromagnetic field that is weak enough that pair creation can be neglected?" It means inequality including the external force, doesn't it?

You contradict yourself in other places too, and it is a serious matter, Arnold, a serious matter.

Strong external field in a bound state is a case of a Hydrogen-like atom (or ion, to be exact) with big value of $Z$. Even though the pair states are not excited (pairs are not born), the external field may impact the motion of the pairs in their "ground state" (a quasi-adiabatic polarization).

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The field of the hydrogen atom is by far not strong enough for pair production! One would need a nucleus with Z>137 and a single atom. But the fine splitting is observed already for ordinary hydrogen (Z=1).

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All splittings are derived from the usual Dirac equation. Form-factors too.

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The usual Dirac equation has neither form factors nor self-energies. The whole point of my answer is to write a generalized Dirac equation where form factors and self-energies are present. And the hyperfine splitting is not explained by form factors alone but needs self-energy.

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In other words, it needs interaction. Because self-energy is a result of self-interaction. And with interaction the usual Dirac equation "gives everything", as we know. Empty talk.

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Interaction is just a word for a term in the Hamiltonian. it is empty talk to give it a deeper meaning.

Note that my equation is not QED, but a single particle equation for an electron in an external electromagnetic field. That it can be approximately derived from QED is a different matter.

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When I was a student, I was taught that electron form-factors were strictly derived from QED equations. So I was surprised to read that you wrote about Dirac equation.

Also, in the course of electrodynamics of hadrons, such form-factors were "postulated" and experiments were proposed to measure them in different regions of the transferred momentum $q$. Today the hardonic form-factors are also derived and calculated (in principle) from QCD or from the Standard model. Interaction is not an empty word, but the essence of physics.

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I wrote about the generalization of the Dirac equation that is actually used (and special cases of which were always used) to compute the hyperfine splitting. There is no other known way to derive the hyperfine splitting except by using the above equations. The functions entering this generalized Dirac equations (the form factors and self-energies) are either fitted to experiment (for hadrons) or derived from QED (for electrons) or QCD (for hadrons). 

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Interaction is important physics, but it is only the total Hamiltonian that defines the interacting system. Nature knows no division into free and interacting. How this Hamiltonian is split into an exactly solvable part and a residual part called the interaction is therefore a matter of choice, and the naive choice given in introductory textbooks is usually not the best choice. 

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I agree with your last comment, except for some subtleties on free and interacting things. Some strongly interacting system may have its natural modes that are "free" and non-interacting in the linear approximation.

"A matter of choice" matters: roughly, the closer the initial approximation to the exact solution, the better is the perturbative series convergence. I demonstrated this in many papers.

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Ok, here: http://www.physicsoverflow.org/37863/the-quartic-anharmonic-quantum-oscillator

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Let us discuss the anharmonic oscillator case in another thread, please. Let us decide who will start the new discussion. I can start with giving your texts and with adding my complements, if you like.