In all text books (and papers for that matter) about QFT and the classical limit of relativistic equations, one comes across the "minimal substitution" to introduce the magnetic potential into the equation (Schrödinger/Dirac/Klein-Gordon) through: $$ \hat{p}^2 \rightarrow (\hat{p} - e \hat{A})^2$$

The wording suggests that this is an approximation for small? electromagnetic fields (or at least not strongly coupled). I understand this was chosen such that the "classical" Lorentz force is retrieved from the Hamiltonian, but not why exactly this form and not another that leads to the same result.

  1. To what extent is this "approximation" valid?

  2. How can one improve this "minimal substitution"? Is there a more general expression, intuitively something like a series of the vector potential?


As explained on Wikipedia, the reason this procedure is called "minimal coupling" is that it effectively ignores all but the first multipole moment (i.e. the charge) of the charged particle. But as long as you stick to monopoles, it's an exact expression, not an approximation.

You can get it from the Lagrangian for a relativistic charged particle in an electromagnetic field:

$$L = -\frac{mc^2}{\gamma} + q\mathbf{v}\cdot\mathbf{A} - q\phi$$

by taking $\frac{\partial L}{\partial \mathbf{v}}$. The second term which produces the adjustment to the momentum corresponds to the magnetic term in the Lorentz force law, $\mathbf{F} = q\mathbf{v}\times\mathbf{B}$.

Alternatively, you can get it from the $U(1)$ gauge transformation which quantum electrodynamics is based on. In order to preserve gauge invariance, a little bit of math shows that the derivative operator has to change from $D_\mu = \partial_\mu$ to

$$D_\mu = \partial_\mu - ieA_\mu$$

Since the momentum operator is $p_\mu = -iD_\mu$ (in units where $\hbar = 1$), this becomes

$$p_\mu \to p_\mu - eA_\mu$$

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  • $\begingroup$ Aha, gauge invariance it is indeed. Now the name "minimal substitution" fits even less than I thought... $\endgroup$ – rubenvb Oct 6 '11 at 17:03
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    $\begingroup$ @David: +1, this is true, but the reason people often say "approximation" is because if the particle has substructure, like an intrinsic magnetic moment, or higher multipole moments, you get extra interactions with the electromagnetic field. The standard explanation for why these structure constants are zero is that the particle is small, and this is ultimately an appeal to renormalizability and a big gap between the Planck scale and the particle scale. $\endgroup$ – Ron Maimon Oct 6 '11 at 17:22
  • $\begingroup$ Vladimir: if I understand correctly, you're saying what I initially thought: if $A$ is $A_\text{ext}$ and large, this is an approximation. Then how would one improve the equation (not the solution), so that one could perhaps get an improved numerical solution? $\endgroup$ – rubenvb Oct 6 '11 at 17:29
  • $\begingroup$ @Ron: So, the sense of "minimal" is jsut lacking substructure? Then I tend to think it is forced upon us by gauge invariance, once we decide to disregard particle's internal degrees of freedom (see my related question at TheoreticalPhysics.SE [theoreticalphysics.stackexchange.com/q/22/97]). $\endgroup$ – Slaviks Oct 6 '11 at 17:50
  • $\begingroup$ @Ron: true, I was actually thinking about mentioning something about substructure and then I forgot ;-) So I guess I'll edit that in. $\endgroup$ – David Z Oct 6 '11 at 17:52

It is approximate if you mean an external force vector potential $A_{ext}$ (Lagrangian $\propto j\cdot A_{ext}$) because you do not take into account the radiation resistance force. And it is partially wrong if you mean the self-field (the field $A$ created with this charge itself). To make it right they discard (via renormalizations) self-action contributions and leave interaction contributions to the solutions. In case of non minimal interaction even discarding does not help repair solutions.

Non minimal term in Lagrangian like $\propto \sigma_{\mu \nu}F_{\mu\nu}$ is also porblem-less for external fields $F_{ext}$ but it is non renormalizable in the self-action approach.

Renormalizations are done perturbatively and result in correct solutions (expansions). A very simplified explanation of what we do wrong and why renormalizations may work is given here. I think the internal structure (multipole moments) can be taken reasonably into account in the frame of interaction rather than self-action ansatz. I have not finished developing the corresponding approach yet.

EDIT: It does not "follow" from gauge invariance but complaint with it, of course. The local gauge invariance "principle" reproduces the wrong self-action term most of which is later removed via renormalizations. Without renormalizations this "exact expression" yields rubbish.

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