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I know in classical field theory we have the electromagnetic field. And Maxwell's equations show how electromagnetic radiation can propagate through empty space.

I also have been reading about QED and I gather the electric repulsion between two electrons is mediated by a virtual photon.

Also, as I understand it, in quantum field theory we speak of particles as manifestation of an underling field. For instance, a photon is a manifestation of a photon field.

Two Questions:

  1. Are quantum fields like electron fields or photon fields one big field (like we assume gravity to be one field) or are there separate ones? Meaning, can I have several electron fields?

  2. I often here the term electromagnetism and people say they are the same force. Are electron fields and photons fields part of the same underlying field or are they separate fields that just interact?

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2 Answers 2

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In our modern understanding, every electron is thought to be a localized excitation of the electron (or Dirac) (spinor) field $\Psi(x^\mu)$, while every photon is considered to be an excitation of the photon (vector) field $A^\nu(x^\mu)$, which is the quantum field-theoretic counterpart of the classical four-potential.

Thus, the answer to your questions are:

  1. All particles of the same type (e.g. photons or electrons) is understood to be 'coming from' one all-permeating quantum field. It should be noted that these fields also give rise to the corresponding anti-particles, so the positron field is the same as the electron field.

  2. The different particle types are truly separated in quantum field theory: Each type is represented by one field, and the fields interact. These interactions are quantified by the Lagrangian (density), which essentially determines everything about the theory. In pure electrodynamics, the quantum field-theoretic Lagrangian density is (using 'mostly minus' sign convention for the metric)

$$\mathcal{L}_{\text{QED}}= \bar\Psi(i\gamma^\mu D_\mu-m)\Psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} =\bar\Psi(i\gamma^\mu (\partial_\mu+ieA_\mu)-m)\Psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} $$ where $F_{\mu\nu}\equiv \partial_\mu A_\nu-\partial_\nu A_\mu$ is the electromagnetic field strength tensor. The 'covariant derivative' $D_\mu\equiv \partial_\mu+ie A_\mu$ encodes the interaction between the two fields $A_\mu$ and $\Psi$, and the 'strength' of the interaction is given by $e$, the charge of the electron.

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  • $\begingroup$ +1 Nice, complete answer. Wow, I didn't realize that. So the electron field is $\Psi$? I didn't realize that was the symbol for it. I thought $\Psi$ stood for a wave function. Also, this isn't the same covariant derivative from Riemannian geometry right? This is something called the gauge covariant derivative. I don't really know much about it, but I recently learned from my book Quantum Field Theory in a Nutshell that it can somehow restore some kind of symmetry or something along those lines, right? $\endgroup$
    – Stan Shunpike
    Feb 23, 2015 at 21:23
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    $\begingroup$ @StanShunpike well, the symbol $\Psi$ is very likely taken exactly because we're all used to $\Psi$ describing electrons from using the Schrodinger equation... And yes, this is exactly the differentiation from Riemannian geometry. It is introduced (and with it, the gauge field $A_\mu$ which describes electromagnetism) to maintain local $U(1)$ invariance of the Lagrangian. There is a rich theory of geometry behind gauge theories: The buzzword is Yang-Mills theory. $\endgroup$
    – Danu
    Feb 23, 2015 at 21:26
  • $\begingroup$ That's interesting. I was just saying to myself I should learn more about Yang-Mills theory. I haven't studied it yet. My text Quantum Field Theory in a Nutshell doesn't cover it. Is there a recommended beginner's text that covers Yang-Mills well? A Zee is too advanced for me. I haven't really tried Peskin and Schroeder because I have been happy with my text, but this Yang-Mills seems to be a topic omitted now that I think about it. $\endgroup$
    – Stan Shunpike
    Feb 23, 2015 at 21:29
  • $\begingroup$ @StanShunpike I know a number of texts that discuss it, but I can't say I'm a big fan of any particular textbook. I am personally also looking for a monograph on the mathematics of Yang-Mills theory, but haven't been able to find anything yet. If you want to learn about the mathematics of it too, you would have to study differential geometry (and Riemannian geometry) first, of course. $\endgroup$
    – Danu
    Feb 23, 2015 at 21:31
  • $\begingroup$ I have studied the Riemannian geometry, that's why I'm surprised I haven't yet understood what a gauge covariant derivative is yet. Maybe The H Bar would have some suggestions. I'll try there and see what I find. $\endgroup$
    – Stan Shunpike
    Feb 23, 2015 at 21:36
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For what it's worth, I showed in my recent article http://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-013-2371-4.pdf (published in European Phys. J. C) that one can eliminate the Dirac field from the Dirac-Maxwell electrodynamics after introduction of a complex electromagnetic 4-potential (producing the same electromagnetic field as the real 4-potential), so modified Maxwell equations can describe both electrons and photons.

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