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After reading through a number of questions on SE including What are field quanta? and What are quantum fields mathematically?, I am still struggling with what specific operators are associated to quantum fields like the electron or electromagnetic field.

From my understanding, a quantum field in QFT is an operator-valued distribution. This means that it associates an operator to each distribution in spacetime, and in particular to single points in spacetime given by Dirac delta distributions. My understanding is also that operators are associated to observables, for instance that of position/momentum/energy.

Now take the electron field in particular (or the electromagnetic field if easier). What are the operators given by the electron field? What is the associated observable? Is it the creation operator, or the annihilation operator, or both, or the particle number operator, or the energy operator?

My current thought process is that it should be something like the creation operator, after reading that electrons are quanta of the electron field and photons are quanta of the electromagnetic field... Since then it would make sense to call 'electrons' the first application of the electron field's corresponding operator (the creation operator) on the vacuum state $\left|0\right>$.

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  • $\begingroup$ Your thinking is very roughly correct (though I believe fields aren't exactly distributions except in free theories...but that get's into renormalization and would be besides the point I think). These things are, however, laid out in any introductory text on quantum field theory. For example, essentially all will contain an early chapter on QED. $\endgroup$
    – Richard Myers
    Commented Mar 11, 2021 at 22:30
  • $\begingroup$ @RichardMyers Even in interacting theories, the local observables are distributions. Renormalization doesn't change that. The complication here is that the gauge fields are not local observables. Only gauge invariant combinations of them like $F_{\mu\nu}$. $\endgroup$
    – user1504
    Commented Mar 11, 2021 at 23:01
  • $\begingroup$ @user1504 Yeah that bit about the gauge fields is fine, but my impression about operators being distributions followed from the Wightman axioms, which are known to not really say anything about interacting theories and part of the issue is that in interacting theories the products of operators is actually worse than the product of distributions. $\endgroup$
    – Richard Myers
    Commented Mar 11, 2021 at 23:15
  • $\begingroup$ @RichardMyers There are interacting theories that satisfy the Wightman axioms. In all of them, the local observables are distributions. Basically all of the super-renormalizable theories have been constructed. Is your claim that local observables in marginally renormalizable theories can't be represented by distributions? $\endgroup$
    – user1504
    Commented Mar 12, 2021 at 0:52
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    $\begingroup$ Peskin & Shroeder ch 4/5. Zee II.5 and II.6. Weinberg ch 8. Nair ch6/7. David Tong's notes ch 6. Schwartz and Srednicki will also have this but I don't have copies in front of me. You're asking about QED which is the standard second example after scalar $\phi^4$ theory. The only point often glossed is that correlators involving the vector potential vanish due to gauge invariance, so Wilson lines are the object to consider, this is also not exactly the question you posed. See Banks' book or Nair's for comments on that. $\endgroup$
    – Richard Myers
    Commented Mar 18, 2021 at 20:58

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To avoid the problem of questions answered only in the comments, I will post the following answer as Community wiki. Richard Myers helpfully points out that my question would be answered in most introductory QED texts including Peskin & Shroeder ch 4/5, Zee II.5 and II.6, Weinberg ch 8, Nair ch6/7, David Tong's notes ch 6.

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