I have a question about the structure of the QED lagrangian, in particular the free photon lagrangian which is contained in it. My premise is: I only know how to exploit canonical quantization in order to quantize a theory; I don't know how to use the path integral formulation.

The QED lagrangian is: $$ \mathcal{L}=-\frac{1}{4}F^{\mu \nu}F_{\mu \nu}+\bar{\psi}(i\gamma^{\mu}D_{\mu}-m)\psi, $$ so I assume that the free photon theory exploited here is $$ \mathcal{L_{free}}=-\frac{1}{4}F^{\mu \nu}F_{\mu \nu}. $$ However, I also learnt that $\mathcal{L_{free}}$ jointed with Lorenz's gauge cannot give us a covariant quantization for the electromagnetic field (by means of the canonical quantization, at least). In fact, we introduce the following lagrangian: $$ \mathcal{L_{feyn}}=-\frac{1}{4}F^{\mu \nu}F_{\mu \nu}-\frac{1}{2 \xi}(\partial_{\mu}A^{\mu})^2 $$ with Feynman gauge choice $\xi=1$. This, jointed with Gupta-Bleuer constraint, gives us the physical states of the electromagnetism.

So: why do we adopt $\mathcal{L_{free}}$ instead of $\mathcal{L_{feyn}}$? I know that the latter is not gauge-invariant, but the covariant quantization of the theory is achieved through that, so this point is not clear to me.

  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/147394/2451 , physics.stackexchange.com/q/139475/2451 , physics.stackexchange.com/q/372594/2451 , physics.stackexchange.com/q/75981/2451 and links therein. $\endgroup$
    – Qmechanic
    Commented Jan 2, 2019 at 20:59
  • 1
    $\begingroup$ I think that my question is slightly different: I understand the benefits of using $\mathcal{L}_{feyn}$ instead of $\mathcal{L}_{free}$. What I don't understand is: why do we adopt $\mathcal{L}_{free}$ as the free photon lagrangian in qed? $\endgroup$
    – E. Marc.
    Commented Jan 2, 2019 at 21:30
  • $\begingroup$ @Qmechanic The question that you refer to as a duplicate is different. Unless this question has been asked and answered adequately it should be reopened. $\endgroup$
    – my2cts
    Commented Jan 2, 2019 at 23:18
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    $\begingroup$ What do you mean "we adapt $\mathcal{L}_{free}$"? The Lagrangian $\mathcal{L}_{free}$ is just an incomplete formulation before gauge-fixing. The Lagrangian should ultimately be gauge-fixed. Who are "we"? "Adapt" in which context? $\endgroup$
    – Qmechanic
    Commented Jan 2, 2019 at 23:59
  • $\begingroup$ To adopt: it's just a synonym for "to write, to use". When we write down the QED lagrangian we use the Maxwell lagrangian instead of the lagrangian with the gauge fixing term. We: it's just an impersonal pronoun I used in order to describe a wide-spread procedure or habit in theoretical physics. $\endgroup$
    – E. Marc.
    Commented Jan 3, 2019 at 8:41


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