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Is raising and lowering indices in quantum field theory works the same as in the general theory of relativity?

By means of this metric tensor?

$$g^{μν}= \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1\\ \end{pmatrix}.$$

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  • $\begingroup$ No. In GR, the metric becomes another dynamical field so it is not fixed in the form you have mentioned. It has a totally generic form where each component is an arbitrary function of the coordinates. All the components put together satisfy Einstein's equations. $\endgroup$
    – Prahar
    Commented Apr 15, 2021 at 19:54
  • $\begingroup$ Hi Peter. Welcome to Phys.SE. Do you mean QFT in curved space? $\endgroup$
    – Qmechanic ♦
    Commented Apr 15, 2021 at 19:58
  • $\begingroup$ @PraharMitra I understood the question as asking if that metric (the Minkowski metric) is used in QFT to raise and lower spacetime indices, just as one does with an arbitrary metric in GR. $\endgroup$
    – Eletie
    Commented Apr 15, 2021 at 19:59
  • $\begingroup$ @Qmechanic nope, in flat space. $\endgroup$
    – Peter
    Commented Apr 15, 2021 at 20:16
  • $\begingroup$ I asked this because I wanted to understand how it all works in the Lagrangian of quantum field theory. For example gluon field stress tensor $G_{μν}G^{μν}$ $\endgroup$
    – Peter
    Commented Apr 15, 2021 at 20:25

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Lorentz indices are indeed raised and lowered with the metric tensor. But note that in QFT there are also other indices than Lorentz indices and these are raised an lowered with the appropriate tensor.

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