I read that for quantization of the EM-Field, you demand the canonical equal-time commutation relations:

$$[A^\mu(\vec{x},t), \pi^\nu(\vec{y},t)] = i \hbar g^{\mu \nu} \delta^3(\vec{x} - \vec{y}). $$

Is this equality covariant? My first intuition is that it is not, because $\delta^3(\vec{x}-\vec{y})$ is not a Lorenz invariant quantity. On the other hand, $\pi^\nu(y)$ isn't one either (although it is denoted as being one). Could those quantities behave in a certain way, so that the whole statement is covariant and holds in every reference frame?

If it's not covariant, then what does this mean for the quantization formalism of Bleuler and Gupta, since they make use of this commutation relation?

  • $\begingroup$ I don't think that it makes sense to talk about Lorentz transformation properties of equal-time commutation relations. These relations hold only at the same time $t$. $\endgroup$ – Darkseid Sep 24 '17 at 20:03

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