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According to wikipedia, the operator of the electromagnetic field is given by: $$E \propto \sum_k \sqrt{k}(a_k^{+}e^{-ikr}+a_ke^{ikr})$$

wouldn't that mean that the expactation value of $E^2$ is infinite for the vacuum? $\langle 0|E^2|0\rangle = \infty $? And what is meant by vacuum fluctuations in general? Does it just mean that $\langle 0|E^2|0 \rangle \neq 0$?

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    $\begingroup$ That is why you renormalize the expectation value of the Hamiltonian, as with other quantum fields. $\endgroup$ – Slereah Jul 20 '18 at 7:15
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    $\begingroup$ Also the vacuum fluctuations don't influence the expectation value of the energy, only the standard deviation of its measurement. $\endgroup$ – Slereah Jul 20 '18 at 7:17
  • $\begingroup$ @Slereah but $\langle 0|E^2|0 \rangle \neq 0$ means that we have vacuum fluctuations or what do we exactly mean by it? $\endgroup$ – yasalami Jul 20 '18 at 12:28

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