# Expectation value of the electromagnetic field

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 $\langle 0|E^2|0 \rangle \neq 0$?

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