I am trying to self study QED so I apologize if my question seems silly. As I realize, all physical processes should stem from some "hermitian" operator in the quantum language. As it is well known, both in the QM region (SHO) and the QED area, the creation and annihilation operators are non-hermitian (actually these pair are complex adjoints to eachother). Now, my question is how could photon creation, which is known to occur in multiple circumstances (e.g. spontaneous emission from an atoms excited state, or radiation due to a single electron having non-vanishing acceleration), correspond to a non-hermitian operator.
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You are right that the photon creation operator $a^\dagger$ and the photon annihilation operator $a$ are non-hermitian. You are also right that all physical processes (like for example the electron-photon interaction) need to be hermitian operators.
The solution is that the physical processes always come as hermitian combinations of the creation and annihilation operators.
Example: The electron-photon interaction (taken from this answer to How do electrons and photons interact?) can be represented by $$ \hat{V}_\text{int}=e\hat{\mathbf{r}}\cdot\left(\mathbf{E}_0\hat a+\mathbf{E}_0^\ast\hat a^\dagger\right).$$
Although operators $\hat{a}$ and $\hat{a}^\dagger$ are non-hermitian, the combination $\hat{V}_\text{int}$ is hermitian.
Thomas Fritsch
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