The quantum field resultant from the quantization of a real classical field is hermitian, but why the quantum field corresponding to a complex classical field should be non-hermitian?
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2Write the complex field as the sum of two real fields. Both are Hermitian. Apply now complex conjugation to the complex scalar field as their sum. Observe that it is not Hermitian. – ACuriousMind Aug 29 '14 at 14:05
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by the sum of two real fields do you mean one real field as the real part and the other as the imaginary part? then both the real and imaginary parts are hermitian after quantization,but the full complex field will have an extra imaginary unit 'i', making the field non-hermitian. – M. Zeng Aug 30 '14 at 02:16
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Yes, that is exactly what I meant. – ACuriousMind Aug 30 '14 at 14:05
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Note that the spectrum ${\rm Spec}(\hat{A}) \subseteq \mathbb{R}$ of a Hermitian/self-adjoint operator $\hat{A}$ belongs to the real axis $\mathbb{R}\subseteq \mathbb{C}$, cf. e.g. this Phys.SE post.
It is therefore not surprising that a reality condition for a classical field naturally translates into a Hermiticity/self-adjointness condition for the corresponding operator, and vice-versa.
