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I am trying to demonstrate that the slashed covariant derivative $$ \gamma^\mu D_\mu = \gamma^\mu(\partial_\mu -iA_\mu) $$ has real eigenvalues: $$ \gamma^\mu D_\mu \varphi_m(x)=\lambda_m \varphi_m(x) \Rightarrow \lambda_m \in \mathbb{R} $$ However I do not know how to prove this.

I am a bit confused and I would really appreciate some help!

Ghoster
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Gorga
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    Why do you think it should be Hermitian? Also, Hermitian on what vector space? – Prahar Feb 26 '24 at 15:08
  • In the lecture notes of an subject that I am studying it is said that is hermitian so its eigenvalues are real, and this is used to proof some properties of anomalies. Thank you! – Gorga Feb 26 '24 at 15:11
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    Specify the underying space (of functions) and the associated scalar product. Otherwise your question does not make sense. – Hyperon Feb 26 '24 at 16:17
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  • The Dirac operator in Lorentzian signature is not (on the most natural space to consider it on) Hermitian, see also this answer of mine. 2. QFT does not have a scalar product on the space of fields - the fields are operators, not states, and the scalar product of quantum theories is on the space of states, so it's not really clear what you're asking.
    • – ACuriousMind Feb 26 '24 at 16:31
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  • Your first displayed equation makes no sense since it has mismatched indices - on the l.h.s. is the Dirac operator with no free indices, but on the r.h.s. is the covariant derivative with a free $\mu$
    • – ACuriousMind Feb 26 '24 at 16:34
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    @ACuriousMind That is my concern also. – Lewis Miller Feb 26 '24 at 16:48
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    I have edited the question to bind the index on the r.h.s. of the first equation. I think my edit is what OP was trying to write... – hft Feb 26 '24 at 16:55