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I happened to read Witten SU(2) anomaly paper (1982), and come back to digest again what I said the hermitian of the Dirac operator.

  1. According to Prahar https://physics.stackexchange.com/a/701287/310987, Prahar said that I should not consider or ask

Is the Dirac operator $i \not D$ or $\gamma^0 i \not D$ and nonabelian gauge field $A_\mu = A_\mu^\alpha T^\alpha$ hermitian?

  1. But according to Witten (1982), he said the hermitian operator $i \not D$ in his first page.

So how do we reconcile with Prahar and Witten opinion on the hermitian of the Dirac operator?

  • Is Witten paper in Euclidean signature so $i \not D$ is hermitian?
  • If it is in Lorentz signature, only $\gamma^0 i \not D$ is hermitian?

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Qmechanic
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Марина Marina S
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1 Answers1

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This is a totally different context than the one I described earlier. There, you were thinking of a Hilbert space and $\psi(x)$ was an operator and $i\!\!\not\!\!D$ is NOT an operator on the Hilbert space.

In the path integral, $\psi(x)$ is a function which we are integrating over. In this formula, $i\!\!\not\!\!D$ is an operator acting on the space of functions and it is possible to ask whether the operator is Hermitian or not.

Let me say it again. Before you can ask about hermiticity, you must first ask what your Hilbert space is and what is the norm with respect to which you are defining the adjoint. Once you have defined the adjoint, you can now ask if the operator is Hermitian or not.

$i\!\!\not\!\!D$ is not an operator on the Hilbert space (where $\psi(x)$ is an operator), but it is an operator in the path integral (where $\psi(x)$ is a function).

Prahar
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