John Greger
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- TL;DR
- You want your action to be hermitian, how would you check this quickly?
Hi!
In QFT we are usually interested in actions that are hermitian. Say we are looking at scattering of Dirac fermions with a real coupling constant g, whose Lagrangian is given by:
$$L= \bar{\psi}(i \gamma_{\mu} \partial^{\mu} -m) \psi - \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2}M^2 \phi^2 - g \phi \bar{\psi} \psi$$
It's fairly straight forward to show that the lagrangian is hermitian but how would I show that the action is hermitian as well? Is there a theorem or something saying that if the lagrangian is hermitian, so is the action?
What is your go-to method for checking that the action is hermitian?
($\phi$ is scalar field with mass M)
In QFT we are usually interested in actions that are hermitian. Say we are looking at scattering of Dirac fermions with a real coupling constant g, whose Lagrangian is given by:
$$L= \bar{\psi}(i \gamma_{\mu} \partial^{\mu} -m) \psi - \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2}M^2 \phi^2 - g \phi \bar{\psi} \psi$$
It's fairly straight forward to show that the lagrangian is hermitian but how would I show that the action is hermitian as well? Is there a theorem or something saying that if the lagrangian is hermitian, so is the action?
What is your go-to method for checking that the action is hermitian?
($\phi$ is scalar field with mass M)