Graduate Understanding Hermiticity of Actions in QFT for Checking and Confirming

[John Greger]

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)

 

[mitochan]

I am not a good learner of QFT. Is action an operator not just a real number ?

 

[hilbert2]

The Lagrangian should be hermitian if the corresponding Hamiltonian is, but I'm not sure if there's any point in taking a time integral of it over some $[t_1 ,t_2 ]$ after the time variable in classical Lagrangian function $L$ has disappeared upon it being converted to an operator. Google search revealed some discussions about path integrals being needed in Lagrangian QM.

 

[vanhees71]

An alternative is the Schwinger action principle. See his quantum mechanics textbook:

J. Schwinger, Quantum Mechanics, Symbolism of Atomic Measurements, Springer, Berlin, Heidelberg, New York (2001).

 

[hilbert2]

If the Lagrangian of a one-particle system is an analytical function, would it be possible to write its time integral as some kind of power series containing arbitrarily high order derivatives of $p$ and $x$ and then convert them to operators like done when forming the acceleration operator in the thread linked below? The Cauchy integral formula allows writing derivatives as integrals, and the opposite seems to be even more easy.

https://www.physicsforums.com/threads/acceleration-operator.381084/

Also, here's something about quantization of higher time derivatives of the classical position variable:

http://bdigital.unal.edu.co/63666/1...and possible stabilization. Juan Valencia.pdf