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joneall

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- TL;DR Summary
- Problem understanding Lagrangian

I’m reading Lancaster & Blundell, Quantum field theory for the gifted amateur (even tho I”m only an amateur...) and have a problem with their explanation of symmetry breaking from page 242. They start with this Lagrangian:

##

\mathcal{L} =

(\partial_{\mu} \psi^{\dagger} - iq A^{\mu}\psi^{\dagger})

(\partial_{\mu} \psi + iq A^{\mu}\psi) +

\mu^{2} (\psi^{\dagger} \psi) - \lambda^{2} (\psi^{\dagger} \psi)^2 -

\tfrac{1}{4\pi} F^{\mu \nu} F_{\mu \nu}

##

which along with

## A_{\mu} \rightarrow A_{\mu} -\tfrac{1}{q} \partial_{\mu} \alpha(x) ##

is locally invariant under the transformation

## \psi \rightarrow \psi e^{i \alpha(x)} ##

So far, so good. But then they state that this Lagrangian describes “… a world made up of two sorts of oppositely charged massive scalar particles with energies ##E_p = (p^2 + \mu^2 )^{\tfrac{1}{2}} ## and two sorts of transversely polarized photons with energies ##E_p = |p| ##, which are, of course, massless.”

I think they may be using something defined several pages earlier

## \psi = \psi_1 + i \psi_2 ## and ## \psi^{\dagger} = \psi_1 - i \psi_2 ##

which could explain the two scalars. But I do not at all see two photons in there. Can anyone point them out to me?

Thanks in advance.

Btw, how do you make a dagger (for a Hermitian adjoint, e.g.) in your version of latex?

##

\mathcal{L} =

(\partial_{\mu} \psi^{\dagger} - iq A^{\mu}\psi^{\dagger})

(\partial_{\mu} \psi + iq A^{\mu}\psi) +

\mu^{2} (\psi^{\dagger} \psi) - \lambda^{2} (\psi^{\dagger} \psi)^2 -

\tfrac{1}{4\pi} F^{\mu \nu} F_{\mu \nu}

##

which along with

## A_{\mu} \rightarrow A_{\mu} -\tfrac{1}{q} \partial_{\mu} \alpha(x) ##

is locally invariant under the transformation

## \psi \rightarrow \psi e^{i \alpha(x)} ##

So far, so good. But then they state that this Lagrangian describes “… a world made up of two sorts of oppositely charged massive scalar particles with energies ##E_p = (p^2 + \mu^2 )^{\tfrac{1}{2}} ## and two sorts of transversely polarized photons with energies ##E_p = |p| ##, which are, of course, massless.”

I think they may be using something defined several pages earlier

## \psi = \psi_1 + i \psi_2 ## and ## \psi^{\dagger} = \psi_1 - i \psi_2 ##

which could explain the two scalars. But I do not at all see two photons in there. Can anyone point them out to me?

Thanks in advance.

Btw, how do you make a dagger (for a Hermitian adjoint, e.g.) in your version of latex?

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