# A Longitudinal and transverse response function

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1. Jul 16, 2017

### Kurret

I was reading the book "finite temperature field theory" (https://www.amazon.com/Finite-Temperature-Field-Theory-Applications-Mathematical/dp/0521820820) and encountered a problem on page 111 about linear response theory. Consider a system with some conserved baryon matter perturbed by a source $J_\mu$, coupled to the baryon current $J_B^\mu$ (so the Hamiltonian is perturbed by a term $\int d^3x J_B^\mu J_\mu$). The corresponding response function, or retarded Green's function, is

$$iB_R^{\mu\nu}=\langle [J_B^\mu(x,t),J_B^\nu(x',t')]\rangle \theta(t-t')$$

Now, they claim that "since baryon number is conserved the most general form of the response function is

$$B_R^{\mu\nu}=B_L P_L^{\mu\nu}+B_T P_T^{\mu\nu}$$

where $B_L$ and $B_T$ are longitudinal and transverse response functions".

My question is, I don't understand what is meant by longitudinal and transverse response functions. Is it transverse with respect to the current or to the momentum, or something else? How are $P_L$ and $P_R$ defined? Also, I do not understand why this decomposition can only be done when baryon number is conserved?

Moreover, they also claim that the longitudinal response function is essentially the same as the time-time component of the full response function ($B_R^{00}$). Why is that?

2. Jul 21, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.