- #1
Kurret
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I was reading the book "finite temperature field theory" (https://www.amazon.com/dp/0521820820/?tag=pfamazon01-20) and encountered a problem on page 111 about linear response theory. Consider a system with some conserved baryon matter perturbed by a source [itex]J_\mu[/itex], coupled to the baryon current [itex]J_B^\mu[/itex] (so the Hamiltonian is perturbed by a term [itex]\int d^3x J_B^\mu J_\mu[/itex]). 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 [itex]B_L[/itex] and [itex]B_T[/itex] 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 [itex]P_L[/itex] and [itex]P_R[/itex] 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 ([itex]B_R^{00}[/itex]). Why is that?
$$iB_R^{\mu\nu}=\langle [J_B^\mu(x,t),J_B^\nu(x',t')]\rangle \theta(t-t')$$
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 [itex]B_L[/itex] and [itex]B_T[/itex] 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 [itex]P_L[/itex] and [itex]P_R[/itex] 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 ([itex]B_R^{00}[/itex]). Why is that?