Longitudinal and transverse response function

[Kurret]

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 $J_\mu$, 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?

 

[AndyW]

I can address these questions and provide some clarification on the topic of linear response theory.

Firstly, the term "longitudinal" and "transverse" in this context refer to the direction of the response with respect to the perturbing source. In this case, the current J_\mu acts as the source and the response function B_R^{\mu\nu} describes the response of the system to this source. The longitudinal response function refers to the response of the system in the same direction as the source, while the transverse response function refers to the response perpendicular to the source.

The longitudinal and transverse response functions are defined in terms of the projection operators P_L and P_T, respectively. These projection operators are used to decompose the full response function into its longitudinal and transverse components. P_L and P_T are defined in such a way that P_L + P_T = 1, and they project onto the longitudinal and transverse directions, respectively.

Now, why is this decomposition only valid when baryon number is conserved? This is because in a system where baryon number is not conserved, the response function B_R^{\mu\nu} would also have off-diagonal terms, which cannot be decomposed into longitudinal and transverse components. In other words, the conservation of baryon number restricts the form of the response function and allows for this decomposition.

Finally, the claim that the longitudinal response function is essentially the same as the time-time component of the full response function (B_R^{00}) is based on the fact that in many physical systems, the time-time component of the response function dominates over other components. This is because the time-time component is related to the energy density of the system, which is often the most important quantity in a perturbed system.

I hope this explanation helps to clarify the concepts of longitudinal and transverse response functions and their relation to the conservation of baryon number. It is important to note that these concepts are specific to the topic of linear response theory and may vary in other contexts.