I'm interested in the retarded propagator for a free massless Dirac fermion, i.e. solutions $ψ$ to the inhomogeneous PDE
$$ (∂_t- \nabla·\vec σ) ψ(x,t) = f(x,t) $$
with boundary conditions
$$\quad ψ(x,t) \to 0 \text{ for } t \to -∞$$
where $\vec σ = (σ_1,σ_2,σ_3)^T$ are the three Pauli matrices. (The boundary conditions can be even more restrictive, I just want the solution to decay sufficiently quickly at infinity so that it becomes unique and has a well-defined Fourier transform.)
Now, solving the Dirac equation is a standard exercise in virtually every QFT book, but all the books I've looked at only consider the Fourier transform of the propagator.
However, I am interested in the real space formula for the retarded propagator
Using the retarded propagator for the wave equation in $3+1$ dimensions, we can write
$$ ψ(x,t) = (∂_t + \nabla·\vec σ)(∂_t^2 – \nabla^2)^{-1} f(x,t) $$
$$ = (∂_t + \nabla·\vec σ) \frac1{4π·\text{something}}∫d^3x'dt' \frac1{|x-x'|}\delta(|x-x'|-|t-t'|) f(x',t')$$
but this formula strikes me as seriously weird: carrying out the differentiation with respect to $x$ and $t$ will differentiate the $\delta$-function in the integral, which means that the solution depends on the derivatives of the function $f$. This goes against my intuition that a linear first-order PDE should depend on the initial values directly, and not on their time and space derivatives!
Is there a reference where I can find a discussion of the retarded propagator of the (massless) Dirac equation in real space?
Best Answer
maybe you already thought about it, but I'll ask anyway: why don't you try to tackle directly the differential operator: $\partial_t -\vec{\sigma}\cdot\nabla$ ? It's a first order PDE so you would get something of the kind: $$ \psi(x,t) = \psi_0(x,t) + \int G(x,x',t,t')f(x',t')\, dx'dt' \qquad \text{(1)} $$ Where $\psi_0(x,t)$ is a solution of the homogeneous equation and $G(x,x',t,t')$ is the solution of $$ (\partial_t -\vec{\sigma}\cdot\nabla)G(x,x',t,t') = \delta(x-x')\delta(t-t') $$
Now, it can be shown that equation 1 can be cast into the form (see Ref.): $$ \psi(x,t) = \psi_0(x,t) + \int dx' \int_{-\infty}^t G_1(x,x',t,t')f(x',t')\, dt' \qquad \text{(2)} $$
where $G_1(x,x',t,t')$ is an object that satisfy: $$ (\partial_t -\vec{\sigma}\cdot\nabla)G_1(x,x',t,t') = 0 $$
and therefore should be less clumsy to solve.
Reference: Mathematical Method of Classical and Quantum Physics, F. W. Byron and R. W. Fuller. Chapter 7: Time-dependent Green's functions: First Order