In condensed matter physics, we like to talk about quasiparticle scattering and lifetimes - but what machinery needs to be built up to get there? The answer goes back to Green's functions used to solve wave equations and diffusion equations 1. Green’s functions are a handy mathematical tool to help handle tricky partial differential equation (PDE) problems - namely, for a differential operator $L$, the Green’s function satisfies $L G = \delta$, where $\delta$ is the Dirac delta function. At the core, we can use the old faithful “guess and check” method “ to determine solutions to nontrivial differential operators.

Suppose we are trying to find the solution $u(x)$ to the equation $L u(x) = f(x)$. Using the properties of the Green’s function and the fact that our operator $L=L(x)$, we can express the solution $u(x$) as: $$ \int L G(x-s) \;f(s) \;ds = \int \delta(x-s) \; f(s) \; ds = f(x) ; \qquad u(x) = \int G(x-s)\; f(s) \; ds $$

Thus, if we can find a suitable Green’s function, we have the solution! We can put on our physics/engineering hat and just stick a bunch of Green’s functions into the equation until one fits. Smarter physicists than I would check the dimensional analysis to get more insight what the Green’s function solution should be. For the various common differential operators in physics, there are known classes of Green’s functions. I closed the Wikipedia article when I started seeing Bessel functions $J(\nu)$ and the scarier modified Bessel functions, so if you’re interested you’ll have to go look for yourself. I promise that the Green’s functions have all be figured out for all of the hard differential operators you’ll come across in physics.

For me, my exposure to Green’s functions was first through electrostatics in solving the Poisson equation for charge $\nabla^2 \Phi = \frac{\rho}{\epsilon_0}$ with Dirchlet boundary conditions (the potential $\Phi \rightarrow 0$ as $x \rightarrow \infty $). The typical $1/r$ form for the electrical potential that any undergrad physics major should be able to rattle off is from the Green’s function:

$$ G(\bf{x},\bf{x’}) = \frac{1}{|\bf{x}-\bf{x’}|} $$

where $\bf{x}$ is the vector position of interest and the source charge is at $\bf{x’}$ 2.

The lifetime of a quasiparticles in Landau-Fermi liquid theory 3 can be determined from the associated (many-body) Green’s function, which typically has a form:

$$ G(p,\omega) = \frac{1}{\omega - (\epsilon(p) - \mu + \mathrm{Re}\Sigma(p,\omega)) - i \mathrm{Im}\Sigma(p,\omega)} $$

where $p$ is the momentum, $\omega$ is the frequency, $\epsilon, \mu$ are the energy and chemical potential and $\Sigma(p,\omega)$ is the self energy. The decay rate ($\Gamma = 1/\tau$), the inverse of the lifetime $\tau$ is related to the imaginary part of the self energy:

$$ \frac{1}{\tau}(p,\omega) = \mathrm{Im}\Sigma(p,\omega) $$

If I have some spectroscopic peak in frequency space ($\omega$) or equivalently energy space ($\hbar \omega$), the Fourier transform of the peak will give the lifetime - a wider peak corresponds a short lifetime and vice versa. This can be measured via multiple spectroscopic probes - optics, scanning tunnelling spectroscopy (STS) or Angle resolved photoemission (ARPES).

  1. If you took Jackson E&M, I am sorry for bringing up the trauma. Up to this point I had relegated Green’s functions to a forgotten corner of my brain. ↩

  2. See USCD Physics Notes or Stonybrook notes for more details on Green’s functions and their relevance to E&M. ↩

  3. See the nice and condensed Fermi liquid theory notes by Eduardo Fradkin ↩