Scalar quantum field theory in disordered media

A free massive scalar field in inhomogeneous random media is investigated. The coefficients of the Klein-Gordon equation are taken to be random functions of the spatial coordinates. The case of a disordered medium modeled by centered stationary and Gaussian processes is analyzed. After performing the averages over the random functions, we obtain the two-point and the four-point Green’s functions of the model up to one loop. The disordered scalar quantum field theory becomes qualitatively similar to a $\lambda {\phi }^4$ self-interacting theory with a frequency-dependent coupling.

Figure 1

Perturbative expansion of $G$ in terms of the disorder. The wavy lines represent generically the random functions $\mu$ and $\xi$.

Figure 2

The one-loop correction to the two-point causal function due to the random inhomogeneities.

Figure 3

The four-point Green function up to one loop, after noise averaging. “1PR graphs” means one-particle reducible graphs.

Figure 4

The disorder-induced, tree-level four-point Green’s function.

Figure 5

One-loop corrections of the first kind to the four-point Green’s function.

Figure 6

One-loop corrections of the second kind to the four-point Green’s function.

Figure 7

Graphical representation of a generic term in Eq. (a4).