A. Zee, Quantum Field Theory in a Nutshell · Problem VII.4.5

Field Theory Redux

Physics — QFT

Problem

For hermitean random matrices ϕ\phi, calculate Gc(z,w)≡⟨1Ntr1z−ϕ1Ntr1w−ϕ⟩−⟨1Ntr1z−ϕ⟩⟨1Ntr1w−ϕ⟩G_c(z, w) \equiv \langle \frac{1}{N} \text{tr} \frac{1}{z - \phi} \frac{1}{N} \text{tr} \frac{1}{w - \phi} \rangle - \langle \frac{1}{N} \text{tr} \frac{1}{z - \phi} \rangle \langle \frac{1}{N} \text{tr} \frac{1}{w - \phi} \rangle for V(ϕ)=12m2ϕ2V(\phi) = \frac{1}{2}m^2\phi^2 using Feynman diagrams. [Note that this is a much simpler object to study than the object we need to study in order to learn about localization (see exercise VI.6.1).] Show that by taking suitable imaginary parts we can extract the correlation of the density of eigenvalues with itself. For help, see E. Brézin and A. Zee, Phys. Rev. E51: p. 5442, 1995.

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