Idea

We found that the direct perturbation theory in the Ginzburg–Landau–Wilson theory breaks down below the upper critical dimension because the perturbation parameter grows with the correlation length, and so becomes arbitrarily large as the critical point is approached. If the system were finite, on the other hand, the correlation length would be bound by its size, and perturbation theory could succeed. Singular thermodynamic behavior near the critical point comes from the thermodynamic limit, or, more precisely, from those modes that have arbitrary low energies in an infinitely large system. This is called the infrared singularity. This observation suggests the following strategy to avoid the problem of direct perturbation theory.

First, note that mass m only provides the energy scale in Eq. (2.36). It is practical to rescale it out by absorbing it into the chemical potential, rede- fined as 2mμ/ħ2 → μ, and into the interaction coupling, as 2mλ/ħ2 → λ. Similarly, near the critical point, the temperature T ≈ Tc may be replaced by the critical temperature, and then eliminated by rescaling the action as 2mκBTcS/ħ2 → S.