An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 11.1

Renormalization and Symmetry — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · spin-wave-theory-goldstone-modes

Problem

11.1 Spin-wave theory. (a) Prove the following wonderful formula: Let ϕ(x)\phi(x) be a free scalar field with propagator ⟨Tϕ(x)ϕ(0)⟩=D(x)\langle T\phi(x)\phi(0)\rangle = D(x). Then ⟨Teiϕ(x)e−iϕ(0)⟩=e[D(x)−D(0)].\left\langle Te^{i\phi(x)}e^{-i\phi(0)}\right\rangle = e^{[D(x)-D(0)]}. (The factor D(0)D(0) gives a formally divergent adjustment of the overall normalization.) (b) We can use this formula in Euclidean field theory to discuss correlation functions in a theory with spontaneously broken symmetry for T<TCT < T_C. Let us consider only the simplest case of a broken O(2)O(2) or U(1)U(1) symmetry. We can write the local spin density as a complex variable s(x)=s1(x)+is2(x).s(x) = s^1(x) + is^2(x). The global symmetry is the transformation s(x)→e−iαs(x).s(x) \to e^{-i\alpha}s(x). If we assume that the physics freezes the modulus of s(x)s(x), we can parametrize s(x)=Aeiϕ(x)s(x) = Ae^{i\phi(x)} and write an effective Lagrangian for the field ϕ(x)\phi(x). The symmetry of the theory becomes the translation symmetry ϕ(x)→ϕ(x)−α.\phi(x) \to \phi(x) - \alpha. Show that (for d>0d > 0) the most general renormalizable Lagrangian consistent with this symmetry is the free field theory L=12ρ(∇⃗ϕ)2.\mathcal{L} = \tfrac{1}{2}\rho(\vec{\nabla}\phi)^2. In statistical mechanics, the constant ρ\rho is called the *spin wave modulus*. A reasonable hypothesis for ρ\rho is that it is finite for T<TCT < T_C and tends to 0 as T→TCT \to T_C from below. (c) Compute the correlation function ⟨s(x)s∗(0)⟩\langle s(x)s^*(0)\rangle. Adjust AA to give a physically sensible normalization (assuming that the system has a physical cutoff at the scale of one atomic spacing) and display the dependence of this correlation function on xx for d=1,2,3,4d = 1, 2, 3, 4. Explain the significance of your results.

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