An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 13.3

Critical Exponents and Scalar Field Theory — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · cpn-model-nonlinear-sigma

Problem

**The CPNCP^N model.** The nonlinear sigma model discussed in the text can be thought of as a quantum theory of fields that are coordinates on the unit sphere. A slightly more complicated space of high symmetry is complex projective space, CPNCP^N. This space can be defined as the space of (N+1)(N+1)-dimensional complex vectors (z1,…,zN+1)(z_1, \ldots, z_{N+1}) subject to the condition ∑j∣zj∣2=1,\sum_j |z_j|^2 = 1, with points related by an overall phase rotation identified, that is, (eiαz1,…,eiαzN+1)identified with(z1,…,zN+1).(e^{i\alpha}z_1, \ldots, e^{i\alpha}z_{N+1}) \quad \text{identified with} \quad (z_1, \ldots, z_{N+1}). In this problem, we study the two-dimensional quantum field theory whose fields are coordinates on this space. (a) One way to represent a theory of coordinates on CPNCP^N is to write a Lagrangian depending on fields zj(x)z_j(x), subject to the constraint, which also has the local symmetry zj(x)→eiα(x)zj(x),z_j(x) \to e^{i\alpha(x)} z_j(x), independently at each point xx. Show that the following Lagrangian has this symmetry: L=1g2[∣∂μzj∣2+∣zj∗∂μzj∣2].\mathcal{L} = \frac{1}{g^2}[|\partial_\mu z_j|^2 + |z_j^* \partial_\mu z_j|^2]. To prove the invariance, you will need to use the constraint on the zjz_j, and its consequence zj∗∂μzj=−(∂μzj∗)zj.z_j^* \partial_\mu z_j = -(\partial_\mu z_j^*) z_j. Show that the nonlinear sigma model for the case N=3N=3 can be converted to the CPNCP^N model for the case N=1N=1 by the substitution ni=z∗σiz,n^i = z^* \sigma^i z, where σi\sigma^i are the Pauli sigma matrices. (b) To write the Lagrangian in a simpler form, introduce a scalar Lagrange multiplier λ\lambda which implements the constraint and also a vector Lagrange multiplier AμA_\mu to express the local symmetry. More specifically, show that the Lagrangian of the CPNCP^N model is obtained from the Lagrangian L=1g2[∣Dμzj∣2−λ(∣zj∣2−1)],\mathcal{L} = \frac{1}{g^2}[|D_\mu z_j|^2 - \lambda(|z_j|^2 - 1)], where Dμ=(∂μ+iAμ)D_\mu = (\partial_\mu + iA_\mu), by functionally integrating over the fields λ\lambda and AμA_\mu. (c) We can solve the CPNCP^N model in the limit N→∞N \to \infty by integrating over the fields zjz_j. Show that this integral leads to the expression Z=∫DADλexp⁡[−N trlog⁡(−D2−λ)+ig2∫d2x λ],Z = \int \mathcal{D}A\mathcal{D}\lambda \exp\left[-N\,\text{tr}\log(-D^2 - \lambda) + \frac{i}{g^2}\int d^2 x\,\lambda\right], where we have kept only the leading terms for N→∞N \to \infty, g2Ng^2 N fixed. Using methods similar to those we used for the nonlinear sigma model, examine the conditions for minimizing the exponent with respect to λ\lambda and AμA_\mu. Show that these conditions have a solution at Aμ=0A_\mu = 0 and λ=m2>0\lambda = m^2 > 0. Show that, if g2g^2 is renormalized at the scale MM, mm can be written as m=Mexp⁡[−2πg2N].m = M \exp\left[-\frac{2\pi}{g^2 N}\right]. (d) Now expand the exponent about Aμ=0A_\mu = 0. Show that the first nontrivial term in this expansion is proportional to the vacuum polarization of massive scalar fields. Evaluate this expression using dimensional regularization, and show that it yields a standard kinetic energy term for AμA_\mu. Thus the strange nonlinear field theory we started with is finally transformed into a theory of (N+1)(N+1) massive scalar fields interacting with a massless photon.

Browse the problem library