A. Zee, Quantum Field Theory in a Nutshell · Problem VI.6.1

Field Theory Redux

Physics — QFT

Problem

An exercise in elementary geometry: Draw a straight line tilted at an angle θ\theta with respect to the horizontal. The line represents a small segment of the surface at time tt. Now draw a number of circles of radius dd tangent to and on top of this line. Next draw another line tilted at angle θ\theta with respect to the horizontal and lying on top of the circles, namely tangent to them. This new line represents the segment of the surface some time later (see fig. VI.6.1). Note that Δh=d/cos⁡θ≃d(1+12θ2)\Delta h = d/\cos \theta \simeq d(1 + \frac{1}{2} \theta^2). Show that this generates the nonlinear term (λ/2)(∇h)2(\lambda/2)(\nabla h)^2 in the KPZ equation (1). For applications of the KPZ equation, see for example, T. Halpin–Healy and Y.-C. Zhang, Phys. Rep. 254: 215, 1995; A. L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth.

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