An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 15.4

Non-Abelian Gauge Invariance — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · non-abelian-gauge-invariance

Problem

Scalar propagator in a gauge theory. Consider the equation for the Green's function of the Klein-Gordon equation: (∂2+m2)DF(x,y)=−iδ(4)(x−y).(\partial^2 + m^2)D_F(x,y) = -i\delta^{(4)}(x-y). We can find an interesting representation for this Green's function by writing DF(x,y)=∫0∞dT D(x,y,T),D_F(x,y) = \int_0^\infty dT\, D(x,y,T), where D(x,y,T)D(x,y,T) satisfies the Schrödinger equation [i∂∂T−(∂2+m2)]D(x,y,T)=iδ(T)δ(4)(x−y).\left[i\frac{\partial}{\partial T} - (\partial^2 + m^2)\right] D(x,y,T) = i\delta(T)\delta^{(4)}(x-y). Now, represent D(x,y,T)D(x,y,T) using the functional integral solution of the Schrödinger equation presented in Section 9.1. (a) Using the explicit formula of the propagator of the Schrödinger equation, show that this integral formula gives the standard expression for the Feynman propagator. (b) Using the method just described, show that the expression DF(x,y)=∫0∞dT∫Dxexp⁡[i∫dt 12((dxμdt)2−m2)−ie∫dt dxμdtAμ(x)]D_F(x,y) = \int_0^\infty dT \int \mathcal{D}x \exp\left[i\int dt\, \frac{1}{2}\left(\left(\frac{dx^\mu}{dt}\right)^2 - m^2\right) - ie \int dt\, \frac{dx^\mu}{dt} A_\mu(x)\right] is a functional integral representation for the scalar field propagator in an arbitrary background electromagnetic field. Show, in particular, that the functional integral satisfies the relevant Schrödinger equation. Notice that this integral depends on AμA_\mu through the Wilson line. (c) Generalize this expression to a non-Abelian gauge theory. Show that the functional integral solves the relevant Schrödinger equation only if the group matrices in the exponential for the Wilson line are path-ordered.

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