An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 15.3

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

Physics — QFT · non-abelian-gauge-invariance

Problem

Coulomb potential. (a) Using functional integration, compute the expectation value of the Wilson loop in pure quantum electrodynamics without fermions. Show that ⟨UP(z,z)⟩=exp⁡[−e2∮Pdxμ∮Pdyν gμν18π2(x−y)2],\langle U_P(z,z) \rangle = \exp\left[-e^2 \oint_P dx^\mu \oint_P dy^\nu \, g_{\mu\nu} \frac{1}{8\pi^2(x-y)^2}\right], with xx and yy integrated around the closed curve PP. (b) Consider the Wilson loop of a rectangular path of (spacelike) width RR and (timelike) length TT, T≫RT \gg R. Compute the expectation value of the Wilson loop in this limit and compare to the general expression for time evolution, ⟨UP⟩=exp⁡[−iE(R)T],\langle U_P \rangle = \exp[-iE(R)T], where E(R)E(R) is the energy of the electromagnetic sources corresponding to the Wilson loop. Show that the potential energy of these sources is just the Coulomb potential, V(R)=−e2/4πRV(R) = -e^2/4\pi R. (c) Assuming that the propagator of the non-Abelian gauge field is given by the Feynman gauge expression ⟨Aμa(x)Aμb(y)⟩=∫d4p(2π)4−igμνδabp2e−ip⋅(x−y),\langle A^a_\mu(x) A^b_\mu(y) \rangle = \int \frac{d^4p}{(2\pi)^4} \frac{-ig_{\mu\nu}\delta^{ab}}{p^2} e^{-ip\cdot(x-y)}, compute the expectation value of a non-Abelian Wilson loop to order g2g^2. The result will depend on the representation rr of the gauge group in which one chooses the matrices that appear in the exponential. Show that, to this order, the Coulomb potential of the non-Abelian gauge theory is V(R)=−g2C2(r)/4πRV(R) = -g^2 C_2(r)/4\pi R.

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