In
quantum field theory, Gauss's integral definition arises when computing the expectation value of the
Wilson loop observable in U(1)
Chern–Simons gauge theory. Explicitly, the abelian Chern–Simons action for a gauge potential one-form A on a three-
manifold M is given by : S_{CS} = \frac{k}{4\pi} \int_M A \wedge dA We are interested in doing the
Feynman path integral for Chern–Simons in M = \mathbb{R}^3 : : Z[\gamma_1, \gamma_2] = \int \mathcal{D} A_\mu \exp \left( \frac{ik}{4\pi} \int d^3 x \varepsilon^{\lambda \mu \nu} A_\lambda \partial_\mu A_\nu + i \int_{\gamma_1} dx^\mu \, A_\mu + i \int_{\gamma_2} dx^\mu \, A_\mu \right) Here, \epsilon is the antisymmetric symbol. Since the theory is just Gaussian, no ultraviolet
regularization or
renormalization is needed. Therefore, the topological invariance of right hand side ensures that the result of the path integral will be a topological invariant. The only thing left to do is provide an overall normalization factor, and a natural choice will present itself. Since the theory is Gaussian and abelian, the path integral can be done simply by solving the theory classically and substituting for A. The classical equations of motion are : \varepsilon^{\lambda \mu \nu} \partial_\mu A_\nu = \frac{2\pi}{k} J^\lambda Here, we have coupled the Chern–Simons field to a source with a term -J_\mu A^\mu in the Lagrangian. Obviously, by substituting the appropriate J, we can get back the Wilson loops. Since we are in 3 dimensions, we can rewrite the equations of motion in a more familiar notation: : \vec{\nabla} \times \vec{A} = \frac{2\pi}{k} \vec{J} Taking the curl of both sides and choosing
Lorenz gauge \partial^\mu A_\mu = 0 , the equations become : \nabla^2 \vec{A} = - \frac{2\pi}{k} \vec{\nabla} \times \vec{J} From electrostatics, the solution is : A_\lambda(\vec{x}) = \frac{1}{2k} \int d^3 \vec{y} \, \frac{\varepsilon_{\lambda \mu \nu} \partial^\mu J^\nu (\vec{y})} The path integral for arbitrary J is now easily done by substituting this into the Chern–Simons action to get an effective action for the J field. To get the path integral for the Wilson loops, we substitute for a source describing two particles moving in closed loops, i.e. J = J_1 + J_2 , with : J_i^\mu (x) = \int_{\gamma_i} dx_i^\mu \delta^3 (x - x_i (t)) Since the effective action is quadratic in J, it is clear that there will be terms describing the self-interaction of the particles, and these are uninteresting since they would be there even in the presence of just one loop. Therefore, we normalize the path integral by a factor precisely cancelling these terms. Going through the algebra, we obtain : Z[\gamma_1, \gamma_2] = \exp{ \left( \frac{2\pi i}{k} \Phi[\gamma_1, \gamma_2] \right) }, where : \Phi[\gamma_1, \gamma_2] = \frac{1}{4\pi} \int_{\gamma_1} dx^\lambda \int_{\gamma_2} dy^\mu \, \frac{(x - y)^\nu}{|x - y|^3} \varepsilon_{\lambda \mu \nu}, which is simply Gauss's linking integral. This is the simplest example of a
topological quantum field theory, where the path integral computes topological invariants. This also served as a hint that the nonabelian variant of Chern–Simons theory computes other knot invariants, and it was shown explicitly by
Edward Witten that the nonabelian theory gives the invariant known as the Jones polynomial. The Chern-Simons gauge theory lives in 3 spacetime dimensions. More generally, there exists higher dimensional topological quantum field theories. There exists more complicated multi-loop/string-braiding statistics of 4-dimensional gauge theories captured by the link invariants of exotic
topological quantum field theories in 4 spacetime dimensions. ==Generalizations==