Editing Linking number (section)

==In quantum field theory==
In [[quantum field theory]], Gauss's integral definition arises when computing the expectation value of the [[Wilson loop]] observable in <math>U(1)</math> [[Chern–Simons]] [[gauge theory]]. Explicitly, the abelian Chern–Simons action for a gauge potential one-form <math>A</math> on a three-[[manifold]] <math>M</math> is given by

: <math> 
S_{CS} = \frac{k}{4\pi} \int_M A \wedge dA
</math>

We are interested in doing the [[Feynman path integral]] for Chern–Simons in <math> M = \mathbb{R}^3 </math>:  

: <math>
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) 
</math>

Here, <math>\epsilon</math> is the antisymmetric symbol. Since the theory is just Gaussian, no ultraviolet [[regularization (physics)|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 <math>A</math>. 

The classical equations of motion are

: <math>
\varepsilon^{\lambda \mu \nu} \partial_\mu A_\nu = \frac{2\pi}{k} J^\lambda 
</math> 

Here, we have coupled the Chern–Simons field to a source with a term <math>-J_\mu A^\mu</math> in the Lagrangian. Obviously, by substituting the appropriate <math>J</math>, 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:

: <math>
\vec{\nabla} \times \vec{A} = \frac{2\pi}{k} \vec{J}
</math>

Taking the curl of both sides and choosing [[Lorenz gauge]] <math> \partial^\mu A_\mu = 0 </math>, the equations become 

: <math> 
\nabla^2 \vec{A} = - \frac{2\pi}{k} \vec{\nabla} \times \vec{J} 
</math>

From electrostatics, the solution is

: <math>
A_\lambda(\vec{x}) = \frac{1}{2k} \int d^3 \vec{y} \, \frac{\varepsilon_{\lambda \mu \nu} \partial^\mu J^\nu (\vec{y})}{|\vec{x} - \vec{y}|}
</math>

The path integral for arbitrary <math>J</math> is now easily done by substituting this into the Chern–Simons action to get an effective action for the <math>J</math> field. To get the path integral for the Wilson loops, we substitute for a source describing two particles moving in closed loops, i.e. <math> J = J_1 + J_2 </math>, with 

: <math>
J_i^\mu (x) = \int_{\gamma_i} dx_i^\mu \delta^3 (x - x_i (t))
</math> 

Since the effective action is quadratic in <math>J</math>, 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 

: <math>
Z[\gamma_1, \gamma_2] = \exp{ \left( \frac{2\pi i}{k} \Phi[\gamma_1, \gamma_2] \right) },
</math>

where 

: <math> 
\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}, 
</math>

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. <ref name="Witten1989">{{cite journal  |last=Witten |first=E. |year=1989 |title=Quantum field theory and the Jones polynomial|url=https://projecteuclid.org/euclid.cmp/1104178138 |journal=Comm. Math. Phys. |volume=121 |issue=3 |pages=351–399|mr=0990772|zbl=0667.57005 |doi=10.1007/bf01217730|bibcode=1989CMaPh.121..351W}}</ref>

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. <ref name="1612.09298">{{cite journal | arxiv=1612.09298  | last=Putrov | first=Pavel| last2=Wang | first2=Juven| last3=Yau | first3=Shing-Tung|title=Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions| doi=10.1016/j.aop.2017.06.019 |volume=384C|journal=Annals of Physics|pages=254–287|bibcode=2017AnPhy.384..254P|date=September 2017}}</ref>