The Physics Travel Guide

Yang-Mills Equations

Prerequisites: Gauge Models. Full path (17 pages)

Best explanations

∂μFμνa−gfabcAbμFμνc=0\partial^{\mu}F_{\mu\nu}^a-gf_{abc}A^{\mu}_bF_{\mu\nu}^c=0

see also Gauge Models and Gauge Symmetry

Intuitive

The Yang-Mills equations generalize the Maxwell equations. In electromagnetism there is one kind of charge, electric charge, and the field that charges create, the electromagnetic field, is itself uncharged. In classical electromagnetism, two light beams pass through each other without noticing each other. This is why the Maxwell equations are linear.

The strong and the weak interaction work with a richer kind of charge. In the strong interaction a quark carries one of three "colors", and the interaction can change a quark's color. Instead of one photon there are several force particles, eight gluons in the case of the strong interaction, and these carry color charge themselves. So the field acts as a source of itself: gluons attract and scatter other gluons. In the equations this shows up as extra terms in which the field appears multiplied by itself, which makes the Yang-Mills equations nonlinear. Mathematically, the extra terms exist because the symmetry transformations behind the theory are described by matrices that do not commute, so the order in which two transformations are applied matters.

The self-interaction has large consequences in the quantum theory. For the strong interaction, the force becomes weak at very short distances (asymptotic freedom) and strong at large distances, which is the accepted explanation of why quarks and gluons are never seen in isolation, although a proof is missing. Proving that the quantum theory of pure Yang-Mills fields has a mass gap, meaning that its lightest states have a nonzero mass, is an open problem.

L=1/4g²∫TrF∧∗F

Concrete

Derivation of the Yang-Mills equation from the Yang-Mills Lagrangian

Let L=−14FaμνFμνa\begin{equation} \mathcal{L}=-\frac 14 F^{\mu\nu}_aF_{\mu\nu}^a \end{equation} and AaμA^{\mu}_a be real functions.

The Euler-Lagrange equations for L=L(Aμ,∂νAμ)\mathcal{L}=\mathcal{L}(A^{\mu},\partial^{\nu}A^{\mu}) are ∂ν(L∂(∂νAaμ))=∂L∂Aaμ\begin{equation} \partial^{\nu}\left( \frac {\mathcal{L}}{\partial\left(\partial^{\nu}A_a^{\mu}\right)}\right)=\frac {\partial \mathcal{L}}{\partial A_a^{\mu}} \end{equation}

Then

∂Fdμν∂Aaμ=−gfdacAcν∂Fμνd∂Aaμ=−gfdacAcνgμμgνν∂L∂Aaμ=12gfabcAbνFμνc∂Fdμν∂(∂νAaμ)=−δad∂Fμνd∂(∂νAaμ)=δad(gμνgνμ−gμμgνν)∂L∂(∂νAaμ)=12Fμνa\begin{align} &\frac {\partial F^{\mu\nu}_d}{\partial A_a^{\mu}}=-gf_{dac}A_c^{\nu}\\ &\frac {\partial F^d_{\mu\nu}}{\partial A_a^{\mu}}=-gf_{dac}A_c^{\nu}g_{\mu\mu}g_{\nu\nu}\\ &\frac {\partial \mathcal{L}}{\partial A_a^{\mu}}=\frac 12 gf_{abc}A_b^{\nu}F^c_{\mu\nu}\\ &\frac {\partial F^{\mu\nu}_d}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=-\delta_{ad}\\ &\frac {\partial F^d_{\mu\nu}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=\delta_{ad}(g_{\mu\nu}g_{\nu\mu}-g_{\mu\mu}g_{\nu\nu})\\ &\frac {\partial \mathcal{L}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=\frac 12 F_{\mu\nu}^a \end{align}

Directly computing ∂Fdμν∂Aaμ=∂∂Aaμ(−gfdbcAbμAcν)=−gδabfdbcAcν=−gfdacAcν\begin{equation} \begin{gathered} \frac {\partial F^{\mu\nu}_d}{\partial A_a^{\mu}}=\frac {\partial}{\partial A_a^{\mu}}\left( -gf_{dbc}A^{\mu}_bA^{\nu}_c\right) =-g\delta_{ab}f_{dbc}A_c^{\nu}= -gf_{dac}A_c^{\nu} \end{gathered} \end{equation}

∂Fμνd∂Aaμ=∂∂Aaμ(−gfdbcAμbAνc)=∂∂Aaμ(−gfdbcgμαAbαAνc)=−gfdacgμαδαμδabAνc=−gfdacAνcgμμ=−gfdacgναAcαgμμ=−gfdacAcνgμμgνν\begin{equation} \begin{gathered} \frac {\partial F^d_{\mu\nu}}{\partial A_a^{\mu}}=\frac {\partial}{\partial A_a^{\mu}}\left( -gf_{dbc}A^b_{\mu}A^c_{\nu}\right) =\frac {\partial}{\partial A_a^{\mu}}\left( -gf_{dbc}g_{\mu\alpha}A^{\alpha}_bA^c_{\nu}\right)\\ =-gf_{dac}g_{\mu\alpha}\delta_{\alpha\mu}\delta_{ab}A^c_{\nu}= -gf_{dac}A^c_{\nu}g_{\mu\mu}\\ =-gf_{dac}g_{\nu\alpha}A^{\alpha}_cg_{\mu\mu}= -gf_{dac}A_c^{\nu}g_{\mu\mu}g_{\nu\nu} \end{gathered} \end{equation}

∂L∂Aaμ=−14((∂Fdμν∂Aaμ)Fμνd+Fdμν(∂Fμνd∂Aaμ))\begin{equation} \frac {\partial \mathcal{L}}{\partial A_a^{\mu}}=-\frac 14 \left(\left(\\ \frac {\partial F^{\mu\nu}_d}{\partial A_a^{\mu}}\right)F^d_{\mu\nu}\\ +F^{\mu\nu}_d\left(\frac {\partial F^d_{\mu\nu}}{\partial A_a^{\mu}}\right)\right) \end{equation}

=14gfdac(AcνFμνd+FdμνAcνgμμgνν)=14gfdac(AcνFμνd+AcνgμαgνβFdαβ)=14gfdac(AcνFμνd+AcνFμνd)=12gfdacAcνFμνd=12gfacdAcνFμνd=12gfabcAbνFμνc\begin{gather*} =\frac 14 gf_{dac}\left(A_c^{\nu}F^d_{\mu\nu}+F^{\mu\nu}_d A^{\nu}_cg_{\mu\mu}g_{\nu\nu}\right)\\ =\frac 14 gf_{dac}\left(A_c^{\nu}F^d_{\mu\nu}+A^{\nu}_cg_{\mu\alpha}g_{\nu\beta}F^{\alpha\beta}_d \right)\\ =\frac 14 gf_{dac}\left(A_c^{\nu}F^d_{\mu\nu}+A^{\nu}_cF^d_{\mu\nu} \right)\\ =\frac 12 gf_{dac}A_c^{\nu}F^d_{\mu\nu}=\frac 12 gf_{acd}A_c^{\nu}F^d_{\mu\nu}\\ =\frac 12 gf_{abc}A_b^{\nu}F^c_{\mu\nu} \end{gather*}

∂Fdμν∂(∂νAaμ)=∂∂(∂νAaμ)(∂μAdν−∂νAdμ)=−δad\begin{gather*} \frac {\partial F^{\mu\nu}_d}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}=\frac {\partial}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}\left(\partial^{\mu}A^{\nu}_d-\partial^{\nu}A^{\mu}_d\right)=-\delta_{ad} \end{gather*}

∂Fμνd∂(∂νAaμ)=∂∂(∂νAaμ)(∂μAνd−∂νAμd)=∂∂(∂νAaμ)(gμαgνβ(∂αAdβ−∂βAdα))=gμαgνβδadδμβδνα−gμαgνβδadδνβδμα=−δad(gμνgνμ−gμμgνν)\begin{gather*} \frac {\partial F^d_{\mu\nu}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)} =\frac {\partial}{\partial \left(\partial^{\nu}A_a^{\mu}\right)}\left(\partial_{\mu}A^d_{\nu}-\partial_{\nu}A^d_{\mu}\right)\\ =\frac {\partial}{\partial \left(\partial^{\nu}A_a^{\mu}\right)} \left(g_{\mu\alpha}g_{\nu\beta}\left(\partial^{\alpha}A^{\beta}_d-\partial^{\beta}A^{\alpha}_d\right)\right)\\ =g_{\mu\alpha}g_{\nu\beta}\delta_{ad}\delta_{\mu\beta}\delta_{\nu\alpha} -g_{\mu\alpha}g_{\nu\beta}\delta_{ad}\delta_{\nu\beta}\delta_{\mu\alpha}\\ =-\delta_{ad}\left( g_{\mu\nu}g_{\nu\mu}-g_{\mu\mu}g_{\nu\nu}\right) \end{gather*}

∂L∂(∂νAaμ)=−14(−δadFμνd−gμμgννδadFdμν)=14(Fμνa+gμαgνβFaαβ)=14(Fμνa+gμαgνβFaαβ)=14(Fμνa+Fμνa)=12Fμνa\begin{gather*} \frac {\partial \mathcal{L}}{\partial \left(\partial^{\nu}A_a^{\mu}\right)} =-\frac 14\left(-\delta_{ad}F_{\mu\nu}^d-g_{\mu\mu}g_{\nu\nu}\delta_{ad}F_d^{\mu\nu}\right)\\ =\frac 14\left(F_{\mu\nu}^a+g_{\mu\alpha}g_{\nu\beta}F_a^{\alpha\beta}\right)\\ =\frac 14\left(F_{\mu\nu}^a+g_{\mu\alpha}g_{\nu\beta}F_a^{\alpha\beta}\right)\\ =\frac 14\left(F_{\mu\nu}^a+F^a_{\mu\nu}\right)=\frac 12F_{\mu\nu}^a \end{gather*}

Abstract

The Yang-Mills equations can be expressed with the Hodge star operator as dA∗F=0F=dA+A∧A\begin{equation} d_A *F = 0 \quad F=dA+A\wedge A \end{equation} where dAd_A is the gauge-covariant extension of the exterior derivative. The curvature also satisfies the Bianchi identity dAF=0d_A F = 0, which holds for every AA. The gauge field AA is a one-form A(x)=Aμa(x)tadxμ\begin{equation} A(x)=A_{\mu}^a(x)t^adx^{\mu} \end{equation} with the values on the Lie algebra of a compact simple Lie group GG. The curvature is a two-form F=dA+A∧AF=12Fμνatadxμ∧dxνFμνa=∂μAνa−∂νAμa+fabcAμbAνc\begin{equation} \begin{gathered} F=dA+A\wedge A \\ F=\frac12 F_{\mu\nu}^at^adx^{\mu}\wedge dx^{\nu}\\ F_{\mu\nu}^a=\partial_{\mu}A_{\nu}^a-\partial_{\nu}A_{\mu}^a+f^{abc}A_{\mu}^bA_{\nu}^c \end{gathered} \end{equation} with [ta,tb]=fabctc[t^a,t^b]=f^{abc}t^c and the coupling constant gg absorbed into AA.

Why is it interesting?

In physics, of course, Maxwell's equations of electromagnetism are linear partial differential equations. Their counterparts, the famous Yang-Mills equations, are non-linear equations which are supposed to govern the forces involved in the structure of matter. The equations are non-linear, because the Yang-Mills equations are essentially matrix versions of Maxwell's equations, and the fact that matrices do not commute is what produces the non-linear term in the equations.

The Aleph of Space by Luciano Boi

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