Classical pure SU(2) Yang–Mills model

Each instance of the adopted classical pure Yang–Mills model has spacetime $M={\mathbb{R}}^{1,3}$, coordinates $x^0,x^1,x^2,x^3$ of length, orientation $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, time orientation of increasing $x^0$, and Lorentz metric $\eta =\mathrm{diag}(-1,1,1,1)$. Use $c=\mathrm{\hslash }=1$, so $x^0=t$, the coordinate and spacetime lengths have unit $\mathrm{m}$, and the natural action $S\mathrm{/}\mathrm{\hslash }$ is dimensionless.

Fix the global structure group $G=SU(2)$, not its quotient $SO(3)$. Each instance selects a smooth right principal $SU(2)$-bundle $\pi :P\to M$ and admits only its bundle sector $\mathrm{\Sigma }=\{[P]\}$. The matter-representation label set is empty, so there are no matter fields or current source. The Lie algebra is $\mathfrak{s}\mathfrak{u}(2)$ with basis $T_a=-i{\sigma }_a\mathrm{/}2$ and positive $\mathrm{Ad}$-invariant form $\kappa (X,Y)=-2{\mathrm{Tr}}_2(XY)$, normalized by $\kappa (T_a,T_b)={\delta }_{ab}$. The positive coupling $g_{\mathrm{Y}\mathrm{M}}$ is a fixed dimensionless model parameter.

A smooth right principal bundle here has a free and transitive smooth right $SU(2)$ action on each fibre and equivariant local trivializations; a global section is not part of the datum. The sector $[P]$ is the nonempty minimum-rank set code of its smooth equivariant bundle-isomorphism type over $M$, in the supplied non-Abelian gauge-data convention. The gauge group $\mathcal{G}(P)$ consists of smooth diffeomorphisms $\mathrm{\Phi }:P\to P$ with $\pi \circ \mathrm{\Phi }=\pi$ and $\mathrm{\Phi }(pg)=\mathrm{\Phi }(p)g$. A gauge function $U_i\to SU(2)$ is only a local description unless a global trivialization has been specified.

A principal connection is a smooth right-invariant horizontal complement $TP=VP\oplus H$. A gauge field is its connection form $\omega$ with $\omega (Z_P)=Z$ and $R_g^{\ast }\omega ={\mathrm{Ad}}_{g^{-1}}\omega$, where $R_g(p)=pg$ and $Z_P(p)={\frac{d}{dt}\mid }_{t=0}p{\exp }_{SU(2)}(tZ)$. In a local section $s_i:U_i\to P$, let $A_i=s_i^{\ast }\omega$ and let $F_i=d{A}_i+\frac{1}{2}[A_i\wedge A_i]$ be its curvature. Here the bracket is the alternating-form extension of the Lie bracket, and the local forms on overlaps satisfy the right-action transition convention $A_j={\mathrm{Ad}}_{g_{ij}^{-1}}{A}_i+g_{ij}^{-1}d{g}_{ij}$ and $F_j={\mathrm{Ad}}_{g_{ij}^{-1}}{F}_i$ when $s_j=s_i{g}_{ij}$. Define the real adjoint vector bundle by $\mathrm{ad}(P):=(P\times \mathfrak{s}\mathfrak{u}(2))\mathrm{/}\sim ,\text{where }\sim \text{ is generated by }(pg,Z)\sim (p,{\mathrm{Ad}}_{g}Z)\mathrm{.}$ For each local section $s_i$, use the smooth vector-bundle chart $[s_i(x)h,Z]\mapsto (x,{\mathrm{Ad}}_{h}Z)$. Consequently, the components of a smooth section $\eta$ are maps ${\eta }_i:U_i\to \mathfrak{s}\mathfrak{u}(2)$ and obey ${\eta }_j={\mathrm{Ad}}_{g_{ij}^{-1}}{\eta }_i$ when $s_j=s_i{g}_{ij}$. Thus $F_i$ represents one global section of ${\mathrm{\Lambda }}^2{T}^{\ast }M\otimes \mathrm{ad}(P)$, and the invariant form $\kappa$ makes $\kappa (F_i\wedge {\ast }_{\eta }{F}_i)$ a global scalar four-form.

Write ${\mathrm{\Omega }}^k(M)$ for the space of smooth real $k$-forms on $M$. For $\alpha ,\beta \in {\mathrm{\Omega }}^k(M)$, write $\alpha =\frac{1}{k!}{\alpha }_{{\mu }_1\dots {\mu }_k}d{x}^{{\mu }_1}\wedge \cdots \wedge d{x}^{{\mu }_k}$, raise indices using ${\eta }^{-1}$ and set $⟨\alpha ,\beta {⟩}_{\eta }:=\frac{1}{k!}{\alpha }_{{\mu }_1\dots {\mu }_k}{\beta }^{{\mu }_1\dots {\mu }_k},{\mathrm{vol}}_{\eta }:=d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3\mathrm{.}$ For $0\le k\le 4$, define the Lorentzian Hodge map ${\ast }_{\eta }:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{4-k}(M)$ as the unique smooth fibrewise-linear map satisfying $\alpha \wedge {\ast }_{\eta }\beta =⟨\alpha ,\beta {⟩}_{\eta }{\mathrm{vol}}_{\eta }$ for all $\alpha ,\beta \in {\mathrm{\Omega }}^k(M)$. The nondegenerate metric-induced pairing and the wedge pairing give this unique map pointwise, and the constant coefficients make it smooth. On $\mathfrak{s}\mathfrak{u}(2)$-valued forms it acts on the spacetime-form factor.

Adopt the local Lorentzian action density

$${\mathcal{L}}_{\mathrm{Y}\mathrm{M}}(A_i)d^{4}x=-\frac{1}{2g_{\mathrm{Y}\mathrm{M}}^2}\kappa (F_i\wedge {\ast }_{\eta }{F}_i),[A_i]={\mathrm{m}}^{-1},[F_i]={\mathrm{m}}^{-2}\mathrm{.}$$

For any relatively compact region on which the density is integrable, the action is its integral. On all of $M$, the action is defined when that integral converges. Local stationarity is tested through relative action differences under smooth compactly supported adjoint-bundle-valued one-form variations; these differences have compact support and are finite. No boundary values are imposed at infinity. On a finite region, admissible variations vanish near its boundary. This is an adopted classical model and normalization, not a theorem, measured law, or claim that a quantum continuum theory exists.

Matthijs Vákár, Principal Bundles and Gauge Theories

Karl-Hermann Neeb, Differential Topology of Fiber Bundles (full lecture notes, 2024)

David Tong, Gauge Theory, Chapter 2, Yang-Mills Theory