Yang-Mills functional

Photons appear as the quanta of Maxwell's classical electromagnetic theory, an Abelian theory in the sense that the circle group embodies the phase factor. The aim of quantum field theory is to treat other elementary particles by quantizing appropriate classical non-Abelian field theories, phrased as gauge theories. These were invented by C.N. Yang and R.L. Mills [a2]. The circle group is thereby replaced by a non-Abelian compact Lie group dictated by the (observed classical) symmetries and the Yang–Mills equation (cf. Yang–Mills field) generalizes the Maxwell equations (in vacuum). The quantization of non-Abelian gauge theories is still in its infancy.

On the mathematical side, gauge theory is a well established branch of differential geometry known as the theory of fibre bundles with connection (cf. also Connection). The Yang–Mills equations or field equations are derived by an action principle, reflecting Einstein's point of view that the basic laws of physics should all be combined in geometrical form: Consider a principal fibre bundle $\xi : P \rightarrow M$ over a smooth oriented Riemannian manifold $M$ with compact structure group $G$, and consider the affine space $\mathcal{A} ( \xi )$ of connections (gauge potentials in physics terminology); for a connection $A$, let $F _ { A }$ be its curvature (gauge field or field strength in physics terminology). The curvature can be thought of as the distortion produced by an external field, or it can be identified with the field when one thinks of a field of force measured by its local effect. This distortion does not take place in the geometry of "space-time" (or what corresponds to it: $M$), though, but in the geometry of some state-space of internal structure superimposed to $M$. Given an invariant scalar product $\langle \, .\, ,\, . \, \rangle$ on the Lie algebra of $G$ (e.g., the negative of the Killing form when $G$ is semi-simple), the Yang–Mills functional $\mathcal{L}$ on $\mathcal{A} ( \xi )$ assigns the real number $\mathcal{L} ( A ) = \int _ { M } \langle F _ { A } \wedge * F _ { A } \rangle$ to any connection $A$ on $\xi $; here $*$ refers to the Hodge star operator (cf. also Laplace operator) determined by the data. This functional is also called the action of the (gauge) theory; this way of writing it and the resulting equations exhibits clearly its invariance and covariance properties.

One way of attempting to develop the quantum theory is to use the Feynman functional integral approach, which involves the function $\operatorname{exp}( i \mathcal{L} )$. Critical values of the latter will then occur at those connections $A$ which are critical for the action $\mathcal{L}$, and one is led to determine those connections or classical field configurations which are stationary for $\mathcal{L}$. These connections $A$ satisfy the equation

\begin{equation} \tag{a1} \nabla _ { A } * F _ { A } = 0, \end{equation}

the Euler–Lagrange equation of the corresponding variational principle or Yang–Mills equation (cf. Yang–Mills field), and they are called Yang–Mills connections; here, $\nabla$ refers to the covariant derivative operator. The field equations are the equation (a1) together with the Bianchi identity

\begin{equation} \tag{a2} \nabla _ { A } F _ { A } = 0. \end{equation}

Only the equation (a1) imposes a condition on the connection or potential. The non-uniqueness of the potential has its counterpart in the form of bundle automorphisms or gauge transformations, and the Yang–Mills functional is clearly invariant under gauge transformations. Its true solution space is the moduli space of Yang–Mills connections, the space of gauge equivalence classes of Yang–Mills connections. On this space, the problem of gauge fixing is that of choosing continuously a potential in each gauge equivalence class.

For example, in Maxwell's theory, $G$ is the circle group, $M$ is at first $\mathbf{R} ^ { 4 }$, the coefficients of the curvature are the components of the electric and magnetic fields, and among the potentials for which the action $\mathcal{L}$ is finite one looks for those which minimize the action. To achieve that the action is finite one imposes appropriate asymptotic conditions and is thus led to consider bundles $\xi $ having as base $M$ the $4$-sphere, viewed as a (conformal) compactification of space-time $\mathbf{R} ^ { 4 }$.

In non-Abelian gauge theories on closed manifolds $M$ like $S ^ { 4 }$, when the group $G$ is connected and simply connected, the corresponding principal bundles fall into distinct topological types (these correspond to the elements of the fourth integral cohomology group of $M$); when the bundle is topologically non-trivial, gauge fixing is impossible. This is sometimes referred to as the Gribov ambiguity. Suitably normalized, the value of the absolute minimum of the Yang–Mills functional just amounts to the corresponding cohomology class. An analogous formula in dimension two is Gauss' classical theorem expressing the Euler characteristic as the integral of the scalar curvature. This problem does not occur in ordinary Maxwell theory over $\mathbf{R} ^ { 4 }$ (or $S ^ { 4 }$); yet it occurs in the Maxwell theory over a manifold $M$ having second integral cohomology group non-zero. This indicates that, mathematically, Yang–Mills theory leads to global questions incorporating both topology and analysis, as opposed to the purely local theory of classical differential geometry. See also Yang–Mills functional, geometry of the.

References

[a1]	M.F. Atiyah, "Geometry of Yang–Mills fields" , Lezioni Fermiane , Accad. Nazionale dei Lincei Scuola Norm. Sup. Pisa (1979)
[a2]	R.L. Mills, C.N. Yang, "Conservation of isotopic spin and isotopic gauge invariance" Phys. Rev. , 96 (1954) pp. 191