Editing Yang–Mills theory (section)

== Mathematical overview ==
{{See also|Yang–Mills equations}}
{{multiple image
 |perrow = 2 |total_width=280 |left
 | image1 = -y-(x^2+y^2+1) plot; BPST instanton.png
 | image2 = X-(x^2+y^2+1) plot; BPST instanton.png
 | image3 = Curvature of BPST Instanton.png
 | image4 = BPST on sphere.png
 | footer = The {{math|d''x''<sup>1</sup>⊗''σ''<sub>3</sub>}} coefficient of a [[BPST instanton]] on the {{math|(''x''<sup>1</sup>,''x''<sup>2</sup>)}}-slice of {{math|ℝ<sup>4</sup>}} where {{math|''σ''<sub>3</sub>}} is the third [[Pauli matrix]] (top left). The {{math|d''x''<sup>2</sup>⊗''σ''<sub>3</sub>}} coefficient (top right). These coefficients determine the restriction of the BPST instanton {{mvar|A}} with {{nobr|{{math|''g''{{=}}2, ''ρ''{{=}}1, ''z''{{=}}0}}}} to this slice. The corresponding field strength centered around {{math|''z''{{=}}0}} (bottom left). A visual representation of the field strength of a BPST instanton with center {{mvar|z}} on the [[compactification (mathematics)|compactification]] {{math|''S''<sup>4</sup>}} of {{math|ℝ<sup>4</sup>}} (bottom right). The BPST instanton is a classical [[instanton]] solution to the [[Yang–Mills equations]] on {{math|ℝ<sup>4</sup>}}.
}}

Yang–Mills theories are special examples of gauge theories with a non-abelian symmetry group given by the [[Lagrangian (field theory)|Lagrangian]]

:<math>\ \mathcal{L}_\mathrm{gf} = -\tfrac{1}{2}\operatorname{tr}(F^2) = - \tfrac{1}{4} F^{a\mu \nu} F_{\mu \nu}^a\ </math>

with the generators <math>\ T^a\ </math> of the [[Lie algebra]], indexed by {{mvar|a}}, corresponding to the {{mvar|F}}-quantities (the [[curvature]] or field-strength form) satisfying
:<math>\ \operatorname{tr}\left( T^a\ T^b \right) = \tfrac{1}{2} \delta^{ab}\ , \qquad \left[ T^a,\ T^b \right] = i\ f^{abc}\ T^c ~.</math>

Here, the {{mvar|f&thinsp;<sup>abc</sup>}} are [[structure constant]]s of the Lie algebra (totally antisymmetric if the generators of the Lie algebra are normalised such that <math>\ \operatorname{tr}(T^a\ T^b)\ </math> is proportional to <math>\ \delta^{ab}\ </math>), the [[covariant derivative]] is defined as

:<math>\ D_\mu = I\ \partial_\mu - i\ g\ T^a\ A^a_\mu\ ,</math>

{{mvar|I}} is the [[identity matrix]] (matching the size of the generators), <math>\ A^a_\mu\ </math> is the [[Four-vector|vector]] potential, and {{mvar|g}} is the [[coupling constant]]. In four dimensions, the coupling constant {{mvar|g}} is a pure number and for a {{math|SU(''n'')}} group one has <math>\ a, b, c = 1 \ldots n^2-1 ~.</math>

The relation

:<math>\ F_{\mu \nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g\ f^{abc}\ A_\mu^b\ A_\nu^c\ </math>

can be derived by the [[commutator]]

:<math>\ \left[ D_\mu, D_\nu \right] = -i\ g\ T^a\ F_{\mu\nu}^a ~.</math>

The field has the property of being self-interacting and the equations of motion that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. This means that one can manage this theory only by [[perturbation theory]] with small nonlinearities.{{Citation needed|reason=The use of the word "only" needs clarification and a proper citation will give contextual meaning that is evidently missing.|date=June 2023}}

Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components is trivial for ''a'' indices (e.g. <math>\ f^{abc} = f_{abc}\ </math>), whereas for μ and ν it is nontrivial, corresponding e.g. to the usual Lorentz signature, <math>\ \eta_{\mu \nu } = {\rm diag}(+---) ~.</math>

From the given Lagrangian one can derive the equations of motion given by

:<math>\ \partial^\mu F_{\mu\nu}^a + g\ f^{abc}\ A^{\mu b}\ F_{\mu\nu}^c = 0 ~.</math>

Putting <math>\ F_{\mu\nu}=T^aF^a_{\mu\nu}\ ,</math> these can be rewritten as

:<math>\ \left( D^\mu F_{\mu\nu} \right)^a = 0 ~.</math>

A [[Bianchi identity]] holds

:<math>\ \left( D_\mu\ F_{\nu \kappa} \right)^a + \left( D_\kappa\ F_{\mu \nu} \right)^a + \left( D_\nu\ F_{\kappa \mu} \right)^a = 0\ </math>

which is equivalent to the [[Jacobi identity]]

:<math>\ \left[ D_{\mu}, \left[ D_{\nu},D_{\kappa} \right] \right] + \left[ D_{\kappa}, \left[ D_{\mu},D_{\nu} \right] \right] + \left[ D_{\nu}, \left[ D_{\kappa},D_{\mu} \right] \right] = 0\ </math>

since <math>\ \left[ D_{\mu},F^a_{\nu\kappa} \right] = D_{\mu}\ F^a_{\nu\kappa} ~.</math> Define the [[Hodge star operator|dual]] strength tensor
<math>\ \tilde{F}^{\mu\nu} = \tfrac{1}{2}\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}\ ,</math> then the Bianchi identity can be rewritten as

:<math>\ D_{\mu}\tilde{F}^{\mu\nu} = 0 ~.</math>

A source <math>\ J_\mu^a\ </math> enters into the equations of motion as

:<math>\ \partial^\mu F_{\mu\nu}^a + g\ f^{abc}\ A^{b\mu}\ F_{\mu\nu}^c = -J_\nu^a ~.</math>

Note that the currents must properly change under gauge group transformations.

We give here some comments about the physical dimensions of the coupling. In {{mvar|D}} dimensions, the field scales as <math>\ \left[A\right]=\left[ L^{\left(\tfrac{2-D}{2}\right)} \right]\ </math> and so the coupling must scale as <math>\ \left[g^2\right] = \left[L^{\left(D-4\right)}\right] ~.</math> This implies that Yang–Mills theory is not [[renormalization|renormalizable]] for dimensions greater than four. Furthermore, for {{nobr|{{math|''D'' {{=}} 4}} ,}} the coupling is dimensionless and both the field and the square of the coupling have the same dimensions of the field and the coupling of a massless quartic [[scalar field theory]]. So, these theories share the [[scale invariance]] at the classical level.