User:PatentPhysicist/sandbox

Respectfully suggested improvements for Yang–Mills_theory#Mathematical_overview

Having the reviews made public by Symmetry is an author choice. As the author of this paper, because of my belief in the importance of transparency, I opted for this Open Review. If someone believes the reviews were inadequate, perhaps they should invest the effort to study and offer substantive comments on the paper itself, rather than on a) me personally, b) another person who made favorable comments about the paper and about Symmetry, c) the journal Symmetry, and / or d) the adequacy of the peer reviews from Symmetry. I also point out that bulleted at the very top of the page Talk:Yang–Mills_theory, is the statement: "Yang–Mills theory has been listed as a level-5 vital article in Science, Physics. If you can improve it, please do." And the please do link says to "be bold," and offers some suggestions about being so, as well as pointing to Wikipedia:Assume_good_faith and Wikipedia:Civility. Perhaps I was too bold, but it is clear that the Wikipedia editors themselves believe that the article at Yang–Mills theory needs to be improved, and that its present rating is only B-Class. Therefore, a bit less boldly, I have made some suggestions for improvement at the talk page, here. I also note from its talk page that "[t]his article [on Yang-Mills existence and the mass gap] has been rated as C-Class on the project's quality scale," so there is work to do there also.

The Mathematical overview for Yang–Mills_theory is not wrong in any way I can see, but I respectfully suggest the following improvements:

1) This whole subsection does not contain a single external source citation. It would be helpful to have one or more citations to support the various formulas in this subsection. One good source which has been my bible since 1984 is Halzen, F.; Martin, A.D. (1984). Quarks and Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 978-0-471-88741-6., although not all of the Mathematical overview formulas are in there. If others are aware of some good references to establish the mathematics in this subsection, they should be added.

Now as to some specifics regarding formula discussions which I believe need to be improved:

2) Although true and correct, the relation $F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c}$ and the commutator $[D_{\mu },D_{\nu }]=-igT^{a}F_{\mu \nu }^{a}$ are merely stated, with no more. These should be better developed, and their derivation explicitly shown. Indented below is in the nature of what I respectfully suggest:

The commutator

$\hbar c\left[{{D}^{\mu }},{{D}^{\nu }}\right]=-ig{{F}^{\mu \nu }}=-ig{{T}^{a}}{{F}^{a}}^{\mu \nu }$

can be derived as follows: With $\hbar =c=1$ and using $\left[{{\partial }^{\mu }},{{\partial }^{\nu }}\right]=0,$ apply the commutator $\left[{{D}^{\mu }},{{D}^{\nu }}\right]$ to operate on any field $\phi \left(t,\mathbf {x} \right).$ Attentive to the product rule, one may obtain:

${\begin{aligned}&\left[{{D}^{\mu }},{{D}^{\nu }}\right]\phi =\left[\left({{\partial }^{\mu }}-ig{{A}^{\mu }}\right),\left({{\partial }^{\nu }}-ig{{A}^{\nu }}\right)\right]\phi \\&\quad \quad \quad \quad \ \ =-ig{{A}^{[\mu }}{{\partial }^{\nu ]}}\phi -ig{{\partial }^{[\mu }}\left({{A}^{\nu ]}}\phi \right)-{{g}^{2}}\left[{{A}^{\mu }},{{A}^{\nu }}\right]\phi \\&\quad \quad \quad \quad \ \ =-ig{{\partial }^{[\mu }}{{A}^{\nu ]}}\phi -{{g}^{2}}\left[{{A}^{\mu }},{{A}^{\nu }}\right]\phi =-ig{{F}^{\mu \nu }}\phi \\\end{aligned}}.$

Given that $\left[{{A}^{\mu }},{{A}^{\nu }}\right]=\left[{{T}_{i}},{{T}_{j}}\right]{{A}_{i}}^{\mu }{{A}_{j}}^{\nu }=i{{f}_{ijk}}{{T}_{k}}{{A}_{i}}^{\mu }{{A}_{j}}^{\nu }\neq 0,$ this includes the relation

${{F}^{\mu \nu }}={{T}_{c}}{{F}_{c}}^{\mu \nu }={{\partial }^{[\mu }}{{A}^{\nu ]}}-ig\left[{{A}^{\mu }},{{A}^{\nu }}\right]={{T}_{c}}\left\{{{\partial }^{\mu }}A_{c}^{\nu }-{{\partial }^{\nu }}A_{c}^{\mu }+g{{f}_{abc}}{{A}_{a}}^{\mu }{{A}_{b}}^{\nu }\right\}$

which is further seen to contain:

${{F}_{a}}^{\mu \nu }={{\partial }^{\mu }}A_{a}^{\nu }-{{\partial }^{\nu }}A_{a}^{\mu }+g{{f}_{abc}}{{A}_{b}}^{\mu }{{A}_{c}}^{\nu }.$

3) The following existing material is important, but needs improvement:

A Bianchi identity holds

$(D_{\mu }F_{\nu \kappa })^{a}+(D_{\kappa }F_{\mu \nu })^{a}+(D_{\nu }F_{\kappa \mu })^{a}=0$

which is equivalent to the Jacobi identity

$[D_{\mu },[D_{\nu },D_{\kappa }]]+[D_{\kappa },[D_{\mu },D_{\nu }]]+[D_{\nu },[D_{\kappa },D_{\mu }]]=0$

since $[D_{\mu },F_{\nu \kappa }^{a}]=D_{\mu }F_{\nu \kappa }^{a}$.

Specifically, derivation of the identity $[D_{\mu },F_{\nu \kappa }^{a}]=D_{\mu }F_{\nu \kappa }^{a}$ which underlies other important identities (including the above magnetic monopole identity) should be explicitly shown. It would also be helful to connect together $g{{D}^{\sigma }}{{F}^{\mu \nu }}=g\left[{{D}^{\sigma }},{{F}^{\mu \nu }}\right]=i\hbar c\left[{{D}^{\sigma }},\left[{{D}^{\mu }},{{D}^{\nu }}\right]\right].$

4) It should also be pointed out that, as it is for the Electromagnetic tensor in Maxwell’s electrodynamics, the field strength trace is zero:

$F={{F}^{\mu }}_{\mu }={{\partial }^{\mu }}{{A}_{\mu }}-{{\partial }_{\mu }}{{A}^{\mu }}-ig\left[{{A}^{\mu }},{{A}_{\mu }}\right]=i\hbar c\left[{{D}^{\mu }},{{D}_{\mu }}\right]/g=0.$

5) Finally, continuity equations underlie charge conservation, and so are very important to include. Specifically, just as ${{\partial }_{\nu }}{{\partial }_{\mu }}{{F}^{\mu \nu }}=0$ is an identity central to charge conservation in Maxwell's equations, so too, ${{D}_{\nu }}{{D}_{\mu }}{{F}^{\mu \nu }}=0$ is an identity central to charge conservation in any Yang-Mills gauge theory. Accordingly, I respectfully suggest adding material to show the identity ${{D}_{\nu }}{{D}_{\mu }}{{F}^{\mu \nu }}=0,$ along the lines of what is below:

We may obtain a Yang-Mills continuity identity as follows: Start with the relations $i\hbar c\left[{{D}^{\mu }},{{D}^{\nu }}\right]/g={{F}^{\mu \nu }}$ and ${{D}^{\sigma }}{{F}^{\mu \nu }}=\left[{{D}^{\sigma }},{{F}^{\mu \nu }}\right]$, then write:

${\begin{aligned}&{{D}_{\nu }}{{D}_{\mu }}{{F}^{\mu \nu }}={{D}_{\nu }}\left[{{D}_{\mu }},{{F}^{\mu \nu }}\right]\\&=i\hbar c{{D}_{\nu }}\left[{{D}_{\mu }},\left[{{D}^{\mu }},{{D}^{\nu }}\right]\right]/g=i\hbar c\left[{{D}_{\nu }}{{D}_{\mu }}\left[{{D}^{\mu }},{{D}^{\nu }}\right]-{{D}_{\nu }}\left[{{D}^{\mu }},{{D}^{\nu }}\right]{{D}_{\mu }}\right]/g\\&={{D}_{\nu }}{{D}_{\mu }}{{F}^{\mu \nu }}-{{D}_{\nu }}{{F}^{\mu \nu }}{{D}_{\mu }}\\\end{aligned}},$

from which we deduce that ${{D}_{\nu }}{{F}^{\mu \nu }}{{D}_{\mu }}=0$ and thus ${{D}_{\mu }}{{F}^{\mu \nu }}{{D}_{\nu }}=0.$ The Jacobian identity $\left[{{D}_{\nu }},\left[{{D}_{\mu }},\left[{{D}^{\mu }},{{D}^{\nu }}\right]\right]\right]=0$ combined with the foregoing relations further implies that $\left[{{D}_{\nu }},\left[{{D}_{\mu }},{{F}^{\mu \nu }}\right]\right]=\left[{{D}_{\nu }},{{D}_{\mu }}{{F}^{\mu \nu }}\right]={{D}_{\nu }}{{D}_{\mu }}{{F}^{\mu \nu }}-{{D}_{\mu }}{{F}^{\mu \nu }}{{D}_{\nu }}=0$. Further combining with ${{D}_{\mu }}{{F}^{\mu \nu }}{{D}_{\nu }}=0$, we finally deduce that:

${{D}_{\nu }}{{D}_{\mu }}{{F}^{\mu \nu }}=0.$

This is the Yang-Mills generalization of the continuity identity ${{\partial }_{\nu }}{{\partial }_{\mu }}{{F}^{\mu \nu }}=0$ from electrodynamics.

Wiki Physics response

I will not respond to the negative personal remarks made about me above. What I will say is this: Yes, I am the author of this paper. I only just now, for the first time, as a result of the above, saw the 2016 Woit post which takes issue with a retracted Bell’s Theorem paper by a different individual who happens to be a friend and online colleague, but which post also mentions me. I also happen to be a friend and online colleague of the person who led the effort to have Annals of Physics retract that Bell paper, and have been asked by both, at times, to objectively mediate their scientific disputes. I participated extensively in the discussion of all this at Retraction Watch. And yes, I have a non-traditional background for someone involved in theoretical physics: I am an MIT alum, but I have made my living securing patents for a variety of scientists and engineers. Physics is a passion not a livelihood, so this frees me from the usual pressures imposed upon a professional physicist in academe (which is an advantage), but it also isolates me from the resources and contacts that a professional physicist can access (which is a disadvantage). Thankfully, I have become financially independent as a result of my patent work, so I can focus my efforts entirely on my physics work, with full independence and no material restraints. The only restraints I respect and esteem, are the physical truths of the natural world. And that is all I will say about my personal background because a) this is immaterial to the physics we all prefer to study and discuss and b) it is likely that not many people will give a darn.

The above from User:Tazerenix (to whom thanks and credit is due for contributing to many important physics articles on Wikipedia) is the first time I have seen a physicist refer to Maxwell’s equations as “esotertic [sic] mathematics.” Because at bottom, mathematically and physically, the referenced paper is really just about Maxwell’s equations, and how to systematically extend them to the non-commuting gauge fields of Yang-Mills (YM) theory. Just as we can invert Maxwell’s charge equations (with suitable gauge conditions and / or the introduction of a Proca mass) to obtain photon propagators, and use the current density and the continuity equation to identify Dirac wavefunctions for electrons, we need to meticulously follow all the same steps for the generalization of Maxwell’s equations to Yang Mills. Of course, the mathematics is more complicated, because it becomes highly non-linear (and for my friends with a computer science background, infinitely recursive).

In this regard, all I am really doing in the paper, and suggesting be appropriately exposited on Wikipedia, is expanding upon what is laid out in section 1 of Jaffee and Witten: You start with a field strength two-form $F=dA$ and the source-free Maxwell equations $0=d*F=dF$. And you note that with sources, the charge equation becomes $c{{\mu }_{0}}*j=d*F$. You then form the gauge-covariant extension of the exterior derivative ${{d}_{A}}=d-ieA$ and modify the field strength to ${{F}_{\text{YM}}}=dA+A\wedge A$ with non-commuting gauge fields, and extend Maxwell’s source-free equations to $0={{d}_{A}}*{{F}_{\text{YM}}}={{d}_{A}}{{F}_{\text{YM}}}$. And you note that with sources, the charge equation will become $c{{\mu }_{0}}*{{j}_{\text{YM}}}={{d}_{A}}*{{F}_{\text{YM}}}$. If you look at my sections 2-4, you will see that that is really all I have done there, but with the tensor representation of all this as well. Then you systematically do all the things to these new extension equations that you do to Maxwell’s: invert to find propagators, use a continuity equation to connect with fermions, and so on. And in the end, just by following the math, you find a lot of interesting things. If I wanted an alternative, more prosaic title for the paper, I might call it "Study of Maxwell's equations for non-commuting gauge fields."

In the upcoming days I will prepare a clear and concise proposed replacement for what I wrote earlier, in the form of an exposition of Jaffee and Witten’s section 1 along the lines of the preceding paragraph, using differential forms and tensors, and post it to the Yang-Mills theory talk page for feedback, before looking to do anything more.

To conclude, let me make one more thing clear: I always value substantive feedback. Experimental method is the foundation of modern science. Often, experimental results arrive in the form of other people of knowledge and goodwill pointing out conflicts with established theory or observations, in which case adjustment and change is mandatory. We should have it no other way. PatentPhysicist (talk) 01:58, 8 March 2021 (UTC)

Test of boxes

Name	Forms equations	Tensor equations
Electric charge equation	$A$	$B$
Magnetic charge equation	$C$	$D$
Continuity equation	$E$	$F$

Commuting versus non-commuting gauge fields

Studying the physics of Yang-Mills gauge theory requires understanding what happens to Maxwell’s electrodynamics, and U(1) quantum electrodynamics (QED), when Maxwell’s commuting (abelian) gauge fields ${{A}^{\mu }}$ become non-commuting (nonabelian) gauge fields ${{G}^{\mu }}={{\tau }_{i}}G_{i}^{\mu }$ covariantly transforming, for example, under the compact simple Yang-Mills gauge group SU(N) with NxN Hermitian generators ${{\tau }_{i}}={{\tau }_{i}}^{\dagger }$ and a commutator $\left[{{\tau }_{i}},{{\tau }_{j}}\right]=i{{f}_{ijk}}{{\tau }_{k}}$ typically normalized such that ${\text{tr}}\left({{\tau }_{i}}^{2}\right)={\tfrac {1}{2}}$ for each $i=1...{{N}^{2}}-1$. Whereas electrodynamics is a linear theory in which the gauge fields do not interact with one another, as noted in the above mathematical overview and vertex diagrams, Yang-Mills theory is highly nonlinear with mutual interactions amongst the gauge fields.

Commuting (abelian) gauge fields

In flat spacetime, in classical electrodynamics, a gauge-invariant field strength ${{F}^{\mu \nu }}$ is related to the gauge fields ${{A}^{\mu }}$ by ${{F}^{\mu \nu }}={{\partial }^{\mu }}{{A}^{\nu }}-{{\partial }^{\nu }}{{A}^{\mu }}$. Using the gauge-covariant derivative ${{D}^{\mu }}={{\partial }^{\mu }}-ie{{A}^{\mu }}/\hbar c$, this may, however, be written more generally as

${{F}^{\mu \nu }}={{D}^{\mu }}{{A}^{\nu }}-{{D}^{\nu }}{{A}^{\mu }}={{\partial }^{\mu }}{{A}^{\nu }}-{{\partial }^{\nu }}{{A}^{\mu }}-ie\left[{{A}^{\mu }},{{A}^{\nu }}\right]={{\partial }^{\mu }}{{A}^{\nu }}-{{\partial }^{\nu }}{{A}^{\mu }},$

because the commutator for the commuting gauge fields is $\left[{{A}^{\mu }},{{A}^{\nu }}\right]=0$.

With ${{c}^{2}}{{\mu }_{0}}{{\varepsilon }_{0}}=1$ and Coulomb constant ${{k}_{\text{e}}}=1/4\pi {{\varepsilon }_{0}}$, the classical Maxwell field equation for electric charge density is:

$c{{\mu }_{0}}{{j}^{\nu }}={{\partial }_{\mu }}{{F}^{\mu \nu }}=\left({{g}^{\mu \nu }}{{\partial }_{\sigma }}{{\partial }^{\sigma }}-{{\partial }^{\nu }}{{\partial }^{\mu }}\right){{A}_{\mu }},$

which spacetime-covariantly includes Gauss’ electricity and Ampere’s current laws. The classical field equation for magnetic charge density is

$c{{\mu }_{0}}{{p}^{\sigma \mu \nu }}={{\partial }^{\sigma }}{{F}^{\mu \nu }}+{{\partial }^{\mu }}{{F}^{\nu \sigma }}+{{\partial }^{\nu }}{{F}^{\sigma \mu }}=0,$

which spacetime-covariantly includes Gauss’ magnetism and Faraday’s induction laws. See Maxwell's equations in covariant form. The zero in the monopole equation and thus the non-existence of magnetic monopoles (setting aside possible Dirac charge quantization, see also https://encyclopediaofmath.org/wiki/Dirac_monopole) arises from the flat spacetime commutator of ordinary derivatives being $\left[{{\partial }_{\mu }},{{\partial }_{\nu }}\right]=0$. And these monopoles continue to be zero in curved spacetime, because of the first Bianchi identity, ${{R}_{\alpha \left[\sigma \mu \nu \right]}}=0$, of the Riemann curvature tensor. In integral form, the Gauss's law for magnetism component of the above becomes $\;\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$, whereby there is no net flux of magnetic fields across closed spatial surfaces. (Note: The point of various “bag models” of QCD quark confinement, is that there is similarly no net flux of color charge across the closed spatial surfaces of color-neutral baryons, see, e.g., section 18.3 of .)

Summing the four-gradient ${{\partial }_{\nu }}$ with the electric charge strength above to obtain the four-dimensional spacetime divergence, we readily obtain:

$c{{\mu }_{0}}{{\partial }_{\nu }}{{j}^{\nu }}={{\partial }_{\nu }}{{\partial }_{\mu }}{{F}^{\mu \nu }}={{\partial }_{\nu }}{{\partial }_{\sigma }}{{\partial }^{\sigma }}{{A}^{\nu }}-{{\partial }_{\nu }}{{\partial }_{\sigma }}{{\partial }^{\nu }}{{A}^{\sigma }}=0,$

which is the continuity equation governing the conservation of electric charge. This becomes zero, once again, because of flat spacetime commutator $\left[{{\partial }_{\mu }},{{\partial }_{\nu }}\right]=0$.

In QED, the Dirac current charge density becomes related to the Dirac wavefunctions $\psi$ for individual fermions by ${{j}^{\nu }}=e{\overline {\psi }}Q{{\gamma }^{\nu }}\psi$ where $e$ is the electric charge strength related to the running "fine structure" coupling ${{\alpha }_{e}}\left(\mu =0\right)=1/137.036...$ by ${{k}_{\text{e}}}{{e}^{2}}=\hbar c{{\alpha }_{e}}$, and $Q=-1,+{\tfrac {2}{3}},-{\tfrac {1}{3}}$ for the electron, up and down fermions, and their higher-generational counterparts. Meanwhile the propagators for the individual photons which form the gauge fields are obtained by inverting the electric charge equation and converting from configuration into momentum space using the substitution $i\hbar {{\partial }^{\mu }}\to {{q}^{\mu }}$ and the $+i\varepsilon$ prescription. Because the charge equation is not invertible without taking some further steps, it is customary to utilize the gauge condition ${{\partial }_{\sigma }}{{A}^{\sigma }}=0$ to obtain

${{A}_{\alpha }}={{\hbar }^{2}}c{{\mu }_{0}}{\frac {-{{g}_{\alpha \nu }}}{{{q}_{\sigma }}{{q}^{\sigma }}+i\varepsilon }}{{j}^{\nu }}$

which includes the photon propagator up to a factor of $i$. Alternatively, one can introduce a Proca mass by hand into the charge equation, which then becomes $c{{\mu }_{0}}{{j}^{\nu }}=\left({{g}^{\mu \nu }}\left({{\partial }_{\sigma }}{{\partial }^{\sigma }}+{{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right)-{{\partial }^{\mu }}{{\partial }^{\nu }}\right){{A}_{\mu }},$ see Proca field equation. Then, ${{\partial }_{\sigma }}{{A}^{\sigma }}=0$ is no longer a gauge condition but a requirement to maintain continuity (charge conservation), and with $i\hbar {{\partial }^{\mu }}\to {{k}^{\mu }}$ we arrive at the inverse:

${{A}_{\alpha }}={{\hbar }^{2}}c{{\mu }_{0}}{\frac {-{{g}_{\alpha \nu }}+{{k}_{\nu }}{{k}_{\alpha }}/{{m}^{2}}{{c}^{2}}}{{{k}_{\sigma }}{{k}^{\sigma }}-{{m}^{2}}{{c}^{2}}+i\varepsilon }}{{j}^{\nu }}$

which includes a massive vector boson propagator up to $i$. Of course, adding a mass by hand destroys renormalizability, so it is necessary to find a way that this can be restored. The Higgs mechanism used for the Electroweak interaction is best-known example of how to obtain a non-zero vector boson mass without sacrificing renormalizability.

Non-commuting (nonabelian) gauge fields

In Yang-Mills (YM) Gauge Theory, as distinct from a gauge theory with commuting gauge fields, ${{A}^{\mu }}\to {{G}^{\mu }}={{\tau }_{i}}G_{i}^{\mu }$ becomes a non-commuting gauge field, $\left[{{G}^{\mu }},{{G}^{\nu }}\right]=\left[{{\tau }_{i}},{{\tau }_{j}}\right]{{G}_{i}}^{\mu }{{G}_{j}}^{\nu }=i{{f}_{ijk}}{{\tau }_{k}}{{G}_{i}}^{\mu }{{G}_{j}}^{\nu }\neq 0.$ The gauge-covariant derivative is now denoted by ${{D}^{\mu }}={{\partial }^{\mu }}-ig{{G}^{\mu }}/\hbar c$, and the field strength therefore graduates to the gauge-covariant, not gauge-invariant:

${{F}_{\text{YM}}}^{\mu \nu }={{\tau }_{k}}{{F}_{k}}^{\mu \nu }={{D}^{\mu }}{{G}^{\nu }}-{{D}^{\nu }}{{G}^{\mu }}={{\partial }^{\mu }}{{G}^{\nu }}-{{\partial }^{\nu }}{{G}^{\mu }}-ig\left[{{G}^{\mu }},{{G}^{\nu }}\right]={{\tau }_{k}}\left\{{{\partial }^{\mu }}G_{k}^{\nu }-{{\partial }^{\nu }}G_{k}^{\mu }+g{{f}_{ijk}}{{G}_{i}}^{\mu }{{G}_{j}}^{\nu }\right\},$

which is an NxN Hermitian matrix for SU(N). With ${{A}^{\mu }}$ replaced by ${{G}^{\mu }}$, it will be seen that this contains the equation $F_{\mu \nu }^{a}=\partial _{\mu }G_{\nu }^{a}-\partial _{\nu }G_{\mu }^{a}+gf^{abc}G_{\mu }^{b}G_{\nu }^{c}$ from the mathematical overview above. Using differential forms, as pointed out at at pages 1 and 2, this is equivalently represented by the fact that "the curvature arising from the connection must be modified to $F=dG+G\wedge G$." The non-linearity of Yang-Mills gauge theories becomes apparent if one uses the above to advance the source-free Lagrangian from the mathematical overview to:

${\mathcal {L}}=-{\tfrac {1}{2}}{\text{Tr}}\left({{F}_{\text{YM}}}^{\mu \nu }{{F}_{\text{YM}}}_{\mu \nu }\right)=-{\tfrac {1}{4}}{{F}_{i}}^{\mu \nu }{{F}_{i}}_{\mu \nu }=-{\tfrac {1}{4}}{{\partial }^{[\mu }}{{G}_{i}}^{\nu ]}{{\partial }_{[\mu }}{{G}_{i}}_{\nu ]}-{\tfrac {1}{2}}g{{f}_{ijk}}{{\partial }^{[\mu }}{{G}_{i}}^{\nu ]}{{G}_{j}}_{\mu }{{G}_{k}}_{\nu }-{\tfrac {1}{4}}{{g}^{2}}{{f}_{ijk}}{{f}_{ilm}}{{G}_{j}}^{\mu }{{G}_{k}}^{\nu }{{G}_{l}}_{\mu }{{G}_{m}}_{\nu },$

which includes the three- and four-gauge boson interaction vertices illustrated in the above section on quantization.

Yang-Mills gauge theory differs from the abelian gauge theory of U(1) electrodynamics, by the mathematical and physical consequences of what happens when the gauge fields go from commuting to non-commuting in this way.

Canonic versus dynamic Yang-Mills field and continuity equations

In Yang-Mills gauge theory, the field equations which generalize Maxwell’s equations for electrodynamics may be cast in one of two interrelated forms: canonic, and dynamic.

Canonic field equation

In canonic form, one starts with the two spacetime-covariant Maxwell equations $c{{\mu }_{0}}{{j}^{\nu }}={{\partial }_{\mu }}{{F}^{\mu \nu }}$ and $c{{\mu }_{0}}{{p}^{\sigma \mu \nu }}={{\partial }^{\sigma }}{{F}^{\mu \nu }}+{{\partial }^{\mu }}{{F}^{\nu \sigma }}+{{\partial }^{\nu }}{{F}^{\sigma \mu }}=0$ reviewed in the last section, uses the non-abelian field strength ${{F}_{\text{YM}}}^{\mu \nu }={{D}^{\mu }}{{G}^{\nu }}-{{D}^{\nu }}{{G}^{\mu }}={{\partial }^{\mu }}{{G}^{\nu }}-{{\partial }^{\nu }}{{G}^{\mu }}-ig\left[{{G}^{\mu }},{{G}^{\nu }}\right]$ with non-commuting $\left[{{G}^{\mu }},{{G}^{\nu }}\right]\neq 0$ rather than $\left[{{A}^{\mu }},{{A}^{\nu }}\right]=0$ as also just reviewed, and in addition, advances all the remaining ordinary derivatives to gauge-covariant derivatives, ${{\partial }^{\mu }}\to {{D}^{\mu }}={{\partial }^{\mu }}-ig{{G}^{\mu }}/\hbar c$. It is also helpful to use the uppercase notation ${{j}^{\nu }}\to {{J}^{\nu }}$ and ${{p}^{\sigma \mu \nu }}\to {{P}^{\sigma \mu \nu }}$ to denote the electric and magnetic charge densities in the canonic equations, retaining the lowercase notation for the dynamic form to be reviewed below. With the foregoing, the Yang-Mills extensions of Maxwell’s electric and magnetic equations, in canonic form, are as follows:

$c{{\mu }_{0}}{{J}^{\nu }}={{D}_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }=\left({{g}^{\mu \nu }}{{D}_{\sigma }}{{D}^{\sigma }}-{{D}^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}$

$c{{\mu }_{0}}{{P}^{\sigma \mu \nu }}={{D}^{\sigma }}{{F}_{\text{YM}}}^{\mu \nu }+{{D}^{\mu }}{{F}_{\text{YM}}}^{\nu \sigma }+{{D}^{\nu }}{{F}_{\text{YM}}}^{\sigma \mu }=0$

The Yang-Mills canonic magnetic charge density, although generalized above, remains equal to zero just like the magnetic charge density in Maxwell’s electrodynamics. This is no longer because of the flat spacetime commutator $\left[{{\partial }_{\mu }},{{\partial }_{\nu }}\right]=0$, but rather because of the Jacobi identity $\left[{{D}_{\sigma }},\left[{{D}_{\mu }},{{D}_{\nu }}\right]\right]+\left[{{D}_{\mu }},\left[{{D}_{\nu }},{{D}_{\sigma }}\right]\right]+\left[{{D}_{\nu }},\left[{{D}_{\sigma }},{{D}_{\mu }}\right]\right]=0$ combined with the further identities $i\hbar c\left[{{D}^{\sigma }},\left[{{D}^{\mu }},{{D}^{\nu }}\right]\right]=g\left[{{D}^{\sigma }},{{F}_{\text{YM}}}^{\mu \nu }\right]=g{{D}^{\sigma }}{{F}^{\text{YM}}}^{\mu \nu },$ all reviewed in the Mathematical overview.

Canonic continuity equation

Applying ${{D}_{\nu }}$ to the canonic charge density above, the identity ${{D}_{\nu }}{{D}_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }=0$ reviewed in the Mathematical overview enables us to calculate the Yang-Mills canonic continuity relation:

${\begin{aligned}&c{{\mu }_{0}}{{D}_{\nu }}{{J}^{\nu }}={{D}_{\nu }}{{D}_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }=\left({{g}^{\mu \nu }}{{D}_{\nu }}{{D}_{\sigma }}{{D}^{\sigma }}-{{D}_{\nu }}{{D}^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}\\&={{\partial }_{\nu }}{{\partial }_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }-\left(ig\left({{G}_{\nu }}{{\partial }_{\mu }}+{{\partial }_{\nu }}{{G}_{\mu }}\right)+{{g}^{2}}{{G}_{\nu }}{{G}_{\mu }}\right){{F}_{\text{YM}}}^{\mu \nu }=\left({{\partial }_{\nu }}{{\partial }_{\mu }}-{{V}_{\nu \mu }}\right){{F}_{\text{YM}}}^{\mu \nu }=0\\\end{aligned}},$

which includes a perturbation tensor defined by:

${{V}_{\mu \nu }}\equiv ig\left({{G}_{\mu }}{{\partial }_{\nu }}+{{\partial }_{\mu }}{{G}_{\nu }}\right)+{{g}^{2}}{{G}_{\mu }}{{G}_{\nu }}.$

The trace $V={{V}^{\sigma }}_{\sigma }=ig\left({{G}^{\sigma }}{{\partial }_{\sigma }}+{{\partial }^{\sigma }}{{G}_{\sigma }}\right)+{{g}^{2}}{{G}^{\sigma }}{{G}_{\sigma }}$ of the above is the standard expression for the perturbation in the Klein–Gordon (relativistic Schrödinger) equation. This starts out in the free-particle form $\left({{\partial }_{\sigma }}{{\partial }^{\sigma }}+{{m}^{\sigma }}\right)\phi =0$, uses gauge-covariant derivatives to promote to the canonic form $\left({{D}_{\sigma }}{{D}^{\sigma }}+{{m}^{\sigma }}\right)\phi =0$ thereby introducing interactions of the particle with gauge fields, and is then readily restructured into the dynamic form $\left({{\partial }_{\sigma }}{{\partial }^{\sigma }}+{{m}^{\sigma }}\right)\phi =V\phi$ to show the observed particle behavior, no longer free because of its gauge field interactions.

Dynamic field equation

In dynamic form, one still begins with Maxwell’s electrodynamic equations $c{{\mu }_{0}}{{j}^{\nu }}={{\partial }_{\mu }}{{F}^{\mu \nu }}$ and $c{{\mu }_{0}}{{p}^{\sigma \mu \nu }}={{\partial }^{\sigma }}{{F}^{\mu \nu }}+{{\partial }^{\mu }}{{F}^{\nu \sigma }}+{{\partial }^{\nu }}{{F}^{\sigma \mu }}=0$ reviewed above, and still uses the non-commuting field strength ${{F}_{\text{YM}}}^{\mu \nu }={{D}^{\mu }}{{G}^{\nu }}-{{D}^{\nu }}{{G}^{\mu }}={{\partial }^{\mu }}{{G}^{\nu }}-{{\partial }^{\nu }}{{G}^{\mu }}-ig\left[{{G}^{\mu }},{{G}^{\nu }}\right]$ with $\left[{{G}^{\mu }},{{G}^{\nu }}\right]\neq 0$, but does nothing further. That is, one keeps the remaining derivatives ordinary and keeps the source density notation in lowercase. Consequently, in dynamic form, the Yang-Mills generalization of Maxwell's electric charge equation (Gauss and Ampere) is:

${\begin{aligned}&c{{\mu }_{0}}{{j}^{\nu }}={{\partial }_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }=\left({{g}^{\mu \nu }}{{\partial }_{\sigma }}{{D}^{\sigma }}-{{\partial }^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}\\&\quad \quad \ \ =\left({{g}^{\mu \nu }}\left({{\partial }_{\sigma }}{{\partial }^{\sigma }}-ig{{\partial }_{\sigma }}{{G}^{\sigma }}/\hbar c\right)-\left({{\partial }^{\mu }}{{\partial }^{\nu }}-ig{{\partial }^{\mu }}{{G}^{\nu }}/\hbar c\right)\right){{G}_{\mu }}\\\end{aligned}}$

while the dynamic Yang-Mills generalization of Maxwell's magnetic charge equation (Gauss and Faraday) is:

${\begin{aligned}&c{{\mu }_{0}}{{p}^{\sigma \mu \nu }}={{\partial }^{\sigma }}{{F}_{\text{YM}}}^{\mu \nu }+{{\partial }^{\mu }}{{F}_{\text{YM}}}^{\nu \sigma }+{{\partial }^{\nu }}{{F}_{\text{YM}}}^{\sigma \mu }\\&\quad \quad \quad \ \ =-ig\left({{\partial }^{\sigma }}\left[{{G}^{\mu }},{{G}^{\nu }}\right]+{{\partial }^{\mu }}\left[{{G}^{\nu }},{{G}^{\sigma }}\right]+{{\partial }^{\nu }}\left[{{G}^{\sigma }},{{G}^{\mu }}\right]\right)\neq 0\\\end{aligned}}$

These are simply Maxwell's equations without change, aside from the promotion of ${{F}^{\mu \nu }}={{D}^{\mu }}{{A}^{\nu }}-{{D}^{\nu }}{{A}^{\mu }}$ with $\left[{{A}^{\mu }},{{A}^{\nu }}\right]=0$, to ${{F}_{\text{YM}}}^{\mu \nu }={{D}^{\mu }}{{G}^{\nu }}-{{D}^{\nu }}{{G}^{\mu }}={{\partial }^{\mu }}{{G}^{\nu }}-{{\partial }^{\nu }}{{G}^{\mu }}-ig\left[{{G}^{\mu }},{{G}^{\nu }}\right]$ with $\left[{{G}^{\mu }},{{G}^{\nu }}\right]\neq 0$. That is, these are Maxwell’s equations for non-commuting gauge fields, with nothing else changed. However, the identity which causes the uppercase-denoted magnetic source density to vanish in the canonic equation, ${{P}^{\sigma \mu \nu }}=0$, does not operate to vanish the magnetic source density to vanish from the dynamic equation, ${{p}^{\sigma \mu \nu }}\neq 0$. Instead, using the “zero” from the canonic magnetic charge equation, we are able to calculate in the above that the dynamic ${{p}^{\sigma \mu \nu }}\neq 0$ differs from zero in proportion to the index-cyclic derivatives ${{\partial }^{\sigma }}\left[{{G}^{\mu }},{{G}^{\nu }}\right]+{{\partial }^{\mu }}\left[{{G}^{\nu }},{{G}^{\sigma }}\right]+{{\partial }^{\nu }}\left[{{G}^{\sigma }},{{G}^{\mu }}\right]$ of the non-zero Yang-Mills gauge field commutator. The magnetic monopoles of electrodynamics (again, aside from perhaps Dirac's monopoles), are vanishing for the precise reason that the gauge fields of Maxwell's electrodynamics are commuting, $\left[{{A}^{\mu }},{{A}^{\nu }}\right]=0$ (and in curved spacetime, additionally because of the first Bianchi identity).

Dynamic continuity equation

To obtain the dynamic Yang-Mills continuity equation, we apply the ordinary derivative to the dynamic Yang-Mills electric charge equation above. Using the zero from the above canonic continuity equation, it is straightforward to find that:

$c{{\mu }_{0}}{{\partial }_{\nu }}{{j}^{\nu }}={{\partial }_{\nu }}{{\partial }_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }={{V}_{\nu \mu }}{{F}_{\text{YM}}}^{\mu \nu }\neq 0$

This dynamic continuity equation is also not equal to zero. Rather, this differs from zero by the double-contracted product ${{V}_{\nu \mu }}{{F}_{\text{YM}}}^{\mu \nu }$ of the perturbation tensor with the Yang-Mills field strength tensor.

Free, canonic, and dynamic field and motion equations in physics

As noted above, the Klein Gordon equation begins as an equation for a free particle, is promoted to a canonic equation by maintaining its form while simply replacing ordinary with gauge-covariant derivatives, and is finally reformulated as a dynamic equation by segregating out the original form of the free equation containing only ordinary derivatives. This promotion of ordinary to covariant derivatives while otherwise maintaining the original form of a "free" equation is a common technique in modern physics: Another example is the free Dirac equation $\left(i{{\gamma }^{\sigma }}{{\partial }_{\sigma }}-m\right)\psi =0$ which is promoted to the canonic form $\left(i{{\gamma }^{\sigma }}{{D}_{\sigma }}-m\right)\psi =0$, then rendered dynamic by isolating the original equation in the form of $\left(i{{\gamma }^{\sigma }}{{\partial }_{\sigma }}-m\right)\psi =-{{\gamma }^{0}}{{V}_{D}}\psi$ where ${{V}_{D}}\equiv g{{\gamma }^{0}}{{\gamma }^{\sigma }}{{G}_{\sigma }}$ is defined as the Dirac perturbation. A final example is Newton’s first law for straight-line inertial motion written as $d{{u}^{\alpha }}/d\tau =0$ with a four velocity ${{u}^{\alpha }}=d{{x}^{\alpha }}/d\tau$, which uses the gravitationally-covariant derivative ${{\partial }_{;\nu }}{{u}^{\mu }}={{\partial }_{\nu }}{{u}^{\mu }}+{{\Gamma }^{\mu }}_{\alpha \nu }{{u}^{\alpha }}$ to promote to the canonic form $D{{u}^{\mu }}/D\tau ={{\partial }_{;\nu }}{{u}^{\mu }}{{\mu }^{\nu }}=d{{u}^{\mu }}/d\tau +{{\Gamma }^{\mu }}_{\alpha \nu }{{u}^{\alpha }}{{u}^{\nu }}=0$ of the derivative along a curve. Then, by isolating the original Newtonian derivative, we obtain the dynamic equation $d{{u}^{\mu }}/d\tau =-{{\Gamma }^{\mu }}_{\alpha \nu }{{u}^{\alpha }}{{u}^{\nu }}$, which is the equation for gravitational geodesic motion along “straight lines” in the curved spacetime of general relativity. In all cases, it is the dynamic equations which, via the explicit showing of ${{\partial }_{\nu }}=\left(\partial /c\partial t,\nabla \right)$ and the original free equations, directly describe the observed physical evolution through time and space brought about by interactions with external fields. This section, most simply stated, has shown how to obtain the dynamic forms of Maxwell's equations and the continuity (conservation) equation in the circumstance where the commuting gauge fields $\left[{{A}^{\mu }},{{A}^{\nu }}\right]=0$ of U(1) electrodynamics reviewed in the last section are replaced by non-commuting gauge fields $\left[{{G}^{\mu }},{{G}^{\nu }}\right]\neq 0$ for -- without restriction -- "any compact simple gauge group G" of Yang-Mills theory.

Spacetime-invariant Yang-Mills gauge condition

In order to analytically calculate quantum propagators for the Yang-Mills gauge fields ${{G}^{\mu }}={{\tau }_{i}}G_{i}^{\mu },$ it is necessary to calculate the inverse of the dynamic Yang-Mills electric charge equation reviewed here. However, as with Maxwell’s classical electric charge equation reviewed here, this dynamic Yang-Mills charge equation is not invertible without taking affirmative steps to make it so.

Accordingly, we first introduce a nonzero Proca mass $m>0$ by hand for the gauge boson in the canonic electric charge equation reviewed here, by which this equation advances to:

$c{{\mu }_{0}}{{J}^{\nu }}=\left({{g}^{\mu \nu }}\left({{D}_{\sigma }}{{D}^{\sigma }}+{{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right)-{{D}^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}.$

The canonic continuity equation reviewed here therefore becomes, in pertinent part:

$c{{\mu }_{0}}{{D}_{\nu }}{{J}^{\nu }}=\left({{g}^{\mu \nu }}{{D}_{\nu }}\left({{D}_{\sigma }}{{D}^{\sigma }}+{{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right)-{{D}_{\nu }}{{D}^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}=0+\left({{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right){{D}_{\mu }}{{G}^{\mu }},$

with the 0 arising from the mathematical identity ${{D}_{\nu }}{{D}_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }=0$ reviewed in the Mathematical overview and used in the canonic continuity equation.

In the above, to ensure that ${{D}_{\nu }}{{J}^{\nu }}=0$ and so maintain continuity for the sources, and given that ${{D}_{\mu }}={{\partial }_{\mu }}-ig{{G}_{\mu }}/\hbar c,$ it follows that a gauge condition required for continuity is now:

${{D}_{\sigma }}{{G}^{\sigma }}={{\partial }_{\sigma }}{{G}^{\sigma }}-ig{{G}_{\sigma }}{{G}^{\sigma }}/\hbar c=0\quad {\text{i}}{\text{.e}}{\text{.}}\quad {{\partial }_{\sigma }}{{G}^{\sigma }}=ig{{G}_{\sigma }}{{G}^{\sigma }}/\hbar c.$

The above is a spacetime-invariant (and by implication Lorentz-invariant) Yang-Mills gauge condition which removes one of the four spacetime degrees of freedom from each $G_{i}^{\mu }$ in ${{G}^{\mu }}={{\tau }_{i}}G_{i}^{\mu },$ and so gives these gauge fields the polarization attributes of massive vector bosons. This is a canonic Yang-Mills extension of the spacetime-scalar gauge condition ${{\partial }_{\mu }}{{A}^{\mu }}=0$ required to maintain continuity for the Proca field equation of classical electrodynamics, which in classical electrodynamics results from the identity ${{\partial }_{\nu }}{{\partial }_{\mu }}{{F}^{\mu \nu }}=0$ coupled with the continuity requirement ${{\partial }_{\nu }}{{j}^{\nu }}=0$ for sources interacting with the massive vector boson.

The above also implies that ${{\partial }_{\sigma }}{{D}^{\sigma }}={{\partial }_{\sigma }}{{\partial }^{\sigma }}-ig{{\partial }_{\sigma }}{{G}^{\sigma }}/\hbar c={{\partial }_{\sigma }}{{\partial }^{\sigma }}+{{g}^{2}}{{G}_{\sigma }}{{G}^{\sigma }}/{{\hbar }^{2}}{{c}^{2}},$ which enables the dynamic Yang-Mills electric charge equation to be rewritten to include both the Proca mass and the above continuity condition which that mass necessitates, as:

${\begin{aligned}&c{{\mu }_{0}}{{j}^{\nu }}={{\partial }_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }=\left({{g}^{\mu \nu }}\left({{\partial }_{\sigma }}{{D}^{\sigma }}+{{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right)-{{\partial }^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}\\&\quad \quad \ \ =\left({{g}^{\mu \nu }}\left({{\partial }_{\sigma }}{{\partial }^{\sigma }}+{{g}^{2}}{{G}_{\sigma }}{{G}^{\sigma }}/{{\hbar }^{2}}{{c}^{2}}+{{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right)-{{\partial }^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}\\\end{aligned}}$

Because of the Proca mass, the above loses its renormalizability, which must be restored. However, because of the gauge condition ${{D}_{\sigma }}{{G}^{\sigma }}=0$ necessitated by this Proca mass, the above also naturally acquires a term ${{g}^{2}}{{G}_{\sigma }}{{G}^{\sigma }}$ which is identical to the term in the Lagrangian ${\mathcal {L}}={{D}_{\sigma }}*\phi *{{D}^{\sigma }}\phi ={{\partial }_{\sigma }}\phi *{{\partial }^{\sigma }}\phi +{{g}^{2}}{{G}_{\sigma }}{{G}^{\sigma }}\phi *\phi$ through which renormalizable gauge bosons masses are introduced by weak and electroweak spontaneous symmetry breaking.

Spacetime-invariant Yang-Mills gauge condition

In order to analytically calculate quantum propagators for the Yang-Mills gauge fields ${{G}^{\mu }}={{\tau }_{i}}G_{i}^{\mu },$ it is necessary to calculate the inverse of the dynamic Yang-Mills electric charge equation reviewed here. However, as with Maxwell’s classical electric charge equation reviewed here, this dynamic Yang-Mills charge equation is not invertible without taking affirmative steps to make it so.

Accordingly, we first introduce a nonzero Proca mass $m>0$ by hand for the gauge boson in the canonic electric charge equation reviewed here, by which this equation advances to:

$c{{\mu }_{0}}{{J}^{\nu }}=\left({{g}^{\mu \nu }}\left({{D}_{\sigma }}{{D}^{\sigma }}+{{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right)-{{D}^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}.$

The canonic continuity equation reviewed here therefore becomes, in pertinent part:

$c{{\mu }_{0}}{{D}_{\nu }}{{J}^{\nu }}=\left({{g}^{\mu \nu }}{{D}_{\nu }}\left({{D}_{\sigma }}{{D}^{\sigma }}+{{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right)-{{D}_{\nu }}{{D}^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}=0+\left({{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right){{D}_{\mu }}{{G}^{\mu }},$

with the 0 arising from the mathematical identity ${{D}_{\nu }}{{D}_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }=0$ reviewed in the Mathematical overview and used in the canonic continuity equation.

In the above, to ensure that ${{D}_{\nu }}{{J}^{\nu }}=0$ and so maintain continuity for the sources, and given that ${{D}_{\mu }}={{\partial }_{\mu }}-ig{{G}_{\mu }}/\hbar c,$ it follows that a gauge condition required for continuity is now:

${{D}_{\sigma }}{{G}^{\sigma }}={{\partial }_{\sigma }}{{G}^{\sigma }}-ig{{G}_{\sigma }}{{G}^{\sigma }}/\hbar c=0\quad {\text{i}}{\text{.e}}{\text{.}}\quad {{\partial }_{\sigma }}{{G}^{\sigma }}=ig{{G}_{\sigma }}{{G}^{\sigma }}/\hbar c.$

The above is a spacetime-invariant (and by implication Lorentz-invariant) Yang-Mills gauge condition which removes one of the four spacetime degrees of freedom from each $G_{i}^{\mu }$ in ${{G}^{\mu }}={{\tau }_{i}}G_{i}^{\mu },$ and so gives these gauge fields the polarization attributes of massive vector bosons. This is a canonic Yang-Mills extension of the spacetime-scalar gauge condition ${{\partial }_{\mu }}{{A}^{\mu }}=0$ required to maintain continuity for the Proca field equation of classical electrodynamics, which in classical electrodynamics results from the identity ${{\partial }_{\nu }}{{\partial }_{\mu }}{{F}^{\mu \nu }}=0$ coupled with the continuity requirement ${{\partial }_{\nu }}{{j}^{\nu }}=0$ for sources interacting with the massive vector boson.

The above also implies that ${{\partial }_{\sigma }}{{D}^{\sigma }}={{\partial }_{\sigma }}{{\partial }^{\sigma }}-ig{{\partial }_{\sigma }}{{G}^{\sigma }}/\hbar c={{\partial }_{\sigma }}{{\partial }^{\sigma }}+{{g}^{2}}{{G}_{\sigma }}{{G}^{\sigma }}/{{\hbar }^{2}}{{c}^{2}},$ which enables us to rewrite the dynamic Yang-Mills electric charge equation to include both the Proca mass and the above continuity condition which that mass necessitates, as:

${\begin{aligned}&c{{\mu }_{0}}{{j}^{\nu }}={{\partial }_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }=\left({{g}^{\mu \nu }}\left({{\partial }_{\sigma }}{{D}^{\sigma }}+{{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right)-{{\partial }^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}\\&\quad \quad \ \ =\left({{g}^{\mu \nu }}\left({{\partial }_{\sigma }}{{\partial }^{\sigma }}+{{g}^{2}}{{G}_{\sigma }}{{G}^{\sigma }}/{{\hbar }^{2}}{{c}^{2}}+{{m}^{2}}{{c}^{2}}/{{\hbar }^{2}}\right)-{{\partial }^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}\\\end{aligned}}$

Because of the Proca mass, the above loses its renormalizability, which must be restored. However, because of the gauge condition ${{D}_{\sigma }}{{G}^{\sigma }}=0$ necessitated by this Proca mass, the above also naturally acquires a term ${{g}^{2}}{{G}_{\sigma }}{{G}^{\sigma }}$ which is identical to the term in the Lagrangian ${\mathcal {L}}={{D}_{\sigma }}*\phi *{{D}^{\sigma }}\phi ={{\partial }_{\sigma }}\phi *{{\partial }^{\sigma }}\phi +{{g}^{2}}{{G}_{\sigma }}{{G}^{\sigma }}\phi *\phi$ through which renormalizable gauge bosons masses are introduced by weak and electroweak spontaneous symmetry breaking.

Canonic versus dynamic Yang-Mills field equations

In Yang-Mills gauge theory, the field equations which generalize Maxwell’s equations for electrodynamic may be cast in one of two interrelated forms: canonic, and dynamic.

In canonic form, one starts with the two spacetime-covariant equations $c{{\mu }_{0}}{{j}^{\nu }}={{\partial }_{\mu }}{{F}^{\mu \nu }}$ and $c{{\mu }_{0}}{{p}^{\sigma \mu \nu }}={{\partial }^{\sigma }}{{F}^{\mu \nu }}+{{\partial }^{\mu }}{{F}^{\nu \sigma }}+{{\partial }^{\nu }}{{F}^{\sigma \mu }}=0$ reviewed in the last section, uses the non-abelian field strength ${{F}_{\text{YM}}}^{\mu \nu }={{D}^{\mu }}{{G}^{\nu }}-{{D}^{\nu }}{{G}^{\mu }}={{\partial }^{\mu }}{{G}^{\nu }}-{{\partial }^{\nu }}{{G}^{\mu }}-ig\left[{{G}^{\mu }},{{G}^{\nu }}\right]$ with non-commuting $\left[{{G}^{\mu }},{{G}^{\nu }}\right]\neq 0$ rather than $\left[{{A}^{\mu }},{{A}^{\nu }}\right]=0$ as also just reviewed, and in addition, advances all of the remaining ordinary derivatives to gauge-covariant derivatives, ${{\partial }^{\mu }}\to {{D}^{\mu }}={{\partial }^{\mu }}-ig{{G}^{\mu }}/\hbar c$. It is also helpful to use the uppercase notation ${{j}^{\nu }}\to {{J}^{\nu }}$ and ${{p}^{\sigma \mu \nu }}\to {{P}^{\sigma \mu \nu }}$ to denote for the electric and magnetic charge densities in the canonic equations, retaining the lowercase notation for the dynamic form to be reviewed below. With the foregoing, Maxwell’s Yang-Mills “electric” and “magnetic” equations in canonic form are as follows:

$c{{\mu }_{0}}{{J}^{\nu }}={{D}_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }=\left({{g}^{\mu \nu }}{{D}_{\sigma }}{{D}^{\sigma }}-{{D}^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}$

$c{{\mu }_{0}}{{P}^{\sigma \mu \nu }}={{D}^{\sigma }}{{F}_{\text{YM}}}^{\mu \nu }+{{D}^{\mu }}{{F}_{\text{YM}}}^{\nu \sigma }+{{D}^{\nu }}{{F}_{\text{YM}}}^{\sigma \mu }=0$

The Yang-Mills canonic magnetic charge density, although generalized above, remains equal to zero just like the magnetic charge density in Maxwell’s electrodynamics. This is no longer because of the commutator $\left[{{\partial }_{\mu }},{{\partial }_{\nu }}\right]=0$, but rather because of the Jacobi identity $\left[{{D}_{\sigma }},\left[{{D}_{\mu }},{{D}_{\nu }}\right]\right]+\left[{{D}_{\mu }},\left[{{D}_{\nu }},{{D}_{\sigma }}\right]\right]+\left[{{D}_{\nu }},\left[{{D}_{\sigma }},{{D}_{\mu }}\right]\right]=0$ combined with the further identity $\left[{{D}_{\sigma }},{{F}_{\text{YM}}}_{\mu \nu }\right]={{D}_{\sigma }}{{F}_{\text{YM}}}_{\mu \nu }$, both reviewed in the Mathematical overview.

Applying ${{D}_{\nu }}$ to the above charge density, the identity ${{D}_{\nu }}{{D}_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }=0$ reviewed in the Mathematical overview enables us to calculate the Yang-Mills canonic continuity relation:

${\begin{aligned}&c{{\mu }_{0}}{{D}_{\nu }}{{J}^{\nu }}={{D}_{\nu }}{{D}_{\mu }}{{F}_{\text{YM}}}^{\mu \nu }=\left({{g}^{\mu \nu }}{{D}_{\nu }}{{D}_{\sigma }}{{D}^{\sigma }}-{{D}_{\nu }}{{D}^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}\\&={{\partial }_{\nu }}{{\partial }_{\mu }}{{F}^{\mu \nu }}-\left(ig\left({{G}_{\nu }}{{\partial }_{\mu }}+{{\partial }_{\nu }}{{G}_{\mu }}\right)+{{g}^{2}}{{G}_{\nu }}{{G}_{\mu }}\right){{F}^{\mu \nu }}=\left({{\partial }_{\nu }}{{\partial }_{\mu }}-{{V}_{\nu \mu }}\right){{F}^{\mu \nu }}=0\\\end{aligned}}$ ,

which includes a perturbation tensor defined by:

${{V}_{\mu \nu }}\equiv ig\left({{G}_{\mu }}{{\partial }_{\nu }}+{{\partial }_{\mu }}{{G}_{\nu }}\right)+{{g}^{2}}{{G}_{\mu }}{{G}_{\nu }}$.

The trace $V={{V}^{\sigma }}_{\sigma }=ig\left({{G}^{\sigma }}{{\partial }_{\sigma }}+{{\partial }^{\sigma }}{{G}_{\sigma }}\right)+{{g}^{2}}{{G}^{\sigma }}{{G}_{\sigma }}$ of the above is the standard expression for the perturbation in the Klein–Gordon (relativistic Schrödinger) equation.

In dynamic form, one still begins with Maxwell’s electrodynamic equations $c{{\mu }_{0}}{{j}^{\nu }}={{\partial }_{\mu }}{{F}^{\mu \nu }}$ and $c{{\mu }_{0}}{{p}^{\sigma \mu \nu }}={{\partial }^{\sigma }}{{F}^{\mu \nu }}+{{\partial }^{\mu }}{{F}^{\nu \sigma }}+{{\partial }^{\nu }}{{F}^{\sigma \mu }}=0$, and still uses the non-commuting field strength ${{F}_{\text{YM}}}^{\mu \nu }={{D}^{\mu }}{{G}^{\nu }}-{{D}^{\nu }}{{G}^{\mu }}={{\partial }^{\mu }}{{G}^{\nu }}-{{\partial }^{\nu }}{{G}^{\mu }}-ig\left[{{G}^{\mu }},{{G}^{\nu }}\right]$ with $\left[{{G}^{\mu }},{{G}^{\nu }}\right]\neq 0$, but does nothing further. That is, one keeps the remaining derivatives ordinary and keeps the source density notation in lowercase. Consequently, Maxwell’s Yang-Mills “electric” and “magnetic” equations in dynamic form are as follows:

$c{{\mu }_{0}}{{j}^{\nu }}={{\partial }_{\sigma }}{{F}_{\text{YM}}}^{\sigma \nu }=\left({{g}^{\mu \nu }}{{\partial }_{\sigma }}{{D}^{\sigma }}-{{\partial }^{\mu }}{{D}^{\nu }}\right){{G}_{\mu }}$

${\begin{aligned}&c{{\mu }_{0}}{{p}^{\sigma \mu \nu }}={{\partial }^{\sigma }}{{F}_{\text{YM}}}^{\mu \nu }+{{\partial }^{\mu }}{{F}_{\text{YM}}}^{\nu \sigma }+{{\partial }^{\nu }}{{F}_{\text{YM}}}^{\sigma \mu }\\&\quad \quad \quad \ \ =-ig\left({{\partial }^{\sigma }}\left[{{G}^{\mu }},{{G}^{\nu }}\right]+{{\partial }^{\mu }}\left[{{G}^{\nu }},{{G}^{\sigma }}\right]+{{\partial }^{\nu }}\left[{{G}^{\sigma }},{{G}^{\mu }}\right]\right)\neq 0\\\end{aligned}}$

These are simply Maxwell equations without change, aside from the promotion of ${{F}^{\mu \nu }}={{\partial }^{\mu }}{{A}^{\nu }}-{{\partial }^{\nu }}{{A}^{\mu }}$ with $\left[{{A}^{\mu }},{{A}^{\nu }}\right]=0$, to ${{F}_{\text{YM}}}^{\mu \nu }={{D}^{\mu }}{{G}^{\nu }}-{{D}^{\nu }}{{G}^{\mu }}={{\partial }^{\mu }}{{G}^{\nu }}-{{\partial }^{\nu }}{{G}^{\mu }}-ig\left[{{G}^{\mu }},{{G}^{\nu }}\right]$ with $\left[{{G}^{\mu }},{{G}^{\nu }}\right]\neq 0$. That is, these are Maxwell’s equations for non-commuting gauge fields, with nothing else changed. However, the identity which causes the uppercase-denoted magnetic source density to vanish in the canonic equation, ${{P}^{\sigma \mu \nu }}=0$, do not operate to vanish the lowercase-denoted magnetic source density to vanish from the dynamic equation, ${{p}^{\sigma \mu \nu }}\neq 0$. Instead, using the “zero” from the canonic magnetic charge equation, we are able to calculate in the above that the dynamic ${{p}^{\sigma \mu \nu }}\neq 0$ differs from zero by the index-cyclic derivatives ${{\partial }^{\sigma }}\left[{{G}^{\mu }},{{G}^{\nu }}\right]+{{\partial }^{\mu }}\left[{{G}^{\nu }},{{G}^{\sigma }}\right]+{{\partial }^{\nu }}\left[{{G}^{\sigma }},{{G}^{\mu }}\right]$ of the non-zero Yang-Mills gauge field commutator.

Commuting versus non-commuting gauge fields

Studying the physics of Yang-Mills gauge theory requires understanding what happens to Maxwell’s electrodynamics, and U(1) quantum electrodynamics (QED), when Maxwell’s commuting (abelian) gauge fields ${{A}^{\mu }}$ become non-commuting (nonabelian) gauge fields ${{G}^{\mu }}={{\tau }_{i}}G_{i}^{\mu }$ covariantly transforming, for example, under the compact simple Yang-Mills gauge group SU(N) with NxN Hermitian generators ${{\tau }_{i}}={{\tau }_{i}}^{\dagger }$ and a commutator $\left[{{\tau }_{i}},{{\tau }_{j}}\right]=i{{f}_{ijk}}{{\tau }_{k}}$ typically normalized such that ${\text{tr}}\left({{\tau }_{i}}^{2}\right)={\tfrac {1}{2}}$ for each $i=1...{{N}^{2}}-1$. Whereas electrodynamics is a linear theory in which the gauge fields do not interact with one another, Yang-Mills theory is highly nonlinear with mutual interactions amongst the gauge fields.

In flat spacetime, in classical electrodynamics, a gauge-invariant field strength ${{F}^{\mu \nu }}$ is related to the gauge fields ${{A}^{\mu }}$ by:

${{F}^{\mu \nu }}={{\partial }^{\mu }}{{A}^{\nu }}-{{\partial }^{\nu }}{{A}^{\mu }}$.

This may also be written more generally as ${{F}^{\mu \nu }}={{D}^{\mu }}{{A}^{\nu }}-{{D}^{\nu }}{{A}^{\mu }}$ using the gauge-covariant derivative ${{D}^{\mu }}={{\partial }^{\mu }}-ie{{A}^{\mu }}/\hbar c$, because the commutator $\left[{{A}_{\mu }},{{A}_{\nu }}\right]=0$. With ${{c}^{2}}{{\mu }_{0}}{{\varepsilon }_{0}}=1$ and Coulomb constant ${{k}_{\text{e}}}=1/4\pi {{\varepsilon }_{0}}$, the classical Maxwell equation for electric charge strength is:

$c{{\mu }_{0}}{{j}^{\nu }}={{\partial }_{\sigma }}{{F}^{\sigma \nu }}=\left({{g}^{\mu \nu }}{{\partial }_{\sigma }}{{\partial }^{\sigma }}-{{\partial }^{\nu }}{{\partial }^{\mu }}\right){{A}_{\mu }}$,

which spacetime-covariantly includes Gauss’ electricity and Ampere’s current laws. The classical equation for magnetic charge strength is

$c{{\mu }_{0}}{{p}^{\sigma \mu \nu }}={{\partial }^{\sigma }}{{F}^{\mu \nu }}+{{\partial }^{\mu }}{{F}^{\nu \sigma }}+{{\partial }^{\nu }}{{F}^{\sigma \mu }}=0$,

which spacetime-covariantly includes Gauss’ magnetism and Faraday’s induction laws. The zero in the monopole equation and thus the non-existence of magnetic monopoles (setting aside possible Dirac charge quantization) arises from the flat spacetime commutator of ordinary derivatives being $\left[{{\partial }_{\mu }},{{\partial }_{\nu }}\right]=0$. In integral form, the Gauss’ magnetism law component of the above becomes $\;\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$, whereby there is no net flux of magnetic fields across closed spatial surfaces. (Note: The point of various “bag models” of QCD quark confinement, is that there is similarly no net flux of color charge across the closed spatial surfaces of color-neutral baryons, see, e.g., section 18.3 of .)

Summing the four-gradient ${{\partial }_{\nu }}$ with the electric charge strength above to obtain the four-dimensional spacetime divergence, we readily obtain:

$c{{\mu }_{0}}{{\partial }_{\nu }}{{j}^{\nu }}={{\partial }_{\nu }}{{\partial }_{\sigma }}{{\partial }^{\sigma }}{{A}^{\nu }}-{{\partial }_{\nu }}{{\partial }_{\sigma }}{{\partial }^{\nu }}{{A}^{\sigma }}=0$,

which is the continuity equation governing the conservation of electric charge. This becomes zero, once again, because of flat spacetime commutator $\left[{{\partial }_{\mu }},{{\partial }_{\nu }}\right]=0$.

In QED, the charge density becomes related to the Dirac wavefunctions $\psi$ for individual fermions by ${{j}^{\nu }}=e{\overline {\psi }}Q{{\gamma }^{\nu }}\psi$ where $e$ is the electric charge strength related to the running "fine structure" coupling ${{\alpha }_{e}}\left(\mu =0\right)=1/137.036...$ by ${{k}_{\text{e}}}{{e}^{2}}=\hbar c{{\alpha }_{e}}$, and $Q=-1,+{\tfrac {2}{3}},-{\tfrac {1}{3}}$ for the electron, up and down fermions, and their higher-generational counterparts. Meanwhile the propagators for the individual photons which form the gauge fields are obtained by inverting the electric charge equation and converting from configuration into momentum space using the substitution $i\hbar {{\partial }^{\mu }}\to {{q}^{\mu }}$ and the $+i\varepsilon$ prescription. Because the charge equation is not invertible without taking some further steps, it is customary to utilize the gauge condition ${{\partial }_{\sigma }}{{A}^{\sigma }}=0$ to obtain

${{A}_{\alpha }}={{\hbar }^{2}}c{{\mu }_{0}}{\frac {-{{g}_{\alpha \nu }}}{{{q}_{\sigma }}{{q}^{\sigma }}+i\varepsilon }}{{j}^{\nu }}$

which includes the photon propagator up to a factor of $i$. Alternatively, one can introduce a Proca mass by hand into the charge equation. Then, ${{\partial }_{\sigma }}{{A}^{\sigma }}=0$ is no longer a gauge condition but a requirement to maintain continuity (charge conservation), and with $i\hbar {{\partial }^{\mu }}\to {{k}^{\mu }}$ we arrive at the inverse:

${{A}_{\alpha }}={{\hbar }^{2}}c{{\mu }_{0}}{\frac {-{{g}_{\alpha \nu }}+{{k}_{\nu }}{{k}_{\alpha }}/{{m}^{2}}}{{{k}_{\sigma }}{{k}^{\sigma }}-{{m}^{2}}+i\varepsilon }}{{j}^{\nu }}$

which includes a massive vector boson propagator up to $i$. Of course, adding a mass by hand destroys renormalizability, so it is necessary to find a way that this can be restored.

In Yang-Mills Gauge Theory, as distinct from a gauge theory with commuting gauge fields, ${{A}^{\mu }}\to {{G}^{\mu }}={{\tau }_{i}}G_{i}^{\mu }$ becomes a non-commuting gauge field, $\left[{{G}_{\mu }},{{G}_{\nu }}\right]=\left[{{\tau }_{i}},{{\tau }_{j}}\right]{{G}_{i}}_{\mu }{{G}_{j}}_{\nu }=i{{f}_{ijk}}{{\tau }_{k}}{{G}_{i}}_{\mu }{{G}_{j}}_{\nu }\neq 0$, and the field strength therefore graduates to the gauge-covariant, not gauge-invariant:

${{F}^{\mu \nu }}={{\tau }_{k}}{{F}_{k}}^{\mu \nu }={{D}^{\mu }}{{G}^{\nu }}-{{D}^{\nu }}{{G}^{\mu }}={{\partial }^{\mu }}{{G}^{\nu }}-{{\partial }^{\nu }}{{G}^{\mu }}-ig\left[{{G}^{\mu }},{{G}^{\nu }}\right]={{\tau }_{k}}\left\{{{\partial }^{\mu }}G_{k}^{\nu }-{{\partial }^{\nu }}G_{k}^{\mu }+g{{f}_{ijk}}{{G}_{i}}^{\mu }{{G}_{j}}^{\nu }\right\}$.

With A replaced by G, it will be seen that this contains the equation $F_{\mu \nu }^{a}=\partial _{\mu }G_{\nu }^{a}-\partial _{\nu }G_{\mu }^{a}+gf^{abc}G_{\mu }^{b}G_{\nu }^{c}$ from the Mathematical overview above. Using differential forms, this may be written as the curvature $F=dG+G\wedge G$ arising from the gauge connection, see at pages 1 and 2. The non-linearity of Yang-Mills gauge theories becomes apparent if one uses the above to advance the source-free Lagrangian from the Mathematical overview to:

${\mathcal {L}}_{\mathrm {gf} }=-{\tfrac {1}{2}}{\text{Tr}}\left({{F}^{\mu \nu }}{{F}_{\mu \nu }}\right)=-{\tfrac {1}{4}}{{F}_{i}}^{\mu \nu }{{F}_{i}}_{\mu \nu }=-{\tfrac {1}{4}}{{\partial }^{[\mu }}{{G}_{i}}^{\nu ]}{{\partial }_{[\mu }}{{G}_{i}}_{\nu ]}-{\tfrac {1}{2}}g{{f}_{ijk}}{{\partial }^{[\mu }}{{G}_{i}}^{\nu ]}{{G}_{j}}_{\mu }{{G}_{k}}_{\nu }-{\tfrac {1}{4}}{{g}^{2}}{{f}_{ijk}}{{f}_{ilm}}{{G}_{j}}^{\mu }{{G}_{k}}^{\nu }{{G}_{l}}_{\mu }{{G}_{m}}_{\nu }$,

which includes the three- and four-gauge boson interaction vertices illustrated earlier.

Yang-Mills gauge theory differs from the abelian gauge theory of U(1) electrodynamics, by the mathematical and physical consequences of what happens when the gauge fields go from commuting to non-commuting in this way.

Test

The commutator relation

$\hbar c\left[{{D}^{\mu }},{{D}^{\nu }}\right]=-ig{{F}^{\mu \nu }}=-ig{{T}^{a}}{{F}^{a}}^{\mu \nu }$

can be derived as follows: With $\hbar =c=1$ and using $\left[{{\partial }^{\mu }},{{\partial }^{\nu }}\right]=0,$ apply the commutator $\left[{{D}^{\mu }},{{D}^{\nu }}\right]$ to operate on any field $\phi \left(t,\mathbf {x} \right).$ Attentive to the product rule, one may obtain:

${\begin{aligned}&\left[{{D}^{\mu }},{{D}^{\nu }}\right]\phi =\left[\left({{\partial }^{\mu }}-ig{{A}^{\mu }}\right),\left({{\partial }^{\nu }}-ig{{A}^{\nu }}\right)\right]\phi \\&\quad \quad \quad \quad \ \ =-ig{{A}^{[\mu }}{{\partial }^{\nu ]}}\phi -ig{{\partial }^{[\mu }}\left({{A}^{\nu ]}}\phi \right)-{{g}^{2}}\left[{{A}^{\mu }},{{A}^{\nu }}\right]\phi \\&\quad \quad \quad \quad \ \ =-ig{{\partial }^{[\mu }}{{A}^{\nu ]}}\phi -{{g}^{2}}\left[{{A}^{\mu }},{{A}^{\nu }}\right]\phi =-ig{{F}^{\mu \nu }}\phi \\\end{aligned}}.$

Given that $\left[{{A}^{\mu }},{{A}^{\nu }}\right]=\left[{{T}_{i}},{{T}_{j}}\right]{{A}_{i}}^{\mu }{{A}_{j}}^{\nu }=i{{f}_{ijk}}{{T}_{k}}{{A}_{i}}^{\mu }{{A}_{j}}^{\nu }\neq 0,$ this includes the relation

${{F}^{\mu \nu }}={{T}_{c}}{{F}_{c}}^{\mu \nu }={{\partial }^{[\mu }}{{A}^{\nu ]}}-ig\left[{{A}^{\mu }},{{A}^{\nu }}\right]={{T}_{c}}\left\{{{\partial }^{\mu }}A_{c}^{\nu }-{{\partial }^{\nu }}A_{c}^{\mu }+g{{f}_{abc}}{{A}_{a}}^{\mu }{{A}_{b}}^{\nu }\right\}$

which is further seen to contain:

${{F}_{a}}^{\mu \nu }={{\partial }^{\mu }}A_{a}^{\nu }-{{\partial }^{\nu }}A_{a}^{\mu }+g{{f}_{abc}}{{A}_{b}}^{\mu }{{A}_{c}}^{\nu }.$

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