Talk:Yang–Mills theory

Integrable solutions of classical Yang-Mills equations and QFT (Disputed section)

Disputed section

One of the main difficulties that one meets on managing Yang-Mills theory at low energies is that Hamiltonian homogeneous equations of the theory admit essentially chaotic solutions and nobody is able to formulate a quantum field theory starting from such solutions. But, beside chaotic solutions, this theory also admits integrable solutions that can be used for these aims. These solutions can be found with the so called Smilga's choice that permits to fully map Yang-Mills theory on a $\lambda \phi ^{4}$ massless field theory and, for this case, the infrared theory has been recently formulated . So, one can compute the propagator and the spectrum . The results appear to be in agreement with computations with lattice field theory yielding a zero momentum propagator

$\ \Delta (\omega )=\sum _{n=0}^{\infty }{\frac {B_{n}}{\omega ^{2}-\omega _{n}^{2}+i\epsilon }}$

being

$\ B_{n}=(2n+1){\frac {\pi ^{2}}{K^{2}(i)}}{\frac {(-1)^{n+1}e^{-(n+{\frac {1}{2}})\pi }}{1+e^{-(2n+1)\pi }}}$,

and a mass spectrum

$\ \omega _{n}=\left(n+{\frac {1}{2}}\right){\frac {\pi }{K(i)}}\left({\frac {Ng^{2}}{2}}\right)^{\frac {1}{4}}\Lambda$

proper to an harmonic oscillator. The ghost field has the same propagator as a free particle, i.e. it decouples from the gluon field.

An example of such classical solutions can be yielded for SU(2) gauge group (but it is also easy to obtain it for SU(3))

$\ A_{\mu }^{a}=\eta _{\mu }^{a}\Lambda \left({\frac {1}{g^{1 \over 2}}}\right){\rm {sn}}(p\cdot x+\phi ,i)$

being a Jacobi elliptic function, $\Lambda$ and $\phi$ integration constants and $\eta _{\mu }^{a}=((0,1,0,0),(0,0,1,0),(0,0,0,1))$. This solution for Yang-Mills equations holds only if the following dispersion relation holds

$\ p^{2}=g\Lambda ^{2}$

showing in this way as a massless theory can have classical massive solutions.

Computation of the gluon propagator is essential because it permits to obtain, starting from quantum chromodynamics, a Nambu-Jona-Lasinio model and all hadronic phenomenology seen at lower energies can be derived. Research in this field is currently very active and important results are expected shortly.

Discussion, edit warring. Comment

Could we please discuss the passages and parts regarding known or unknown quantities which seem to inflame our anonymous editor so much? I'm afraid I don't know much about theoretical physics, but I'm willing to try to learn. — Javert2113 (talk) 15:05, 29 April 2018 (UTC)

Sorry, forgot to add @Pra1998: thoughts? — Javert2113 (talk) 15:11, 29 April 2018 (UTC)

It is quite simple. Wherever this guy reads "Frasca" he aims to remove the material without mercy. The point is that the cited paper is on the same foot of the others in that section and in the preceding one and so, based on his principles, we should remove both of them. The paper in question is published in one of the most important journals of particle physics, Physics Letters B (It is the one where CERN published its work on the discovery of the Higgs particle). Besides, it is a collaboration with one of the Editors of the European Physical Journal C that, as importance, is on the same foot of the other.--Pra1998 (talk) 15:13, 29 April 2018 (UTC)

There is no need to deal with the complex scientific issues here. Pra1998=Marco Frasca, and my understanding is that Wikipedia policy does not allow people to add references to their own work to Wikipedia pages. Peterwoit (talk) 14:50, 30 April 2018 (UTC)

Commuting versus non-commuting gauge fields

I would like to add a new section to this article, with the above section title, following the section "Quantization." My proposed section is below. Are there any objections or suggestions, prior to my doing so?

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 to 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 ${{D}^{\mu }}={{\partial }^{\mu }}-ig{{A}^{\mu }}/\hbar c$, 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.)

Summing the four-gradient ${{\partial }_{\nu }}$ with the above electric charge strength, 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 quantum electrodynamics, 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, ${{A}^{\mu }}\to {{G}^{\mu }}={{\tau }_{i}}G_{i}^{\mu }$ 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 three- and four-gauge boson interaction vertices.

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. PatentPhysicist (talk) 17:34, 21 February 2021 (UTC)

I think this could be a valuable addition to the article. Chanacya (talk) 18:41, 21 February 2021 (UTC)

PS: I noticed that this is a level 5 vital priority article, but only B class rated. I have studied Yang-Mill gauge theories for over 15 years, and would like to try to contribute to raising this. PatentPhysicist (talk) 02:51, 22 February 2021 (UTC)

I just added the section as written above, to the main article. PatentPhysicist (talk) 18:54, 23 February 2021 (UTC)

Canonic versus dynamic Yang-Mills field equations

I have just finished drafting a new section for this article, below. I will leave this in talk for a few days before posting, for review and comment.

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.

Canonic

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, the Yang-Mills extension 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 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.

Dynamic

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, in dynamic form, the Yang-Mills generalization of Maxwell's electric charge equation (Gauss and Ampere) is:

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

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 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. PatentPhysicist (talk) 20:28, 28 February 2021 (UTC)