Quantum Chromodynamics

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Quantum Field Theory
... no spooky action at a distance (Einstein)
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Relativistic Quantum Mechanics	Klein-Gordon Equation Dirac Equation
The Dawn of QFT	Spinors Spin Feynman Slash Notation Antimatter Klein-Gordon Field Dirac Field Renormalisation Grassman Variable Conformal Field Theory
Countdown to the Standard Model
From a framework to a model	Yang-Mills Theory Quantum Electrodynamics Quantum Chromodynamics Electroweak Theory Higgs Mechanism Standard Model
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Beyond the Standard Model	Beyond the Standard Model Quantum Gravity Theory of Everything
Related
Related	De Donder-Weyl Theory

Quantum Chromodynamics is a Quantum Field Theory that describes Quarks, Gluons, and their interactions through the Strong Force. It is a strongly-coupled theory, which means that there is the need of Renormalisation.

Free Quarks

Free Quarks clearly should obey the Free (i.e. Potential-less) Dirac Equation:

$\left( i\hbar {{\gamma }^{\mu }}{{\partial }_{\mu }}-m{{c}_{0}} \right)\psi =0$

Applying the Euler-Lagrange Equations, we see that the Lagrangian Density would then be:

$\mathsf{\mathcal{L}}={{c}_{0}}\bar{\psi }\left( i\hbar {{\gamma }^{\mu }}{{\partial }_{\mu }}-m{{c}_{0}} \right)\psi$

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Sidenote: This is the Lagrangian Density for a Free Quark. For the entire Quantum Chromodynamics Lagrangian Density, one also needs to find the Lagrangian Density \ for only Gluons, and the Lagrangian Density for Quark-Gluon interaction.

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This Lagrangian Density ensures invariance under $SU(3)$ transformations. I.e. transforming the quark field $\psi$ as $\psi \mapsto Q\psi$ where $Q\in SU\left( 3 \right)$ does not change the Lagrangian Density.

Quark Confinement

In experiment, Quarks have never been observed as free, they are always interacting through the Strong Force, with Gluons and other Quarks. This phenomenon is known as Quark Confinement.

Gluons and the strong force

Let us now introduce 8 Gluon Potentials $A_{a}^{\mu }$ where $a$ goes from 1 to 8. The Quarks will no longer be free. The interaction between the Quarks and the Gluons is known as the Strong Force. The Lagrangian Density due to the Strong Force is then given by the following expression:

$\mathcal L=-\hbar {{g}_{QCD}}{{\gamma }^{\mu }}A_{a}^{\mu }\frac{{{\lambda }^{a}}}{2}$

The Gluons themselves are Yang-Mills Fields, and have the Yang-Mills Lagrangian Density. This is as follows:

$\mathcal L=-\frac{1}{4}{{H}^{\mu \nu \rho }}{{H}_{\mu \nu \rho }}$

With these Gluon Fields, the covariant derivative can be defined as follows:

$\mathcal L={{\nabla }_{\mu }}_{\operatorname{QCD}}={{\partial }_{\mu }}+i{{g}_{QCD}}A_{a}^{\mu }\frac{{{\lambda }^{a}}}{2}$

Which invites correct comparisons with General Relativity, including bundle curvatures, etc.

Total Lagrangian Density

Adding up the three individual Lagrangian Densityies previously discussed, one obtains the total Lagrangian Density of Quantum Chrodynamics:

Lagrangian Density of Quantum Chromodynamics ${{\mathsf{\mathcal{L}}}_{QCD}}=-\frac{1}{4}{{H}^{\mu \nu \rho }}{{H}_{\mu \nu \rho }}+{{c}_{0}}\bar{\psi }\left( i\hbar {{{\not{\nabla }}}_{\operatorname{QCD}}}-m{{c}_{0}} \right)\psi$

Note that we do not immediately observe the interaction term, but this is merely because we have quietly replaced $\partial$ with $\nabla$.

Gauge Invariance

Quantum Chromodynamics is then invariant under $SU(3)$ transformations if the Gluon Potentials simultaneously transform as:

$A_{a}^{\mu }\mapsto {{U}^{\dagger }}\left( A_{a}^{\mu }+\frac{i}{{{g}_{QCD}}}{{\partial }_{\mu }} \right)U$

This is a Gauge Transformation, and therefore, Quantum Chromodynamics is a Gauge Theory of with a Gauge Group of $SU(3)$.