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Quantum Chromodynamics

Scattered off Strong Force

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Quantum Field Theory
... no spooky action at a distance (Einstein)
Early Results
Relativistic Quantum Mechanics Klein-Gordon Equation
Dirac Equation
The Dawn of QFT Spinors
Feynman Slash Notation
Klein-Gordon Field
Dirac Field
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
Semi-Classical Gravity and the Dark Age Hawking Radiation
Chandrashekhar Limit
Problems with the Standard Model
Beyond the Standard Model Beyond the Standard Model
Quantum Gravity
Theory of Everything
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 QuarksEdit this section

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

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.

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 ConfinementEdit this section

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 forceEdit this section

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 DensityEdit this section

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 InvarianceEdit this section

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).

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