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

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Dirac fields are spin-1/2 fields (often Fermions), with the Lagrangian Density:

$ \mathcal L = \bar\Psi\left(i\hbar c_0\not\nabla-mc_0^2\right)\Psi = \bar\Psi\left(i\hbar c_0\gamma^\mu\nabla_\mu-mc_0^2\right)\Psi $

These fields are Spin-1/2 Fields. Applying the Principle of Least Action, one obtains the Dirac Equation:

$ \left(i\hbar c_0\not\nabla-mc_0^2\right)\Psi=0 $

The free Dirac Fields take the general form:

$ \psi = \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2E_{p}}}\sum_{s} \left( a^{s}_{\textbf{p}}u^{s}(p)e^{-ip \cdot x}+b^{s \dagger}_{\textbf{p}}v^{s}(p)e^{ip \cdot x}\right).\, $

Quantum Field Theory and the Standard ModelEdit this section

In Quantum Chromodynamics and Electroweak Theory, the Lagrangian Densityies can be written as [1] :

$ \mathcal L = -\frac14F^2+\bar\Psi\left(i\hbar c_0\not\nabla-mc_0^2\right)\Psi = -\frac{F^2}4 +\bar\Psi\left(i\hbar c_0\gamma^\mu\nabla_\mu-mc_0^2\right)\Psi $

I.e. as sums of Yang-Mills interaction Field Strength Lagrangian Densityies and Dirac Lagrangian Densityies. The quarks and electrons, along with the quark-gluon or electron-photon interactions, for example, are Dirac Fields, whereas the Gluons alone are the Yang-Mills Fields here..

Dirac Fields are in contrast to Klein-Gordon Fields for spin-0 particles, such as the Higgs Field.

String TheoryEdit this section

In RNS String Theory, the Polyakov Lagrangian Density is added to the Dirac Lagrangian Density, obtaining the RNS, or Ramond-Neveu-Schwarz Lagrangian Density:[2].

$ {{\mathsf{\mathcal{L}}}_{RNS}}=\frac{T}{2} h^{\alpha \beta} \left( \partial_\alpha X^\mu \partial_\beta X^\nu +i\hbar c_0 \bar{\psi_\mu} \not\partial \psi^\mu \right) g_{\mu\nu} $:

ReferencesEdit this section