Gluon field

 ( Gauge Theories ) 

In theoretical particle physics, the gluon field is a four-vector field characterizing the propagation of in the strong interaction between . It plays the same role in quantum chromodynamics as the electromagnetic four-potential in quantum electrodynamics the gluon field constructs the gluon field strength tensor.

Throughout this article, Latin indices take values 1, 2, ..., 8 for the eight gluon , while Greek indices take values 0 for timelike components and 1, 2, 3 for spacelike components of four-dimensional vectors and tensors in spacetime. Throughout all equations, the summation convention is used on all color and tensor indices, unless explicitly stated otherwise.

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Introduction Gluons can have eight so there are eight fields, in contrast to photons which are neutral and so there is only one photon field.

The gluon fields for each color charge each have a "timelike" component analogous to the electric potential, and three "spacelike" components analogous to the magnetic vector potential. Using similar symbols:

$\backslash boldsymbol\{\backslash mathcal\{A\}\}^n(\backslash mathbf\{r\},\; t)\; =\; =\; \backslash phi^n$

where are not but enumerate the eight gluon color charges, and all components depend on the position vector of the gluon and time t. Each $\backslash mathcal\{A\}^a\_\backslash alpha$ is a scalar field, for some component of spacetime and gluon color charge.

The Gell-Mann matrices are eight 3 × 3 matrices which form matrix representations of the SU(3) group. They are also generators of the SU(3) group, in the context of quantum mechanics and field theory; a generator can be viewed as an operator corresponding to a symmetry transformation (see symmetry in quantum mechanics). These matrices play an important role in QCD as QCD is a gauge theory of the SU(3) gauge group obtained by taking the color charge to define a local symmetry: each Gell-Mann matrix corresponds to a particular gluon color charge, which in turn can be used to define color charge operators. Generators of a group can also form a basis for a vector space, so the overall gluon field is a "superposition" of all the color fields. In terms of the Gell-Mann matrices (divided by 2 for convenience),

$t\_a\; =\; \backslash frac\{\backslash lambda\_a\}\{2\}\backslash ,,$

the components of the gluon field are represented by 3 × 3 matrices, given by:

$\backslash mathcal\{A\}\_\{\backslash alpha\}\; =\; t\_a\; \backslash mathcal\{A\}^a\_\backslash alpha\; \backslash equiv\; t\_1\; \backslash mathcal\{A\}^1\_\backslash alpha\; +\; t\_2\; \backslash mathcal\{A\}^2\_\backslash alpha\; +\; \backslash cdots\; +\; t\_8\; \backslash mathcal\{A\}^8\_\backslash alpha$

or collecting these into a vector of four 3 × 3 matrices:

$\backslash boldsymbol\{\backslash mathcal\{A\}\}(\backslash mathbf\{r\},\; t)\; =\; \backslash mathcal\{A\}\_0(\backslash mathbf\{r\},$

the gluon field is:

$\backslash boldsymbol\{\backslash mathcal\{A\}\}\; =\; t\_a\; \backslash boldsymbol\{\backslash mathcal\{A\}\}^a\; \backslash ,.$

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Gauge covariant derivative in QCD Below the definitions (and most of the notation) follow K. Yagi, T. Hatsuda, Y. Miake

(2025). 9780521561082, Cambridge University Press. . ISBN 9780521561082

and Greiner, Schäfer.

(1994). 9783540571032, Springer. . ISBN 9783540571032

The gauge covariant derivative is required to transform quark fields in manifest covariance; the partial derivatives that form the four-gradient alone are not enough. The components which act on the color triplet quark fields are given by:

$D\_\backslash mu\; =\backslash partial\_\backslash mu\; \backslash pm\; ig\_s\; t\_a\; \backslash mathcal\{A\}^a\_\backslash mu\backslash ,,$

wherein is the imaginary unit, and

$g\_s\; =\; \backslash sqrt\{4\backslash pi\; \backslash alpha\_s\}$

is the dimensionless coupling constant for QCD, and $\backslash alpha\_s$ is the strong coupling constant. Different authors choose different signs. The partial derivative term includes a 3 × 3 identity matrix, conventionally not written for simplicity.

The quark fields in triplet representation are written as :

$\backslash psi=\backslash begin\{pmatrix\}\backslash psi\_\{1\}\backslash \backslash$

\psi_{2}\\ \psi_{3} \end{pmatrix},\overline{\psi}=\begin{pmatrix}\overline{\psi}^*_{1}\\ \overline{\psi}^*_{2}\\ \overline{\psi}^*_{3} \end{pmatrix}

The quark field belongs to the fundamental representation ( 3) and the antiparticle field belongs to the complex conjugate representation ( 3^*), complex conjugate is denoted by (not overbar).

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Gauge transformations The gauge transformation of each gluon field $\backslash mathcal\{A\}^n\_\backslash alpha$ which leaves the gluon field strength tensor unchanged is;

$\backslash mathcal\{A\}^n\_\backslash alpha\backslash rightarrow\; e^\{i\backslash bar\{\backslash theta\}(\backslash mathbf\{r\},t)\}\; \backslash left(\backslash mathcal\{A\}^n\_\backslash alpha\; +\; \backslash frac\{i\}\{g\_s\}\backslash partial\_\backslash alpha\backslash right)e^\{-i\backslash bar\{\backslash theta\}(\backslash mathbf\{r\},t)\}$

where

$\backslash bar\{\backslash theta\}(\backslash mathbf\{r\},t)\; =\; t\_n\; \backslash theta^n(\backslash mathbf\{r\},t)\backslash ,,$

is a 3 × 3 matrix constructed from the matrices above and are eight Gauge theory dependent on spatial position and time t. Matrix exponentiation is used in the transformation. The gauge covariant derivative transforms similarly. The functions here are similar to the gauge function when changing the electromagnetic four-potential , in spacetime components:

$A\text{'}\_\backslash alpha\; (\backslash mathbf\{r\},t)\; =\; A\_\backslash alpha\; (\backslash mathbf\{r\},t)\; -\; \backslash partial\_\backslash alpha\; \backslash chi\; (\backslash mathbf\{r\},t)\; \backslash ,$

leaving the electromagnetic tensor invariant.

The quark fields are invariant under the gauge transformation;

$\backslash psi(\backslash mathbf\{r\},t)\; \backslash rightarrow\; e^\{ig\backslash bar\{\backslash theta\}(\backslash mathbf\{r\},t)\}\backslash psi(\backslash mathbf\{r\},t)$

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See also

• Quark confinement
• Gell-Mann matrices
• Field (physics)
• Einstein tensor
• Symmetry in quantum mechanics
• Wilson loop
• Wess–Zumino gauge

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Notes

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Further reading

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Books

• (2025). 9781139462211, Cambridge University Press. . ISBN 9781139462211
• H. Fritzsch (1982). 9780713915334, Allen lane. ISBN 9780713915334
• (2025). 9783642022852, Springer. . ISBN 9783642022852
• (1987). 9782863320488, Atlantica Séguier Frontières. . ISBN 9782863320488
• (1995). 9783540601371, Springer. . ISBN 9783540601371
• K. Chung (2025). 9780549597742 . ISBN 9780549597742
• J. Collins (2025). 9780521855334, Cambridge University Press. . ISBN 9780521855334
• (1998). 9780521588324, Cambridge University Press. . ISBN 9780521588324

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Selected papers

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External links

Categories: Gauge Theories, Quantum Chromodynamics, Field

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