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{{Short description|Local symmetry principle in quantum field theory that determines fundamental interactions through gauge invariance}}

{{Quantum book backlink|Quantum field theory}}
'''Gauge symmetry in quantum field theory''' is a fundamental principle stating that certain transformations of the fields leave the physical predictions of a theory unchanged.<ref name="weinberg">Weinberg, S. ''The Quantum Theory of Fields'' (1995).</ref> These symmetries determine the form of interactions and require the existence of gauge fields that mediate forces.

<div style="float:right; border:1px solid #ccc; padding:4px; background:#fff8dc; margin:0 0 1em 1em; width:420px;">
[[File:Gauge_symmetry_transformation.jpg|400px]]
<div style="font-size:90%;">Gauge symmetry: local transformations of fields requiring compensating gauge fields to preserve invariance</div>
</div>

== Global and local symmetry ==
A symmetry transformation changes the fields without affecting observable quantities.

A '''global symmetry''' uses the same transformation everywhere:
<math>
\psi(x) \rightarrow e^{i\alpha} \psi(x)
</math>

A '''local symmetry''' allows the transformation parameter to vary with space-time:
<math>
\psi(x) \rightarrow e^{i\alpha(x)} \psi(x)
</math>

Local symmetry is much more restrictive and leads directly to interactions.<ref name="peskin">Peskin, M. E.; Schroeder, D. V. ''An Introduction to Quantum Field Theory'' (1995).</ref>

== Emergence of gauge fields ==
A naive local transformation introduces extra terms in derivatives:
<math>
\partial_\mu \psi(x) \rightarrow (\partial_\mu + i\,\partial_\mu \alpha(x)) \psi(x)
</math>

To restore invariance, a gauge field <math>A_\mu(x)</math> is introduced and the derivative is replaced by the covariant derivative:
<math>
D_\mu = \partial_\mu + i g A_\mu
</math>

This ensures that the theory remains invariant under local transformations.<ref name="schwartz">Schwartz, M. D. ''Quantum Field Theory and the Standard Model'' (2014).</ref>

== Gauge invariance ==
The gauge field transforms simultaneously:
<math>
A_\mu(x) \rightarrow A_\mu(x) - \frac{1}{g} \partial_\mu \alpha(x)
</math>

This compensates the change in the matter field, preserving the symmetry of the Lagrangian.

Gauge invariance is therefore not just a mathematical property but a principle that determines the structure of interactions.

== Example: Quantum electrodynamics ==
In QED, the symmetry group is <math>U(1)</math>. The requirement of local gauge invariance leads directly to the electromagnetic interaction.

The interaction term:
<math>
\bar{\psi}\gamma^\mu A_\mu \psi
</math>

arises naturally from imposing gauge symmetry.<ref name="weinberg"/>

== Non-Abelian gauge theories ==
More complex gauge symmetries involve non-commuting groups such as:

* <math>SU(2)</math>
* <math>SU(3)</math>

These lead to non-Abelian gauge theories, where the gauge fields themselves interact.

This structure underlies:

* the weak interaction
* the strong interaction

and forms the basis of the Standard Model.<ref name="peskin"/>

== Field strength tensor ==
The dynamics of gauge fields are described by the field strength tensor:
<math>
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu
</math>

In non-Abelian theories, additional terms appear due to field self-interactions.

The Lagrangian includes:
<math>
-\frac{1}{4} F_{\mu\nu} F^{\mu\nu}
</math>

which governs the propagation of gauge fields.<ref name="schwartz"/>

== Physical interpretation ==
Gauge symmetry implies that certain degrees of freedom are not physically observable, but instead reflect redundancy in the mathematical description.

Observable quantities depend only on gauge-invariant combinations of fields.

== Conceptual importance ==
Gauge symmetry is one of the central organizing principles of modern physics. It explains:

* the existence of force-carrying particles
* the structure of interactions
* the unification of fundamental forces

All known fundamental interactions (except gravity in its classical form) are described by gauge theories.

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}
{{Author|Harold Foppele}}

{{Sourceattribution|Quantum field theory (QFT) core|1}}

Physics:Quantum Field Theory Gauge symmetry - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template