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{{Quantum book backlink|Quantum field theory}}
'''Quantum field theory''' (QFT) is the theoretical framework that combines quantum mechanics with special relativity by describing physical systems in terms of fields defined over space-time.<ref name="peskin">Peskin, M. E.; Schroeder, D. V. ''An Introduction to Quantum Field Theory'' (1995).</ref> Particles appear as quantized excitations of these fields.
<div style="float:right; border:1px solid #ccc; padding:4px; background:#fff8dc; margin:0 0 1em 1em; width:420px;">
[[File:Quantum_field_theory_core.jpg|400px]]
<div style="font-size:90%;">Core structure of quantum field theory: Lagrangian, fields, symmetries, and operators</div>
</div>

== Fields and quantization ==
In QFT, classical fields such as scalar fields <math>\phi(x)</math>, spinor fields <math>\psi(x)</math>, and gauge fields <math>A_\mu(x)</math> are promoted to operators acting on a Hilbert space.<ref name="weinberg">Weinberg, S. ''The Quantum Theory of Fields'' (1995).</ref>

Quantization replaces classical variables with operator-valued distributions satisfying commutation or anticommutation relations:
<math>
[\phi(x), \pi(y)] = i \delta^{(3)}(x - y)
</math>

for bosonic fields, and
<math>
\{\psi_\alpha(x), \psi^\dagger_\beta(y)\} = \delta_{\alpha\beta} \delta^{(3)}(x - y)
</math>

for fermionic fields.<ref name="schwartz">Schwartz, M. D. ''Quantum Field Theory and the Standard Model'' (2014).</ref>

== Lagrangian formulation ==
The dynamics of a quantum field theory are determined by a Lagrangian density <math>\mathcal{L}</math>, from which the equations of motion follow via the principle of least action:
<math>
S = \int d^4x \, \mathcal{L}
</math>

A typical interacting theory is described by:
<math>
\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu}
</math>

where:
* <math>\psi</math> is a fermion field
* <math>D_\mu</math> is the covariant derivative
* <math>F_{\mu\nu}</math> is the field strength tensor

This structure encodes both particle dynamics and interactions.<ref name="peskin"/>

== Symmetry and gauge structure ==
Symmetries play a central role in QFT. Continuous symmetries lead to conserved quantities via Noether’s theorem.<ref name="noether">Noether, E. (1918). Invariant variation problems.</ref>

Gauge symmetries define the fundamental interactions:
* <math>U(1)</math> → electromagnetism
* <math>SU(2)</math> → weak interaction
* <math>SU(3)</math> → strong interaction

These symmetries require the introduction of gauge fields and determine the interaction terms in the Lagrangian.<ref name="weinberg"/>

== Operators and states ==
Physical states are constructed in a Fock space, where creation and annihilation operators act on the vacuum:
<math>
a^\dagger_{\mathbf{p}} |0\rangle
</math>

creates a particle with momentum <math>\mathbf{p}</math>. Observables correspond to operators acting on these states.

Correlation functions and expectation values encode measurable quantities:
<math>
\langle 0 | T\{\phi(x)\phi(y)\} | 0 \rangle
</math>

which describe propagation and interactions.<ref name="schwartz"/>

== Interactions and Feynman diagrams ==
Perturbative expansions allow interaction processes to be represented diagrammatically using Feynman diagrams.<ref name="feynman">Feynman, R. P. (1949). Space-time approach to quantum electrodynamics.</ref>

These diagrams correspond to terms in a series expansion of the S-matrix and provide a practical computational tool for scattering amplitudes.

== Renormalization ==
Quantum field theories often produce divergent integrals. Renormalization systematically absorbs these divergences into redefined parameters such as mass and charge.<ref name="peskin"/>

Renormalizable theories yield finite, predictive results and form the basis of the Standard Model of particle physics.

== See also ==
{{:Physics:Quantum basics/See also}}
* [[Physics:Quantum electrodynamics]]
* [[Physics:Quantum chromodynamics]]
* [[Physics:Standard Model]]

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

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

Physics:Quantum field theory (QFT) core - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template