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* <math>F_{\mu\nu}</math> is the field strength tensor   
* <math>F_{\mu\nu}</math> is the field strength tensor   


This structure encodes both particle dynamics and interactions.<ref name="peskin"/>
This structure encodes both particle dynamics and interactions.<ref name="peskin">Peskin, M. E.; Schroeder, D. V. ''An Introduction to Quantum Field Theory'' (1995).</ref>


== Symmetry and gauge structure ==
== Symmetry and gauge structure ==
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{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}


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


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

Latest revision as of 00:31, 24 May 2026

← Back to Quantum field theory Book I
← Previous : Field theory (QFT) basics
Next : Fields and Particles →

field theory (QFT) core 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. Particles appear as quantized excitations of these fields. Core structure of quantum field theory: Lagrangian, fields, symmetries, and operators 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. Particles appear as quantized excitations of these fields. Quantization replaces classical variables with operator-valued distributions satisfying commutation or anticommutation relations: A typical interacting theory is described by: This structure encodes both particle dynamics and interactions.

Quantum field theory (QFT) core.

Fields and quantization

In QFT, classical fields such as scalar fields ϕ(x), spinor fields ψ(x), and gauge fields Aμ(x) are promoted to operators acting on a Hilbert space.[1]

Quantization replaces classical variables with operator-valued distributions satisfying commutation or anticommutation relations: [ϕ(x),π(y)]=iδ(3)(x−y)

for bosonic fields, and {ψα(x),ψβ†(y)}=δαβδ(3)(x−y)

for fermionic fields.[2]

Lagrangian formulation

The dynamics of a quantum field theory are determined by a Lagrangian density ℒ, from which the equations of motion follow via the principle of least action: S=∫d4xℒ

A typical interacting theory is described by: ℒ=ψ¯(iγμDμ−m)ψ−14FμνFμν

where:

  • ψ is a fermion field
  • Dμ is the covariant derivative
  • Fμν is the field strength tensor

This structure encodes both particle dynamics and interactions.[3]

Symmetry and gauge structure

Symmetries play a central role in QFT. Continuous symmetries lead to conserved quantities via Noether’s theorem.[4]

Gauge symmetries define the fundamental interactions:

  • U(1) → electromagnetism
  • SU(2) → weak interaction
  • SU(3) → strong interaction

These symmetries require the introduction of gauge fields and determine the interaction terms in the Lagrangian.[1]

Operators and states

Physical states are constructed in a Fock space, where creation and annihilation operators act on the vacuum: a𝐩†|0⟩

creates a particle with momentum 𝐩. Observables correspond to operators acting on these states.

Correlation functions and expectation values encode measurable quantities: ⟨0|T{ϕ(x)ϕ(y)}|0⟩

which describe propagation and interactions.[2]

Interactions and Feynman diagrams

Perturbative expansions allow interaction processes to be represented diagrammatically using Feynman diagrams.[5]

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.[3]

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

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ 1.0 1.1 Weinberg, S. The Quantum Theory of Fields (1995).
  2. ↑ 2.0 2.1 Schwartz, M. D. Quantum Field Theory and the Standard Model (2014).
  3. ↑ 3.0 3.1 Peskin, M. E.; Schroeder, D. V. An Introduction to Quantum Field Theory (1995).
  4. ↑ Noether, E. (1918). Invariant variation problems.
  5. ↑ Feynman, R. P. (1949). Space-time approach to quantum electrodynamics.
Author: Harold Foppele


Source attribution: Physics:Quantum field theory (QFT) core

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