Physics:Quantum field theory (QFT) core

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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.^[1] Particles appear as quantized excitations of these fields.

Core structure of quantum field theory: Lagrangian, fields, symmetries, and operators

Fields and quantization

In QFT, classical fields such as scalar fields $ϕ (x)$ , spinor fields $ψ (x)$ , and gauge fields $A_{μ} (x)$ are promoted to operators acting on a Hilbert space.^[2]

Quantization replaces classical variables with operator-valued distributions satisfying commutation or anticommutation relations: $[ϕ (x), π (y)] = i δ^{(3)} (x - y)$

for bosonic fields, and ${ψ_{α} (x), ψ_{β}^{†} (y)} = δ_{α β} δ^{(3)} (x - y)$

for fermionic fields.^[3]

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: $S = \int d^{4} x ℒ$

A typical interacting theory is described by: $ℒ = \bar{ψ} (i γ^{μ} D_{μ} - m) ψ - \frac{1}{4} F_{μ ν} F^{μ ν}$

where:

$ψ$ is a fermion field
$D_{μ}$ is the covariant derivative
$F_{μ ν}$ is the field strength tensor

This structure encodes both particle dynamics and interactions.^[1]

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
$S U (2)$ → weak interaction
$S U (3)$ → strong interaction

These symmetries require the introduction of gauge fields and determine the interaction terms in the Lagrangian.^[2]

Operators and states

Physical states are constructed in a Fock space, where creation and annihilation operators act on the vacuum: $a_{𝐩}^{†} | 0 ⟩$

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.^[3]

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.^[1]

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

Index

Core theory Foundations Conceptual and interpretations Mathematical structure and systems Atomic and spectroscopy Wavefunctions and modes Quantum dynamics and evolution Measurement and information Quantum information and computing

Applications and extensions Quantum optics and experiments Open quantum systems Quantum field theory Statistical mechanics and kinetic theory Condensed matter and solid-state physics Plasma and fusion physics Timeline Advanced and frontier topics

Quantum Book II

Matter by scale

Quantum Book III

Methods and tools

Quantum Book IV

Data Analysis Techniques

Full contents

Measurement and information

Quantum information and computing

Physics:Quantum information theory
Physics:Quantum Qubit
Physics:Quantum Entanglement
Physics:Quantum Bell state
Physics:Quantum Gates and circuits
Physics:Quantum BB84
Physics:Quantum No-cloning theorem
Physics:Quantum Computing Algorithms in the NISQ Era
Physics:Quantum Noisy Qubits
Physics:Quantum error correction
Physics:Quantum Boson sampling
Physics:Quantum random access code
Physics:Quantum pseudo-telepathy
Physics:Quantum network
Physics:Quantum money

Quantum optics and experiments

Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
Physics:Quantum Stern-Gerlach experiment
Physics:Quantum Experimental quantum physics
Physics:Quantum optics
Physics:Quantum optics beam splitter experiments
Physics:Quantum Mach-Zehnder interferometer
Physics:Quantum Hong-Ou-Mandel effect
Physics:Quantum eraser experiment
Physics:Quantum delayed-choice quantum eraser
Physics:Quantum Ultra fast lasers
Template:Quantum optics operators

Open quantum systems

Quantum field theory

Physics:Quantum field theory (QFT) basics
Physics:Quantum field theory (QFT) core
Physics:Quantum Fields and Particles
Physics:Quantum Second quantization
Physics:Quantum Fock space
Physics:Quantum Harmonic Oscillator field modes
Physics:Quantum Creation and annihilation operators
Physics:Quantum vacuum fluctuations
Physics:Quantum Casimir effect
Physics:Quantum Propagators in quantum field theory
Physics:Quantum Feynman diagrams
Physics:Quantum Path integral formulation
Physics:Quantum Renormalization in field theory
Physics:Quantum Renormalization group
Physics:Quantum Field Theory Gauge symmetry
Physics:Quantum Spontaneous symmetry breaking
Physics:Quantum Non-Abelian gauge theory
Physics:Quantum Electrodynamics (QED)
Physics:Quantum chromodynamics (QCD)
Physics:Quantum Electroweak theory
Physics:Quantum Standard Model
Physics:Quantum triviality
Physics:Quantum confinement problem

Statistical mechanics and kinetic theory

Physics:Quantum Statistical mechanics
Physics:Quantum Partition function
Physics:Quantum Distribution functions
Physics:Quantum Liouville equation
Physics:Quantum Kinetic theory
Physics:Quantum Boltzmann equation
Physics:Quantum BBGKY hierarchy
Physics:Quantum Relaxation and thermalization
Physics:Quantum Thermodynamics

Condensed matter and solid-state physics

Physics:Quantum Band structure
Physics:Quantum Fermi surfaces
Physics:Quantum Landau levels
Physics:Quantum fractional Hall effect
Physics:Quantum Semiconductor physics
Physics:Quantum Phonons
Physics:Quantum Electron-phonon interaction
Physics:Quantum Exchange interaction
Physics:Quantum Superconductivity
Physics:Quantum Topological phases of matter
Physics:Quantum anyon
Physics:Quantum well
Physics:Quantum spin liquid
Physics:Quantum spin Hall effect
Physics:Quantum phase transition
Physics:Quantum critical point
Physics:Quantum dot

Plasma and fusion physics

Physics:Quantum Fusion reactions and Lawson criterion
Physics:Quantum Plasma (fusion context)
Physics:Quantum Magnetic confinement fusion
Physics:Quantum Inertial confinement fusion
Physics:Quantum Plasma instabilities and turbulence
Physics:Quantum Tokamak core plasma
Physics:Quantum Tokamak edge physics and recycling asymmetries
Physics:Quantum Stellarator

Timeline

Physics:Quantum mechanics/Timeline
Physics:Quantum mechanics/Timeline/Pre-quantum era
Physics:Quantum mechanics/Timeline/Old quantum theory
Physics:Quantum mechanics/Timeline/Modern quantum mechanics
Physics:Quantum mechanics/Timeline/Quantum field theory era
Physics:Quantum mechanics/Timeline/Quantum information era
Physics:Quantum mechanics/Timeline/Quantum technology era
Physics:Quantum mechanics/Timeline/Quiz

Advanced and frontier topics

Physics:Quantum topology
Physics:Quantum battery
Physics:Quantum Supersymmetry
Physics:Quantum Black hole thermodynamics
Physics:Quantum Holographic principle
Physics:Quantum gravity
Physics:Quantum De Sitter invariant special relativity
Physics:Quantum Doubly special relativity
Physics:Quantum arithmetic geometry
Physics:Quantum unsolved problems
Physics:Quantum Yang-Mills mass gap
Physics:Quantum gravity problem
Physics:Quantum black hole information paradox
Physics:Quantum dark matter problem
Physics:Quantum neutrino mass problem
Physics:Quantum matter-antimatter asymmetry problem

References

↑ ^1.0 ^1.1 ^1.2 Peskin, M. E.; Schroeder, D. V. An Introduction to Quantum Field Theory (1995).
↑ ^2.0 ^2.1 Weinberg, S. The Quantum Theory of Fields (1995).
↑ ^3.0 ^3.1 Schwartz, M. D. Quantum Field Theory and the Standard Model (2014).
↑ Noether, E. (1918). Invariant variation problems.
↑ Feynman, R. P. (1949). Space-time approach to quantum electrodynamics.

Author: Harold Foppele

Source attribution: Quantum field theory (QFT) core

[peskin-1] 1.0 ^1.1 ^1.2 Peskin, M. E.; Schroeder, D. V. An Introduction to Quantum Field Theory (1995).

[weinberg-2] 2.0 ^2.1 Weinberg, S. The Quantum Theory of Fields (1995).

[schwartz-3] 3.0 ^3.1 Schwartz, M. D. Quantum Field Theory and the Standard Model (2014).

[noether-4] Noether, E. (1918). Invariant variation problems.

[feynman-5] Feynman, R. P. (1949). Space-time approach to quantum electrodynamics.

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Contents

Fields and quantization

Lagrangian formulation

Symmetry and gauge structure

Operators and states

Interactions and Feynman diagrams

Renormalization

See also

Index

Full contents

Foundations

Conceptual and interpretations

Mathematical structure and systems

Atomic and spectroscopy

Wavefunctions and modes