Book I

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.

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.^[1]

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.^[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: $S = \int d^{4} x ℒ$

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.^[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
$S U (2)$ → weak interaction
$S U (3)$ → strong interaction

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

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.^[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]

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Physics:Quantum field theory (QFT) core

Fields and quantization

Lagrangian formulation

Symmetry and gauge structure

Operators and states

Interactions and Feynman diagrams

Renormalization

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