15.4 Introduction to quantum field theory

Quantum field theory unites quantum mechanics and special relativity, addressing particle creation and annihilation. It treats particles as excitations of underlying fields, explaining interactions through field exchanges and virtual particles.

Second quantization promotes classical fields to quantum operators, allowing particle number fluctuations. Creation and annihilation operators act on quantum states, while commutation relations determine particle statistics. This framework applies to various field types, including scalar, fermionic, and gauge fields.

Foundations of Quantum Field Theory

Motivation for quantum field theory

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• Non-relativistic quantum mechanics falls short in describing particle creation and annihilation processes (radioactive decay)
• Incompatibility with special relativity led to need for theory unifying quantum mechanics and relativity (electron-positron pair production)
• Fundamental interactions in nature require field-based approach electromagnetic force mediates through photon exchange, strong nuclear force binds quarks via gluons
• Quantum field theory treats particles as excitations of underlying fields explains interactions through field exchanges (virtual particles)

Concept of second quantization

• Classical fields promoted to quantum operators allow for particle number fluctuations
• Creation and annihilation operators act on quantum states to add or remove particles
• Quantization applied to various field types scalar fields (Higgs boson), fermionic fields (electrons), gauge fields (photons)
• Fock space represents multi-particle states built from vacuum state
• Commutation relations for bosonic fields $[a(k), a^\dagger(k')] = \delta(k-k')$ determine particle statistics
• Anticommutation relations for fermionic fields ${a(k), a^\dagger(k')} = \delta(k-k')$ ensure Pauli exclusion principle

Formulations and Applications

Principles of canonical quantization

• Hamiltonian formulation of classical field theory provides framework for quantization
• Canonical variables promoted to operators maintain classical Poisson bracket structure
• Equal-time commutation relations preserve causality in quantum theory
• Lagrangian density and action $S = \int d^4x \mathcal{L}$ form basis for field equations
• Euler-Lagrange equations derived from action principle yield field equations of motion

Applications in particle physics

• Quantum Electrodynamics (QED) quantizes electromagnetic field describes electron-photon interactions
• Dirac equation governs behavior of electrons and positrons in QED
• Feynman diagrams visually represent QED processes (electron-positron annihilation)
• Standard Model unifies fundamental particles quarks, leptons, gauge bosons
• Electroweak theory combines electromagnetic and weak interactions
• Quantum Chromodynamics (QCD) describes strong nuclear force between quarks
• Gauge theories incorporate local symmetries (U(1) for QED, SU(3) for QCD)
• Spontaneous symmetry breaking explains origin of particle masses
• Higgs mechanism generates masses for W and Z bosons
• Renormalization handles infinities in quantum field theory calculations
• Running coupling constants describe strength of interactions at different energy scales