Quantum field theory is the foundation of particle physics, blending quantum mechanics and special relativity. It describes particles as excitations of fields, explaining their behavior and interactions. This framework is crucial for understanding the quantum nature of fundamental forces and particles.

In quantum electrodynamics, these principles are applied to electromagnetic interactions. QED uses second quantization to treat photons and electrons as field excitations, allowing for precise calculations of electromagnetic processes and providing a deep understanding of light-matter interactions.

Quantum Field Theory Principles

Fundamental Concepts and Particle Behavior

• Quantum field theory (QFT) merges quantum mechanics with special relativity describing subatomic particle behavior and interactions
• Fields in QFT function as operator-valued functions of spacetime representing fundamental entities rather than particles
• Locality principle in QFT dictates particle interactions occur at specific spacetime points preserving causality
• Virtual particles in QFT explain force mediation between real particles through force carrier exchange (photons for electromagnetic force)

Symmetries and Mathematical Techniques

• Symmetries play crucial role in QFT structure (Lorentz invariance, gauge invariance)
• Lorentz invariance ensures physical laws remain consistent across all inertial reference frames
• Gauge invariance allows for redundant degrees of freedom in field descriptions without affecting physical observables
• Renormalization technique in QFT handles calculation infinities enabling meaningful predictions of observable quantities
  • Involves introducing a cutoff scale and redefining parameters to absorb divergences
  • Allows for consistent treatment of quantum corrections to particle properties (mass, charge)

Field Quantization and Particle Interpretation

• Fields undergo quantization in QFT leading to particle interpretation as field excitations
• Quantum harmonic oscillator analogy helps understand field quantization
  • Energy levels of oscillator correspond to particle number states in QFT
  • Ground state of field represents vacuum state with zero particles
• Particle-wave duality naturally emerges from field quantization
  • Particles arise as localized excitations of fields
  • Wave-like behavior results from field's extended nature in space

Second Quantization in QED

Electromagnetic Field Quantization

• Second quantization treats particles as excitations of underlying quantum fields
• Electromagnetic field quantization in QED leads to photon concept as field quanta
• Photons emerge as massless, spin-1 particles mediating electromagnetic interactions
• Quantized electromagnetic field describes both wave-like and particle-like aspects of light

Electron Field Quantization

• Electron field quantization in QED allows creation and annihilation of electron-positron pairs
• Dirac field describes electrons and positrons as excitations of a single quantum field
• Quantized electron field incorporates spin-1/2 nature and negative energy solutions (positrons)
• Field quantization naturally accounts for indistinguishability of identical particles (electrons)

Multi-Particle Systems and Interactions

• Second quantization formalism describes multi-particle systems using field operators acting on vacuum state
• Enables calculation of transition amplitudes and cross-sections for electromagnetic processes
  • Example: Electron-positron annihilation into two photons
  • Example: Compton scattering of photons by electrons
• Normal ordering in second quantization defines observables and avoids infinite vacuum energies
  • Rearranges creation and annihilation operators to place annihilation operators to the right
  • Subtracts vacuum expectation values to remove divergent terms

Creation and Annihilation Operators

Operator Properties and Commutation Relations

• Creation operators increase particle number in quantum state, annihilation operators decrease it
• Operators satisfy specific commutation relations reflecting particle statistics
  • Bosons (photons): $[a, a^\dagger] = 1$ (commutation)
  • Fermions (electrons): $\{a, a^\dagger\} = 1$ (anticommutation)
• Vacuum state defined as state annihilated by all annihilation operators: $a_k|0\rangle = 0$
• Creation and annihilation operators construct field operators, fundamental QFT objects
  • Example: Scalar field operator $\phi(x) = \int \frac{d^3k}{(2\pi)^3} \frac{1}{\sqrt{2\omega_k}} (a_k e^{-ikx} + a_k^\dagger e^{ikx})$

Fock Space and Multi-Particle States

• Fock space (QFT Hilbert space) constructed using creation operators acting on vacuum state
• Basis states in Fock space represent multi-particle configurations
  • Example: Two-photon state $|k_1, k_2\rangle = a_{k_1}^\dagger a_{k_2}^\dagger |0\rangle$
• Creation and annihilation operators enable description of particle interactions through action on multi-particle states
  • Example: Photon absorption by an electron $a_k^\dagger |e^-\rangle \rightarrow |e^-\gamma\rangle$

Observables and Normal Ordering

• Normal-ordered product of creation and annihilation operators defines physical observables in QFT
• Normal ordering places creation operators to the left of annihilation operators
  • Example: Number operator $N = a^\dagger a$ (already in normal order)
• Wick's theorem relates time-ordered products to normal-ordered products plus contractions
  • Facilitates calculation of correlation functions and scattering amplitudes

Lagrangian and Hamiltonian Formulations of QED

QED Lagrangian Density

• QED Lagrangian density incorporates terms for free electron field, free electromagnetic field, and their interaction
• Dirac Lagrangian describes free electron field: $\mathcal{L}_D = \bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi$
• Maxwell Lagrangian describes free electromagnetic field: $\mathcal{L}_M = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$
• Interaction term couples electron current to electromagnetic potential: $\mathcal{L}_\text{int} = -e\bar{\psi}\gamma^\mu\psi A_\mu$
• Complete QED Lagrangian: $\mathcal{L}_\text{QED} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}$
  • Covariant derivative $D_\mu = \partial_\mu + ieA_\mu$ ensures gauge invariance

Gauge Invariance and Conservation Laws

• Gauge invariance crucial symmetry of QED Lagrangian leading to electric charge conservation
• Local gauge transformation: $\psi \rightarrow e^{-ie\alpha(x)}\psi, A_\mu \rightarrow A_\mu + \partial_\mu\alpha(x)$
• Noether's theorem relates gauge invariance to conserved current: $\partial_\mu j^\mu = 0, j^\mu = e\bar{\psi}\gamma^\mu\psi$

Hamiltonian Formulation and Path Integrals

• QED Hamiltonian derived from Lagrangian using Legendre transformation and canonical quantization
• Hamiltonian formulation allows calculation of energy spectrum and time evolution of QED systems
  • Example: Hydrogen atom energy levels including QED corrections (Lamb shift)
• Path integral formulation provides alternative approach to deriving QED Lagrangian and calculating observables
  • Feynman path integral: $Z = \int \mathcal{D}\psi \mathcal{D}\bar{\psi} \mathcal{D}A_\mu e^{iS[\psi,\bar{\psi},A_\mu]}$
  • Generates perturbation series and Feynman diagrams for scattering amplitudes