🔬Quantum Field Theory Unit 4 – Interacting Fields & Perturbation Theory

Key Concepts

• Quantum field theory (QFT) provides a framework for describing the behavior of quantum systems with an infinite number of degrees of freedom
• Interacting fields introduce coupling between different quantum fields, allowing for the description of particle interactions and decays
• Perturbation theory is a method for approximating solutions to complex interacting field equations by treating interactions as small perturbations to free field solutions
• Feynman diagrams are pictorial representations of mathematical expressions describing the behavior of interacting particles
  • Consist of lines representing particle propagators and vertices representing interaction points
  • Enable the calculation of scattering amplitudes and cross-sections (e.g., for electron-positron annihilation)
• Renormalization is a process of redefining the parameters of a theory to absorb divergences that arise in perturbative calculations
• Gauge theories (e.g., quantum electrodynamics and quantum chromodynamics) are a class of QFTs that describe the interactions of matter fields with gauge fields
• The Standard Model of particle physics is a gauge theory that unifies the electromagnetic, weak, and strong interactions

Mathematical Foundations

• Lagrangian formalism is used to derive the equations of motion for quantum fields
  • The Lagrangian density is a function of the fields and their derivatives
  • The action is the integral of the Lagrangian density over spacetime
• Canonical quantization promotes classical fields to quantum operators satisfying commutation or anticommutation relations
• Path integral formulation expresses quantum amplitudes as integrals over all possible field configurations weighted by the exponential of the action
  • Provides a non-perturbative approach to QFT
  • Enables the derivation of Feynman rules for perturbative calculations
• Symmetries play a crucial role in QFT, with conserved quantities (e.g., charge, energy, and momentum) arising from continuous symmetries via Noether's theorem
• Gauge invariance is a local symmetry that requires the introduction of gauge fields (e.g., the photon field in quantum electrodynamics)

Interacting Fields Basics

• Interacting fields are described by Lagrangian densities that include interaction terms coupling different fields
  • For example, the interaction term in quantum electrodynamics is $-e\bar{\psi}\gamma^{\mu}A_{\mu}\psi$, where $\psi$ is the electron field, $A_{\mu}$ is the photon field, and $e$ is the electron charge
• The strength of interactions is characterized by coupling constants (e.g., the fine-structure constant $\alpha \approx 1/137$ in quantum electrodynamics)
• Interacting field equations are generally non-linear and cannot be solved exactly
• The interaction picture is a formulation of quantum mechanics that separates the time evolution of a system into free and interaction parts
  • Useful for perturbative calculations in QFT
• The S-matrix (scattering matrix) relates the initial and final states of a quantum system and encodes the probability amplitudes for various scattering processes
  • Calculated using perturbation theory in interacting field theories

Perturbation Theory Overview

• Perturbation theory is a method for finding approximate solutions to problems that cannot be solved exactly
• In QFT, perturbation theory is used to calculate scattering amplitudes and cross-sections for interacting particles
• The perturbative expansion expresses the S-matrix as a power series in the coupling constant
  • Each term in the series corresponds to a specific order in the perturbative expansion (e.g., tree-level, one-loop, two-loop)
• Feynman rules are a set of prescriptions for translating Feynman diagrams into mathematical expressions for scattering amplitudes
  • Derived from the Lagrangian density of the theory using the path integral formalism
• Divergences can arise in perturbative calculations, requiring regularization techniques (e.g., dimensional regularization) to isolate and remove the infinities
• Renormalization is the process of absorbing the divergences into redefined parameters of the theory (e.g., mass, charge, and field strength)

Feynman Diagrams

• Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of interacting particles
• External lines represent initial and final state particles, while internal lines represent virtual particles that mediate the interaction
• Vertices represent interaction points where particles are created, annihilated, or scattered
  • The type of vertex depends on the specific interaction (e.g., electron-photon vertex in quantum electrodynamics)
• Propagators are internal lines that describe the propagation of virtual particles between interaction points
  • Represented by a line with an arrow indicating the direction of propagation
  • Different particle types have different propagators (e.g., fermion propagator, photon propagator)
• Feynman diagrams can be translated into mathematical expressions using Feynman rules
  • Each line and vertex is associated with a specific mathematical factor
  • The overall amplitude is obtained by multiplying these factors and integrating over the momenta of internal lines
• Higher-order diagrams involve more interaction vertices and virtual particle exchanges, leading to more complex calculations

Renormalization

• Renormalization is the process of redefining the parameters of a theory to absorb the divergences that arise in perturbative calculations
• Bare parameters (e.g., mass, charge) are the parameters that appear in the original Lagrangian density
• Renormalized parameters are the physically observable quantities that include the effects of quantum corrections
  • Related to bare parameters through renormalization constants (e.g., $m_{\text{renormalized}} = Z_{m} m_{\text{bare}}$)
• Counterterms are additional terms added to the Lagrangian density to cancel the divergences in perturbative calculations
• Renormalization group equations describe how the renormalized parameters change with the energy scale of the process
  • Lead to the concept of running coupling constants (e.g., the running of the fine-structure constant in quantum electrodynamics)
• Renormalizability is a property of a theory that ensures divergences can be consistently absorbed into redefined parameters at all orders in perturbation theory
  • The Standard Model of particle physics is a renormalizable theory

Applications in Particle Physics

• Quantum electrodynamics (QED) is the QFT describing the electromagnetic interaction between charged particles
  • Successfully predicts the anomalous magnetic moment of the electron and the Lamb shift in atomic spectra
• Quantum chromodynamics (QCD) is the QFT describing the strong interaction between quarks and gluons
  • Explains the confinement of quarks inside hadrons (e.g., protons and neutrons) and the asymptotic freedom of the strong interaction at high energies
• Electroweak theory unifies the electromagnetic and weak interactions into a single framework
  • Predicts the existence of the W and Z bosons, which mediate the weak interaction
• The Higgs mechanism is a process by which particles acquire mass through their interaction with the Higgs field
  • Confirmed by the discovery of the Higgs boson at the Large Hadron Collider in 2012
• Perturbative QCD calculations are used to predict cross-sections for various processes in high-energy particle collisions (e.g., jet production, Higgs boson production)

Advanced Topics & Current Research

• Non-perturbative methods, such as lattice QFT, are used to study strong-coupling phenomena in QCD (e.g., quark confinement, hadron masses)
• Effective field theories (EFTs) are used to describe low-energy physics in terms of a simplified set of degrees of freedom
  • Examples include chiral perturbation theory for low-energy QCD and the Fermi theory of weak interactions
• Supersymmetry (SUSY) is a proposed symmetry between bosons and fermions that could solve various problems in the Standard Model (e.g., hierarchy problem, dark matter)
  • Supersymmetric QFTs are an active area of research
• String theory is a framework that attempts to unify all fundamental interactions, including gravity, by describing particles as vibrations of one-dimensional objects called strings
  • Involves the concept of extra spatial dimensions and has inspired various developments in QFT (e.g., AdS/CFT correspondence)
• Quantum gravity aims to develop a QFT that consistently incorporates general relativity
  • Approaches include loop quantum gravity and causal set theory
• Quantum computing and quantum information theory have led to new insights and applications of QFT (e.g., quantum error correction, holographic entanglement entropy)