Fundamentals of Quantum Field Theory

• physical system: a physical system consists of one or more quantum fields and their associated excitations (particles). These fields are defined over spacetime and interact with each other according to specific rules.
  • physical systems in QFT:
    • electromagnetic field and its excitations (photons)
    • electron-positron (Dirac field) and its excitations (electron and positions)
    • system of interacting fields, e.g. Higgs field coupled to the electromagnetic field
  • physical system is characterized by its Lagrangian or Hamiltonian, which encodes the dynamics of the fields and its interactions
• state of the physical system: described by a quantum state in the Hilbert space associated with the system. This state encodes all the information about the system’s configuration and properties.
  • the state is typically represented as a state vector (e.g., $\ket{\psi}$) in Fock space, which is a space that accommodates variable numbers of particles (quanta of the fields).
  • states in QFT:
    • vacuum state $\ket{0}$, which represents the state with no particles (lowest energy state of the fields).
    • single-particle state $\ket{p}$, representing a single quantum of a field with momentum $p$.
    • multi-particle state $\ket{p_1, p_2, …}$ representing multiple quanta of the fields.
    • coherent state, which describes a classical-like configuration of a field (e.g., a classical electromagnetic wave).
  • The state of the system evolves over time according to the Schrödinger equation or the Heisenberg picture equations of motion.
• physical law of QFT: the fundamental principles and equations that govern the behavior of quantum fields and their interactions. These laws are derived from the Lagrangian density or Hamiltonian of the system, which encapsulates the dynamics and symmetries of the fields.
  • Key physical laws in QFT include:
    • The equations of motion for the fields
      • the Klein-Gordon equation for scalar fields
      • the Dirac equation for fermionic fields
      • Maxwell’s equations for the electromagnetic field
    • The symmetry principles (e.g., gauge invariance, Lorentz invariance) that constrain the form of interactions and lead to conservation laws (e.g., conservation of charge, energy, and momentum).
    • The quantization rules (e.g., canonical quantization or path integral quantization) that define how classical fields are promoted to quantum operators.
    • The interaction terms in the Lagrangian, which describe how fields couple to each other (e.g., the interaction between the electron field and the electromagnetic field in Quantum Electrodynamics (QED)).
  • Physical laws in QFT are subject to conditions of applicability. For example:
    • QFT is typically valid at energy scales where quantum effects dominate but gravitational effects are negligible.
    • Specific laws (e.g., the Standard Model of particle physics) are valid within certain energy ranges and under specific symmetry constraints.