PART 3: Covariant Hamilton Field Theory under Fixed Spacetime

The following sections are included:

• Chapter 7 Covariant Canonical Field Equations

  • Euler-Lagrange equations of field theory

  • Non-uniqueness of the Lagrangian

  • Covariant canonical field equations

  • Canonical energy-momentum tensor

  • Non-uniqueness of the conjugate momentum fields ${\pi }_I^{\mu }$

  • Hybrid Hamiltonians

• Chapter 8 Canonical Transformations in Covariant Hamiltonian Field Theory

  • Generating functions of type F₁(ϕ_I, Φ_I, x)

  • Canonical transformation rule for the energy-momentum tensor

  • Generating functions of type F₂(ϕ_I, Π_I, x)

  • Generating functions of type F₃(Φ_I, π_I, x)

  • Generating functions of type F₄(π_I, Π_I, x)

  • Consistency checks of canonical transformation rules

  • Poisson brackets

  • Canonical invariance of Poisson brackets

  • Liouville’s theorem of covariant Hamiltonian field theory

  • Jacobi’s identity theorem in canonical field theory

  • Poisson’s theorem in canonical field theory

  • Hamilton-Jacobi equation

• Chapter 9 Examples of Hamiltonians in Covariant Field Theory

  • Ginzburg-Landau Hamiltonian

  • Klein-Gordon Hamiltonian for a real scalar field

  • Klein-Gordon Hamiltonian for complex fields

  • Maxwell Hamiltonian

  • Proca Hamiltonian

  • Coupled Klein-Gordon-Maxwell system

  • Covariant Dirac Hamiltonian

    • Conventional Dirac Lagrangian

    • Lorentz invariance of the Dirac equation

    • Regularized Dirac Lagrangian, Dirac Hamiltonian

• Chapter 10 Examples of Canonical Transformations in Covariant Hamiltonian Field Theory

  • Point transformations

  • Canonical shift of the conjugate momentum vector field ${\pi }_I^{\mu }$

  • Global and local gauge transformation of the fields ϕ_I

  • Interchange of canonical spinor variables

  • Generalized Noether theorem in the realm of field theories

  • Canonical transformation inducing an infinitesimal spacetime step

  • Gauge invariance of the electromagnetic 4-potential

  • Symmetry transformation of the coupled Klein-Gordon-Maxwell field and the pertaining conserved Noether current

• Chapter 11 U(1) and SU(N ) Gauge Theories in the Hamiltonian Formulation

  • Gauge theories as canonical transformations

  • U(1) gauge theory

  • SU(N) gauge theory

    • External gauge field

    • Inclusion of the gauge field dynamics

    • Hamiltonian of the free (uncoupled) gauge field

    • Inserting the gauge-invariant Hamiltonian ℋ₃ into the action integral

    • Locally gauge-invariant Lagrangian, Legendre transformation for a general system Hamiltonian

    • Klein-Gordon type system Hamiltonian

    • Hamiltonian of the Dirac system

    • Comparison with Pauli’s amended Lagrangian

  • Noether current of the SU(N) gauge transformation