Course Assignments

• Deriving the Hamiltonian of a Quantized Scalar Field How is the Hamiltonian of a free scalar field expressed in terms of creation and annihilation operators?

  My solution starts from the free Klein–Gordon field Hamiltonian density and rewrites the total Hamiltonian using a_p and a_p^†, including the vacuum-energy term.

• Deriving the Momentum Operator for a Quantized Scalar Field How is the momentum operator of a free scalar field expressed in terms of creation and annihilation operators?

  The solution expands the scalar field and its conjugate momentum into Fourier modes and shows that the total momentum is a momentum-weighted number operator, proportional to ∫ d³p p a_p^†a_p.

• How do Dirac spin generators satisfy the Lorentz algebra?
• Wick's Theorem Check for Four Scalar Fields
• Why do Dirac bilinear operators commute at spacelike separation? How does fermionic anticommutation imply that local Dirac bilinears commute at spacelike separation?

  Microcausality in quantum field theory: I'm showing that operators of the form ψ(x)Γ¹ψ(x) and ψ(y)Γ²ψ(y) commute when x and y are spacelike separated. Although the individual Dirac fields anticommute, each bilinear contains an even number of fermion fields, so the minus signs cancel when the two bilinears are exchanged.