Foundations of Relativistic Quantum Fields

Derive the Klein-Gordon equation from relativistic principles and define its Lagrangian density.

Apply Noether's theorem to the Klein-Gordon field to derive its conserved energy-momentum tensor and charge current.

Apply canonical quantization to the Klein-Gordon field, imposing equal-time commutation relations.

Expand the quantum scalar field in terms of creation and annihilation operators and derive their commutation relations.

Construct the Fock space for scalar particles and define the vacuum state.

Calculate the Hamiltonian and momentum operators in terms of creation and annihilation operators, identifying the zero-point energy.

Derive the Feynman propagator for the scalar field as the vacuum expectation value of a time-ordered product of fields.

Demonstrate how imposing commutation relations ensures the locality of observables and preserves causality outside the light cone.