A derivation of the Källén–Lehmann spectral representation for the exact two-point function, its interpretation in terms of the physical particle spectrum, and the constraints unitarity and causality impose on the spectral density.
A derivation of the Seeley–DeWitt coefficients for a Laplace-type operator, and their use in extracting the one-loop effective action, renormalization, and curvature corrections in quantum field theory.
A derivation of the Schwinger-Dyson equations from the functional integral, their relation to the quantum effective action, and the structure of the 1PI hierarchy.
How reflection positivity encodes unitarity in Euclidean quantum field theory, and why the Osterwalder–Schrader reconstruction is the rigorous bridge between Euclidean correlators and a Wightman QFT.
A rigorous treatment of spontaneous symmetry breaking in quantum field theory: Goldstone's theorem, the Higgs mechanism, the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism, and the structure of the Standard Model scalar sector.
A rigorous treatment of the S-matrix program: LSZ reduction, the optical theorem and Cutkosky rules, dispersion relations, and modern on-shell methods including BCFW recursion and color-kinematics duality.
A rigorous treatment of chiral anomalies and the Atiyah-Singer index theorem, with emphasis on anomaly cancellation conditions, topological invariants, and the structural constraints these impose on consistent quantum field theories.
A rigorous geometric treatment of causal structure in general relativity: conformal compactification, Penrose diagrams, global hyperbolicity, the Raychaudhuri equation, the focusing theorem, and the Penrose-Hawking singularity theorems.
A rigorous treatment of the worldsheet formulation of string theory.
A rigorous treatment of Noether currents, Ward identities, and their role as symmetry constraints in quantum field theory: classical and quantum conservation laws, Ward-Takahashi and Slavnov-Taylor identities, anomaly matching, and applications to non-perturbative physics.