https://physicslly.github.io/physicsllyAdvanced notes on theoretical physics, quantum field theory, gravity, cosmology, mathematical physics, and the search for unification. 2026-07-02T18:20:18+08:00 physicslly https://physicslly.github.io/ Jekyll © 2026 physicslly /assets/img/favicons/favicon.ico /assets/img/favicons/favicon-96x96.png The Källén–Lehmann Spectral Representation: How the Full Propagator Encodes the Physical Spectrum2026-07-01T00:01:00+08:00 2026-07-01T00:01:00+08:00 https://physicslly.github.io/posts/kallen-lehmann-spectral-representation-propagator-spectrum/ physicslly 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. The Heat Kernel Expansion and Seeley–DeWitt Coefficients: Computing the One-Loop Effective Action2026-06-30T00:01:00+08:00 2026-06-30T00:01:00+08:00 https://physicslly.github.io/posts/heat-kernel-expansion-seeley-dewitt-coefficients-one-loop-effective-action/ physicslly 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. Schwinger-Dyson Equations and the Structure of the Quantum Effective Action2026-06-28T00:01:00+08:00 2026-06-28T19:09:45+08:00 https://physicslly.github.io/posts/schwinger-dyson-equations-structure-quantum-effective-action/ physicslly 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. Reflection Positivity and the Osterwalder–Schrader Axioms: Reconstructing Unitary QFT from Euclidean Correlators2026-06-28T00:01:00+08:00 2026-06-28T00:01:00+08:00 https://physicslly.github.io/posts/osterwalder-schrader-reflection-positivity-euclidean-qft/ physicslly 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. Spontaneous Symmetry Breaking, the Higgs Mechanism, and Goldstone's Theorem2026-06-26T00:01:00+08:00 2026-06-28T19:09:45+08:00 https://physicslly.github.io/posts/spontaneous-symmetry-breaking-higgs-mechanism-goldstone-theorem/ physicslly 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.