A research-depth graduate course in Poincaré representations, Weyl and Dirac fields, Majorana systems, Clifford algebras, covariant scattering, spinor-helicity, instantons, anomalies, topological response and geometric fermions. Twelve independently authored units now combine detailed textbook exposition, two full intermediate derivations, extended worked calculations, reproducible numerical baselines, authentic credited scientific figures where available, 12 guided primary-paper reproductions, and instructor lectures from MIT and Cambridge. Students must distinguish reproducing established results from proposing and validating genuinely original research.
Published syllabus
The Lorentz group preserves the Minkowski interval; Poincaré symmetry also includes spacetime translations. Irreducible positive-energy unitary representations characterize relativistic particle states by mass and spin or helicity. Representation labels belong to the physical Hilbert space, whereas coordinates are observer choices.
Lorentz spinors transform under representations of the double cover of the proper orthochronous Lorentz group. A four-component Dirac spinor combines inequivalent Weyl representations; mass terms mix chiralities. The spinor representation is not the ordinary four-vector representation, even though bilinears form Lorentz tensors.
The Weyl equation describes massless chiral spin-half fields in two-component language. For massless on-shell particles, helicity and chirality are closely connected under the standard positive-energy identification; they are not generally interchangeable for massive particles. Weyl semimetals realize effective low-energy band crossings with related algebra but different microscopic symmetries.
The Dirac equation linearizes relativistic dispersion using gamma matrices satisfying the Clifford algebra. Its conserved current gives a consistent probability/charge interpretation within field theory. Positive and negative frequency solutions are reinterpreted through antiparticle creation operators after second quantization.
Majorana spinors obey a reality condition up to charge-conjugation conventions and describe neutral fermionic degrees of freedom under suitable representations. In superconductors, Majorana zero modes emerge as self-conjugate Bogoliubov quasiparticle operators, a related algebraic property without necessarily being elementary Majorana particles.
Clifford algebras are generated by elements whose anticommutator is fixed by a quadratic form. Gamma matrices provide a concrete representation for Lorentzian metrics and organize spinor bilinears and projection operators. Different matrix bases yield equivalent physics after suitable similarity transformations.
Lorentz-invariant scattering uses four-momenta constrained to mass shells and conservation laws. Mandelstam variables s,t,u simplify 2-to-2 kinematics and crossing relations. Physical cross sections require incoming flux normalization, final-state phase space and treatment of identical particles.
Twistors encode spacetime and null geometric relations using complex projective structures. Spinor-helicity variables factorize lightlike four-momenta into two-component spinors and simplify gauge-theory amplitudes. Their utility follows from Lorentz representations and massless on-shell kinematics, not from an automatic replacement of ordinary quantum theory.
Instantons are finite-action Euclidean solutions contributing nonperturbative sectors, while magnetic monopoles carry topological magnetic charge in appropriate gauge-Higgs theories. Their classification often involves homotopy groups of vacuum manifolds, but instanton number and monopole winding are different invariants.
Quantum anomalies occur when a classical symmetry fails under regularization of the quantum theory. A chiral anomaly is related to gauge fields and spectral flow, and anomaly cancellation is necessary for consistency of gauged symmetries. Boundary modes may compensate anomalies through inflow from a higher-dimensional bulk.
Theta terms, Chern–Simons terms and related topological actions capture global field configuration information. Their quantization and symmetry properties depend on dimension, manifold topology and boundary terms. Observable response can emerge from a term that is a total derivative in a local coordinate description.
Geometric approaches to field theory organize gauge connections, characteristic classes and spin structures on manifolds. Spin structure choices can change fermionic boundary conditions and spectra. Quantum-gravity-motivated geometric proposals should be separated from established predictions of gauge theories on fixed backgrounds.