N.S.B. Cosmic Center

Introduction: Generalized quantum theory. The functional integral and the effective action. Quantum field divergences and renormalization. Gauge field quantization. Chiral anomalies.

Dirac Quantization: Constrained systems and the Dirac technique. Relativistic particles, strings, and membranes. Vector fields. Spin-2 field. Fermionic fields.

The Effective Quantum Action: Definition. Iterative expansion in Planck's constant. Gaussian integral and mean values. Two-loop effective action. Graphical rules for higher orders.

Fermi-Bose Systems: Hyperfield formalism. Iterative expansion for the hyperfield effective action. Generalized gaussian integral. Generalized mean values. The two-loop hypersymmetric effective action. Complex fields. Quantum electrodynamics.

The Effective Action for Singular Gauge Systems: Non-Abelian gauge fields. Conventional approach. Dirac brackets and the functional space metric. Projective space approach. Effective Feynman rules.

The Renormalized Effective Action: General formalism. Two-loop contributions to the components of the effective action. The renormalized effective action for a scalar field.

The Regular Effective Action: Functional metric regularization. Gauge-field regularizing metric. Symmetrical sums. The photon self-energy.

The Regular Effective Action for Weyl Fermions: Introduction. Chiral transformations and the functional integral. The regular effective action for Weyl fermions. Extension to massive fermions. Conclusions. Appendix: The gauge-invariant triangle amplitude.

Appendix Four-dimensional integrals with gaussian cutoff. Formulae for symmetrical integration. Expanding the logarithm of an operator product. Useful expansions.