Abstract. In contrast with
QFT, classical field theory can be formulated
in a strict mathematical way if one defines even classical
fields as sections of smooth fiber bundles. Formalism of jet manifolds provides the
conventional language of dynamic systems (nonlinear differential equations
and operators) on fiber bundles.
Lagrangian theory on fiber bundles is
algebraically formulated in terms of the variational bicomplex of exterior
forms on jet manifolds where the Euler–Lagrange operator is present as a
coboundary operator. This formulation is generalized to Lagrangian theory of
even and odd fields on graded manifolds.
Cohomology of the variational bicomplex provides a solution of the global
inverse problem of the calculus of variations, states the first variational formula and Noether’s first theorem in a very general setting of supersymmetries depending on higher-order
derivatives of fields. A theorem on the
Koszul–Tate complex of reducible Noether identities and Noether’s inverse
second theorem extend an original field theory to
prequantum field-antifield BRST theory. Particular field models, jet techniques and some quantum outcomes are
discussed.
Contents
I. Introduction
II. ACFT. The general framework
1. The main postulate, 2. Jet manifolds, 3. Jets and connections, 4.
Lagrangian theory
of even fields, 5. Odd fields, 6. The algebra of even and odd fields, 7. Lagrangian theory
of even and odd fields, 8. Noether’s
first theorem in a general
setting, 9. The Koszul–Tate complex of Noether identities, 10. Noether’s
inverse second theorem, 11. BRST extended field theory, 12.
Local BRST cohomology.
III. Particular models
13. Gauge theory of principal connections, 14. Topological Chern–Simons
theory, 15.
Topological BF theory, 16. SUSY gauge theory, 17. Field theory on
composite bundles,
18. Symmetry breaking and Higgs fields, 19. Dirac
spinor fields, 20. Natural and gauge natural
bundles. 21. Gauge gravitation theory, 22. Covariant Hamiltonian field theory,
23. Time-dependent mechanics, 24. Jets of submanifolds, 25. Relativistic
mechanics, 26. String theory.
IV. Quantum outcomes
27. Quantum master equation, 28. Gauge fixing procedure, 28. Green
function identities