My review “Axiomatic classical (prequantum) field theory”

G. Sardanashvily, Axiomatic classical (prequantum) field theory. Jet formalism
arXiv:hep-th/0612182

  

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