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Thursday, 15 November 2012

My review “Axiomatic classical (prequantum) field theory”

G. Sardanashvily, Axiomatic classical (prequantum) field theory. Jet formalism
Abstract. In contrast with QFT, classical eld theory can be formulated in a strict mathematical way if one denes even classical elds as sections of smooth ber bundles. Formalism of jet manifolds provides the conventional language of dynamic systems (nonlinear dierential equations and operators) on ber bundles. Lagrangian theory on ber 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 elds on graded manifolds. Cohomology of the variational bicomplex provides a solution of the global inverse problem of the calculus of variations, states the rst variational formula and Noether’s rst theorem in a very general setting of  supersymmetries depending on higher-order derivatives of elds. A theorem on the Koszul–Tate complex of reducible Noether identities and Noether’s inverse second theorem extend an original eld theory to prequantum eld-antield BRST theory. Particular eld models, jet techniques and some quantum outcomes are discussed.


I. Introduction

II. ACFT. The general framework
1. The main postulate, 2. Jet manifolds, 3. Jets and connections, 4. Lagrangian theory
of even elds, 5. Odd elds, 6. The algebra of even and odd elds, 7. Lagrangian theory
of even and odd elds, 8. Noether’s rst theorem in a general setting, 9. The Koszul–Tate complex of Noether identities, 10. Noether’s inverse second theorem, 11. BRST extended eld 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 elds, 19. Dirac spinor elds, 20. Natural and gauge natural bundles. 21. Gauge gravitation theory, 22. Covariant Hamiltonian eld theory,
23. Time-dependent mechanics, 24. Jets of submanifolds, 25. Relativistic mechanics, 26. String theory.

IV. Quantum outcomes
27. Quantum master equation, 28. Gauge xing procedure, 28. Green function identities

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