Foundations of Modern Physics 3: Classical field theory

Classical field theory admits a comprehensive formulation in terms of fibre bundles and graded manifolds.

Observable classical fields are even (commutative) electromagnetic and gravitational fields and odd (anti-commutative) Dirac spinor fields. One also considers classical non-Abelian gauge fields and Higgs fields. Classical gauge and gravitation field theories are conventionally formulated in the geometric terms of fibre bundles. Generalizing this geometric formulation, one comes to the following.

Axiom I. Even classical fields are sections of smooth fibre bundles over smooth manifolds.

As a consequence, it is essential that classical fields on a smooth manifold X represented by sections of a fibre bundle over X form a projective module of finite rank over the ring C(X) of smooth real functions on X in accordance with the well-known Serre – Swan theorem.

There are different descriptions of odd fields either on graded manifolds or supermanifolds. Note that both graded manifolds and supermanifolds are described in terms of sheaves of graded commutative algebras, but graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves on supervector spaces. In order to treat odd and even fields on the same level, one can follow the above mentioned Serre – Swan theorem, which states that, if an anti-commutative algebra is generated by a projective C(X)-module of finite rank, it is isomorphic to the algebra of graded functions on a graded manifold whose body is X.

Axiom II. Odd classical fields on a smooth manifold X are elements of the structure algebra of a graded manifold whose body is X.

Dynamic equations of all observable classical fields including electromagnetic, spinor and gravitational fields are Euler – Lagrange equations derived from a Lagrangian. This fact leads us to the following.

Axiom III. Classical field theory is a Lagrangian theory.

Lagrangian theory on fibre bundles and graded manifolds is adequately formulated in algebraic terms of the variational bicomplex of exterior forms on jet manifolds. The Euler – Lagrange operator is a coboundary operator of this bicomplex and its cohomology provides the first variational formula, the first Noether theorem and conservation laws. Thus classical field theory is formulated in a complete way, but we obtain something more, namely, its prequantization.

Quantization of Lagrangian field theory essentially depends on its degeneracy characterized by a family of non-trivial reducible Noether identities. These Noether identities can obey first-stage Noether identities, which in turn are subject to the second-stage ones, and so on. This hierarchy of Noether identities is described by the exact Koszul - Tate chain complex of antifields. The second Noether theorem associates to this Koszul--Tate complex the cochain sequence of ghosts with the ascent gauge operator, whose components are gauge and higher-stage gauge symmetries of Lagrangian field theory. If gauge symmetries are algebraically closed, this gauge operator admits a nilpotent BRST prolongation. Thus we come to the BRST extension of original Lagrangian field theory which is a first step towards its quantization.

References:

G.Giachetta, L. Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (WS, 2009) (#)

G.Sardanashvily, Classical field theory. Advanced mathematical formulation, Int. J. Geom. Methods Modern Physics, 5 (2008) 1163 (#) 

WikipediA: Covariant classical field theory

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