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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