The generalized Serre – Swan theorem is a cornerstone of classical field theory

Classical field theory admits the adequate geometric formulation in the terms of fiber bundles and graded manifolds (Archive). Why? A cornerstone of classical field theory is the generalized Serre – Swan theorem.

Let X be a compact smooth manifold and C(X) the ring of smooth functions on X. The original Serre – Swan theorem states that a C(X)-module is a projective module of finite rank if and only if it is isomorphic to a module of sections of some vector bundle over X.  This theorem has been extended to an arbitrary smooth manifold due to the fact that any smooth manifold admits a finite manifold atlas.

It follows from the Serre – Swan theorem that, if classical fields are assumed to constitute a projective C(X)-module of finite rank, they are represented by sections of a vector bundle.

In a general setting, theory of Grassman-graded even and odd classical fields is considered. There are different models of odd classical fields in the terms of graded manifolds and supermanifolds. Combination of the well-known Batchelor theorem and the above mentioned Serre – Swan theorem results in a generalization of the Serre – Swan theorem to graded manifolds as follows.

Given a smooth manifold X, a graded commutative C(X)-algebra is isomorphic to the structure ring of a graded manifold with a body X if and only if it is the exterior algebra of some projective C(X)-module of finite rank.

References:

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

G.Sardanashvily,  Classical field theory. Advanced mathematical formulation, arXiv: 0811.0331

Metric gravity as a non-quantized Higgs field

If gravity is a pseudo-Riemannian metric, it is a Higgs field. Being a Higgs field, a metric gravitational field is non-quantized.

Classical field theory is adequately described in terms of fibre bundles (Archiv). Classical gravitation theory formulated in these terms is metric-affine theory whose dynamic variables are a pseudo-Riemannian metric, treated as a metric gravitational field, and a general linear connection on a world manifold X.

We concentrate our attention to a metric gravitation field.  A pseudo-Riemannian metric on a world manifold X is defined as a global section of the quotient LX/SO(1,3) of the linear frame bundle LX by the Lorentz group SO(1,3). Therefore, it exemplifies a Higgs field in classical field theory on fibre bundles.

Its Higgs character is displayed as follows. Given different pseudo-Riemannian metrics g and g', the representations of the holonomic coframes dx by the Dirac matrices acting on Dirac spinor fields are not equivalent.

It follows that a Dirac spinor field can not be considered in the case of a superposition of different metric gravitational fields. Therefore, quantization of a metric gravitational field fails to satisfy the superposition principle, and we think that it is non-quantized.

References:

G. Sardanashvily, Metric gravity as a non-quantized Higgs field (2010).

What are gauge symmetries?

In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial.

In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory. It comes from gauge theory on principal bundles whose vertical automorphisms, called the gauge transformations, are gauge symmetries of the Yang – Mills Lagrangian of gauge fields. Gauge symmetries of gravitation theory are general covariant transformations.

A gauge symmetry of a Lagrangian L is defined as a differential operator on some vector bundle E taking its values in the linear space of (variational or exact) symmetries of L. Therefore, a gauge symmetry of L depends on sections of E and their partial derivatives. For instance, this is the case of gauge symmetries in classical field theory.

Gauge symmetries possess the following two peculiarities.

(i) Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy first Noether’s theorem, but the corresponding conserved current J^μ takes a particular superpotential form J^μ = W^μ + d_νU^νμ where the first term W^μ vanishes on solutions of the Euler – Lagrange equations and the second one is a boundary term, where U^νμ is called a superpotential.

(ii) In accordance with second Noether’s theorem there is one-to-one correspondence between the gauge symmetries of a Lagrangian and the Noether identities which the Euler–Lagrange operator satisfies. Consequently, gauge symmetries characterize the degeneracy of a Lagrangian system.

Note that, in quantum field theory, a generating functional fail to be invariant under gauge transformations, and gauge symmetries are replaced with the BRST symmetries, depending on ghosts and acting both on fields and ghosts.

References:

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

G.Gaichetta, L.Mangiarotti, G.Sardanashvily, On the notion of gauge symmetries of generic Lagrangian field theory, arXiv: 0807.3003

Why connections in classical mechanics?

The main reasons why connections play a prominent role in many theoretical models field models lie in the fact that they enable us to deal with well (globally, invariantly) defined objects.

Connections in classical field theory have been discussed (Why connections in classical field theory?).

Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles Q->R over the time axis R (Mechanics as particular classical field theory). Its velocity phase space is the first order jet bundle JQ->R. Its momentum phase space is the vertical cotangent bundle V*Q of Q. The concept of a connection is the central ingredient in this geometric formulation as follows.

(i) An essential difference between classical mechanics and field theory lies in the fact that connections on a fibre bundle Q->R are flat and, therefore, they fail to be dynamic variables. They describe non-relativistic reference frames. This fact enables us to define relative velocities and accelerations, and describe non-relativistic mechanics with respect to different reference frames.

In particular, one can define a free motion equation and the geodesic reference frame for it which is called the inertial reference frame. However, an absolute inertial frame fails to be defined.

(ii) Equations of motion of non-relativistic mechanics almost always are of second order. Second order dynamic equations on a fiber bundle Q->R are conventionally defined as the holonomic connections on the jet bundle JQ->R. These equations also are represented by connections on the jet bundle JQ->Q and, due to the canonical imbedding of JQ to the tangent bundle TQ, they are proved to be equivalent to non-relativistic geodesic equations on TQ.

(iii) In Hamiltonian non-relativistic mechanics on the momentum phase space V*Q,

Hamiltonian connections on V*Q->R define the Hamilton equations.

References:

L.Mangiarotti, G.Sardanashvily, Gauge Mechanics (WS, 1998)

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced mechanics. Mathematical introduction, arXiv: 0911.0411