# Geometry of Classical Physics in terms of Gauge Theory geometry?

• A
jordi
Reading the interesting book "Groups_and_Manifolds__Lectures_for_Physicists_with_Examples_in_Mathematica", in the introduction it is stated:

(...) we have, within our contemporary physical paradigm, a rather simple and universal scheme of interpretation of the Fundamental Interactions and of the Fundamental Constituents of Matter based on the following few principles:

A)The categorical reference frame is provided by Field Theory defined by some action [Action] wherehttps://www.physicsforums.com/ms-local-stream%3A//EpubReader_3406DD30C1D59CF1682EF5E2E9D121CBA2E20E005093C4AF90FB236C1C2718/Content/OPS/graphic/07_Preface_fig_02.png [Lagrangian] denotes some Lagrangian depending on a set of fields Φ(x).

B)All fundamental interactions are described by connections A on principal fiber-bundles P(G,M) where G is a Lie group and the base manifold M is some space-time in d = 4 or in higher dimensions.

C)All the fields Φ describing fundamental constituents are sections of vector bundles B(G, V,M), associated with the principal one P(G,M) and determined by the choice of suitable linear representations D(G) ∶ V → V of the structural group G.

D)The spin zero particles, described by scalar fields ϕI , have the additional feature of admitting non-linear interactions encoded in a scalar potential V (ϕ) for whose choice general principles, supported by experimental confirmation, have not yet been determined.

E)Gravitational interactions are special among the others and universal since they deal with the tangent bundle TM →M to space-time. The relevant connection is in this case the Levi-Civita connection (or some of its generalizations with torsion) which is determined by a metric g on M.

Of course, nothing here is new, being completely standard (even though it is nice that it is explicitly stated).

But then I have thought: what about classical mechanics (both galilean and relativistic)?

Sometimes I have read that Classical Mechanics is just a field theory on 1+0 dimensions, which of course is right. The four-vector potential couples with the momentum in the "right way", as analogously with gauge fields.

But then there is the complexity of the quantization of both classical, non-relativistic particles and electromagnetism, as in the last chapter of "Lectures on QM", by Weinberg. So, we are mixing particles with fields. Also, classical mechanics is endowed with a symplectic geometry, which I do not know if it plays a role here or not.

My question is: is there a way to define particles as some kind of (possibly trivial) section on a bundle, such as the principles A) to E) above still hold, even in the case of combining particles with electromagnetism? In fact, in General Relativity I guess this is done in some way, since gravity couples with particles.

Classical mechanics is often described in terms of symplectic structures on tangent and cotangent bundles.
Do you know V. I. Arnold's book: Mathematical methods of classical mechanics, Springer ?

jordi
jordi
Yes, I know Arnold book, thank you.

My question can be rephrased as:

When we quantize at the same time particles and light, as Weinberg does in the last chapter of his QM lectures, at the classical level there are two different geometrical structures: on one hand, particles with a symplectic structure. On the other hand, light with a principal bundle structure. And these two structures are not connected (for example, if adding matter to Yang-Mills, matter becomes a section of the same principal bundle structure of light, so the geometric structure is "the same" for both matter and light, in this "field" case).

Is this not a problem? Maybe not.

But at least for "aesthetical" reasons, I would like to see if the symplectic structure could be put, in some way, "inside" the principal bundle structure of light.