Spectulations on Lagrangians

[booker]

General relativity is premised on the Einstein Equivalence Principle wedding Newton's equivalence of inertial and gravitatinal mass with special relativity.

 

One result is that particles not subject to an external force follow geodesics, where the geodesic is fully determied by the initial conditions; the tangent vectors of displacement and velocity.

 

So rather, instead, one can begin with geodesics as the 'fundemental premise', where inertial and gravitation mass become two names for the same gravitational charge.

 

Generalizing on geodesics, I'm would be temped to say "World lines are completely determined by the initial conditions; the tangent vectors of displacement and velocity."

 

(It would be getting a little ahead of the game to talk about particle fields rather than classical particles.)

 

In other words, as in general relativity where gravity is not a force, neither should the strong nor the electroweak force be forces, but described by curvatures on some R^N psuedo Riemann manifold.

 

Does this notion pertain to any legitimate study in physics?

[ajb]

The closest I can think of are Kaluza-Klein theories. These however are know to be in trouble when when quantises.

 

You should be aware that gauge theories (including gravity) are very geometric in nature. The fundamental geometric construction here is a (principle or associated vector) fibre bundle. For gravity the important bundle is the frame bundle of the space-time manifold and for Yang-Mills type theories it is a principle bundle over the space-time manifold. Matter are sections of anassociated vector bundle. The gauge field is recognised as a connection and the gauge transformations are nothing but changing local bundle coordinates. From a connection you can build a curvature.

 

The actions of gauge theories (no matter for now) are built from the curvatures. So yes, curvatures are important. But what I should stress is that one talks about a connection and a curvature on a Riemannian manifold, but what you really mean is connection in the frame bundle associated with the tangent bundle of a Riemannian manifold and the curvature of the said frame bundle. It is because the bundle structures are canonical that ones does not need to specify the frame bundle.

 

Some of this is outlined in my first year report for my PhD. Section 2 The Geometric Setting of Classical Field Theory is what you should pay attention to.

[booker]

The closest I can think of are Kaluza-Klein theories. These however are know to be in trouble when when quantises.

 

Could you be more specific? I'm a little bit familiar with Kaluza-Klein, having arrived at it somewhat circuitously, regauging electromagnetism.

 

Some of this is outlined in my first year report for my PhD. Section 2 The Geometric Setting of Classical Field Theory is what you should pay attention to.

 

I get a message "This file is damaged and could not be repaired," trying to open your thesis.

[iNow]

I get a message "This file is damaged and could not be repaired," trying to open your thesis.

 

The problem is local to your machine or network connection. It works fine when tested from here.

[ajb]

There is a couple of problems with KK-theories.

 

1) Fermions can not be accommodated.

2) The theory is non-remormalisable.

 

To get round this you need supersymmetry and strings.

[booker]

Thanks for the many links. I'll see where they might lead.