The math of physics - Calculus of Variation?

[Mike2]

We escape the problems of particle physics by exploring the higher dimensions of String theory. When we have questions about String theory, we jump to the higher dimensions of M-theory to answer them. And some have purposed to use the higher dimensions of F-theory to answer questions about M-theory.

Yet, we do have principles that apply no matter what dimensionality we go to. Path integrals, the action of the Lagrangians, Noether's theorem, various kinds of symmetry, the geodesics of General Relativity. All these principles are applicable at every level of dimensionality we explore, and they are all expressible in terms of the functional calculus of variation. Least action is where the functional derivative is zero. The path integral is a functional integral integrated over the variation of a function.

But functional calculus is not well understood yet. Integrating over function spaces that include the function and how it may vary is not well defined. It has yet to be developed whether even functional differentiation is the inverse of functional integration. I think more study needs to be given this subject.

I am attempting to develop physics from logic. Your insights are
appreciated.

More at:
http://www.sirus.com/users/mjake/StringTh.html

 

[HallsofIvy]

Do you mean YOU don't understand functional analysis? I believe it's a fairly well developed field.

Or perhaps you mean physicists don't understand functional analysis. That might well be true.

 

[Mike2]

The Feynman path integral used in quantum mechanics is an integral over the function space of admissible functions. What is the inverse operation of this path integral? Is it the variation with respect to how a function may vary? References please.

 

[eljose79]

Problems with functional analysis..

In fact the main problem with this approach is that you can use functional differentiation easily but you can not make a functional integration...is this could be made our problems would be over as the Feynman Path integral is only a functional integral to define the propagator if you could do it then all the problems with normalization and infinites would be over..

Perhaps you could use the bernoulli formula that relates an integral with an infinite series of derivatives...

Inty(x)dx=c-xy+x**2y -x**3y´+... and so on

 

[Mike2]

I'm not real clear on what Feynman is accomplishing with a path integral. He integrates over every possible path. Is this the same as integrating over a volume through which the paths may travel? Each path is weighted by the exponent of the action integral. Is this an average characteristic of all paths? Or maybe this is another way of finding some topological invariant of the space of the paths. Any clues?