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So can we consider that the path integral is an integral over manifolds?
Paths are manifold.Feynman said:So can we consider that the path integral is an integral over manifolds?
I've taken to re-reading about path integrals to see if I can get a better intuition of what's going on. Hatfield's presentation seems to be the best I've seen so far - not so many things pulled out of the hat.selfAdjoint said:Mike means that a path (a one-dimensional continuum) is a manifold. A one-dimensional manifold. But I am sure that wasn't what you meant. But I can't figure out what you did mean.
So each "path" is weighted by the same magnitude but different phase?selfAdjoint said:Mike, e to the power i times something is always a complex number of modulus (i.e. magnitude) 1.
I'm sure there is a little more to then that. The path integral does not result in a classical path; for then there would be no need for path integral in the first place. I think it means that the classical path simply contributes most to the path integral than for the far fetched paths, right?selfAdjoint said:So then you add up (integrate) all the complex values along each path and then integrate the sums across all the paths, and what happens is that all the different pointing arrows wash each other out and only the classical path comes out of the integration.
I can't make head or tail of this. Clarify please?Feynman said:So gentelman ,
we are taking about the Path wich mean (maybe) manifolds, and not complex .
My question is why the path can be consider that is a manifold and can be consider the path integral is an integral over a manifold?
Thanks
String theory uses a type of path integral, only it sums up 2D "paths", on 1D paths. There they do say that the Feynman path integral is "summed over manifolds". I suppose the same thing can be said of the 1D case.Feynman said:So gentelman ,
we are taking about the Path wich mean (maybe) manifolds, and not complex .
My question is why the path can be consider that is a manifold and can be consider the path integral is an integral over a manifold?
Thanks
Feynman said:so?
what do you mean selfAdjoint
As I recall, [tex]\phi (x,t)[/tex] is the wave function of 1st quantization. And I believe [tex]\Psi (x,t)[/tex] is called the "field" of second quantization. Are there any other names for [tex]\Psi (x,t)[/tex], for example, amplitude of something, field of something? Thanks.Feynman said:Thank you Mike2, [tex]\Psi (x,t)[/tex] is the solution of SCHROD equation's,
But the problem is HOW WE CAN DEFINE [tex]D \hat \phi [/tex]? mathematically and physically
thx