May be f(→x),→g(→x) an arbitrary functions dependent on the coordinates →x=(x,y,z)T. Defining the following function dependent on a 3-dimensional curve →γ(t) parametrized by t∈[0,1]:
S:=∫R3∫R3(f(→x)exp(∫10→g(→γ(t))→γ′(t)dt)f(→x′))d3xd3x′.
It was defined →x=→γ(0),→x′=→γ(1) and it was integrated over all possible endpoints of the curve →γ(t). How I can compute
I:=∫Sd[→γ(t)] ([→γ(t)] denotes that every possible path with endpoints →x,→x′ that is connected must be integrated up)?
This Integration is similar to Feynman's path-integral over all possible paths in Quantum field theory. The exponential I can expand into a taylor series, but how can I evaluate the products ∫→γ′(t1)⊗...⊗→γ′(tn)d[→γ(t)] where every →γ(ti),i∈{1,...,n} has the same endpoints (there occur both open and closed paths)?
Every help would be greatly appreciated.
This post imported from StackExchange Physics at 2015-03-10 13:00 (UTC), posted by SE-user kryomaxim