Help with this path integral.

mhill

I am having troubles to solve the functional integral:

[tex] \int D( X) e^{i(\dot X)^{2}+ a\delta (X-1)+ b\delta (X-3) [/tex]

if a and b were 0 the integral is just a Gaussian integral but i do not know how to deal with the Delta distribution inside some may help ??

jostpuur

I have difficulty understanding what X is. Is it some variable, or is it a function, like X(x). When you write

[tex]
\int\mathcal{D}X
[/tex]

it looks like X is a function, for example [itex]X:\mathbb{R}\to\mathbb{R}[/itex] or [itex]X:[-L,L]\to\mathbb{R}[/itex] or something similar. But when you write

[tex]
\delta(X-3)
[/tex]

it looks like X is just some parameter, like [itex]X\in\mathbb{R}[/itex].

Or is the number 3 a constant function [itex]\mathbb{R}\to\mathbb{R}[/itex], [itex]3(x)=3[/itex], and the delta function an infinite dimensional delta function, like [itex]\delta^{\mathbb{R}}[/itex]?

mjsd

if that is in path integral repreesetation, shouldn't there be an integral in the exponent.?,
[tex]e^{S(q)}[/tex] where S(q) is the action which is an integration over relevant time period of the Lagrangian of system.