Path Integral Troubleshooting: Dealing with Delta Distributions in the Exponent

[mhill]

I am having troubles to solve the functional integral:

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

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

<br /> \int\mathcal{D}X<br />

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

<br /> \delta(X-3)<br />

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

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

 

[mjsd]

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