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

$$\int\mathcal{D}X$$

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

$$\delta(X-3)$$

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.