Woher kommt das Delta von Null δ(0)δ(0)\delta(0)?

Diese Formel folgt den üblichen heuristischen Diskretisierungsregeln (hier in 1D geschrieben):

$$\begin{align} i\in\{1, \ldots,N\}, ~~x_i=i\Delta \qquad\longrightarrow\qquad&~x~\in~[0,L]\cr \text{discrete var.}\qquad\qquad\qquad &\text{cont. var.},\end{align} \tag{1}$$

$$\text{sum}\qquad \sum_{i=1}^N \qquad\longrightarrow\qquad \int_0^L \! \frac{dx}{\Delta} \qquad\text{integral},\tag{2}$$

$$\text{"volume" of unit cell:}\qquad \Delta ~=~\frac{L}{N}, \tag{3}$$

$$\begin{align} \frac{1}{\Delta} \delta_{i,j} \qquad\longrightarrow\qquad& \delta(x_i-x_j) \cr \text{Kronecker delta fct}\qquad\qquad\qquad& \text{Dirac delta fct},\end{align} \tag{4}$$

$$\frac{1}{\Delta}\qquad\longrightarrow\qquad\delta(0), \tag{5}$$

für$N\to \infty$. Also formell

$$f(x_j)~=~\sum_{i=1}^N \delta_{i,j} ~f(x_i) ~~\longrightarrow~~ \int_0^L \! dx~\delta(x-x_j) ~f(x), \tag{6}$$

Und

$$\begin{align} \prod_{i=1}^N \exp\left[f(x_i)\right] ~=~& \exp\left[\sum_{i=1}^Nf(x_i)\right] \cr\cr ~\downarrow~&\cr\cr \exp\left[\int_0^L \! \frac{dx}{\Delta}f(x)\right]~=~&\exp\left[\delta(0)\int_0^L \! dx~f(x)\right] .\end{align}\tag{7}$$