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I am reading about the S-matrix in QFT (Standard Model book by Burgess and Moore) and I came across the energy-conserving delta function, which is factored out of the S-matrix. I would greatly appreciate if someone would explain me, or give a reference to an explaination of what is energy-conserving delta function and why do we factor it out.

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It's just a way of saying that the S-matrix only connects initial states to final states that have the same energy and momentum. With finitely many states the S-matrix is a finite matrix, and the $m$:th element in the $n$:th row is non-zero if the time evolution of the $n$:th initial state has an $m$:th state component. In an energy conserving theory, it is necessary for this that $E(n) = E(m)$, thus we can say that $$S_{nm} \propto \delta_{E(n) E(m)}$$ where this is the Kronecker delta $\delta_{xy} =\begin{cases}1 & x = y \\ 0 & x \neq y\end{cases}$.

The generalization to the case of a continuous spectrum of states is $$S(i \to f) \propto \delta(E(i) - E(f))$$ where this is the Dirac delta, $$\int_D \delta(x)f(x) \, dx = \begin{cases}f(0) & 0 \in D \\ 0 & 0 \notin D\end{cases}. $$

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