Consider a field theory of a scalar field $\phi$ described by an action $\mathcal{S[\phi]}$. Is there a way to determine the transition amplitude $\langle \phi(x,t)'|\phi(x,0)\rangle$?
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In a scalar field theory the scattering amplitude between an initial state $\vert i \rangle$ of $n$ incoming particles and a final state $\vert f \rangle$ of $n'$ outgoing particles is given by the LSZ (Lehmann-Symanzik-Zimmermann) reduction formula
$\langle f \vert i \rangle = i^{n + n'} \int d^4 x_1 e^{i k_1 x_1} (-\partial_1 ^2 + m^2) ... d^4 x'_1 e^{-i k'_1 x'_1} (-\partial_{1'}^2 + m^2) ... \langle 0 \vert T \varphi(x_1) ... \varphi(x'_1) ... \vert 0 \rangle$
provided that the fields obey the normalization conditions:
$\langle 0 \vert \varphi(x) \vert 0 \rangle = 0$
$\langle k \vert \varphi(x) \vert 0 \rangle = e^{-i k x}$
where $\vert k \rangle = a^\dagger (k) \vert 0 \rangle$ and $a^\dagger$ is a creation operator.
Michele Grosso
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