Editing S-matrix (section)

==== In states expressed as out states ====
The initial states can be expanded in a basis of final states (or vice versa). Using the completeness relation,
<math display="block">\Psi_\alpha^- = \int d\beta (\Psi_\beta^+,\Psi_\alpha^-)\Psi_\beta^+ = 
\int d\beta |\Psi_\beta^+\rangle\langle\Psi_\beta^+|\Psi_\alpha^-\rangle = 
\sum_{n_1\sigma_1n_2\sigma_2\cdots} \int d^3p_1d^3p_2\cdots(\Psi_\beta^+,\Psi_\alpha^-)\Psi_\beta^+ ,</math>
<math display="block">\Psi_\alpha^- = \left| \mathrm{i}, k_1\ldots k_n \right\rangle = C_0 \left| \mathrm{f}, 0\right\rangle\ + \sum_{m=1}^\infty \int{d^4p_1\ldots d^4p_mC_m(p_1\ldots p_m)\left| \mathrm{f}, p_1\ldots p_m \right\rangle} ~,</math>
where {{math|{{!}}''C''<sub>''m''</sub>{{!}}<sup>2</sup>}} is the probability that the interaction transforms
<math display="block">\left| \mathrm{i}, k_1\ldots k_n \right\rangle = \Psi_\alpha^-</math>
into
<math display="block">\left| \mathrm{f}, p_1\ldots p_m \right\rangle = \Psi_\beta^+ .</math>
By the ordinary rules of quantum mechanics,
<math display="block">C_m(p_1\ldots p_m) = \left\langle \mathrm{f}, p_1\ldots p_m \right|\mathrm{i}, k_1\ldots k_n \rangle = (\Psi_\beta^+,\Psi_\alpha^-)</math>
and one may write
<math display="block">\left| \mathrm{i}, k_1\ldots k_n \right\rangle = C_0 \left| \mathrm{f}, 0\right\rangle\ + \sum_{m=1}^\infty \int{d^4p_1\ldots d^4p_m
\left| \mathrm{f}, p_1\ldots p_m \right\rangle}\left\langle \mathrm{f}, p_1\ldots p_m \right|\mathrm{i}, k_1\ldots k_n \rangle ~.</math>
The expansion coefficients are precisely the ''S''-matrix elements to be defined below.