Editing S-matrix (section)

==== Heisenberg picture ====
The [[Heisenberg picture]] is employed henceforth. In this picture, the states are time-independent. A Heisenberg state vector thus represents the complete spacetime history of a system of particles.<ref name=Weinberg_1/> The labeling of the in and out states refers to the asymptotic appearance. A state {{math|Ψ<sub>''α'', in</sub>}} is characterized by that as {{math|''t'' → −∞}} the particle content is that represented collectively by {{mvar|α}}. Likewise, a state {{math|Ψ<sub>''β'', out</sub>}} will have the particle content represented by {{mvar|β}} for {{math|''t'' → +∞}}. Using the assumption that the in and out states, as well as the interacting states, inhabit the same Hilbert space and assuming completeness of the normalized in and out states (postulate of asymptotic completeness<ref name=Greiner_1/>), the initial states can be expanded in a basis of final states (or vice versa). The explicit expression is given later after more notation and terminology has been introduced. The expansion coefficients are precisely the ''S''-matrix elements to be defined below.

While the state vectors are constant in time in the Heisenberg picture, the physical states they represent are ''not''. If a system is found to be in a state {{math|Ψ}} at time {{math|1=''t'' = 0}}, then it will be found in the state {{math|1=''U''(''τ'')Ψ = ''e''<sup>−''iHτ''</sup>Ψ}} at time {{math|1=''t'' = ''τ''}}. This is not (necessarily) the same Heisenberg state vector, but it is an ''equivalent'' state vector, meaning that it will, upon measurement, be found to be one of the final states from the expansion with nonzero coefficient. Letting {{mvar|τ}} vary one sees that the observed {{math|Ψ}} (not measured) is indeed the [[Schrödinger picture]] state vector. By repeating the measurement sufficiently many times and averaging, one may say that the ''same'' state vector is indeed found at time {{math|1=''t'' = τ}} as at time {{math|1=''t'' = 0}}. This reflects the expansion above of an in state into out states.