Editing S-matrix (section)

==== From free particle states ====
For this viewpoint, one should consider how the archetypical scattering experiment is performed. The initial particles are prepared in well defined states where they are so far apart that they don't interact. They are somehow made to interact, and the final particles are registered when they are so far apart that they have ceased to interact. The idea is to look for states in the Heisenberg picture that in the distant past had the appearance of free particle states. This will be the in states. Likewise, an out state will be a state that in the distant future has the appearance of a free particle state.<ref name=Weinberg_1/>

The notation from the general reference for this section, {{harvtxt|Weinberg|2002}} will be used. A general non-interacting multi-particle state is given by
<math display="block">\Psi_{p_1\sigma_1 n_1;p_2\sigma_2 n_2;\cdots},</math>
where
* {{mvar|p}} is momentum,
* {{mvar|σ}} is spin z-component or, in the massless case, [[helicity (particle physics)|helicity]],
* {{mvar|n}} is particle species.
These states are normalized as
<math display="block">\left(\Psi_{p_1'\sigma_1' n_1';p_2'\sigma_2' n_2';\cdots}, \Psi_{p_1\sigma_1 n_1;p_2\sigma_2 n_2;\cdots}\right)
=\delta^3(\mathbf{p}_1' - \mathbf{p}_1)\delta_{\sigma_1'\sigma_1}\delta_{n_1'n_1}
\delta^3(\mathbf{p}_2' - \mathbf{p}_2)\delta_{\sigma_2'\sigma_2}\delta_{n_2'n_2}\cdots \quad \pm \text{ permutations}.</math>
Permutations work as such; if {{math|''s'' ∈ ''S''<sub>''k''</sub>}} is a permutation of {{math|''k''}} objects (for a {{nowrap|{{mvar|k}}-particle}} state) such that
<math display="block">n_{s(i)}' = n_{i}, \quad 1 \le i \le k,</math>
then a nonzero term results. The sign is plus unless {{mvar|s}} involves an odd number of fermion transpositions, in which case it is minus. The notation is usually abbreviated letting one Greek letter stand for the whole collection describing the state. In abbreviated form the normalization becomes
<math display="block">\left(\Psi_{\alpha'}, \Psi_\alpha\right) = \delta(\alpha' - \alpha).</math>
When integrating over free-particle states one writes in this notation
<math display="block"> d\alpha\cdots \equiv \sum_{n_1\sigma_1n_2\sigma_2\cdots} \int d^3p_1 d^3p_2 \cdots,</math>
where the sum includes only terms such that no two terms are equal modulo a permutation of the particle type indices. The sets of states sought for are supposed to be ''complete''. This is expressed as
<math display="block">\Psi = \int d\alpha \ \Psi_\alpha\left(\Psi_\alpha, \Psi\right),</math>
which could be paraphrased as
<math display="block">\int d\alpha \ \left|\Psi_\alpha\right\rangle\left\langle\Psi_\alpha\right| = 1,</math>
where for each fixed {{mvar|α}}, the right hand side is a projection operator onto the state {{mvar|α}}. Under an inhomogeneous Lorentz transformation {{math|(Λ, ''a'')}}, the field transforms according to the rule
{{NumBlk||<math display="block">U(\Lambda ,a)\Psi_{p_1\sigma_1 n_1;p_2\sigma_2 n_2\cdots} = e^{-ia_\mu((\Lambda p_1)^\mu + (\Lambda p_2)^\mu + \cdots)}
\sqrt{\frac{(\Lambda p_1)^0(\Lambda p_2)^0\cdots}{p_1^0p_2^0\cdots}}\sum_{\sigma_1'\sigma_2'\cdots}
D_{\sigma_1'\sigma_1}^{(j_1)}(W(\Lambda, p_1))D_{\sigma_2'\sigma_2}^{(j_2)}(W(\Lambda, p_2))\cdots
\Psi_{\Lambda p_1\sigma_1' n_1;\Lambda p_2\sigma_2' n_2\cdots},</math>|{{EquationRef|1}}}}
where {{math|''W''(Λ, ''p'')}} is the [[Wigner rotation]] and {{math|''D''<sup>(''j'')</sup>}} is the {{nowrap|{{math|(2''j'' + 1)}}-dimensional}} representation of {{math|SO(3)}}. By putting {{math|1=Λ = 1, ''a'' = (''τ'', 0, 0, 0)}}, for which {{mvar|''U''}} is {{math|exp(''iHτ'')}}, in {{EquationNote|1|(1)}}, it immediately follows that
<math display="block">H\Psi = E_\alpha\Psi, \quad E_\alpha = p_1^0 + p_2^0 + \cdots ,</math>
so the in and out states sought after are eigenstates of the full Hamiltonian that are necessarily non-interacting due to the absence of mixed particle energy terms. The discussion in the section above suggests that the in states {{math|Ψ<sup>+</sup>}} and the out states {{math|Ψ<sup>−</sup>}} should be such that
<math display="block">e^{-iH\tau} \int d\alpha g(\alpha)\Psi_\alpha^\pm = \int d\alpha e^{-iE_\alpha \tau} g(\alpha)\Psi_\alpha^\pm</math>
for large positive and negative {{mvar|τ}} has the appearance of the corresponding package, represented by {{math|''g''}}, of free-particle states, {{math|''g''}} assumed smooth and suitably localized in momentum. Wave packages are necessary, else the time evolution will yield only a phase factor indicating free particles, which cannot be the case. The right hand side follows from that the in and out states are eigenstates of the Hamiltonian per above. To formalize this requirement, assume that the full [[Hamiltonian (quantum mechanics)|Hamiltonian]] {{mvar|H}} can be divided into two terms, a free-particle Hamiltonian {{math|''H''<sub>0</sub>}} and an interaction {{mvar|V}}, {{math|1=''H'' = ''H''<sub>0</sub> + ''V''}} such that the eigenstates {{math|Φ<sub>''γ''</sub>}} of {{math|''H''<sub>0</sub>}} have the same appearance as the in- and out-states with respect to normalization and Lorentz transformation properties,
<math display="block">H_0\Phi_\alpha = E_\alpha\Phi_\alpha,</math>
<math display="block">(\Phi_\alpha', \Phi_\alpha) = \delta(\alpha' - \alpha).</math>

The in and out states are defined as eigenstates of the full Hamiltonian,
<math display="block">H\Psi_\alpha^\pm = E_\alpha\Psi_\alpha^\pm,</math>
satisfying
<math display="block">e^{-iH\tau} \int d\alpha \ g(\alpha) \Psi_\alpha^\pm \rightarrow e^{-iH_0\tau}\int d\alpha \ g(\alpha) \Phi_\alpha.</math>
for {{math|''τ'' → −∞}} or {{math|''τ'' → +∞}} respectively. Define
<math display="block">\Omega(\tau) \equiv e^{+iH\tau}e^{-iH_0\tau},</math>
then
<math display="block">\Psi_\alpha^\pm = \Omega(\mp \infty)\Phi_\alpha.</math>
This last expression will work only using wave packages.From these definitions follow that the in and out states are normalized in the same way as the free-particle states,
<math display="block">(\Psi_\beta^+, \Psi_\alpha^+) = (\Phi_\beta, \Phi_\alpha) = (\Psi_\beta^-, \Psi_\alpha^-) = \delta(\beta - \alpha),</math>
and the three sets are unitarily equivalent. Now rewrite the eigenvalue equation,
<math display="block">(E_\alpha - H_0 \pm i\epsilon)\Psi_\alpha^\pm = \pm i\epsilon\Psi_\alpha^\pm + V\Psi_\alpha^\pm,</math>
where the {{math|±''iε''}} terms has been added to make the operator on the LHS invertible. Since the in and out states reduce to the free-particle states for {{math|''V'' → 0}}, put
<math display="block">i\epsilon\Psi_\alpha^\pm = i\epsilon\Phi_\alpha</math>
on the RHS to obtain
<math display="block">\Psi_\alpha^\pm = \Phi_\alpha + (E_\alpha - H_0 \pm i\epsilon)^{-1}V\Psi_\alpha^\pm.</math>
Then use the completeness of the free-particle states,
<math display="block">V\Psi_\alpha^\pm = \int d\beta \ (\Phi_\beta, V\Psi_\alpha^\pm)\Phi_\beta \equiv \int d\beta \ T_{\beta\alpha}^\pm\Phi_\beta,</math>
to finally obtain
<math display="block">\Psi_\alpha^\pm = \Phi_\alpha + \int d\beta \ \frac{T_{\beta\alpha}^\pm\Phi_\beta}{E_\alpha - E_\beta \pm i\epsilon}.</math>
Here {{math|''H''<sub>0</sub>}} has been replaced by its eigenvalue on the free-particle states. This is the [[Lippmann–Schwinger equation]].