Editing S-matrix (section)

=== ''S''-matrix ===
The ''S''-matrix is now defined by<ref name=Weinberg_1/>
<math display="block">S_{\beta\alpha} = \langle\Psi_\beta^-|\Psi_\alpha^+\rangle = \langle \mathrm{f},\beta| \mathrm{i},\alpha\rangle, \qquad |\mathrm{f}, \beta\rangle \in \mathcal H_{\rm f}, \quad |\mathrm{i}, \alpha\rangle \in \mathcal H_{\rm i}.</math>

Here {{mvar|α}} and {{mvar|β}} are shorthands that represent the particle content but suppresses the individual labels. Associated to the ''S''-matrix there is the '''S-operator''' {{mvar|S}} defined by<ref name=Weinberg_1/>
<math display="block">\langle\Phi_\beta|S|\Phi_\alpha\rangle \equiv S_{\beta\alpha},</math>
where the {{math|Φ<sub>''γ''</sub>}} are free particle states.<ref name=Weinberg_1/><ref group=nb>Here it is assumed 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. See {{harvtxt|Weinberg|2002}}, page 110.</ref> This definition conforms with the direct approach used in the interaction picture. Also, due to unitary equivalence,
<math display="block">\langle\Psi_\beta^+|S|\Psi_\alpha^+\rangle = S_{\beta\alpha} = \langle\Psi_\beta^-|S|\Psi_\alpha^-\rangle.</math>

As a physical requirement, {{mvar|S}} must be a [[unitary operator]]. This is a statement of conservation of probability in quantum field theory. But
<math display="block">\langle\Psi_\beta^-|S|\Psi_\alpha^-\rangle = S_{\beta\alpha} = \langle\Psi_\beta^-|\Psi_\alpha^+\rangle.</math>
By completeness then,
<math display="block">S|\Psi_\alpha^-\rangle = |\Psi_\alpha^+\rangle,</math>
so ''S'' is the unitary transformation from in-states to out states.
Lorentz invariance is another crucial requirement on the ''S''-matrix.<ref name=Weinberg_1/><ref group=nb>If {{math|Λ}} is a (inhomogeneous) proper orthochronous Lorentz transformation, then [[Wigner's theorem]] guarantees the existence of a unitary operator {{math|''U''(Λ)}} acting either on {{math|''H''<sub>''i''</sub>}} ''or'' {{math|''H''<sub>''f''</sub>}}. A theory is said to be Lorentz invariant if the same {{math|''U''(Λ)}} acts on {{math|''H''<sub>''i''</sub>}} ''and'' {{math|''H''<sub>''f''</sub>}}. Using the unitarity of {{math|''U''(Λ)}}, {{math|1=''S''<sub>''βα''</sub> = ⟨''i'', ''β''{{!}}''f'', ''α''⟩ = ⟨''i'', ''β''{{!}}''U''(Λ)<sup>†</sup>''U''(Λ){{!}}''f'', ''α''⟩}}. The right-hand side can be expanded using knowledge about how the non-interacting states transform to obtain an expression, and that expression is to be taken as a ''definition'' of what it means for the ''S''-matrix to be Lorentz invariant. See {{harvtxt|Weinberg|2002}}, equation 3.3.1 gives an explicit form.</ref> The S-operator represents the [[Matrix mechanics#Transformation theory|quantum canonical transformation]] of the initial ''in'' states to the final ''out'' states. Moreover, {{mvar|S}} leaves the vacuum state invariant and transforms ''in''-space fields to ''out''-space fields,<ref group=nb>Here the '''postulate of asymptotic completeness''' is employed. The in and out states span the same Hilbert space, which is assumed to agree with the Hilbert space of the interacting theory. This is not a trivial postulate. If particles can be permanently combined into bound states, the structure of the Hilbert space changes. See {{harvnb|Greiner|Reinhardt|1996|loc=section 9.2}}.</ref>
<math display="block">S\left|0\right\rangle = \left|0\right\rangle</math>
<math display="block">\phi_\mathrm{f}=S\phi_\mathrm{i} S^{-1} ~.</math>

In terms of creation and annihilation operators, this becomes
<math display="block">a_{\rm f}(p)=Sa_{\rm i}(p)S^{-1}, a_{\rm f}^\dagger(p)=Sa_{\rm i}^\dagger(p)S^{-1},</math>
hence
<math display="block">\begin{align}
S|\mathrm{i}, k_1, k_2, \ldots, k_n\rangle
&= Sa_{\rm i}^\dagger(k_1)a_{\rm i}^\dagger(k_2) \cdots a_{\rm i}^\dagger(k_n)|0\rangle = 
Sa_{\rm i}^\dagger(k_1)S^{-1}Sa_{\rm i}^\dagger(k_2)S^{-1} \cdots Sa_{\rm i}^\dagger(k_n)S^{-1}S|0\rangle \\[1ex]
&=a_{\rm o}^\dagger(k_1)a_{\rm o}^\dagger(k_2) \cdots a_{\rm o}^\dagger(k_n)S|0\rangle
=a_{\rm o}^\dagger(k_1)a_{\rm o}^\dagger(k_2) \cdots a_{\rm o}^\dagger(k_n)|0\rangle
=|\mathrm{o}, k_1, k_2, \ldots, k_n\rangle.
\end{align}</math>
A similar expression holds when {{mvar|S}} operates to the left on an out state. This means that the ''S''-matrix can be expressed as
<math display="block">S_{\beta\alpha} = \langle \mathrm{o}, \beta|\mathrm{i}, \alpha \rangle = \langle \mathrm{i}, \beta|S|\mathrm{i}, \alpha \rangle = \langle \mathrm{o}, \beta|S|\mathrm{o}, \alpha \rangle.</math>

If {{mvar|S}} describes an interaction correctly, these properties must be also true:
* If the system is made up with ''a single particle'' in momentum eigenstate {{math|{{!}}''k''⟩}}, then {{math|1= ''S''{{!}}''k''⟩ = {{!}}''k''⟩}}. This follows from the calculation above as a special case.
* The ''S''-matrix element may be nonzero only where the output state has the same total [[momentum]] as the input state. This follows from the required Lorentz invariance of the ''S''-matrix.