Editing S-matrix (section)

==== From vacuum ====
If {{math|''a''<sup>†</sup>(''k'')}} is a [[creation operator]], its [[hermitian adjoint]] is an [[annihilation operator]] and destroys the vacuum,
<math display="block">a(k)\left |*, 0\right\rangle = 0.</math>

In [[Dirac notation]], define
<math display="block">|*, 0\rangle</math>
as a [[vacuum state|vacuum quantum state]], i.e. a state without real particles. The asterisk signifies that not all vacua are necessarily equal, and certainly not equal to the Hilbert space zero state {{math|0}}. All vacuum states are assumed [[Poincaré group|Poincaré invariant]], invariance under translations, rotations and boosts,<ref name=Greiner_1/> formally,
<math display="block">P^\mu |*, 0\rangle = 0, \quad M^{\mu\nu} |*, 0\rangle = 0</math>
where {{math|''P''<sup>''μ''</sup>}} is the '''generator of translation''' in space and time, and {{math|''M''<sup>''μν''</sup>}} is the generator of [[Lorentz transformation]]s. Thus the description of the vacuum is independent of the frame of reference. Associated to the in and out states to be defined are the in and out '''field operators''' (aka '''fields''') {{math|Φ<sub>i</sub>}} and {{math|Φ<sub>o</sub>}}. Attention is here focused to the simplest case, that of a '''scalar theory''' in order to exemplify with the least possible cluttering of the notation. The in and out fields satisfy
<math display="block">(\Box^2 + m^2)\phi_{\rm i,o}(x) = 0,</math>
the free [[Klein–Gordon equation]]. These fields are postulated to have the same equal time commutation relations (ETCR) as the free fields,
<math display="block">\begin{align}
{[\phi_{\rm i,o}(x), \pi_{\rm i,o}(y)]}_{x_0 = y_0} &= i\delta(\mathbf{x} - \mathbf{y}),\\
{[\phi_{\rm i,o}(x), \phi_{\rm i,o}(y)]}_{x_0 = y_0} &= {[\pi_{\rm i,o}(x), \pi_{\rm i,o}(y)]}_{x_0 = y_0} = 0,
\end{align}</math>
where {{math|''π''<sub>''i'',''j''</sub>}} is the field '''canonically conjugate''' to {{math|Φ<sub>''i'',''j''</sub>}}. Associated to the in and out fields are two sets of creation and annihilation operators, {{math|''a''<sup>†</sup><sub>i</sub>(''k'')}} and {{math|''a''<sup>†</sup><sub>f</sub> (''k'')}}, acting in the ''same'' [[Hilbert space]],<ref>{{harvnb|Weinberg|2002}} Chapter 3. See especially remark at the beginning of section 3.2.</ref> on two ''distinct'' complete sets ([[Fock space]]s; initial space {{mvar|i}}, final space {{mvar|f}}). These operators satisfy the usual commutation rules,
<math display="block">\begin{align}
{[a_{\rm i,o}(\mathbf{p}), a^\dagger_{\rm i,o}(\mathbf{p}')]} &= i\delta(\mathbf{p} - \mathbf{p'}),\\
{[a_{\rm i,o}(\mathbf{p}), a_{\rm i,o}(\mathbf{p'})]} &= {[a^\dagger_{\rm i,o}(\mathbf{p}), a^\dagger_{\rm i,o}(\mathbf{p'})]} = 0.
\end{align}</math>

The action of the creation operators on their respective vacua and states with a finite number of particles in the in and out states is given by
<math display="block">\begin{align}
\left| \mathrm{i}, k_1\ldots k_n \right\rangle &= a_i^\dagger (k_1)\cdots a_{\rm i}^\dagger (k_n)\left| i, 0\right\rangle,\\
\left| \mathrm{f}, p_1\ldots p_n \right\rangle &= a_{\rm f}^\dagger (p_1)\cdots a_f^\dagger (p_n)\left| f, 0\right\rangle,
\end{align}</math>
where issues of normalization have been ignored. See the next section for a detailed account on how a general {{nowrap|{{mvar|n}}-particle}} state is normalized. The initial and final spaces are defined by
<math display="block">\mathcal H_{\rm i} = \operatorname{span}\{ \left| \mathrm{i}, k_1\ldots k_n \right\rangle = a_{\rm i}^\dagger (k_1)\cdots a_{\rm i}^\dagger (k_n)\left| \mathrm{i}, 0\right\rangle\},</math>
<math display="block">\mathcal H_{\rm f} = \operatorname{span}\{ \left| \mathrm{f}, p_1\ldots p_n \right\rangle = a_{\rm f}^\dagger (p_1)\cdots a_{\rm f}^\dagger (p_n)\left| \mathrm{f}, 0\right\rangle\}.</math>

The asymptotic states are assumed to have well defined Poincaré transformation properties, i.e. they are assumed to transform as a direct product of one-particle states.<ref name=Weinberg_1>{{harvnb|Weinberg|2002}} Chapter 3.</ref> This is a characteristic of a non-interacting field. From this follows that the asymptotic states are all [[eigenstate]]s of the momentum operator {{math|''P<sup>μ</sup>''}},<ref name=Greiner_1/>
<math display="block">P^\mu\left| \mathrm{i}, k_1\ldots k_m \right\rangle = k_1^\mu + \cdots + k_m^\mu\left| \mathrm{i}, k_1\ldots k_m \right\rangle, \quad P^\mu\left| \mathrm{f}, p_1\ldots p_n \right\rangle = p_1^\mu + \cdots + p_n^\mu\left| \mathrm{f}, p_1\ldots p_n \right\rangle.</math>
In particular, they are eigenstates of the full Hamiltonian,
<math display="block">H = P^0.</math>

The vacuum is usually postulated to be stable and unique,<ref name=Greiner_1/><ref group=nb>This is not true if an open system is studied. Under an influence of an external field the in and out vacua can differ since the external field can produce particles.</ref>
<math display="block">|\mathrm{i}, 0\rangle = |\mathrm{f}, 0\rangle = |*,0\rangle \equiv |0\rangle.</math>

The interaction is assumed adiabatically turned on and off.