Editing S-matrix (section)

== Definition in quantum field theory ==

=== Interaction picture ===
A straightforward way to define the ''S''-matrix begins with considering the [[interaction picture]].<ref>{{harvnb|Greiner|Reinhardt|1996}} Section 8.2.</ref> Let the Hamiltonian {{math|''H''}} be split into the free part {{math|''H''<sub>0</sub>}} and the interaction {{math|''V''}}, {{math|1=''H'' = ''H''<sub>0</sub> + ''V''}}. In this picture, the operators behave as free field operators and the state vectors have dynamics according to the interaction {{math|''V''}}. Let
<math display="block">\left|\Psi(t)\right\rangle</math>
denote a state that has evolved from a free initial state
<math display="block">\left|\Phi_{\rm i}\right\rangle.</math>
The ''S''-matrix element is then defined as the projection of this state on the final state
<math display="block">\left\langle\Phi_{\rm f}\right|.</math>
Thus
<math display="block">S_{\rm fi} \equiv \lim_{t \rightarrow +\infty} \left\langle\Phi_{\rm f}|\Psi(t)\right\rangle \equiv \left\langle\Phi_{\rm f}\right|S\left|\Phi_{\rm i}\right\rangle,</math>
where {{math|''S''}} is the '''S-operator'''. The great advantage of this definition is that the '''time-evolution operator''' {{mvar|U}} evolving a state in the interaction picture is formally known,<ref>{{harvnb|Greiner|Reinhardt|1996}} Equation 8.44.</ref>
<math display="block">U(t, t_0) = Te^{-i\int_{t_0}^t d\tau V(\tau)},</math>
where {{mvar|T}} denotes the [[time-ordered product]]. Expressed in this operator,
<math display="block">S_{\rm fi} = \lim_{t_2 \rightarrow +\infty}\lim_{t_1 \rightarrow -\infty}\left\langle\Phi_{\rm f}\right|U(t_2, t_1)\left|\Phi_{\rm i}\right\rangle,</math>
from which
<math display="block">S = U(\infty, -\infty).</math>
[[Matrix exponential|Expanding]] using the knowledge about {{math|''U''}} gives a [[Dyson series]],
<math display="block">S = \sum_{n=0}^\infty \frac{(-i)^n}{n!}\int_{-\infty}^\infty dt_1\cdots \int_{-\infty}^\infty dt_n T\left[V(t_1)\cdots V(t_n)\right],</math>
or, if {{mvar|V}} comes as a Hamiltonian density <math>\mathcal{H}</math>,
<math display="block">S = \sum_{n=0}^\infty \frac{(-i)^n}{n!}\int_{-\infty}^\infty dx_1^4\cdots \int_{-\infty}^\infty dx_n^4 T\left[\mathcal{H}(x_1)\cdots \mathcal{H}(x_n)\right].</math>

Being a special type of time-evolution operator, {{mvar|S}} is unitary. For any initial state and any final state one finds
<math display="block">S_{\rm fi} = \left\langle\Phi_{\rm f}|S|\Phi_{\rm i}\right\rangle = \left\langle\Phi_{\rm f} \left|\sum_{n=0}^\infty \frac{(-i)^n}{n!}\int_{-\infty}^\infty dx_1^4\cdots \int_{-\infty}^\infty dx_n^4 T\left[\mathcal{H}(x_1)\cdots \mathcal{H}(x_n)\right]\right| \Phi_{\rm i}\right\rangle .</math>

This approach is somewhat naïve in that potential problems are swept under the carpet.<ref name=Greiner_1>{{harvnb|Greiner|Reinhardt|1996}} Chapter 9.</ref> This is intentional. The approach works in practice and some of the technical issues are addressed in the other sections.

=== In and out states ===
Here a slightly more rigorous approach is taken in order to address potential problems that were disregarded in the interaction picture approach of above. The final outcome is, of course, the same as when taking the quicker route. For this, the notions of in and out states are needed. These will be developed in two ways, from vacuum, and from free particle states. Needless to say, the two approaches are equivalent, but they illuminate matters from different angles.

==== 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.

==== 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.

==== 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]].

==== 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.

=== ''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.