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{{short description|Matrix representing the effect of scattering on a physical system}}
{{redirect|Scattering matrix|the meaning in linear electrical networks|Scattering parameters}}
{{For|the 1960s approach to particle physics|S-matrix theory}}
{{Italic title|noerror|string=S}}

In [[physics]], the '''''S''-matrix''' or '''scattering matrix''' is a [[Matrix (mathematics)|matrix]] that relates the initial state and the final state of a physical system undergoing a [[scattering|scattering process]]. It is used in [[quantum mechanics]], [[scattering theory]] and [[quantum field theory]] (QFT).

More formally, in the context of QFT, the ''S''-matrix is defined as the [[unitary matrix]] connecting sets of asymptotically free particle states (the ''in-states'' and the ''out-states'') in the [[Hilbert space]] of physical states: a multi-particle state is said to be ''free'' (or non-interacting) if it [[Representation theory of the Lorentz group|transforms]] under [[Lorentz transformation]]s as a [[tensor product]], or ''direct product'' in physics parlance, of ''one-particle states'' as prescribed by equation {{EquationNote|1|(1)}} below. ''Asymptotically free'' then means that the state has this appearance in either the distant past or the distant future.

While the ''S''-matrix may be defined for any background ([[spacetime]]) that is asymptotically solvable and has no [[event horizon]]s, it has a simple form in the case of the [[Minkowski space]]. In this special case, the Hilbert space is a space of irreducible [[unitary representation]]s of the [[inhomogeneous space|inhomogeneous]] [[Lorentz group]] (the [[Poincaré group]]); the ''S''-matrix is the [[evolution operator]] between <math>t= - \infty </math> (the distant past), and <math>t= + \infty </math> (the distant future). It is defined only in the limit of zero energy density (or infinite particle separation distance).

It can be shown that if a quantum field theory in Minkowski space has a [[mass gap]], the [[quantum state|state]] in the asymptotic past and in the asymptotic future are both described by [[Fock space]]s.

== History ==
The initial elements of ''S''-matrix theory are found in [[Paul Dirac]]'s 1927 paper "Über die Quantenmechanik der Stoßvorgänge".<ref>{{Cite journal |last=Dirac |first=Paul |date=1927-08-01 |title=Über die Quantenmechanik der Stoßvorgänge |url=https://doi.org/10.1007/BF01451660 |journal=Zeitschrift für Physik |language=de |volume=44 |issue=8 |pages=585–595 |doi=10.1007/BF01451660 |bibcode=1927ZPhy...44..585D |issn=0044-3328|url-access=subscription }}</ref><ref>{{Cite journal |last1=Sanyuk |first1=Valerii I. |last2=Sukhanov |first2=Alexander D. |date=2003-09-01 |title=Dirac in 20th century physics: a centenary assessment |url=https://ufn.ru/en/articles/2003/9/c/ |journal=Physics-Uspekhi |language=en |volume=46 |issue=9 |pages=937–956 |doi=10.1070/PU2003v046n09ABEH001165 |issn=1063-7869|url-access=subscription }}</ref> The ''S''-matrix was first properly introduced by [[John Archibald Wheeler]] in the 1937 paper "On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure".<ref>John Archibald Wheeler, "[http://link.aps.org/abstract/PR/v52/p1107 On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure]", ''Phys. Rev.'' 52, 1107–1122 (1937).</ref> In this paper Wheeler introduced a ''scattering matrix'' – a unitary matrix of coefficients connecting "the asymptotic behaviour of an arbitrary particular solution [of the integral equations] with that of solutions of a standard form",<ref name = "Mehra">[[Jagdish Mehra]], [[Helmut Rechenberg]], ''The Historical Development of Quantum Theory'' (Pages 990 and 1031) Springer, 2001 {{ISBN|0-387-95086-9}}, {{ISBN|978-0-387-95086-0}}</ref> but did not develop it fully.

In the 1940s, [[Werner Heisenberg]] independently developed and substantiated the idea of the ''S''-matrix. Because of the problematic divergences present in [[quantum field theory]] at that time, Heisenberg was motivated to isolate the ''essential features of the theory'' that would not be affected by future changes as the theory developed. In doing so, he was led to introduce a unitary "characteristic" ''S''-matrix.<ref name = "Mehra"/>

Today, however, exact ''S''-matrix results are important for [[conformal field theory]], [[integrable systems]], and several further areas of quantum field theory and [[string theory]]. ''S''-matrices are not substitutes for a field-theoretic treatment, but rather, complement the end results of such.

== Motivation ==

In high-energy [[particle physics]] one is interested in computing the [[probability]] for different outcomes in [[scattering]] experiments. These experiments can be broken down into three stages:
# Making a collection of incoming [[subatomic particle|particle]]s collide (usually two kinds of particles with high energies).
# Allowing the incoming particles to interact. These interactions may change the types of particles present (e.g. if an [[electron]] and a [[positron]] [[Annihilation|annihilate]] they may produce two [[photon]]s).
# Measuring the resulting outgoing particles.

The process by which the incoming particles are transformed (through their [[Fundamental interaction|interaction]]) into the outgoing particles is called [[scattering]]. For particle physics, a physical theory of these processes must be able to compute the probability for different outgoing particles when different incoming particles collide with different energies.

The ''S''-matrix in quantum field theory achieves exactly this. It is assumed that the small-energy-density approximation is valid in these cases.

== Use ==

The ''S''-matrix is closely related to the transition [[probability amplitude]] in quantum mechanics and to [[cross section (physics)|cross sections]] of various interactions; the [[Matrix element (physics)|elements]] (individual numerical entries) in the ''S''-matrix are known as '''scattering amplitudes'''. [[pole (complex analysis)|Poles]] of the ''S''-matrix in the complex-energy plane are identified with [[bound state]]s, virtual states or [[resonance (particle physics)|resonances]]. [[Branch point|Branch cuts]] of the ''S''-matrix in the complex-energy plane are associated to the opening of a [[scattering channel]].

In the [[Hamiltonian (quantum mechanics)|Hamiltonian]] approach to quantum field theory, the ''S''-matrix may be calculated as a [[time-ordered]] [[matrix exponential|exponential]] of the integrated Hamiltonian in the [[interaction picture]]; it may also be expressed using [[Feynman's path integral]]s. In both cases, the [[perturbation theory (quantum mechanics)|perturbative]] calculation of the ''S''-matrix leads to [[Feynman diagram]]s.

In [[scattering theory]], the '''''S''-matrix''' is an [[operator (physics)|operator]] mapping free particle ''in-states'' to free particle ''out-states'' ([[scattering channel]]s) in the [[Heisenberg picture]]. This is very useful because often we cannot describe the interaction (at least, not the most interesting ones) exactly.

== In one-dimensional quantum mechanics ==
A simple prototype in which the ''S''-matrix is 2-dimensional is considered first, for the purposes of illustration. In it, particles with sharp energy {{math|''E''}} scatter from a localized potential {{math|''V''}} according to the rules of 1-dimensional quantum mechanics. Already this simple model displays some features of more general cases, but is easier to handle.

Each energy {{math|''E''}} yields a matrix {{math|1=''S'' = ''S''(''E'')}} that depends on {{math|''V''}}. Thus, the total ''S''-matrix could, figuratively speaking, be visualized, in a suitable basis, as a "continuous matrix" with every element zero except for {{math|2 × 2}}-blocks along the diagonal for a given {{math|''V''}}.

=== Definition ===
Consider a localized one dimensional [[potential barrier]] {{math|''V''(''x'')}}, subjected to a beam of quantum particles with energy {{math|''E''}}. These particles are incident on the potential barrier from left to right.

The solutions of the [[Schrödinger equation]] outside the potential barrier are [[plane waves]] given by
<math display="block">\psi_{\rm L}(x)= A e^{ikx} + B e^{-ikx}</math>
for the region to the left of the potential barrier, and
<math display="block">\psi_{\rm R}(x)= C e^{ikx} + D e^{-ikx}</math>
for the region to the right to the potential barrier, where
<math display="block">k=\sqrt{2m E/\hbar^{2}}</math>
is the [[wave vector]]. The time dependence is not needed in our overview and is hence omitted. The term with coefficient {{math|''A''}} represents the incoming wave, whereas term with coefficient {{math|''C''}} represents the outgoing wave. {{math|''B''}} stands for the reflecting wave. Since we set the incoming wave moving in the positive direction (coming from the left), {{math|''D''}} is zero and can be omitted.

The "scattering amplitude", i.e., the transition overlap of the outgoing waves with the incoming waves is a linear relation defining the ''S''-matrix,
<math display="block">\begin{pmatrix} B \\ C \end{pmatrix} = \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix} \begin{pmatrix} A \\ D \end{pmatrix}.</math>

The above relation can be written as
<math display="block">\Psi_{\rm out}=S \Psi_{\rm in}</math>
where
<math display="block">\Psi_{\rm out}=\begin{pmatrix}B \\ C \end{pmatrix}, \quad \Psi_{\rm in}=\begin{pmatrix}A \\ D \end{pmatrix}, \qquad S=\begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix}.</math>
The elements of {{math|''S''}} completely characterize the scattering properties of the potential barrier {{math|''V''(''x'')}}.

=== Unitary property ===
The unitary property of the ''S''-matrix is directly related to the conservation of the [[probability current]] in [[quantum mechanics]].

The probability current density {{mvar|J}} of the [[wave function]] {{math|''ψ''(''x'')}} is defined as
<math display="block"> J = \frac{\hbar}{2mi}\left(\psi^* \frac{\partial \psi }{\partial x}- \psi \frac{\partial \psi^* }{\partial x} \right) .</math>
The probability current density <math>J_{\rm L}(x)</math> of <math>\psi_{\rm L}(x)</math> to the left of the barrier is
<math display="block">J_{\rm L}(x)=\frac{\hbar k}{m}\left(|A|^2-|B|^2\right),</math>
while the probability current density <math>J_{\rm R}(x)</math> of <math>\psi_{\rm R}(x)</math> to the right of the barrier is
<math display="block">J_{\rm R}(x)=\frac{\hbar k}{m}\left(|C|^2-|D|^2\right).</math>

For conservation of the probability current, {{math|1=''J''<sub>L</sub> = ''J''<sub>R</sub>}}. This implies the ''S''-matrix is a [[unitary matrix]].
{{Cleanup|proof|reason=Proofs should be in prose|date=January 2025}}
{{math proof|proof=<math display="block">
\begin{align}
 & J_{\rm L} = J_{\rm R} \\
 & |A|^2-|B|^2=|C|^2-|D|^2 \\
 & |B|^2+|C|^2=|A|^2+|D|^2 \\
 & \Psi_\text{out}^\dagger \Psi_\text{out}=\Psi_\text{in}^\dagger \Psi_\text{in} \\
 & \Psi_\text{in}^\dagger S^\dagger S \Psi_\text{in}=\Psi_\text{in}^\dagger \Psi_\text{in} \\
 & S^\dagger S = I\\
\end{align}</math>}}

=== Time-reversal symmetry ===
If the potential {{math|''V''(''x'')}} is real, then the system possesses [[time-reversal symmetry]]. Under this condition, if {{math| ''ψ''(''x'')}} is a solution of the Schrödinger equation, then {{math| ''ψ''*(''x'')}} is also a solution.

The time-reversed solution is given by
<math display="block">\psi^*_{\rm L}(x)= A^* e^{-ikx} + B^* e^{ikx}</math>
for the region to the left to the potential barrier, and
<math display="block">\psi^*_{\rm R}(x)= C^* e^{-ikx} + D^* e^{ikx}</math>
for the region to the right to the potential barrier,
where the terms with coefficient {{math|''B''*}}, {{math|''C''*}} represent incoming wave, and terms with coefficient {{math|''A''*}}, {{math|''D''*}} represent outgoing wave.

They are again related by the ''S''-matrix,
<math display="block">\begin{pmatrix}A^* \\ D^* \end{pmatrix} = \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix}\begin{pmatrix} B^* \\ C^* \end{pmatrix}\,</math>
that is,<br />
<math display="block">\Psi^*_{\rm in}=S \Psi^*_{\rm out}.</math>
Now, the relations
<math display="block">\Psi^*_{\rm in} = S \Psi^*_{\rm out}, \quad \Psi_{\rm out}=S \Psi_{\rm in}</math>
together yield a condition
<math display="block">S^*S=I</math>
This condition, in conjunction with the unitarity relation, implies that the ''S''-matrix is symmetric, as a result of time reversal symmetry,
<math display="block">S^T=S.</math>

By combining the symmetry and the unitarity, the S-matrix can be expressed in the form:
<math display="block">\begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix} = \begin{pmatrix} e^{i\varphi} e^{i\delta} \cdot r & e^{i\varphi} \sqrt{1-r^2}  \\ e^{i\varphi}\sqrt{1-r^2} & -e^{i\varphi} e^{-i\delta} \cdot r  \end{pmatrix} = e^{i\varphi} \begin{pmatrix} e^{i\delta} \cdot r & \sqrt{1-r^2} \\ \sqrt{1-r^2} & -e^{-i\delta} \cdot r  \end{pmatrix}</math> 
with <math>\delta,\varphi \in [0;2\pi]</math> and <math>r\in [0;1]</math>. So the S-matrix is determined by three real parameters.

==== Transfer matrix ====
The ''transfer matrix'' <math>M</math> relates the plane waves <math>C e^{ikx}</math> and <math>D e^{-ikx}</math> on the ''right'' side of scattering potential to the plane waves <math>A e^{ikx}</math> and <math>B e^{-ikx}</math> on the ''left'' side:<ref name="cas.cz">{{cite web |url=https://gemma.ujf.cas.cz/~krejcirik/AAMP13/slides/Mostafazadeh.pdf |website=gemma.ujf.cas.cz |title=Transfer Matrix Formulation of Scattering Theory in Arbitrary Dimensions |access-date=29 October 2022}}</ref>  

<math display="block">\begin{pmatrix}C \\ D \end{pmatrix} = \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix}\begin{pmatrix} A \\ B \end{pmatrix}</math> and its components can be derived from the components of the S-matrix via:<ref name="ucr.edu">{{cite web |url=https://intra.ece.ucr.edu/~korotkov/courses/EE201/EE201-slides/EE201-Lec-6.pdf |title=EE201/MSE207 Lecture 6 |website=intra.ece.ucr.edu |access-date=29 October 2022}}</ref> <math>M_{11}=1/S_{12}^*= 1/S_{21} ^* {,}\  M_{22}= M_{11}^*</math> and <math>M_{12}=-S_{11}^*/S_{12}^* = S_{22}/S_{12} {,}\ M_{21} = M_{12}^*</math>, whereby time-reversal symmetry is assumed.

In the case of time-reversal symmetry, the transfer matrix <math>\mathbf{M}</math> can be expressed by three real parameters:

<math display="block">M = \frac{1}{\sqrt{1-r^2}} \begin{pmatrix} e^{i\varphi} & -r\cdot e^{-i\delta} \\ -r\cdot e^{i\delta} & e^{-i\varphi} \end{pmatrix}</math> 
with <math>\delta,\varphi \in [0;2\pi]</math> and <math>r\in [0;1]</math> (in case {{math|1=''r'' = 1}} there would be no connection between the left and the right side)

==== Finite square well ====
The one-dimensional, non-relativistic problem with time-reversal symmetry of a particle with mass m that approaches a (static) [[finite potential well|finite square ''well'']], has the potential function {{math|''V''}} with
<math display="block">V(x) = \begin{cases}
 -V_0 & \text{for}~~ |x| \le a ~~ (V_0 > 0) \quad\text{and}\\[1ex]
 0 & \text{for}~~ |x|>a
\end{cases}</math>
The scattering can be solved by decomposing the [[wave packet]] of the free particle into plane waves <math>A_k\exp(ikx)</math> with wave numbers <math>k>0</math> for a plane wave coming (faraway) from the left side or likewise <math>D_k\exp(-ikx)</math> (faraway) from the right side.

The S-matrix for the plane wave with wave number {{mvar|k}} has the solution:<ref name="ucr.edu"/>
<math display="block">S_{12}=S_{21}=\frac{\exp(-2ika)}{\cos(2la)-i\sin(2la)\frac{l^2+k^2}{2kl}}</math>
and <math>S_{11}=S_{12}\cdot i\sin(2la)\frac{l^2-k^2}{2kl}</math>  ; hence <math>e^{i\delta}=\pm i</math> and therefore  <math>-e^{-i\delta}=e^{i\delta}</math> and <math>S_{22}=S_{11}</math> in this case.

Whereby <math>l = \sqrt{k^2+\frac{2mV_0}{\hbar^2}}</math> is the (increased) wave number of the plane wave inside the square well, as the energy [[eigenvalue]] <math>E_k</math> associated with the plane wave has to stay constant: <math>E_k = \frac{\hbar^2 k^2}{2m}=\frac{\hbar^2 l^2}{2m}-V_0</math>

The transmission is <math>T_k = |S_{21}|^2=|S_{12}|^2=\frac{1}{(\cos(2la))^2+(\sin(2la))^2\frac{(l^2+k^2)^2}{4k^2l^2}}=\frac{1}{1+(\sin(2la))^2\frac{(l^2-k^2)^2}{4 k^2 l^2}}</math>

In the case of <math>\sin(2la)=0</math> then <math>\cos(2la)=\pm 1</math> and therefore <math>S_{11} = S_{22} = 0</math> and <math>|S_{21}| = |S_{12}| = 1</math> i.e. a plane wave with wave number k passes the well without reflection if <math>k^2+\frac{2mV_0}{\hbar^2}=\frac{n^2 \pi^2}{4a^2}</math> for a <math>n\in\mathbb{N}</math>

==== Finite square barrier ====
The square ''barrier'' is similar to the square well with the difference that <math>V(x)=+V_0 > 0</math> for <math>|x|\le a</math>.

There are three different cases depending on the energy eigenvalue <math>E_k=\frac{\hbar^2 k^2}{2m}</math> of the plane waves (with wave numbers {{mvar|k}} resp. {{math|−''k''}}) far away from the barrier:
{{unordered list
| <math>E_k > V_0</math>: In this case <math>l = \sqrt{k^2-\frac{2mV_0}{\hbar^2}}</math> and the formulas for <math>S_{ij}</math> have the same form as is in the square well case, and the transmission is <math>T_k = |S_{21}|^2 = |S_{12}|^2 = \frac{1}{1+(\sin(2la))^2 \frac{(l^2 - k^2)^2}{4 k^2 l^2}}</math>

| <math>E_k = V_0</math>: In this case <math>\sqrt{k^2-\frac{2mV_0}{\hbar^2}} = 0</math> and the wave function <math>\psi(x)</math> has the property <math>\psi''(x)=0</math> inside the barrier and
{{pb}}
<math>S_{12}=S_{21}=\frac{\exp(-2ika)}{1-ika}</math>  and <math>S_{11} = S_{22} = \frac{-ika\cdot\exp(-2ika)}{1-ika}</math>
{{pb}}
The transmission is: <math>T_k=\frac{1}{1+k^2 a^2}</math>. This intermediate case is not singular, it's the limit (<math>l \to 0</math> resp. <math>\kappa \to 0</math>) from both sides.

| <math>E_k < V_0</math>:In this case <math>\sqrt{k^2-\frac{2mV_0}{\hbar^2}}</math> is an imaginary number. So the wave function inside the barrier has the components <math>e^{\kappa x}</math> and <math>e^{-\kappa x}</math> with <math>\kappa = \sqrt{\frac{2mV_0}{\hbar^2}-k^2}</math>. 
{{pb}}
The solution for the S-matrix is:<ref name="ucsd.edu">{{cite web |url=https://quantummechanics.ucsd.edu/ph130a/130_notes/node152.html |title=The Potential Barrier |access-date=1 November 2022 |website=quantummechanics.ucsd.edu}}</ref> <math>S_{12} = S_{21} = \frac{\exp(-2ika)}{\cosh(2\kappa a)-i\sinh(2\kappa a)\frac{k^2-{\kappa}^2}{2k\kappa}}</math> 
{{pb}}
and likewise: <math>S_{11}=-i\frac{k^2+\kappa^2}{2k\kappa}\sinh(2\kappa a)\cdot S_{12}</math> and also in this case <math>S_{22}=S_{11}</math>.
{{pb}}
The transmission is <math>T_k=|S_{21}|^2=|S_{12}|^2=\frac{1}{1+(\sinh(2\kappa a))^2\frac{(k^2+\kappa^2)^2}{4k^2\kappa^2}}</math>.
}}

=== Transmission coefficient and reflection coefficient ===
The [[transmission coefficient]] from the left of the potential barrier is, when {{math|1=''D'' = 0}},
<math display="block">T_{\rm L}=\frac{|C|^2}{|A|^2} = |S_{21}|^2. </math>

The [[reflection coefficient]] from the left of the potential barrier is, when {{math|1=''D'' = 0}},
<math display="block">R_{\rm L}=\frac{|B|^2}{|A|^2}=|S_{11}|^2.</math>

Similarly, the transmission coefficient from the right of the potential barrier is, when {{math|1=''A'' = 0}},
<math display="block">T_{\rm R}=\frac{|B|^2}{|D|^2}=|S_{12}|^2.</math>

The reflection coefficient from the right of the potential barrier is, when {{math|1=''A'' = 0}},
<math display="block">R_{\rm R}=\frac{|C|^2}{|D|^2}=|S_{22}|^2.</math>

The relations between the transmission and reflection coefficients are
<math display="block">T_{\rm L}+R_{\rm L}=1</math>
and
<math display="block">T_{\rm R}+R_{\rm R}=1.</math>
This identity is a consequence of the unitarity property of the ''S''-matrix.

With time-reversal symmetry, the S-matrix is symmetric and hence <math>T_{\rm L}=|S_{21}|^2=|S_{12}|^2=T_{\rm R}</math> and <math>R_{\rm L} = R_{\rm R}</math>.

=== Optical theorem in one dimension ===
In the case of [[free particle]]s {{math|1=''V''(''x'') = 0}}, the ''S''-matrix is<ref>{{harvnb|Merzbacher|1961}} Ch 6. A more common convention, utilized below, is to have the ''S''-matrix go to the identity in the free particle case.</ref>
<math display="block"> S=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>
Whenever {{math|''V''(''x'')}} is different from zero, however, there is a departure of the ''S''-matrix from the above form, to
<math display="block"> S = \begin{pmatrix} 2ir & 1+2it \\ 1+2it &2ir^* \frac{1+2it}{1-2it^*} \end{pmatrix}.</math>
This departure is parameterized by two [[complex functions]] of energy, {{math|''r''}} and {{math|''t''}}.
From unitarity there also follows a relationship between these two functions,
<math display="block">|r|^2+|t|^2 = \operatorname{Im}(t).</math>

The analogue of this identity in three dimensions is known as the [[optical theorem]].

== 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.

== Evolution operator ''U'' ==
Define a time-dependent creation and annihilation operator as follows,
<math display="block">\begin{align}
a^{\dagger}{\left(k,t\right)} &= U^{-1}(t) \, a^{\dagger}_{\rm i}{\left(k\right)} \, U{\left( t \right)} \\[1ex]
          a{\left(k,t\right)} &= U^{-1}(t) \,           a_{\rm i}{\left(k\right)} \, U{\left( t \right)} \, ,
\end{align}</math>
so, for the fields,
<math display="block">\phi_{\rm f}=U^{-1}(\infty)\phi_{\rm i} U(\infty)=S^{-1}\phi_{\rm i} S~,</math>
where
<math display="block">S= e^{i\alpha}\, U(\infty).</math>

We allow for a phase difference, given by
<math display="block">e^{i\alpha}=\left\langle 0|U(\infty)|0\right\rangle^{-1} ~,</math>
because for {{mvar|S}},
<math display="block">S\left|0\right\rangle = \left|0\right\rangle \Longrightarrow \left\langle 0|S|0\right\rangle = \left\langle 0|0\right\rangle =1 ~.</math>

Substituting the explicit expression for {{mvar|U}}, one has
<math display="block">S=\frac{1}{\left\langle 0|U(\infty)|0\right\rangle}\mathcal T e^{-i\int{d\tau H_{\rm{int}}(\tau)}}~,</math>
where <math> H_{\rm{int}}</math> is the interaction part of the Hamiltonian and <math> \mathcal T </math> is the time ordering.

By inspection, it can be seen that this formula is not explicitly covariant.

== Dyson series ==

{{Main article|Dyson series}}

The most widely used expression for the ''S''-matrix is the Dyson series. This expresses the ''S''-matrix operator as the [[Series (mathematics)|series]]:
<math display="block">S = \sum_{n=0}^\infty \frac{(-i)^n}{n!} \int \cdots \int d^4x_1 d^4x_2 \ldots d^4x_n T [ \mathcal{H}_{\rm{int}}(x_1) \mathcal{H}_{\rm{int}}(x_2) \cdots \mathcal{H}_{\rm{int}}(x_n)] </math>
where:
* <math>T[\cdots]</math> denotes [[time-ordering]],
* <math>\; \mathcal{H}_{\rm{int}}(x)</math> denotes the [[interaction Hamiltonian]] density which describes the interactions in the theory.

== The not-''S''-matrix ==
Since the transformation of particles from black hole to [[Hawking radiation]] could not be described with an ''S''-matrix, Stephen Hawking proposed a "not-''S''-matrix", for which he used the dollar sign ($), and which therefore was also called "dollar matrix".<ref>[[Leonard Susskind]], ''Black Hole War'', chapter 11.</ref>

== See also ==
{{Portal|Mathematics}}
* [[Feynman diagram]]
* [[LSZ reduction formula]]
* [[Wick's theorem]]
* [[Haag's theorem]]
* [[Interaction picture]]
* [[Levinson's theorem]]
* [[Initial and final state radiation]]
* [[Pauli Matrices]]

== Remarks ==
{{reflist|group=nb}}

== Notes ==
{{Reflist}}

== References ==
* {{cite book | author=[[Lev Landau|L.D. Landau]], [[Evgeny Lifshitz|E.M. Lifshitz]] | year=1977 | title=Quantum Mechanics: Non-Relativistic Theory | edition=3rd | volume=3 | publisher=[[Pergamon Press]] | isbn=978-0-08-020940-1 }} §125
* {{citation|last=Weinberg|first=S.|year=2002|title=The Quantum Theory of Fields, vol I|isbn=0-521-55001-7|author-link=Steven Weinberg | publisher = [[Cambridge University Press]] | url-access=registration|url=https://archive.org/details/quantumtheoryoff00stev }}
* {{citation|author-link=Eugen Merzbacher|last=Merzbacher|first=Eugen|title=Quantum Mechanics|at=Ch 13, §3; Ch 19, §6|year=1961 |publisher=Wiley & Sons|isbn=0-471-59670-1}}
* {{cite book|last1=Sakurai|first1=J.J.|author-link1=J. J. Sakurai|last2=Napolitano|first2=J|year=2011|orig-year=1964|title=[[Modern Quantum Mechanics]]|at=Chapter 6|edition=2nd|publisher=[[Addison Wesley]]|isbn=978-0-8053-8291-4}}
* {{cite book |first=Asim Orhan|last=Barut |author-link=Asım Orhan Barut|year=1967 |title=The Theory of the Scattering Matrix for the interactions of fundamental particles|publisher=Macmillan}}
* {{cite book |author=Albert Messiah |title=Quantum Mechanics |year= 1999 |publisher= Dover Publications
|url=https://books.google.com/books?id=mwssSDXzkNcC |isbn=0-486-40924-4 |author-link=Albert Messiah }}
* {{cite web |url=http://www.math.sunysb.edu/~tony/whatsnew/column/feynman-1101/feynman1.html |title=Finite-dimensional Feynman Diagrams |access-date=2007-10-23 |author=Tony Philips |date=November 2001 |work=What's New In Math |publisher=[[American Mathematical Society]]}}
* {{cite book |author=Stephen Gasiorowicz|title=Quantum Physics |year= 1974 |publisher=Wiley & Sons
|url=http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471057002.html |isbn=0-471-29281-8 }}
* {{citation|last1=Greiner|first1=W.|author-link1=Walter Greiner|last2=Reinhardt|first2=J.|year=1996|title=Field Quantization|publisher=[[Springer Publishing]]|isbn=3-540-59179-6|url-access=registration|url=https://archive.org/details/fieldquantizatio0000grei}}
* {{cite journal | doi = 10.1016/0370-1573(92)90047-4| title = Off-critical statistical models: Factorized scattering theories and bootstrap program| journal = Physics Reports| volume = 218| issue = 5–6| pages = 215–379| year = 1992| last1 = Mussardo | first1 = G. |bibcode = 1992PhR...218..215M }}
* {{cite journal | doi = 10.1016/0003-4916(79)90391-9| title = Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models| journal = Annals of Physics| volume = 120| issue = 2| pages = 253–291| year = 1979| last1 = Zamolodchikov | first1 = A. B. |bibcode = 1979AnPhy.120..253Z }}

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[[Category:Quantum field theory]]
[[Category:Scattering theory]]
[[Category:Matrices (mathematics)]]
[[Category:Mathematical physics]]