S-matrix

In physics, the S-matrix or scattering matrix is a matrix that relates the initial state and the final state of a physical system undergoing a 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 transforms under Lorentz transformations as a tensor product, or direct product in physics parlance, of one-particle states as prescribed by equation Template:EquationNote 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 horizons, 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 representations of the 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 state in the asymptotic past and in the asymptotic future are both described by Fock spaces.

History

The initial elements of S-matrix theory are found in Paul Dirac's 1927 paper "Über die Quantenmechanik der Stoßvorgänge".<ref>Template:Cite journal</ref><ref>Template:Cite journal</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, "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 Template:ISBN, Template:ISBN</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:

1. Making a collection of incoming particles collide (usually two kinds of particles with high energies).
2. Allowing the incoming particles to interact. These interactions may change the types of particles present (e.g. if an electron and a positron annihilate they may produce two photons).
3. Measuring the resulting outgoing particles.

The process by which the incoming particles are transformed (through their 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 sections of various interactions; the elements (individual numerical entries) in the S-matrix are known as scattering amplitudes. Poles of the S-matrix in the complex-energy plane are identified with bound states, virtual states or resonances. Branch cuts of the S-matrix in the complex-energy plane are associated to the opening of a scattering channel.

In the Hamiltonian approach to quantum field theory, the S-matrix may be calculated as a time-ordered exponential of the integrated Hamiltonian in the interaction picture; it may also be expressed using Feynman's path integrals. In both cases, the perturbative calculation of the S-matrix leads to Feynman diagrams.

In scattering theory, the S-matrix is an operator mapping free particle in-states to free particle out-states (scattering channels) 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 Template:Math scatter from a localized potential Template:Math 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 Template:Math yields a matrix Template:Math that depends on Template:Math. Thus, the total S-matrix could, figuratively speaking, be visualized, in a suitable basis, as a "continuous matrix" with every element zero except for Template:Math-blocks along the diagonal for a given Template:Math.

Definition

Consider a localized one dimensional potential barrier Template:Math, subjected to a beam of quantum particles with energy Template:Math. 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 Template:Math represents the incoming wave, whereas term with coefficient Template:Math represents the outgoing wave. Template:Math stands for the reflecting wave. Since we set the incoming wave moving in the positive direction (coming from the left), Template:Math 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 Template:Math completely characterize the scattering properties of the potential barrier Template:Math.

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 Template:Mvar of the wave function Template:Math 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, Template:Math. This implies the S-matrix is a unitary matrix. Template:Cleanup Template:Math proof

Time-reversal symmetry

If the potential Template:Math is real, then the system possesses time-reversal symmetry. Under this condition, if Template:Math is a solution of the Schrödinger equation, then Template:Math 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 Template:Math, Template:Math represent incoming wave, and terms with coefficient Template:Math, Template:Math 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,
<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">{{#invoke:citation/CS1|citation |CitationClass=web }}</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">{{#invoke:citation/CS1|citation |CitationClass=web }}</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 Template:Math 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 square well, has the potential function Template:Math 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 Template:Mvar 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 Template:Mvar resp. Template:Math) far away from the barrier: Template:Unordered list</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 Template: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> Template: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>. Template:Pb The solution for the S-matrix is:<ref name="ucsd.edu">{{#invoke:citation/CS1|citation |CitationClass=web }}</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> Template: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>. Template: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 Template:Math, <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 Template:Math, <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 Template:Math, <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 Template:Math, <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 particles Template:Math, the S-matrix is<ref>Template:Harvnb 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 Template:Math 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, Template:Math and Template:Math. 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>Template:Harvnb Section 8.2.</ref> Let the Hamiltonian Template:Math be split into the free part Template:Math and the interaction Template:Math, Template:Math. In this picture, the operators behave as free field operators and the state vectors have dynamics according to the interaction Template:Math. 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 Template:Math is the S-operator. The great advantage of this definition is that the time-evolution operator Template:Mvar evolving a state in the interaction picture is formally known,<ref>Template:Harvnb Equation 8.44.</ref> <math display="block">U(t, t_0) = Te^{-i\int_{t_0}^t d\tau V(\tau)},</math> where Template:Mvar 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> Expanding using the knowledge about Template:Math 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 Template:Mvar 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, Template:Mvar 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>Template:Harvnb 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 Template:Math 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 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 Template:Math. All vacuum states are assumed 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 Template:Math is the generator of translation in space and time, and Template:Math is the generator of Lorentz transformations. 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) Template:Math and Template:Math. 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 Template:Math is the field canonically conjugate to Template:Math. Associated to the in and out fields are two sets of creation and annihilation operators, Template:Math and Template:Math, acting in the same Hilbert space,<ref>Template:Harvnb Chapter 3. See especially remark at the beginning of section 3.2.</ref> on two distinct complete sets (Fock spaces; initial space Template:Mvar, final space Template:Mvar). 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 Template:Nowrap 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>Template:Harvnb Chapter 3.</ref> This is a characteristic of a non-interacting field. From this follows that the asymptotic states are all eigenstates of the momentum operator Template:Math,<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 Template:Math is characterized by that as Template:Math the particle content is that represented collectively by Template:Mvar. Likewise, a state Template:Math will have the particle content represented by Template:Mvar for Template:Math. 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 Template:Math at time Template:Math, then it will be found in the state Template:Math at time Template:Math. 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 Template:Mvar vary one sees that the observed Template:Math (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 Template:Math as at time Template:Math. 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, Template:Harvtxt 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

• Template:Mvar is momentum,
• Template:Mvar is spin z-component or, in the massless case, helicity,
• Template:Mvar 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 Template:Math is a permutation of Template:Math objects (for a Template:Nowrap 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 Template:Mvar 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 Template:Mvar, the right hand side is a projection operator onto the state Template:Mvar. Under an inhomogeneous Lorentz transformation Template:Math, the field transforms according to the rule Template:NumBlk\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>|Template:EquationRef}} where Template:Math is the Wigner rotation and Template:Math is the Template:Nowrap representation of Template:Math. By putting Template:Math, for which Template:Mvar is Template:Math, in Template:EquationNote, 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 Template:Math and the out states Template:Math 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 Template:Mvar has the appearance of the corresponding package, represented by Template:Math, of free-particle states, Template:Math 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 Template:Mvar can be divided into two terms, a free-particle Hamiltonian Template:Math and an interaction Template:Mvar, Template:Math such that the eigenstates Template:Math of Template:Math 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 Template:Math or Template:Math 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 Template:Math 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 Template:Math, 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 Template:Math 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 Template:Math 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 Template:Mvar and Template:Mvar are shorthands that represent the particle content but suppresses the individual labels. Associated to the S-matrix there is the S-operator Template:Mvar defined by<ref name=Weinberg_1/> <math display="block">\langle\Phi_\beta|S|\Phi_\alpha\rangle \equiv S_{\beta\alpha},</math> where the Template:Math are free particle states.<ref name=Weinberg_1/><ref group=nb>Here it is assumed that the full Hamiltonian Template:Mvar can be divided into two terms, a free-particle Hamiltonian Template:Math and an interaction Template:Mvar, Template:Math such that the eigenstates Template:Math of Template:Math have the same appearance as the in- and out-states with respect to normalization and Lorentz transformation properties. See Template:Harvtxt, 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, Template:Mvar 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 Template:Math is a (inhomogeneous) proper orthochronous Lorentz transformation, then Wigner's theorem guarantees the existence of a unitary operator Template:Math acting either on Template:Math or Template:Math. A theory is said to be Lorentz invariant if the same Template:Math acts on Template:Math and Template:Math. Using the unitarity of Template:Math, Template:Math. 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 Template:Harvtxt, equation 3.3.1 gives an explicit form.</ref> The S-operator represents the quantum canonical transformation of the initial in states to the final out states. Moreover, Template:Mvar 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 Template:Harvnb.</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 Template:Mvar 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 Template:Mvar describes an interaction correctly, these properties must be also true:

• If the system is made up with a single particle in momentum eigenstate Template:Math, then Template:Math. 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 Template:Mvar, <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 Template:Mvar, 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

The most widely used expression for the S-matrix is the Dyson series. This expresses the S-matrix operator as the 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

• Feynman diagram
• LSZ reduction formula
• Wick's theorem
• Haag's theorem
• Interaction picture
• Levinson's theorem
• Initial and final state radiation

References

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