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# S-matrix

Scattering matrix redirects here. For the meaning in linear electrical networks, see scattering parameters.

In physics, the (S-matrix) relates the initial state and the final state for an interaction of particles. It is used in quantum mechanics, scattering theory and quantum field theory.

More formally, the S-matrix is defined as the unitary matrix connecting asymptotic particle states in the Hilbert space of physical states (scattering channels). While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no 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 S-matrix is the evolution operator between time equal to minus infinity, and time equal to plus infinity. 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.

## Explanation

### Use of S-matrices

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 Dirac picture; it may be also 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 we cannot describe exactly the interaction (at least, the most interesting ones).

### Mathematical definition

In Dirac notation, we define $left |0rightrangle$ as the vacuum quantum state. If $a^\left\{dagger\right\}\left(k\right)$ is a creation operator, its hermitian conjugate (destruction or annihilation operator) acts on the vacuum as follows:

$a\left(k\right)left |0rightrangle = 0$

Now, we define two kinds of creation/destruction operators, acting on different Hilbert spaces (IN space i, OUT space f), $a_i^dagger \left(k\right)$ and $a_f^dagger \left(k\right)$.

So now

$mathcal H_mathrm\left\{IN\right\} = operatorname\left\{span\right\}\left\{ left| I, k_1ldots k_n rightrangle = a_i^dagger \left(k_1\right)cdots a_i^dagger \left(k_n\right)left| I, 0rightrangle\right\},$
$mathcal H_mathrm\left\{OUT\right\} = operatorname\left\{span\right\}\left\{ left| F, p_1ldots p_n rightrangle = a_f^dagger \left(p_1\right)cdots a_f^dagger \left(p_n\right)left| F, 0rightrangle\right\}.$

It is possible to prove that $left| I, 0rightrangle$ and $left| F, 0rightrangle$ are both invariant under translation and that the states $left| I, k_1ldots k_n rightrangle$ and $left| F, p_1ldots p_n rightrangle$ are eigenstates of the momentum operator $mathcal P^mu$.

In the Heisenberg picture the states are time-independent, so we can expand initial states on a basis of final states (or vice versa) as follows:

$left| I, k_1ldots k_n rightrangle = C_0 + sum_\left\{m=1\right\}^infty int\left\{d^4p_1ldots d^4p_mC_m\left(p_1ldots p_m\right)left| F, p_1ldots p_n rightrangle\right\}$
Where $left|C_mright|^2$ is the probability that the interaction transforms $left| I, k_1ldots k_n rightrangle$ into $left| F, p_1ldots p_n rightrangle$

According to Wigner's theorem, $S$ must be a unitary operator such that $left langle I,beta right |Sleft | I,alpharightrangle = S_\left\{alphabeta\right\} = left langle F,beta | I,alpharightrangle$. Moreover, $S$ leaves the vacuum state invariant and transforms IN-space fields in OUT-space fields:

$Sleft|0rightrangle = left|0rightrangle$

$phi_f=S^\left\{-1\right\}phi_f S$

If $S$ describes an interaction correctly, these properties must be also true:

If the system is made up with a single particle in momentum eigenstate $left| krightrangle$, then $Sleft| krightrangle=left| krightrangle$

The S-matrix element must be nonzero if and only if momentum is conserved.

### S-matrix and evolution operator U

$aleft\left(k,tright\right)=U^\left\{-1\right\}\left(t\right)a_ileft\left(kright\right)Uleft\left(t right\right)$

$phi_f=U^\left\{-1\right\}\left(infty\right)phi_i U\left(infty\right)=S^\left\{-1\right\}phi_i S$

Therefore $S=e^\left\{ialpha\right\}U\left(infty\right)$ where

$e^\left\{ialpha\right\}=leftlangle 0|U\left(infty\right)|0rightrangle^\left\{-1\right\}$

because

$Sleft|0rightrangle = left|0rightrangle.$

Substituting the explicit expression for U we obtain:

$S=frac\left\{1\right\}\left\{leftlangle 0|U\left(infty\right)|0rightrangle\right\}mathcal T e^\left\{-iint\left\{dtau V_i\left(tau\right)\right\}\right\}$

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