S-matrix

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

In physics, the scattering matrix (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 $left\; |0\; ight\; angle$ as the vacuum quantum state. If $a^\{dagger\}(k)$ is a creation operator, its hermitian conjugate (destruction or annihilation operator) acts on the vacuum as follows:

:$a(k)left\; |0\; ight\; angle\; =\; 0$

Now, we define two kinds of creation/destruction operators, acting on different Hilbert spaces (IN space "i", OUT space "f"), $a\_i^dagger\; (k)$ and $a\_f^dagger\; (k)$.

So now

:$mathcal\; H\_mathrm\{IN\}\; =\; operatorname\{span\}\{\; left|\; I,\; k\_1ldots\; k\_n\; ight\; angle\; =\; a\_i^dagger\; (k\_1)cdots\; a\_i^dagger\; (k\_n)left|\; I,\; 0\; ight\; angle\},$:$mathcal\; H\_mathrm\{OUT\}\; =\; operatorname\{span\}\{\; left|\; F,\; p\_1ldots\; p\_n\; ight\; angle\; =\; a\_f^dagger\; (p\_1)cdots\; a\_f^dagger\; (p\_n)left|\; F,\; 0\; ight\; angle\}.$

It is possible to prove that $left|\; I,\; 0\; ight\; angle$ and $left|\; F,\; 0\; ight\; angle$ are both invariant under translation and that the states $left|\; I,\; k\_1ldots\; k\_n\; ight\; angle$ and $left|\; F,\; p\_1ldots\; p\_n\; ight\; angle$ 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:

:$left|\; I,\; k\_1ldots\; k\_n\; ight\; angle\; =\; C\_0\; +\; sum\_\{m=1\}^infty\; int\{d^4p\_1ldots\; d^4p\_mC\_m(p\_1ldots\; p\_m)left|\; F,\; p\_1ldots\; p\_n\; ight\; angle\}$ Where $left|C\_m\; ight|^2$ is the probability that the interaction transforms $left|\; I,\; k\_1ldots\; k\_n\; ight\; angle$ into $left|\; F,\; p\_1ldots\; p\_n\; ight\; angle$

According to Wigner's theorem, $S$ must be a unitary operator such that . Moreover, $S$ leaves the vacuum state invariant and transforms IN-space fields in OUT-space fields:

:$Sleft|0\; ight\; angle\; =\; left|0\; ight\; angle$

:$phi\_f=S^\{-1\}phi\_f\; S$

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 $left|\; k\; ight\; angle$, then $Sleft|\; k\; ight\; angle=left|\; k\; ight\; angle$

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

-matrix and evolution operator "U"

:$aleft(k,t\; ight)=U^\{-1\}(t)a\_ileft(k\; ight)Uleft(\; t\; ight)$

:$phi\_f=U^\{-1\}(infty)phi\_i\; U(infty)=S^\{-1\}phi\_i\; S$

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

:$e^\{ialpha\}=leftlangle\; 0|U(infty)|0\; ight\; angle^\{-1\}$

because

:$Sleft|0\; ight\; angle\; =\; left|0\; ight\; angle.$

Substituting the explicit expression for "U" we obtain:

:$S=frac\{1\}\{leftlangle\; 0|U(infty)|0\; ight\; angle\}mathcal\; T\; e^\{-iint\{d\; au\; V\_i(\; au)$

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

ee also

*Feynman diagram
*LSZ reduction formula
*Wick's theorem

Bibliography