S-matrix

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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.

1 Explanation
2 See also
3 Bibliography

[edit] Explanation

[edit] 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).

[edit] Mathematical definition

In Dirac notation, we define $\left |0\right\rangle$ 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:

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

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

$\mathcal H_\mathrm{IN} = \operatorname{span}\{ \left| I, k_1\ldots k_n \right\rangle = a_i^\dagger (k_1)\cdots a_i^\dagger (k_n)\left| I, 0\right\rangle\},$

$\mathcal H_\mathrm{OUT} = \operatorname{span}\{ \left| F, p_1\ldots p_n \right\rangle = a_f^\dagger (p_1)\cdots a_f^\dagger (p_n)\left| F, 0\right\rangle\}.$

It is possible to prove that $\left| I, 0\right\rangle$ and $\left| F, 0\right\rangle$ are both invariant under translation and that the states $\left| I, k_1\ldots k_n \right\rangle$ and $\left| F, p_1\ldots p_n \right\rangle$ 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:

$\left| I, k_1\ldots k_n \right\rangle = C_0 + \sum_{m=1}^\infty \int{d^4p_1\ldots d^4p_mC_m(p_1\ldots p_m)\left| F, p_1\ldots p_n \right\rangle}$

Where $\left|C_m\right|^2$ is the probability that the interaction transforms $\left| I, k_1\ldots k_n \right\rangle$ into $\left| F, p_1\ldots p_n \right\rangle$

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