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\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:

S\left|0\right\rangle = \left|0\right\rangle

\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\right\rangle, then S\left| k\right\rangle=\left| k\right\rangle

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

S-matrix and evolution operator U

a\left(k,t\right)=U^{-1}(t)a_i\left(k\right)U\left( t \right)

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

Therefore S=e^{i\alpha}U(\infty) where

e^{i\alpha}=\left\langle 0|U(\infty)|0\right\rangle^{-1}

because

S\left|0\right\rangle = \left|0\right\rangle.

Substituting the explicit expression for U we obtain:

S=\frac{1}{\left\langle 0|U(\infty)|0\right\rangle}\mathcal T e^{-i\int{d\tau V_i(\tau)}}

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

See also

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

Bibliography

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