Definitions

S-matrix

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 |0rightrangle 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 |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 (k) and a_f^dagger (k).

So now

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

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_{m=1}^infty int{d^4p_1ldots d^4p_mC_m(p_1ldots p_m)left| F, p_1ldots p_n rightrangle}
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_{alphabeta} = 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^{-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| 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(k,tright)=U^{-1}(t)a_ileft(kright)Uleft(t right)

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)|0rightrangle^{-1}

because

Sleft|0rightrangle = left|0rightrangle.

Substituting the explicit expression for U we obtain:

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

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

See also

Bibliography

(1967). The Theory of the Scattering Matrix. Tony Philips Finite-dimensional Feynman Diagrams. What's New In Math. American Mathematical Society. (2001). Retrieved on 2007-10-23..

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