While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no event horizons, it has a simple form in the case of the Minkowski space.
It is defined only in the limit of zero energy density (or infinite particle separation distance).
The initial elements of S-matrix theory are found in Paul Dirac's 1927 paper "Über die Quantenmechanik der Stoßvorgänge".
[1][2] The S-matrix was first properly introduced by John Archibald Wheeler in the 1937 paper "On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure".
[3] In this paper Wheeler introduced a scattering matrix – a unitary matrix of coefficients connecting "the asymptotic behaviour of an arbitrary particular solution [of the integral equations] with that of solutions of a standard form",[4] but did not develop it fully.
In high-energy particle physics one is interested in computing the probability for different outcomes in scattering experiments.
Branch cuts of the S-matrix in the complex-energy plane are associated to the opening of a scattering channel.
In it, particles with sharp energy E scatter from a localized potential V according to the rules of 1-dimensional quantum mechanics.
Already this simple model displays some features of more general cases, but is easier to handle.
Each energy E yields a matrix S = S(E) that depends on V. Thus, the total S-matrix could, figuratively speaking, be visualized, in a suitable basis, as a "continuous matrix" with every element zero except for 2 × 2-blocks along the diagonal for a given V. Consider a localized one dimensional potential barrier V(x), subjected to a beam of quantum particles with energy E. These particles are incident on the potential barrier from left to right.
Since we set the incoming wave moving in the positive direction (coming from the left), D is zero and can be omitted.
The unitary property of the S-matrix is directly related to the conservation of the probability current in quantum mechanics.
This condition, in conjunction with the unitarity relation, implies that the S-matrix is symmetric, as a result of time reversal symmetry,
(in case r = 1 there would be no connection between the left and the right side) The one-dimensional, non-relativistic problem with time-reversal symmetry of a particle with mass m that approaches a (static) finite square well, has the potential function V with
A straightforward way to define the S-matrix begins with considering the interaction picture.
[9] Let the Hamiltonian H be split into the free part H0 and the interaction V, H = H0 + V. In this picture, the operators behave as free field operators and the state vectors have dynamics according to the interaction V. Let
The great advantage of this definition is that the time-evolution operator U evolving a state in the interaction picture is formally known,[10]
Attention is here focused to the simplest case, that of a scalar theory in order to exemplify with the least possible cluttering of the notation.
See the next section for a detailed account on how a general n-particle state is normalized.
A Heisenberg state vector thus represents the complete spacetime history of a system of particles.
A state Ψα, in is characterized by that as t → −∞ the particle content is that represented collectively by α.
Letting τ vary one sees that the observed Ψ (not measured) is indeed the Schrödinger picture state vector.
The sign is plus unless s involves an odd number of fermion transpositions, in which case it is minus.
The notation is usually abbreviated letting one Greek letter stand for the whole collection describing the state.
so the in and out states sought after are eigenstates of the full Hamiltonian that are necessarily non-interacting due to the absence of mixed particle energy terms.
for large positive and negative τ has the appearance of the corresponding package, represented by g, of free-particle states, g assumed smooth and suitably localized in momentum.
Wave packages are necessary, else the time evolution will yield only a phase factor indicating free particles, which cannot be the case.
To formalize this requirement, assume that the full Hamiltonian H can be divided into two terms, a free-particle Hamiltonian H0 and an interaction V, H = H0 + V such that the eigenstates Φγ of H0 have the same appearance as the in- and out-states with respect to normalization and Lorentz transformation properties,
Here α and β are shorthands that represent the particle content but suppresses the individual labels.
If S describes an interaction correctly, these properties must be also true: Define a time-dependent creation and annihilation operator as follows,