In physics, the Smatrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory.
More formally, the Smatrix is defined as the unitary matrix connecting asymptotic particle states in the Hilbert space of physical states (scattering channels). While the Smatrix may be defined for any background (spacetime) that is asymptotically solvable and has no horizons^{[clarification needed]}, 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^{[clarification needed]}; the Smatrix is the evolution operator between time equal to minus infinity (the distant past), and time equal to plus infinity (the distant future). It is defined only in the limit of zero energy density (or infinite particle separation distance). 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.
Contents
History[edit]
The Smatrix was first introduced by John Archibald Wheeler in the 1937 paper "'On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure'".^{[1]} 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".^{[2]}
In the 1940s, Werner Heisenberg developed, independently, the idea of the Smatrix. Due to the problematic divergences present in quantum field theory at that time Heisenberg was motivated to isolate the essential features of the theory that would not be affected by future changes as the theory developed. In doing so he was led to introduce a unitary "characteristic" Smatrix.^{[2]}
After World War II, the clout of Heisenberg and his attachment to the Smatrix approach may have retarded development of alternative approaches and the closer study of subhadronic physics for a decade or more, at least in Europe: "Pretty much like medieval Scholastic Magisters were extremely inventive in defending the Church Dogmas and blocking the way to experimental science, some great minds in the sixties developed the SMatrix dogma with great perfection and skill before it was buried down in the seventies after discovery of quarks and asymptotic freedom" ^{[3]}
Motivation[edit]
In highenergy particle physics we are interested in computing the probability for different outcomes in scattering experiments. These experiments can be broken down into three stages:
1. Collide together a collection of incoming particles (usually two particles with high energies).
2. Allowing the incoming particles to interact. These interactions may change the types of particles present (e.g. if an electron and a positron annihilate they may produce two photons).
3. Measuring the resulting outgoing particles.
The process by which the incoming particles are transformed (through their interaction) into the outgoing particles is called scattering. For particle physics, a physical theory of these processes must be able to compute the probability for different outgoing particles when we collide different incoming particles with different energies. The Smatrix in quantum field theory is used to do exactly this. It is assumed that the smallenergydensity approximation is valid in these cases.
Use of Smatrices[edit]
The Smatrix is closely related^{[vague]} to the transition probability amplitude in quantum mechanics and to cross sections of various interactions; the elements (individual numerical entries) in the Smatrix are known as scattering amplitudes. Poles of the Smatrix in the complexenergy plane are identified with bound states, virtual states or resonances. Branch cuts of the Smatrix in the complexenergy plane are associated to the opening of a scattering channel.
In the Hamiltonian approach to quantum field theory, the Smatrix may be calculated as a timeordered exponential of the integrated Hamiltonian in the interaction picture; it may also be expressed using Feynman's path integrals. In both cases, the perturbative calculation of the Smatrix leads to Feynman diagrams.
In scattering theory, the Smatrix is an operator mapping free particle instates to free particle outstates (scattering channels) in the Heisenberg picture. This is very useful because often we cannot describe the interaction (at least, not the most interesting ones) exactly.
Mathematical definition[edit]
In Dirac notation, we define as the vacuum quantum state. If is a creation operator, its hermitian conjugate (destruction or annihilation operator) acts on the vacuum as follows:
Now, we define two kinds of creation/destruction operators acting on different Hilbert spaces (initial space i, final space f), and .
So now
It is possible to play the trick assuming that and are both invariant under translation and that the states and are eigenstates of the momentum operator , by adiabatically turning on and off the interaction.
In the Heisenberg picture the states are timeindependent, so we can expand initial states on a basis of final states (or vice versa) as follows:
Where is the probability that the interaction transforms into
According to Wigner's theorem, must be a unitary operator such that . Moreover, leaves the vacuum state invariant and transforms INspace fields in OUTspace fields:
If describes an interaction correctly, these properties must be also true:
 If the system is made up with a single particle in momentum eigenstate , then
 The Smatrix element may be nonzero only where the output state has the same total momentum as the input state.
Smatrix and evolution operator U[edit]
Define a timedependent creation and annihilation operator as follow
Hence
where we have
 .
We allow a phase difference given by
because for :
Substituting the explicit expression for U we obtain:
where is the interaction part of the hamiltonian and is the time ordering. By inspection it can be seen that this formula is not explicitly covariant.
Dyson series[edit]
The most widely used expression for the Smatrix is the Dyson series. This expresses the Smatrix operator as the series:
where:
 denotes timeordering,
 denotes the interaction Hamiltonian density which describes the interactions in the theory.
Smatrix in one dimensional quantum mechanics[edit]
Definition[edit]
Consider a localized one dimensional potential barrier V(x), subjected to a beam of quantum particles with energy E. These particles incident on the potential barrier from left and right. The solution of Schrödinger's equation outside the potential barrier are plane waves and are given by:
for the region left to the potential barrier and
for the region right to the potential barrier, where
is the wave vector,while the terms with coefficients A and D represent the incoming waves,whereas terms with coefficients B and C represent the outgoing waves.The outgoing waves are connected to the incoming waves by a linear relation described by the Smatrix
 .
Elements of this matrix completely characterize the scattering properties of the V(x).
 The above relation can be written as
 where,
 , and .
Unitary property of Smatrix[edit]
The unitary property of Smatrix is directly related the to conservation of probability current in quantum mechanics.The probability current of the wave function is defined as
 .
The current density to the left of the barrier is ,and the current density to the right of the barrier is . According to conservation of probability current density .This implies the Smatrix is a unitary matrix.

Proof
Timereversal symmetry[edit]
If the potential V(x) is real,then the system possesses timereversal symmetry.Under this condition if is a solution of Schrödinger's equation,then is also a solution.The timereversed solution is given by:
for the region left to the potential barrier and
for the region right to the potential barrier, where the terms with coefficient B*,C*represent incoming wave and terms with coefficient A*,D* represent outgoing wave.So they are again related by the Smatrix :
 or .
Now,the relations and together yield a condition . This condition in conjunction with the unitary relation implies that the Smatrix is symmetric as a result of time reversal symmetry:
 .
Transmission coefficient and Reflection coefficient[edit]
Transmission coefficient from the left of the potential barrier is ,when .Thus .
Reflection Coefficient from the left of the potential barrier is ,when .Thus .
Similarly,Transmission Coefficient from the right of the potential barrier is ,when .Thus .
Reflection Coefficient from the right of the potential barrier is ,when .Thus .
The relation between transmission coefficient and reflection coefficient is: and . This relation is the consequence of the unitary property of Smatrix.Thus the elements of Smatrix are basically reflection and transmission coefficients.
Optical theorem in one dimension[edit]
In the case of free particle .The Smatrix is then . Now whenever V(x) is different from zero,there is a departure of Smatrix from the above form.This departure is measured by two complex functions of energy,r and t,which are defined by and .The relation between this two function is given by:
.
The analogue of this identity in three dimension is known as optical theorem.
See also[edit]
Notes[edit]
 ^ John Archibald Wheeler, 'On the Mathematical Description of Light Nuclei by the Method. of Resonating Group Structure' Phys. Rev. 52, 1107–1122 (1937)
 ^ ^{a} ^{b} Jagdish Mehra, Helmut Rechenberg, The Historical Development of Quantum Theory (Pages 990 and 1031) Springer, 2001 ISBN 0387950869, ISBN 9780387950860
 ^ Alexander Migdal, Paradise Lost, Part 1
References[edit]
 Barut, A.O. (1967). The Theory of the Scattering Matrix.
 Tony Philips (November 2001). "Finitedimensional Feynman Diagrams". What's New In Math. American Mathematical Society. Retrieved 20071023.
 Merzbacher Eugene. Quantum Mechanics. John Wiley and Sons.
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