In physics, the S-matrix 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 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[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 S-matrix 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.
- 1 History
- 2 Motivation
- 3 Dyson series
- 4 S-matrix in one dimensional quantum mechanics
- 5 See also
- 6 Notes
- 7 References
The S-matrix 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'". 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".
In the 1940s, Werner Heisenberg developed, independently, the idea of the S-matrix. 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" S-matrix.
After World War II, the clout of Heisenberg and his attachment to the S-matrix approach may have retarded development of alternative approaches and the closer study of sub-hadronic 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 S-Matrix dogma with great perfection and skill before it was buried down in the seventies after discovery of quarks and asymptotic freedom" 
1. Collide together a collection of incoming particles (usually two particles with high energies).
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 S-matrix in quantum field theory is used to do exactly this. It is assumed that the small-energy-density approximation is valid in these cases.
Use of S-matrices
The S-matrix 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 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 interaction picture; it may also be 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 often we cannot describe the interaction (at least, not the most interesting ones) exactly.
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 time-independent, 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
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 S-matrix element may be nonzero only where the output state has the same total momentum as the input state.
S-matrix and evolution operator U
Define a time-dependent creation and annihilation operator as follow
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.
- denotes time-ordering,
- denotes the interaction Hamiltonian density which describes the interactions in the theory.
S-matrix in one dimensional quantum mechanics
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 S-matrix
Elements of this matrix completely characterize the scattering properties of the V(x).
- The above relation can be written as
- , and .
Unitary property of S-matrix
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 S-matrix is a unitary matrix.
If the potential V(x) is real,then the system possesses time-reversal symmetry.Under this condition if is a solution of Schrödinger's equation,then is also a solution.The time-reversed 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 S-matrix :
- or .
Now,the relations and together yield a condition . This condition in conjunction with the unitary relation implies that the S-matrix is symmetric as a result of time reversal symmetry:
Transmission coefficient and Reflection coefficient
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 S-matrix.Thus the elements of S-matrix are basically reflection and transmission coefficients.
Optical theorem in one dimension
In the case of free particle .The S-matrix is then . Now whenever V(x) is different from zero,there is a departure of S-matrix 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.
- John Archibald Wheeler, 'On the Mathematical Description of Light Nuclei by the Method. of Resonating Group Structure' Phys. Rev. 52, 1107–1122 (1937)
- Jagdish Mehra, Helmut Rechenberg, The Historical Development of Quantum Theory (Pages 990 and 1031) Springer, 2001 ISBN 0-387-95086-9, ISBN 978-0-387-95086-0
- Alexander Migdal, Paradise Lost, Part 1
- Barut, A.O. (1967). The Theory of the Scattering Matrix.
- Tony Philips (November 2001). "Finite-dimensional Feynman Diagrams". What's New In Math. American Mathematical Society. Retrieved 2007-10-23.
- Merzbacher Eugene. Quantum Mechanics. John Wiley and Sons.