S-matrix
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 (QFT). More formally, in the context of QFT, the "S"-matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the "in-states" and the "out-states") in the Hilbert space of physical states. A multi-particle state is said to be "free" (non-interacting) if it transforms under Lorentz transformations as a tensor product, or "direct product" in physics  parlance, of "one-particle states" as prescribed by equation below. "Asymptotically free" then means that the state has this appearance in either the distant past or the distant future. 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. In this special case, the Hilbert space is a space of irreducible unitary representations of the inhomogeneous Lorentz group (the Poincaré group); the "S"-matrix is the evolution operator between formula_1 (the distant past), and formula_2 (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. History. 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", but did not develop it fully. In the 1940s, Werner Heisenberg independently developed and substantiated the idea of the "S"-matrix. Because of 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. Today, however, exact "S"-matrix results are a crowning achievement of conformal field theory, integrable systems, and several further areas of quantum field  theory and string theory. "S"-matrices are not substitutes for a field-theoretic treatment, but rather, complement the end results of such. Motivation. In high-energy particle physics one is interested in computing the probability for different outcomes in scattering experiments. These experiments can be broken down into three stages: 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 different incoming particles collide with different energies. The "S"-matrix in quantum  field theory achieves exactly this. It is assumed that the small-energy-density approximation is valid in these cases. Use. 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 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. In one-dimensional quantum mechanics. A simple prototype in which the "S"-matrix is 2-dimensional is considered  first, for the purposes of illustration. In it, particles with sharp energy scatter from a localized potential 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 yields a matrix that depends on . Thus, the total "S"-matrix could, figuratively speaking, be visualized, in a suitable basis, as a "continuous matrix" with every element zero except for -blocks along the diagonal for a given . Definition. Consider a localized one dimensional potential barrier , subjected to a beam of quantum particles with energy . These particles are  incident on the potential barrier from left to right. The solutions of Schrödinger's equation outside the potential barrier are plane waves given by formula_3 for the region to the left of the potential barrier, and formula_4 for the region to the right to the potential barrier, where formula_5 is the wave vector. The time dependence is not needed in our overview and is hence omitted. The term with coefficient represents the incoming wave, whereas term with coefficient represents the outgoing wave. stands for the reflecting wave. Since we set the incoming wave moving in the positive direction (coming from  the left), is zero and can be omitted. The "scattering amplitude", i.e., the transition overlap of the outgoing waves with the incoming waves is a linear relation defining the "S"-matrix, formula_6 The above relation can be written as formula_7 where formula_8 The elements of completely characterize the scattering properties of the potential barrier . Unitary property. The unitary property of "S"-matrix is directly related to the conservation of the probability current in quantum mechanics. The probability current of the wave function is defined as formula_9 The current density to the left of the barrier is formula_10 while the current density to  the right of the barrier is formula_11 For conservation of the probability current density, . This implies the "S"-matrix is a unitary matrix. Time-reversal symmetry. If the potential 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 formula_12 for the region to the left to the potential barrier, and formula_13 for the region to the right to the potential barrier, where the terms with coefficient , represent incoming wave, and terms with coefficient , represent outgoing wave. They are  again related by the "S"-matrix, formula_14 that is, formula_15 Now, the relations formula_16 together yield a condition formula_17 This condition, in conjunction with the unitarity relation, implies that the "S"-matrix is symmetric, as a result of time reversal symmetry, formula_18 Transmission coefficient and reflection coefficient. The transmission coefficient from the left of the potential barrier is, when , formula_19 so the in and out states sough after are eigenstates of the full Hamiltonian that are necessarily non-interacting due to the absence of mixed particle energy terms. The discussion in the section above suggests that the in states and the out  states should be such that formula_20 for large positive and negative has the appearance of the corresponding package, represented by , of free-particle states, 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. The right hand side follows from that the in and out states are eigenstates of the Hamiltonian per above. To formalize this requirement, assume that the full Hamiltonian can be divided into two terms, a free-particle Hamiltonian and an interaction , such that the eigenstates of  have the same appearance as the in- and out-states with respect to normalization and Lorentz transformation properties, formula_21 formula_22 The in and out states are defined as eigenstates of the full Hamiltonian, formula_23 satisfying formula_24 for or respectively. Define formula_25 then formula_26 This last expression will work only using wave packages.From these definitions follow that the in and out states are normalized in the same way as the free-particle states, formula_27 and the three sets are unitarily equivalent. Now rewrite the eigenvalue equation, formula_28 where the terms has been added to make the operator on the LHS  invertible. Since the in and out states reduce to the free-particle states for , put formula_29 on the RHS to obtain formula_30 Then use the completeness of the free-particle states, formula_31 to finally obtain formula_32 Here has been replaced by its eigenvalue on the free-particle states. This is the Lippmann–Schwinger equation. In states expressed as out states. The initial states can be expanded in a basis of final states (or vice versa). Using the completeness relation, formula_33 formula_34 where is the probability that the interaction transforms formula_35 into formula_36. By the ordinary rules of quantum mechanics, formula_37  and one may write formula_38 The expansion coefficients are precisely the "S"-matrix elements to be defined below. The "S"-matrix. The "S"-matrix is now defined by formula_39 Here and are shorthands that represent the particle content but suppresses the individual labels. Associated to the "S"-matrix there is the S-operator defined by formula_40 where the are free particle states. This definition conforms with the direct approach used in the interaction picture. Also, due to unitary equivalence, formula_41 As a physical requirement, must be a unitary operator. This is a statement of conservation of probability in quantum field theory. But formula_42 By  completeness then, formula_43 so "S" is the unitary transformation from in-states to out states. Lorentz invariance is another crucial requirement on the "S"-matrix. The S-operator represents the quantum canonical transformation of the initial "in" states to the final "out" states. Moreover, leaves the vacuum state invariant and transforms "in"-space fields to "out"-space fields, formula_44 formula_45 In terms of creation and annihilation operators, this becomes formula_46 hence formula_47 A similar expression holds when operates to the left on an out state. This means that the "S"-matrix can be expressed as formula_48 If describes an interaction correctly, these properties must  be also true: Evolution operator "U". Define a time-dependent creation and annihilation operator as follows, formula_49 formula_50 so, for the fields, formula_51 where formula_52 We allow for a phase difference, given by formula_53 because for , formula_54 Substituting the explicit expression for , one has formula_55 where formula_56 is the interaction part of the hamiltonian and formula_57 is the time ordering. By inspection, it can be seen that this formula is not explicitly covariant. Dyson series. The most widely used expression for the "S"-matrix is the Dyson series. This expresses the "S"-matrix operator as the series: formula_58 where: The not-"S"-matrix. Since  the transformation of particles from black hole to Hawking radiation could not be described with an "S"-matrix, Stephen Hawking proposed a "not-"S"-matrix", for which he used the dollar sign ($), and which therefore was also called "dollar matrix".