The S-matrix

In order to make the coupling of gluons to coloured quarks invariant imder charge conjugation, from (Al.2.17) we have to demand that 

For a state of a fermion and its antiparticle with orbital angular momentum I and total spin S one has 

It follows from this that the Si state of positronium decays into three photons whereas the state decays into two photons. 

The situation is more complicated for qq — gluons as a consequence of the more involved rule (Al.2.19), and was discussed in Section 11.6. 

The consequences of CPT invariance on the structure of matrix elements is discussed in Appendix 6. 

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In a third step the S-matrix is related to state-selected reaction cross sections a., in principle observable in beam scattering experiments [28, 29, 30, 31, 32, 33, 34 and 35], by the fiindamental equation of scattering theory... 

The S matrix has a number of important properties, one of which is that it is unitary. Mathematically this... 

Another useful property of the S matrix is that it is symmetric. This property follows from conservation of the fluxlike expression... 

Zhang J Z H and Miller W H 1989 Quantum reactive scattering via the S-matrix version of the Kohn variational principle—differential and integral cross sections for D + Hj —> HD + H J. Chem. Phys. 91 1528... 

As shown in Appendix A, the matrix S k2 is formed by finding the eigenvalues Xi and eigenveetors Vij of the S matrix and then eonstrueting ... 

The S matrix contains the overlap elements between basis functions, and the F matrix contains the Fock matrix elements. Each element contains two parts from the Fock operator (eq. (3.36)), integrals involving the one-electron operators, and a sum over... 

Population analysis with semi-empirical methods requires a special comment. These methods normally employ the ZDO approximation, i.e. the overlap S is a unit matrix. The population analysis can therefore be performed directly on the density matrix. In some cases, however, a Mulliken population analysis is performed with DS, which requires an explicit calculation of the S matrix. 

In this case the basis functions (coordinate system) are non-orthogonal, the overlaps are contained in the S matrix. By multiplying from the left by S and inserting a unit matrix written in the form (13.19) may be reformulated as... 

The S matrix is eventually set equal to a unit matrix which leads to tire following equation for A. 

Before embarking on the problem of the interaction of the negaton-positon field with the quantized electromagnetic field, we shall first consider the case of the negaton-positon field interacting with an external, classical (prescribed) electromagnetic field. We shall also outline in the present chapter those aspects of the theory of the S-matrix that will be required for the treatment of quantum electrodynamics. Section 10.4 presents a treatment of the Dirac equation in an external field. 

The square of this matrix element in< at t = — 00 will be found in the state at t = +00. The transformation operator S is the /S-matrix of Heisenberg. We shall return later in this section to the problem of evaluating the -matrix. 

The above rules are readily applied to the case of the electron-positron field interacting with an external field for which... 

If we restrict ourselves to the case of a hermitian U(ia), the vanishing of this commutator implies that the /S-matrix element between any two states characterized by two different eigenvalues of the (hermitian) operator U(ia) must vanish. Thus, for example, positronium in a triplet 8 state cannot decay into two photons. (Note that since U(it) anticommutes with P, the total momentum of the states under consideration must vanish.) Equation (11-294) when written in the form... 

The ri fiiatrix, due to the tune ordering operator in its definition is not invariant under time inversion. The invariance of the theory under tahi ihversidn has the following important consequence for the S-matrix since this operator s matrix elements axe given by ... 

It is calculated in the S-matrix formalism and averaged over impact distances b and velocities v with Maxwellian distribution f(v)... 

One of the simplest examples of line interference is impact broadening of H atom La Stark structure, observed in plasmas [176] (Fig. 4.1.(a)). For a degenerate ground state the impact operator is linear in the S-matrix ... 

As can be seen, the interference in each pair of lines does not disappear even in the lower order of perturbation theory when off-diagonal elements of the S-matrix are linear in V. In the doublet, which represents the absorption spectrum, pi is the quantum equivalent of Gordon s classical... 

The original semiclassical version of the centrifugal sudden approximation (SCS) developed by Strekalov [198, 199] consistently takes into account adiabatic corrections to IOS. Since the orbital angular momentum transfer is supposed to be small, scattering occurs in the collision plane. The body-fixed correspondence principle method (BFCP) [200] was used to write the S-matrix for f — jf Massey parameter a>xc. At low quantum numbers, when 0)zc —> 0, it reduces to the usual non-adiabatic expression, which is valid for any Though more complicated, this method is the necessary extension of the previous one adapted to account for adiabatic corrections at higher excitation... 

We then invert Eq. (15) by integrating over 0, which yields the following expression for the S-matrix elements,... 

The most well-known and dramatic manifestation of an INR is the appearance of a narrow feature in the integral cross-section (ICS), cr(E) at total energy E = Er of width T. Obviously the resonance peak is closely related to the existence of the resonance pole in the S-matrix. Using the normal body-fixed representation for an A + BC v,j) — AB(v, j ) + C reaction, the ICS is related to the S-matrix by... 

Associated with the pole of the S-matrix is a Seigert state, I-Ves, which has purely outgoing boundary conditions and satisfies (with some caveats) the equation, // I res = z les,H being the system Hamiltonian.44 If a square integrable approximation to I res is constructed, then its time evolution, k . (/,), wiH exhibit pure exponential decay after a transient induction period. Of course any L2 state will show quadratic, and hence non-exponential, decay at short times since... 

For an INR, a pole of the S-matrix can lead to a peak in the ICS if the resonance exists for just one partial wave, or if the resonances for each J are well-separated in energy. It is not surprising that similar conclusions can be drawn for the differential cross-sections (DCS). The DCS is defined from the usual body fixed S-matrix for A + BC — AB + C... 

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