Two-particle bound states

In order to discuss the fundamental problems that are connected with the bound states in kinetic theory, we first restrict ourselves to systems with two-particle bound states only. The states of the two-particle system are determined by Eq. (2.12). Furthermore, we remark that to describe the formation of two-particle bound states by a collision, at least three particles are necessary in order to fulfill energy and momentum conservation. Thus, it is necessary to consider the quantum mechanics of three-particle systems. 

Because of the existence of two-particle bound states the three-particle scattering states split into channels that means, we have to take into account scattering processes between free particles and bound pairs. To study the effect of composite particle scattering processes, we pay atten-... 

Let us now consider the problem of the condition of weakening of the initial correlation in systems with two-particle bound states. There are... 

Clearly, it is necessary to generalize the Bogolyubov condition (2.4). It is obvious in generalizing (2.4) to assume that in systems with two-particle bound states, the pair 1 and 2 moved independently of all other particles as t—> -< . Then it is possible to write... 

Because a weakening of the initial correlation for Fbn is absent, a system with two-particle bound states cannot be described only by F, but by F 2, and it is not possible to obtain a closed equation for F,. 

Even the approximation (3.38) for 1234 contains, in a system supporting two-particle bound states, a large variety of scattering processes between subclusters. Among these possibilities, we consider only the scattering between the two (isolated) bound states (12) and (34). This means that... 

With Eq. (3.37) for Fn it is possible to write a kinetic equation for F, that describes the formation and the decay of two-particle bound states in three-particle collisions. Introducing (3.37) into the first equation of the hierarchy (1-29), we obtain in a similar way as in Section III.2 a kinetic equation for the density operator of free particles. This equation may be written in the following form ... 

Obviously we may expect that the simple two- and three-particle collision approximation discussed in the previous sections is not appropriate, because a large number of particles always interact simultaneously. Formally this approximation leads to divergencies. In the previous sections we used in a systematic way cluster expansions for the two- and three-particle density operator in order to include two-particle bound states and their relevant interaction in three- and four-particle clusters. In the framework of that consideration we started with the elementary particles (e, p) and their interactions. The bound states turned out to be special states, and, especially, scattering states were dealt with in a consistent manner. 

By that procedure, an additional factor V l appears in the equation of motion of pep [Eq. (4.29)]. This factor leads to the fact that the four-particle processes accounted for in this manner are not real and may vanish in the thermodynamic limit. At least this is true for four-particle scattering states. However, in the limiting case that we have only two-particle bound states, that is, the neutral gas, we can obtain a kinetic equation for the atoms if we use the special definition of the distribution function of the atoms (4.17) and (4.24). Using the ideas just outlined, the kinetic equation (4.62) was obtained. 

The modification of the three and four-particle system due to the medium can be considered in cluster-mean field approximation. Describing the medium in quasi-particle approximation, a medium-modified Faddeev equation can be derived which was already solved for the case of three-particle bound states in [9], as well as for the case of four-particle bound states in [10]. Similar to the two-particle case, due to the Pauli blocking the bound state disappears at a given temperature and total momentum at the corresponding Mott density. 

The first step toward a practical relativistic many-electron theory in the molecular sciences is the investigation of the two-electron problem in an external field which we meet, for instance, in the helium atom. Salpeter and Bethe derived a relativistic equation for the two-electron bound-state problem [135,170-173] rooted in quantum electrod)mamics, which features two separate times for the two particles. If we assume, however, that an absolute time is a good approximation, we arrive at an equation first considered by Breit [101,174,175]. The Bethe-Salpeter equation as well as the Breit equation hold for a 16-component wave function. From a formal point of view, these 16 components arise when the two four-dimensional one-electron Hilbert spaces are joined by direct multiplication to yield the two-electron Hilbert space. 

The relativistic EOS of nuclear matter for supernova explosions was investigated recently [11], To include bound states such as a-particlcs, medium modifications of the few-body states have to be taken into account. Simple concepts used there such as the excluded volume should be replaced by more rigorous treatments based on a systematic many-particle approach. We will report on results including two-particle correlations into the nuclear matter EOS. New results are presented calculating the effects of three and four-particle correlations. 

One of the most amazing phenomena in quantum many-particle systems is the formation of quantum condensates. Of particular interest are strongly coupled fermion systems where bound states arise. In the low-density limit, where even-number fermionic bound states can be considered as bosons, Bose-Einstein condensation is expected to occur at low temperatures. The solution of Eq. (6) with = 2/j, gives the onset of pairing, the solution of Eq. (7) with EinP = 4/i the onset of quartetting in (symmetric) nuclear matter. At present, condensates are investigated in systems where the cross-over from Bardeen-Cooper-Schrieffer (BCS) pairing to Bose-Einstein condensation (BEC) can be observed, see [11,12], In these papers, a two-particle state is treated in an uncorrelated medium. Some attempts have been made to include the interaction between correlated states, see [7,13]. 

The new bound state equation is constructed for the two-particle Green function defined by the relationship... 

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

Contributions to the energy which depend only on the small parameters a. and Za. are called radiative corrections. Powers of a arise only from the quantum electrodynamics loops, and all associated corrections have a quantum field theory nature. Radiative corrections do not depend on the recoil factor m/M and thus may be calculated in the framework of QED for a bound electron in an external field. In respective calculations one deals only with the complications connected with the presence of quantized fields, but the two-particle nature of the bound state and all problems connected with the description of the bound states in relativistic quantum field theory still may be ignored. 

For a certain range of values of impact parameter b and initial speed vo, the radicand in Eq. 5.70 may actually have three roots. By choosing the largest classical turning point Ro, we have limited our considerations to free, i.e., colliding particles. The other two turning points (if existent) are characteristic of bound states which may be similarly treated [168]. 

Positron annihilation lifetime spectroscopy (PALS) provides a method for studying changes in free volume and defect concentration in polymers and other materials [1,2]. A positron can either annihilate as a free positron with an electron in the material or capture an electron from the material and form a bound state, called a positronium atom. Pnra-positroniums (p-Ps), in which the spins of the positron and the electron are anti-parallel, have a mean lifetime of 0.125 ns. Ortho-positroniums (o-Ps), in which the spins of the two particles are parallel, have a mean lifteime of 142 ns in vacuum. In polymers find other condensed matter, the lifetime of o-Ps is shortened to 1-5 ns because of pick-off of the positron by electrons of antiparallel spin in the surrounding medium. 

Therefore, we have now a two-component system consisting of free particles and atoms. For the description of this system it is necessary to introduce distribution functions for the free particles and atoms. It seems to be obvious that the distribution function of the atoms is connected with the bound-state contribution of Fbu,... 

We will carry out our program in two steps. In this section we will derive the two-particle density operator Fn in a three-particle collision approximation for the application in the collision integral of Fl. As compared with Section II.2, the main difference will be the occurrence of bound states and, especially, the generalization of the asymptotic condition, which now has to account for bound states too. For the purpose of the application in the kinetic equation of the atoms (bound states) we need an approximation of the next-higher-order density matrix, that is, F 23 This quantity will be determined under inclusion of certain four-particle interaction. 

The two-particle Boltzmann collision term if and the three-particle contribution for k = 0 were considered in Section II. It was possible to express those collision integrals in terms of the two- and three-particle scattering matrices. It is also possible to introduce the T matrix in if for the channels k = 1, 2,3, that is, in those cases where three are asymptotically bound states. Here we use the multichannel scattering theory, as outlined in Refs. 9 and 26. 

The channel k = 3 is forbidden because there are no bound states between electrons. In similar way, the channels k=0, 1,2,3 may be found for the other three-particle combinations. Let us now consider the properties of I3a in more detail. In order to discuss this expression in an understandable way, we split up I3a into two parts ... 

The first part [la x is given by the terms with k = 0,1, which means that this term describes collisions in which we have bound states only between the particles 2 and 3. The particle 1 is always free. The second part is given by the contributions with k = 2,3 and describes collisions in which the particle 1 is bound. First let us discuss the structure of [/Jj. For k = 0 we have two contributions. The first one is given by... 

As a consequence of the interaction of the two particles forming a bound state with the surrounding plasma, the effective ionization energy may become zero at a certain density nM for a given temperature T,... 

---