Hence we have to use a z-ordered product to express x in a compact form: should be applied in order of increasing z's. That is what {...}+ means. In any case, the additivity principle is violated: a1 and a2 generated by two sources of the electromagnetic field (at two different positions) are, in general, noncommuting operators and there is no way of adding phase shifts (or, equivalently, multiplying profiles). We can also see that the physical reason for this phenomenon is the coupling between different spin states produced by the term K (81.B1-iỏ1·E1σ2). So, we have to deal with a coupled channels problem. We can also make the following remark: sometimes coupled channels can be decoupled by diagonalization. A remark about "decoupling" channels through a diagonalization procedure Note that "compositeness" of the incident particle is responsible for the existence of more than one channel. For instance, the presence of an anomalous magnetic moment can be looked upon as a mark of "compositeness." Suppose One achieves additivity if the same S diagonalizes all am (r-r;) simultaneously. If it does not, there is no context in which one could talk about additivity of phase shifts. In general the additivity does not occur. Take, e.g., pure Coulomb scattering (B=0, V=Coulomb potential) in eq (2.5): ax -idz where = ax. We have to diagonalize the matrix of eq (2.5) (compare ref. [8]): = Even this soluble case cannot be diagonalized for more than one scattering center if the Dirac particle has an anomalous magnetic moment, K#0. Without going into any details of the calculation let us quote the results. In the case when only one Coulomb potential is present (hence B=0, but E 0), we have Note that since the z dependence is outside the spinor matrix this does commute at different z's: [a (b, z), a (b, z')]=0. Suppose, however, we have two sources of Coulomb field at two different points. Then One could argue that the coupling to the anomalous moment is weak and hence not very relevant. This is true, but one can give some other-though much more complex-examples of scattering from a classical external field in which the "principle of additivity of phase shifts" is also violated. Let us consider a vector particle (hence a very relevant kind of particle to our further analysis). Example 2. Scattering of a charged vector meson in a static field (we shall quote the results, for more details see refs. [9, 10]). Let us allow for our vector particle to have an arbitrary magnetic moment and define the magnetic moment operator (S-spin operator), where x determines the value of the magnetic moment. When x=0 the equations of motion of such a particle are the so-called Proca equations. If, however, #0 some additional terms appear (as in the case of the Dirac equation with anomalous magnetic moment). With 0 we have (in the pseudo-euclidean metric, μ, v = 1, 2, 3, 4), (compare ref. [11]), We shall choose Aμ= idμ4V (r) (just the static Coulomb field). The results of a long and involved analysis [9, 10] are as follows: 1. We recover the principle of additivity of phase shifts only in the case = 1. 2. In all the other cases (including =0), there is no additivity of phase shifts. A general comment on Examples 1 and 2 is in order here. First: terminology. Many authors call * the anomalous magnetic moment of the vector meson [9, 10]. This is presumably so because when one starts with the free vector meson field equations and then introduces the electromagnetic field in the standard way = a (= -ic4.) on -ieA one obtains the дхи x=0 case. (Similarly, if one starts with the free particle Dirac equation and introduces A, in the same way, one obtains the <= 0 case of Example 1). So, from this point of view, the cases κ=0 of Example 1 and x=0 of Example 2 are analogous, and, as we know, in the first case the spin channels decouple in the high energy limit whereas in the second case they do not. K One may ask oneself a question: is there any simple way of telling which values of x and x result in decoupling of various spin states in the high energy limit? The answer seems to be: yes. It is enough to observe that in Example 1 for x=0, the relation between the magnetic moment M and spin S is where e is the charge and m the mass of the particle. Note that eq (2.6) gives the same relation between magnetic moment and spin when x=1! So, in both Examples, when x and x are chosen to make eq (2.7) valid, the spin states decouple in the high energy limit. In order to make the condition (2.7) more plausible, let us consider a charged particle with spin S and magnetic moment M given by (2.7) moving in an almost uniform magnetic field B. This particle follows a circular trajectory with frequency So eq (2.7) makes these two frequencies equal. But that means that the projection of the spin on the direction of particle velocity is a constant of the motion. Hence in this case all helicity spin states are decoupled. Although we have considered a very special case of nonrelativistic motion in a constant magnetic field, the condition (2.7) for the decoupling of spin channels turns out to be very general: The relevant relativistic formulae for precession of the polarization of particles with arbitrary magnetic moments and spins in a slowly varying (in space) electromagnetic field were given in ref. [50]. Their immediate consequence is [51] that in the high energy limit and for the gyromagnetic ratio g=2 (hence when (2.7) is valid because the definition of g is through the equation M = g (e/2m) S) the projection of the polarization on the direction of motion is constant, and hence there is no coupling between various spin channels. To conclude this section we may say that Examples 1 and 2 warn us that if the strong interactions are mediated through vector fields (analogous to the electromagnetic field) one can expect the "principle of additivity of phase shifts" to be violated. 3. Elastic Scattering of Hadrons from Nuclei Let us go back for a moment to scattering of incident particles whose internal structure one can neglect (in particular the internal quantum numbers can be neglected). Let us start with just one scatterer: The incident wave: ekz. The wave immediately behind the scatterer: ≈eikz-y (b) eikz, b=(x,y). The shape of the shadow is given by y(b): As long as 82z/2k21, the z-dependence of the second term in the wave immediately behind the scatterer is given to a good approximation by eik. Otherwise one should realize that k, depends on d which sits in the Fourier transform of the shadow. Hence, away from the scatterer, one would guess the following shape of the wave (compare D. R. Yennie article in [S3]): 1 e ik z d2ô exp (iz√k2-82) exp (id·b)ƒ (8) →e ik z — y (b) e ik z (for small z's). (3.1) 2Tik If the size of y (b), and hence of the scatterer, is a, the representative transverse momentum transfer is d≈a1. We can then estimate the "healing" length, L, of the shadow: Note that (3.1) gives, as r→∞, (compare D. R. Yennie article in [S3]) ↓ (r) = eik z +[ƒ (ke)/r] exp (ikr), (e=component of re,), with f(ke) correctly given by the inverse of y (b). One can see this by shifting the origin of integration to ke1: When the incident wave already has a profile different from unity we get: Let us construct the "shape of the shadow" for a collection of scatterers (nucleons in a nucleus; see e.g., fig. 1): |