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4 4 dztr [ yfJ- <5S(x,(z) Gviz,y) ] =gfJ-p<5(x-y) <5A V x)
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(10-28)
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implying again the infinite renormalization constants necessary to obtain finite results when iterating these equations for the finite Green functions G and S. This is a rather unpleasant situation, except if we return to the perturbative expansion.
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(a) Derive the renormalized forms of Eqs. (10-11) and (10-19). (b) Relate the vertex function to the Bethe-Salpeter kernel and examine the renormalization properties
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of Eq. (10-26).
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(c) Obtain the equivalent equations in the <p4 model and discuss their renormalization properties.
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10-2 RELATIVISTIC BOUND STATES
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In the relativistic approach, bound states and resonances are identified by the occurrence of poles in Green functions. A simple extension of the Schrodinger equation is unfortunately not available, except in limiting cases such as those of static external sources discussed in the framework of Dirac's equation. The general difficulty is in fact twofold. At first we have to face retardation
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QUANTUM FIELD THEORY
effects which introduce an extra relative time variable in the problem. An alternative description uses the mediation of a field. But the quantum aspects of the latter cannot be ignored. Thus it appears that the mere concept of a two-body bound state, for instance, is only an oversimplification of the actual situation. In spite of various artefacts-which may be of great practical importance-it remains true that we have to return to the general field theoretic framework whenever we need a fundamental description. This is also the case if we want to introduce further radiative corrections. A great amount of work has been devoted to the subject of which only a small, but hopefully representative, part will be surveyed here. We recall that we have already discussed aspects of hydrogen-like bound states in Chap. 2 and computed the Lamb-shift corrections to lowest order in Chap. 7.
10-2-1 Homogeneous Bethe-8alpeter Equation
Instead of dealing with the full complexity of the spinor problem, we shall examine for the time being a simpler model of scalar particles interacting through the exchange of another type of scalar particles. This is, of course, a theoretical exercise in order to exhibit some of the features of the real problem. The kernel V has also to be truncated. Moreover, we shall ignore the effects of statistics by assuming that the two "charged" particles are of a different kind. In symbolic notation Eq. (10-26) may be rewritten
(10-29)
where S(l) stands now for the complete propagator of particle 1, soon to be approximated by a free propagator involving the physical mass, and renormalization effects have been discarded. Defining - D as the inverse kernel of the product S(1)S(2) this equation could formally be solved in the form
S(12)
= -(D+ V)-l
(10-30)
showing that poles may occur whenever D + V has a zero eigenvalue. We are thus led to a homogeneous equation describing the properties of bound states. To be precise we define
S(12) = S(1)
<01 T<Pl(Xl)<P2(X2)<pi(Yl)<P~(Y2) 10) <01 T<pl (xd<pi (Yl) 10)
S(12)
(10-31)
A bound-state contribution of mass M to simplicity) will be of the form
(assumed nondegenerate for
with
iP(Yb Y2)
J (2n) 2 P
XP(Xb X2)XP(Yl, Yz)
(10-32)
XP(Xb X2) = <01 T<Pl(Xl)<P2(X2) Ip)
0 <pi T<Pi (yd<P~(Y2) 1 ) = <01 T<Pl (Yl)<P2(Y2) Ip)*
(10-33)
INTEGRAL EQUATIONS AND BOUND-STATE PROBLEMS
Here T denotes antichronological ordering. The generalization to a set of degenerate bound states is straightforward. Approximating S(1) and S(2) by free propagators
(). f
(2n)4 k 2
d k --
mr + ie
e-!'k . ( XI-YI )
(10-34)
we find
(OXI
+ mi)(OX2 + m~)xp(x1o X2) + f
d4z 1 d4z2 V(X1o X2; Zl,
Z2)xP(Zl,
Z2) = 0
(10-35)
Equations (10-35) will be sufficient to illustrate the type of reasoning when dealing with Bethe-Salpeter equations. In spite of some similarity with the nonrelativistic Schrodinger case, this is a very different type of equation. It is reflected in the larger number of configuration variables, in the fact that we deal with fourth-order integro-differential equations, in the presence of a kernel V which has to be specified perturbatively, and in the way the energy momentum of the bound state enters the equation. From translational invariance, it follows that (10-36)
It is natural to introduce the relative space-time coordinate x = Xl - X2. However, the overall configuration variable is a priori arbitrary. We may choose two positive quantities 111 and 112 such that
+ 112 =
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