QUANTUM FIELD THEORY in Visual Studio .NET

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QUANTUM FIELD THEORY
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grams, it violates (s-u) crossing symmetry. It implies a correspondence between the proper time ordering of the relativistic interactions, thus mistreating the relativistic static limit when one of the masses gets large. This is a severe limitation when dealing with realistic problems. For instance, in electrodynamics, unless we find some compelling reasons to use it in a particular gauge (the noncovariant Coulomb gauge for instance), this approximation will not be gauge invariant; nor will it satisfy the reasonable criterion of reducing to the Klein-Gordon (or Dirac in the spin i case) equation when one of the particles becomes very massive.
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To fulfill this criterion it is necessary to include at least the set of crossed-ladder diagrams leading to an infinite kernel V. Find the corresponding approximation in the functional language of Eqs. (10-23) and (10-24).
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With these limitations in mind we return to Eq. (10-52), where we set J12 = o. Wick and Cutkosky, observing the analogy with the hydrogen problem in momentum space, suggested the use of the stereographic map on a unit sphere in five-dimensional space. This method introduced by Fock in the nonrelativistic case enables us to use conformal transformations in a transparent manner and to exhibit the dynamical symmetry of the system. It is convenient to introduce the notation A for the dimensionless quantity (10-53) This choice reminds us that the g are dimensioned cp3 coupling constants.
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It allows a comparison with the electromagnetic case, A being the analog of the
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fine-structure constant. We shall first content ourselves with the equal-mass case ml = m2 = m. Taking m as the energy unit, we find
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. P)2 .P)2 [(p - 12 + 1J[(p + 12 + 1] Xp(p) = nA
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d4 , (p Xp(P') p _ p,)2
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(10-54)
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where P stands for (Po, 0). This may be regarded as an eigenvalue equation for the coupling A and appeal to Fredholm's theory will show that it admits a discrete spectrum. The stereographic projection to be considered in more detail in Chap. 13 amounts to an association of a point p in R4 to a vector z in fivedimensional space on a sphere of radius i 1 - p 2 /4 with four-dimensional pro-
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Figure 10-6 Projection from the four-dimensional p space onto a sphere of diameter 1 - p2/4.
INTEGRAL EQUATIONS AND BOUND-STATE PROBLEMS
jection along p and polar angle' such that tan Q2 = Ip 1 - p2/4 (Fig. 10-6). We use (/3, e, <p) as the additional polar angles of p; hence p' P = Ip II p I cos /3, and
(10-55)
If dQ 4 denotes the solid angle element on the sphere normalized according to SdQ 4 = 8n 2 /3 then
d4 p' _ (1 - p2/4) cos 2 Q2 dQ 4 (p - p,)2 8 cos 6 C/2 1 - cos IX
(10-56)
where IX is the angle between the corresponding vectors z and z', namely [(1 - p2/4)j4J cos IX. Finally, we define a new function K(z) through
K(z)
z z' =
(10-57)
cos6 '/2 X(p)
It follows that
4 +4
K(z) = 8n 3
dQ' 1 _ c~s IX K(z')
(10-58)
The limiting case p2 = 0 admits an 0(5) in variance with K proportional to a fivedimensional spherical harmonic. Let N - 1 stand for the degree of this spherical harmonic, which means that multiplication by Iz IN-1 yields an harmonic function in five-dimensional space. To compute the eigenvalues [with degeneracy N(N + 1)(2N + 1)/6, N = 1,2, ... J it is sufficient to apply (10-58) to special spherical harmonics depending only on the angle with the fifth axis. Those are orthogonal polynomials (generalizing the Legendre polynomials) obtained by expanding in powers of z</z> the elementary Green function
Iz - z'13 -Iz> j3[1 + z~/z;'
- 2(z</z cos IXJ3/2
1 ~ 2 2 J3/2 = N=l CN- 1 (COSIX)t N-1 L.. [1 + t - tCOSIX
(10-59)
Thus, choosing z along the fifth axis, AN is given by
CN-1(1)=~fdQ4 3
3 C N- 1(C) = AN f"d,sin 'CN-1(O 1 - cos C 4n 0 1 - cos,
or, equivalently for
t 1 I-C 1 AN
It I <
1(1)=4n
f" d, sin
1 - cos, (1
+ t2 -
1 1 =--2t cos 0 3/ 2 2n(1 - t)
QUANTUM FIELD THEORY
From its definition (10-59), C N - 1 (1)
AN n
N(N N(N
+ 1)/2; thus + 1)
(10-60)
In the general case 0 < p2 < 4, Eq. (10-58) exhibits an 0(4) in variance very much as in the hydrogenic case. This follows from the fact that only the combination sin ( cos p, proportional to the projection of z on the zeroth axis (i.e., along P), enters the equation, apart from 0(5)-invariant expressions. Thus the 0(4) in variance pertains to rotations leaving the zeroth axis invariant. The unit sphere in R 5 can then be projected back on the euclidean space R4 but now from a point along the zeroth axis. If y is such that cos y = sin ( cos p, let q be the projected four-vector from the unit sphere with JqJ = tan (y/2) and X(q) = cos 6 (y/2) K(z). Performing similar transformations as in (10-55) and (10-56) we find a manifestly 0(4)-invariant equation
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