Equation (10-16) is best understood as giving a meaning to the self-mass operator in the form in .NET framework

Make PDF-417 2d barcode in .NET framework Equation (10-16) is best understood as giving a meaning to the self-mass operator in the form

Equation (10-16) is best understood as giving a meaning to the self-mass operator in the form
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[(i - m - DS] (x, y)
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= 15 4 (x - y)
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(10-18)
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INTEGRAL EQUATIONS AND BOUND-STATE PROBLEMS
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Figure 10-2 Representation of the self-mass operator.
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with (10-19) This formula is illustrated in Fig. 10-2. If we set 1] = if = 0, we may rewrite Eq. (10-4) in the form
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[Ogl'v - (1 - A)ol'ov]AV(x;J) = -JI' - ie tr [YI'S(x, x;J)]
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(10-20)
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In conjunction with (10-15) this yields a closed functional system where the argument is the source J. Alternatively, we may write an equivalent system with an external potential A as argument. For this purpose let Gvp(x, Y; A) be the photon propagator. Differentiating Eq. (10-20) with respect to J we obtain
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+ Ie
d 4 z tr [<5S(X, V A) Gvp(z, y; A) ] YI' <5A x; (z)
5:4( gl'pu X
(10-21)
while (10-15) may be rewritten as
[i x - m - eJ(x)] S(x, y; A) - ie
d z YI'GI'P(x, z; A)
<5S(x, y; A) <5AP(z)
<5 (x - y)
(10-22)
Again this is a complete set to determine Sand G. These remarks are unfortunately of little help due to our inexperience in handling such expressions. Returning to Eq. (10-14) we can obtain further information if we take higher derivatives with respect to spinor sources. Of great interest is the Green function involving the propagation of two charged particles: S(Xio Xl; Yi, Yl; J). Proceeding as before we find
[i x, - m - eJ(xi ;J) - eyl'
ibJ~Xi)Js(xio Xl; Yio Yl;1)
(10-23)
= <5 4 (Xi - Yi)S(Xl, Yl; J) - <5 4 (Xi - Yl)S(Xl, Yi ; J)
QUANTUM FIELD THEORY
Figure 10-3 The kernel V of the Bethe-Salpeter equation. The slashes indicate truncatiol} of the corresponding propagator lines.
The antisymmetric combination on the right-hand side is a reflection of the Pauli principle. Taking into account the relation (10-18) satisfied by the twopoint function, we may act on the variable Xz with the operator i() - m (For different applications we might be interested to act on one of the two remaining variables Yl or yz.) At any rate we find, when setting J = 0,
(i()X2 - m - I)(i()x, - m - Z)S(Xb Xz; Yl, yz)
(j4(Xl - yt}(j4(xz - yz) - (j4(Xl - YZ)(j4(XZ - Yl)
+ (i~X2 -
m - D[eyl'
(jjl'~Xt} -
S(XbYl;XZ,YZ;J)[J=o
(10-24)
The first term on the right-hand side is readily traced out as the contribution of the disconnected amplitude. The second term may best be understood from a graphical analysis. Written as a convolution integral
d4z 1 d4z Z V(Xl, Xz; Zl, ZZ)S(Zl, Zz; Yl, yz)
it involves a kernel V(Xb Xz; Yl, Yz) described in terms of truncated four-fermion diagrams which cannot be disconnected by cutting two fermion lines (Fig. 10-3). To lowest order, and adding superscripts to y matrices to recall on which indices they act, V(Xl, Xz; Yl, yz) = ieZy~l)GI'V(Xl - xzMZ)(j4(Xl - Yl)(j4(XZ - yz) (10-25) With this notation we write the resulting Bethe-Salpeter equation in the form
(i()X2 - m - I)(i()xl - m - Z)S(Xb Xz; Yb yz)
(j4(Xl - yt}(j4(XZ - yz) - (j4(Xl - yz)(j4(xz - Yl)
d 4z 1 d4z Z V(Xb Xz; Zl, ZZ)S(Zl, Zz; Yb Yz)
(10-26)
I +:t+J:+X+]:+
Figure 10-4 Bethe-Salpeter equation.
INTEGRAL EQUATIONS AND BOUND-STATE PROBLEMS
It is represented graphically in Fig. 10-4 where the slashes indicate amputations on the corresponding propagator lines. As it stands this equation has been written in the doubly charged electronelectron channel. It can, of course, be established in the crossed electron-positron channel where it is suited for the investigation of positronium bound states.
10-1-2 Renormalization
The field equations obviously need renormalization. To avoid cumbersome notations we did not introduce the counterterms in the action. They have now to be reinstated. As this is not our main concern here we shall not elaborate these points in great detail. We only sketch the modifications brought upon the equations by taking the multiplicative renormalization of Green functions into account when expressing them in terms of the physical mass and coupling constant. Consider, first, Eq. (10-15) for the electron propagator. In terms of electron wavefunction (Z2) and vertex renormalization (Z d this should properly be written
<5 (x - y) - {ZZ(i - m) - eZ 1 [j(X, J)
+ yfJ-
i<5J~(X)]} S(x, y; J)
(10-27)
The Ward identity requires Zl = Z2, but both are infinite quantities. Similarly, the photon propagator equation (10-21) should read Z3[D gfJ-v - OfJ-OvJGvp(x, y) + A0fJ-0vGvp(x, y)
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