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One way to proceed is to evaluate explicit components, for example,
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[H, L 1 ] = [(l.p,
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= -i l X P)I'
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5.5 Act on the first of eqs. (5.24) with the operator (E + eAo + m),
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If we were able to commute the two operators then, upon using the second of eqs. (5.24), the left-hand side would reduce to (P + eAf. The lack of commu-
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yP- = _(C y 0)yP-*(C y O)-1 = _C y OyP-*y O 1 = -Cyp- TC- 1 CThus, yO = - CyOC- 1, and so (Cyo) = (CyOf implies C = - C T , and (CyOf
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;J;c = 1/J~Yo =
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(Cyo1/J*)t y O = 1/J TC y OyO = _1/J TC- 1
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Adding: 2iipou - 2mu t u = 0, since yOyk = _ykyO. ii(r)u(s) = m u(r)tu(s) = 2m{)r,,' E
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u(s)= N ~ (s) , ( E+ m X
x(s)
-a p E+m
u(S)ii(S)
a'p 5=1,2 ( E+m E+m since LX(s)X(s)t = I and N 2 = E + m.
(E + m
-a. p ) m-E
jJ + m
Answers and Comments on the Exercises
The action of A + on an arbitrary spinor is
C~l aru(r ) = r~l arC~l u(~~(S ) u(r) = r~l aru(r).
II + 2ml + m 2
4m 2
p + 2ml + m 4m 2
I +m
5.12 Particular examples of (5.59) (rotations and Lorentz boosts) are given by Sakurai (1967), but note the different metric and properties of y-matrices. 5.14
whereas
[ l'P + ,8m),a'p]
[a p, l'P + ,8m)].
in the limit E m.
6.1 Start \Yith ufiaP.V(pf - p;)vu; and rewrite the individual terms with the help, of (5.9) so that you can make use of the Dirac equations
Tr(~~) =
-! Tr(~~ + ~~)
-!2gp.va p.bvTr(I)
4a . b.
Tr(~l" '~n) = Tr(~ .. : y5 y 5)
= (-lrTr(y5~1"'~ny5) = (-lrTr(~I'''~n)'
and so, if n is odd, the trace vanishes. 6.4 Since the P is right-handed, the e- must be left-handed (see Fig. 6.8). The e e + from JL + decay is right-handed. See 12 for details.
Answers and Comments on the Exercises
From (2.21),
t - u d~(fJ) = cos () == - - .
f dp6 2P6()(p6) l>(p,2 where p,2
M 2)
p62 - p'2. This result follows from the identity 2 l>(p6 -
a2) = 21al
(l>(p6 -
a) + l>(p6 + a)),
l>(p + q)2 - M 2) = l>(2p. q + q2) =
2~l>(P + 2q~)
1 = 2Ml>[E - E' - EE'(l - cos())/M]. 6.9 Substitute (6.55) into (6.53). Show that the equations for v B and V X E are automatically satisfied and that the remaining two equations can be arranged to give (6.54). JI'JVF!LV = 0 follows on considering JL - P. 6.11 Show that under a rotation axis)
about the photon propagation direction (the z
where AR = 6.13
+ 1 and AL = -1, see (2.12).
( Aq2gl'v + Bql'qv)( - gVAq 2 + qVqA) = A q2( _ l>;q2 + ql'qA)
which cannot be made equal to l>; for any choice of the arbitrary functions A and B. 6.15 See Exercise 6.11; normalized.
e(A=O)
is chosen to satisfy (6.91) and to be suitably
6.16 Equation (6.93) can be checked explicitly component by component or, more elegantly, by writing the sum in its most general Lorentz form Agl'v + BPI' Pv' Then take the scalar product with pI' to show A = - BM 2, and with gl'V to show A = - 1. 6.17 For verification of (6.101) itself, see, for example, Sakurai (1967), page 8, where it is shown that
are Fourier transforms.
Answers and Comments on the Exercises
6.18 To evaluate the
contribution to kI'TP-v, note that
(J + Jt; m)Jt )u(P) = ( -JtjJ +22t p + mJt)u(p) = u(p), ( (p + k) - m 2 p
sincejJJt + JtjJ = 2k p and
= 0 and (jJ - m)u = O.
6.19 The variables (6.109) become
s = (k + p)2 = 2k . p - Q2 = 2k' . p', and so on.
Repeat the derivation of (6.113) and show that 1~112, 1~212 are unchanged but that the interference contribution becomes 4e 4 Q2 t/ su. Use (6.24). 6.20 At high energy, the dominant contribution to a comes from
----
1~12 == 2e
-=-~2) == 4e
2;2 + 1 + cos 0
and the cos 0 integration leads to the log(s/m 2 ) behavior. See Aitchison and Hey (1982), 2, Section 10.
Revision. For further discussion, see Section 7.1.
8.4 For a spherically symmetric potential, we can perform the angular integration in (8.3) and obtain
2 7T P( r ) ( e; q r
~: - ; q r ) r 2 dr.
0) to be
Substitution of p 8.5
Ae- mr and straightforward integration yield the result.
We might expect the most general form (for x
eii(p')(yP-Kl + iaP-v(p' - p).K 2 + iaP-v(p' + pLK3 +(p' - pYK4 +(p' + pYKs)u(p)
where K; == K;(q2). But using the Gordon decomposition, (6.7), we can reexpress the (p' + p)P- terms as linear combinations of the yP- and aP-V (p' - p). terms. Thus, the most general form reduces to