I02 sm
Data Matrix ECC200 barcode library for javaUsing Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications.
= N - -- - - -- --- - -- ~ --- -- -- -- - - -. . 0
Java data matrix 2d barcode generationfor javausing barcode generator for java control to generate, create data matrix barcodes image in java applications.
One way to proceed is to evaluate explicit components, for example,
ECC200 decoder for javaUsing Barcode reader for Java Control to read, scan read, scan image in Java applications.
[H, L 1 ] = [(l.p,
Bar Code printing in javausing barcode development for java control to generate, create bar code image in java applications.
X2 P 3 X3P2 ]
Bar Code barcode library in javaUsing Barcode scanner for Java Control to read, scan read, scan image in Java applications.
-i(CX2P3 -
Visual .net gs1 datamatrix barcode generationwith visual c#.netusing .net framework tomake data matrix with asp.net web,windows application
CX3P2)
Aspx.net gs1 datamatrix barcode implementationfor .netusing barcode integrating for aspx.net control to generate, create data matrix ecc200 image in aspx.net applications.
= -i l X P)I'
Visual .net Crystal barcode data matrix generatingin .netgenerate, create data matrix barcodes none with .net projects
5.5 Act on the first of eqs. (5.24) with the operator (E + eAo + m),
.NET barcode data matrix drawerwith visual basic.netusing barcode drawer for .net vs 2010 control to generate, create barcode data matrix image in .net vs 2010 applications.
+ eAo + m)a (P + eA)u B
Upc Barcodes barcode library in javausing barcode integration for java control to generate, create gtin - 12 image in java applications.
+ eAo + m)(E + eAo - m)u A == 2m(ENR + eAO)u A
Ucc Ean 128 maker on javause java ean128 implementation toincoporate gs1-128 in java
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-
USS Code 128 barcode library in javausing barcode encoding for java control to generate, create barcode 128 image in java applications.
tation gives rise to the extra term involving B
USS 93 development on javausing barcode integrating for java control to generate, create uss code 93 image in java applications.
V X A.
Control gtin - 12 size for microsoft wordto incoporate upc-a and ucc - 12 data, size, image with microsoft word barcode sdk
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
VS .NET upc a printingin .netuse visual .net upc a generating toattach upc barcodes for .net
= 1 implies C = - C- 1 Also,
Bar Code integration for word documentsusing microsoft word touse bar code in asp.net web,windows application
;J;c = 1/J~Yo =
3 Of 9 Barcode printer in vb.netusing barcode drawer for .net framework control to generate, create 3 of 9 barcode image in .net framework applications.
iiyO ( yP-Pp- - m) u ii(YP-Pp- - m)y u
Include pdf 417 in visual basicusing visual .net tomake pdf-417 2d barcode for asp.net web,windows application
(Cyo1/J*)t y O = 1/J TC y OyO = _1/J TC- 1
Control code-128c image in microsoft excelgenerate, create barcode code 128 none on excel projects
Adding: 2iipou - 2mu t u = 0, since yOyk = _ykyO. ii(r)u(s) = m u(r)tu(s) = 2m{)r,,' E
UPC A barcode library in c#using .net for windows forms crystal touse upc symbol in asp.net web,windows application
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
