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APPENDIX
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Metric tensor: 0 -1 0 0 0 0 -1 0
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(A-I)
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Derivatives with respect to contravariant (x ) or covariant (x.) coordinates are sometimes abbreviated as
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(A-2)
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Summation over repeated Lorentz (Greek) or space (Latin) indices is understood unless explicitly stated.
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V'W= v"W = V W. = g.,V W' = g 'V.W, = VOWo-V'W= VOWo- ViW i
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A boldface letter denotes a three-vector or the three-dimensional part of a contravariant fourvector:
v = {Vi, i = 1,2, 3} = {V" Vy, v,,}
The only exception concerns the three-dimensional gradient
(A-4)
(A-5)
The d'alembertian operator is
(A-6)
and the four-momentum operator reads
(A-7)
QUANTUM FIELD THEORY
Totally antisymmetric Levi-Civita tensor:
8"'PO =
if {p, v, p, o-} is an even permutation of {O, 1,2, 3} if it is an odd permutation otherwise
(A-8)
(A-9)
Useful identities:
8"' P0 8""'P'0'
-det (g""'J
CI. = Ct.' =
p, v, p,
ll, v', p',
8"' P0 8""P'0'
-det (g""'J
Ct.'
v, p,
v', p', a'
(A-l0)
Three-dimensional antisymmetric tensor: if (i, j, k) is an even permutation of (1,2, 3)
A-2 DIRAC MATRICES AND SPINORS
The y matrices satisfy
{y", y'}
y"y' + y'y"
2g"'
(A-Il)
with yO hermitian, yi anti hermitian, and are related to the
fJ and C( matrices through
fJoc
(A-12)
i/y'y2 y3 =
yo 8",poY"Y'y P
(A-13)
iYOY1Y2Y3
iy3y2y' / =c. y!
y~ = I
(A-14)
{ys, y"} = 0
Commutator of y matrices:
0-"'
2 [y", y']
(A-15)
y"y' = g"' - io-"'
(A-16)
(A-17)
APPENDIX
Hermitian conjugates:
"10"1""10
"1"1
(A-18)
yOYsyO = -y~ = -"15 yO(YsY")YO y0O""'yO
(Ysy")t (O""')t
For any two spinors 1/11 and 1/12 and any 4 x 4 matrix f',
(1i/ 1f'1/I2)*
Ii!h of't YO )l/Il
(A-19)
while the corresponding identity for two anticommutating spin ~ fields involves an extra minus sign. Charge conjugation matrix:
Cy"c- 1 Cy sC- 1
= -y~ =
(A-20)
Pauli matrices:
0"3 =
(1 0)
(A-21)
Dirac representation:
IZ=O" @G=
(0 G)
y=/3IZ=iO" 2 @G=
(OG )
"15 = "I = I
(0 = I)
(A-22)
"I "I y=I:=I@G=
0" 0' '
(G G 0)
= jO"I
. . 0"' 'iet!
. O"Oi)
cT =
C t =-C
cc t = ctc =
c2 =
(A-23)
QUANTUM FIELD THEORY
Majorana representation:
a3=-(}'@(}3=C~3 _;3)
y' =
(A-24)
i:3 )
Y5 = Y = () @ () =
532(}2
also satisfies (A-23)
Relation with the Dirac representation: with
t U = U =
1 J2 (I
Chiral representation:
o , y=/3=-(}@I=
3 ()
oc=() @()=
i(}2
tT = (
(A-25)
satisfies (A-23)
(}'J
.. =
Gijk
(aX aX 0
APPENDIX
Relation with the Dirac representation: with Contraction identities:
~~ =
(A-26)
a- b - i(J",a"b'
yAy" = 4 yAy"y" = - 2y" y"y"y'y" y"y"y'yPy"
4g"' 2yPy'y"
(A-27)
y"y"y'yPy"y" = 2(y"y"y'yP + yPy'y"y") y"(J"'y" = 0 yA(J"'YPY" = 2YP(J"'
Traces: tr I = 4 tr y" tr yS
(A-28)
The trace of an odd product of y" matrices vanishes: tr (ySy") = 0 tr (y"y') = 4g"' tr ((J"') tr (y"y'yS) tr (y"y'yPy")
= = =
0 0 4(g"'gP" - g"Pg," + g""g'P)
(A-29)
tr (ySy"y'yPy") = -4is"'P" = 4is",p" tr (1/.tl/2 1/.2n) = tr (1/.2n 1/.21/.tl tr (~, ... ~2n)
a, a2 tr (~3 ... ~2n) - a, a3 tr (~2~4 ... ~2n) 4Ls(ai,' aj,) (ai,' ajJ
+ ... + a, a2n tr (~2 ... ~2n- tl
(A-30)
s is the signature of the permutation i,j, ... in)", and the sum runs over the (2n)!/2 nn! different pairings satisfying 1 = i, < i2 < ... < in, ik <).. Dirac spinors u and v solutions of the Dirac equation (p - m)u(")(p) (p
(A-31)
+ m)v(")(p)
are functions of the on-shell momentum p, with pO index (X = 1, 2. Conjugate spinors:
E p ==
J m 2 + p2 and are labeled by a polarization
(A-32)
ii(")(p)(p - m) = 0 v(")(p)(p
+ m) =
(A-33)
QUANTUM FIELD THEORY
Normalization:
ii(aJ(p)dPl(p) = bap v(aJ(p)v(PJ(p) = _bap v(aJ(p)dPJ(p) = u(aJ(p)dPJ(p) = 0
(A-34)
Density:
(A-35)
Projection operators over the positive and negative energy states:
(A-36)
Projectors over a definite polarization state along a space-like four-vector n orthogonal to p,
u(p, n) @ u(p, n) =
-~ --~
p + m 1 + Y5~
2m 2
- v(p, n) @v(p, n) =
m - pI + Y5~ -~ --~ 2m 2
(A-37)
For comments on helicity states, see Sec. 2-2-1. Gordon identities:
(A-38)
In particular:
a(aJ (p)4u(PJ(p) = bap r:!i
(A-39)
daJt(p)ocu(PJ(p) = bap
A-3 NORMALIZATION OF STATES, S MATRIX, UNITARITY, AND CROSS SECTIONS
Normalization of one-boson states:
(A-40)
with wp ==
+ m 2 and polarization indices omitted.
APPENDIX
One-fermion states:
<P [p') =
(2n P(p - p')
(A-41)
(For massless fermions such as neutrinos, it is safer to use a normalization of the form (A-40) in intermediate computations). S matrix and invariant scattering amplitude: S
+ iT
<f[ T[i) =
(A-42)
(2n)4(i4(Pf
P i ): 7fi
Differential cross section for the scattering from an initial state i = {I, 2} involving no massive fermion into a final statef= {3,4, ... ,n}:
1 [,o/'f;[2 dd- (2 )4<4( ) [ 2 2 2 1/2 - - P3'" Pn n U Pi - P f 4 (Pl'P2) - mlm2] S .
(A-43)
The factor S is S
[1 k;!
(A-44)
if there are ki identical particles of species i in the final state. The measure dp generally denotes d3 p dp=--(2n 2wp except for massive fermions for which _ d3 p m dp=-(2n wp (A-4Sb) (A-4Sa)
Accordingly, if the incident particles 1 and/or 2 are massive fermions, the expression (A-43) has to be multiplied by 2ml and/or 2m2' The formula (A-44) may have to be supplemented by an average over the initial polarizations and a summation over the final ones. The decay rate dr = d(r- 1) of a particle of mass M into particles 3,4, ... , n is given in its rest frame by the right-hand side of Eq. (A-43) with the flux factor 1/4[(Pl . P2j2 - mtmD 1/2 replaced by 1/2M. The same modifications as above are to be brought when fermions are present. . Differential cross section for two-body scattering 1 + 2 ---+ 3 + 4 of nonidentical particles:
-= - -
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