Peskin & Schroeder, Chapter 3, Problems 5-8
Problem 3.5 (9 pages)
(a) Express the variation of L as the sum of 4 terms that are total derivatives.
(The second term in δχ may have the wrong sign.)
(b) Verify and then use the identity
X
(χTi σ 2 χj )χk = 0,
perms
where the sum is over permutations of i, j, k.
Problem 3.6 (7 pages)
(a) The normalization condition should be written
tr(ΓA ΓB ) = 4δ AB .
Problem 3.7 (9 pages)
(b) The Fourier expansion for the field operator can be written
φ(~x, t) =
Z
1 d3 p
† −i~
i~
p·~
x−iωp t
p·~
x+iωp t
√
a
e
+
b
e
.
p
~
p
~
(2π)3
2ωp
The operator φ∗ (~x, t) should be denoted by φ† (~x, t). I find that CJ 0 (~x, t)C =
−J 0 (~x, t) only if I ignore a constant proportional to δ 3 (0) that comes from
changing the ordering of operators.
(c) Use the results from part (b) to add a column for φ∗ φ to the Table at the
end of Chapter 2. Notice that an N -index tensor that is a fermion bilinear
transforms under CPT like (−1)N . Use the examples of φ∗ φ and J µ to leap
to an analogous statement about hermitian operators that are bilinear in φ∗
and φ. Show that the statement holds at least for Hermitian Lorentz-scalar
local operators that can be obtained by contracting tensors of the above two
kinds.
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Problem 3.8 (5 pages)
You can assume that a bound state with orbital angular momentum quantum
number ` = 0, 1, 2, . . ., total spin quantum number s = 0 or 1, and total
angular momentum quantum numbers j, mj can be expressed in the form
|`sjmj i =
Z
XX
d3 p
hjmj |`m` sms iY`m` (p̂)
χ`sj (p)
3
(2π)
m` ms
×
XX
m1 m2
hsms | 12 m1 12 m2 i |e− (~p, m1 )e+ (−~p, m2 )i.
(a) Write down the actions of P and C on the |e− e+ i states, taking into
account that particles and antiparticles have opposite intrinsic parities and
that a 2-fermion state changes sign when the order of the fermions is reversed:
|e+ e− i = −|e− e+ i. Deduce the action of P and C on the bound state by
changing the integration variable: p~ → −~p.
(b) Deduce the parity and charge conjugation quantum numbers of the photon from the symmetry of electrodynamics under P and C and from the fact
that photons are created (and annihilated) by the transverse spacial components of the field Aµ (x). The spin-0 and spin-1 ground states of positronium
are the S-wave (` = 0) bound states with s = 0 and s = 1, respectively. Use
charge conjugation symmetry to deduce whether these states can decay into
2 or 3 photons. In the last sentence, interpret “Find the selection rules ...”
as “Find the charge conjugation selection rules ...”
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