Problems
109
(4.126) rather than the usual rule for unitary operators. So from (4.126) we
have
−1
−1
M X =< X, s z |H ˆ |X, s z >
∗ =< X, s z |θ ˆ θ ˆ H ˆ θ ˆ θ ˆ |X, s z >
.
(4.137)
−1
θH ˆ ˆ
ˆ
If the Hamiltonian is CPT invariant, then ˆ θ
= H. Also, we know
the action of P ˆ , C ˆ and T ˆ on the states, from the previous results. Equation
(4.137) then becomes
¯
|H ˆ | ¯
M X =< X, −s z
X, −s z >= M X ¯ ,
(4.138)
stating the equality of particle and antiparticle masses. The most sensitive
¯
test of (4.138) is provided by the K
0
− K
0 system, where the currently quoted
limit for the mass difference is (Nakamura et al. 2010)
|M
0
− M
0
|
K < 8 × 10
−19
K
¯
at 90% C.L.
(4.139)
M average
θ-invariance also implies that the charges of a charged particle and its
antiparticle are equal in magnitude but opposite in sign, as are their magnetic
moments; and in the case of unstable particles it implies that their lifetimes
are equal, to first order in the interaction responsible for the decay (Lee 1981).
All current data support these equalities (Nakamura et al. 2010). Other tests
involve analysis of the implications of θ-invariance as applied to transition
amplitudes. As an example, we refer to a recent analysis of K-decays by
Abouziad et al. (2011), both with and without the assumption of θ-invariance.
The results were consistent with θ-invariance.
Problems
4.1 Consider an infinitesimal boost along the x-axis,
′
t = t − ηx
(4.140)
′
x = x − ηt.
(4.141)
Show that the KG wavefunction transforms according to
φ
′ (x, t) = (1 + iηK ˆ x )φ,
(4.142)
where
ˆ
K x = −i x ∂/∂t − i t ∂/∂x.
(4.143)
ˆ
Defining similar operators K ˆ y , K z for boosts in the y and z directions, show
that
ˆ
[K ˆ x , K y ] = −iL ˆ z .
(4.144)
109
(4.126) rather than the usual rule for unitary operators. So from (4.126) we
have
−1
−1
M X =< X, s z |H ˆ |X, s z >
∗ =< X, s z |θ ˆ θ ˆ H ˆ θ ˆ θ ˆ |X, s z >
.
(4.137)
−1
θH ˆ ˆ
ˆ
If the Hamiltonian is CPT invariant, then ˆ θ
= H. Also, we know
the action of P ˆ , C ˆ and T ˆ on the states, from the previous results. Equation
(4.137) then becomes
¯
|H ˆ | ¯
M X =< X, −s z
X, −s z >= M X ¯ ,
(4.138)
stating the equality of particle and antiparticle masses. The most sensitive
¯
test of (4.138) is provided by the K
0
− K
0 system, where the currently quoted
limit for the mass difference is (Nakamura et al. 2010)
|M
0
− M
0
|
K < 8 × 10
−19
K
¯
at 90% C.L.
(4.139)
M average
θ-invariance also implies that the charges of a charged particle and its
antiparticle are equal in magnitude but opposite in sign, as are their magnetic
moments; and in the case of unstable particles it implies that their lifetimes
are equal, to first order in the interaction responsible for the decay (Lee 1981).
All current data support these equalities (Nakamura et al. 2010). Other tests
involve analysis of the implications of θ-invariance as applied to transition
amplitudes. As an example, we refer to a recent analysis of K-decays by
Abouziad et al. (2011), both with and without the assumption of θ-invariance.
The results were consistent with θ-invariance.
Problems
4.1 Consider an infinitesimal boost along the x-axis,
′
t = t − ηx
(4.140)
′
x = x − ηt.
(4.141)
Show that the KG wavefunction transforms according to
φ
′ (x, t) = (1 + iηK ˆ x )φ,
(4.142)
where
ˆ
K x = −i x ∂/∂t − i t ∂/∂x.
(4.143)
ˆ
Defining similar operators K ˆ y , K z for boosts in the y and z directions, show
that
ˆ
[K ˆ x , K y ] = −iL ˆ z .
(4.144)