104
4. Lorentz Transformations and Discrete Symmetries
is some 20% greater (Nakamura et al. 2010) than the rate for the CPconjugate process
+
B ¯ 0 → K
− π .
(4.105)
(Note that the B ¯ 0 state is conventionally defined as the CP transform of the
B
0 state). So the pion distinguished by being emitted in the higher-yielding
reaction (4.104) defines ‘negatively charged’, and the polarization of the muon
in its decay (4.103) defines what is a right-handed screw sense.
Secondly, CP (and C) violation is one of the three conditions
2 established
by Sakharov (1967) that would enable a universe containing initially equal
amounts of matter and antimatter, when created in the Big Bang, to evolve
into the matter-dominated universe we see today – rather than simply having
the required imbalance as an initial condition. Within the Standard Model,
all known CP violating effects are attributable to the KM mechanism. But
calculations show (Huet and Sather 1995) that the matter-antimatter asymmetry generated from this source is very many orders of magnitude too small.
This is, therefore, one area of physics where the Standard Model fails.
Thirdly, CP violation is directly connected to the violation of another
discrete symmetry, namely time reversal T, because very general principles of
quantum field theory imply that the product CPT (in any order) is conserved
– the CPT theorem. This theorem states (L¨ uders 1954, 1957, Pauli 1957) that
CPT must be an exact symmetry for any Lorentz invariant quantum field
theory constructed out of local fields, with a Hermitian Hamiltonian, and
quantized according to the usual spin-statistics rule (integer spin particles are
bosons, half-odd integer spin particles are fermions). Thus any violation of
CP implies a violation of T if CPT is to be conserved.
We shall return to CPT presently, but first let us deal with T.
4.2.4 Time reversal
The time reversal transformation T is defined by
′
′
T : x → x = x, t → t = −t;
(4.106)
that is, T reverses the direction of time. It follows that T reverses momenta
(p → −p) and angular momenta (x × p → −x × p). Let us also note how
the electromagnetic potentials transform under T: A
0 does not change, being
generated by static charges, while A changes sign, since it is produced by
currents; that is,
0
A T (t
′ ) = A
0 (t) A T (t
′ ) = −A(t).
(4.107)
It follows that the electric field E does not change sign under T, but the
magnetic field B does. It is easily checked that these prescriptions ensure
that the Maxwell equations are covariant under T.
2 The other two are (a) the existence of baryon number violating transitions and (b) a
time when the C, CP and baryon number violating transitions proceeded out of thermal
equilibrium.
4. Lorentz Transformations and Discrete Symmetries
is some 20% greater (Nakamura et al. 2010) than the rate for the CPconjugate process
+
B ¯ 0 → K
− π .
(4.105)
(Note that the B ¯ 0 state is conventionally defined as the CP transform of the
B
0 state). So the pion distinguished by being emitted in the higher-yielding
reaction (4.104) defines ‘negatively charged’, and the polarization of the muon
in its decay (4.103) defines what is a right-handed screw sense.
Secondly, CP (and C) violation is one of the three conditions
2 established
by Sakharov (1967) that would enable a universe containing initially equal
amounts of matter and antimatter, when created in the Big Bang, to evolve
into the matter-dominated universe we see today – rather than simply having
the required imbalance as an initial condition. Within the Standard Model,
all known CP violating effects are attributable to the KM mechanism. But
calculations show (Huet and Sather 1995) that the matter-antimatter asymmetry generated from this source is very many orders of magnitude too small.
This is, therefore, one area of physics where the Standard Model fails.
Thirdly, CP violation is directly connected to the violation of another
discrete symmetry, namely time reversal T, because very general principles of
quantum field theory imply that the product CPT (in any order) is conserved
– the CPT theorem. This theorem states (L¨ uders 1954, 1957, Pauli 1957) that
CPT must be an exact symmetry for any Lorentz invariant quantum field
theory constructed out of local fields, with a Hermitian Hamiltonian, and
quantized according to the usual spin-statistics rule (integer spin particles are
bosons, half-odd integer spin particles are fermions). Thus any violation of
CP implies a violation of T if CPT is to be conserved.
We shall return to CPT presently, but first let us deal with T.
4.2.4 Time reversal
The time reversal transformation T is defined by
′
′
T : x → x = x, t → t = −t;
(4.106)
that is, T reverses the direction of time. It follows that T reverses momenta
(p → −p) and angular momenta (x × p → −x × p). Let us also note how
the electromagnetic potentials transform under T: A
0 does not change, being
generated by static charges, while A changes sign, since it is produced by
currents; that is,
0
A T (t
′ ) = A
0 (t) A T (t
′ ) = −A(t).
(4.107)
It follows that the electric field E does not change sign under T, but the
magnetic field B does. It is easily checked that these prescriptions ensure
that the Maxwell equations are covariant under T.
2 The other two are (a) the existence of baryon number violating transitions and (b) a
time when the C, CP and baryon number violating transitions proceeded out of thermal
equilibrium.