99
4.2. Discrete transformations: P, C and T
implies that the spin of the muon will be polarized along the direction of its
momentum, and furthermore that the angular distribution of positrons in the
subsequent decay
μ
+
→ e
+ + ¯
ν μ + ν e
(4.83)
would (as in the
60 Co experiment) serve as an analyser. This suggestion
was quickly confirmed by Garwin et al. (1957) and by Friedman and Telegdi
(1957); in the rest frame of the pion, the μ
+ spin is aligned opposite to its
momentum, a situation that would be reversed in the parity transformed
frame.
The end result of many years of research was to establish that the currents
responsible for weak interactions of quarks and leptons have precisely the
‘v
μ
− a
μ ’ structure, leading to the observed parity violation (see volume 2).
4.2.2 Charge conjugation
Dirac’s hole theory led him to the remarkable prediction of the positron, and
suggested a new kind of symmetry: to each charged spin-1/2 particle there
must correspond an antiparticle with the opposite charge and the same mass.
Feynman’s interpretation of the negative energy solutions of the KG and Dirac
equations assumes that this symmetry holds for both bosons and fermions.
We now explore the idea of particle-antiparticle symmetry more formally.
We begin with the KG equation for a spin-0 particle of mass m and charge
q in an electromagnetic field A
μ , namely equation (4.1). Inspection of this
equation shows at once that the wave function φ C of a particle with the same
mass and charge −q is related to the original wavefunction φ by
φ C = η C φ
∗
(4.84)
where η C is an arbitrary phase factor which we shall take to be unity. Equation
(4.84) tells us how to connect the solutions of the particle (charge q) and
antiparticle (charge −q) equations. When applied to free-particle solutions of
the KG equation, the transformation (4.84) relates positive and negative 4momentum solutions, as expected in the Feynman interpretation of the latter.
We may extend the transformation (4.84) to a symmetry operation for the
KG equation (4.1) if we introduce an operation which changes the sign of A
μ .
Then the combined operation ‘take the complex conjugate of φ and change A
μ
to −A
μ ’ is a formal symmetry of (4.84), in the sense that the wavefunction φ
∗
in the field −A
μ satisfies exactly the same equation as does the wavefunction
φ in the field A
μ . Of course, we have just seen that φ
∗ is the antiparticle
wavefunction, so it is no surprise that the dynamics of the antiparticle in
a field −A
μ is the same as that of the particle in a field A
μ . Still, this is
symmetry of the KG equation, which we will call charge conjugation, denoted
by C:
μ
C : φ → φ C = φ
∗ , A
μ
→ A = −A
μ .
(4.85)
C
4.2. Discrete transformations: P, C and T
implies that the spin of the muon will be polarized along the direction of its
momentum, and furthermore that the angular distribution of positrons in the
subsequent decay
μ
+
→ e
+ + ¯
ν μ + ν e
(4.83)
would (as in the
60 Co experiment) serve as an analyser. This suggestion
was quickly confirmed by Garwin et al. (1957) and by Friedman and Telegdi
(1957); in the rest frame of the pion, the μ
+ spin is aligned opposite to its
momentum, a situation that would be reversed in the parity transformed
frame.
The end result of many years of research was to establish that the currents
responsible for weak interactions of quarks and leptons have precisely the
‘v
μ
− a
μ ’ structure, leading to the observed parity violation (see volume 2).
4.2.2 Charge conjugation
Dirac’s hole theory led him to the remarkable prediction of the positron, and
suggested a new kind of symmetry: to each charged spin-1/2 particle there
must correspond an antiparticle with the opposite charge and the same mass.
Feynman’s interpretation of the negative energy solutions of the KG and Dirac
equations assumes that this symmetry holds for both bosons and fermions.
We now explore the idea of particle-antiparticle symmetry more formally.
We begin with the KG equation for a spin-0 particle of mass m and charge
q in an electromagnetic field A
μ , namely equation (4.1). Inspection of this
equation shows at once that the wave function φ C of a particle with the same
mass and charge −q is related to the original wavefunction φ by
φ C = η C φ
∗
(4.84)
where η C is an arbitrary phase factor which we shall take to be unity. Equation
(4.84) tells us how to connect the solutions of the particle (charge q) and
antiparticle (charge −q) equations. When applied to free-particle solutions of
the KG equation, the transformation (4.84) relates positive and negative 4momentum solutions, as expected in the Feynman interpretation of the latter.
We may extend the transformation (4.84) to a symmetry operation for the
KG equation (4.1) if we introduce an operation which changes the sign of A
μ .
Then the combined operation ‘take the complex conjugate of φ and change A
μ
to −A
μ ’ is a formal symmetry of (4.84), in the sense that the wavefunction φ
∗
in the field −A
μ satisfies exactly the same equation as does the wavefunction
φ in the field A
μ . Of course, we have just seen that φ
∗ is the antiparticle
wavefunction, so it is no surprise that the dynamics of the antiparticle in
a field −A
μ is the same as that of the particle in a field A
μ . Still, this is
symmetry of the KG equation, which we will call charge conjugation, denoted
by C:
μ
C : φ → φ C = φ
∗ , A
μ
→ A = −A
μ .
(4.85)
C