9 Symmetry Violations and Quark Flavour Physics
521
[4]. The second discrete transformation is particle–antiparticle conjugation C [5].
This transforms each particle into its antiparticle, whereby all additive quantum
numbers change their sign. A third transformation of this kind is time reversal T ,
which reverses momenta and angular momenta [6]. This corresponds formally to an
inversion of the direction of time. According to the CPT theorem of Lüders and
Pauli [7–9], there is a connection between these three transformations such that,
under rather weak assumptions, in a local field theory all processes are invariant
under the combined operation C · P · T .
9.2.1 Discrete Symmetries in Classical Physics
9.2.1.1 Parity P
The parity operation consists in reversing the direction of the position vector
r =
(X, Y, Z) in Cartesian coordinates. This corresponds to reflection in a plane mirror,
followed by a rotation by 180 ◦ . Symmetry under parity operation is therefore also
called mirror symmetry.
The parity operation reverses the direction of all polar vectors derived from the
position vector; in particular, this is the case for the momentum
p = m v = md r/dt
and the acceleration
a = d 2
r/dt 2 . Therefore, the Newtonian force
F = d
p/dt is also
reversed under the parity operation.
This must be also the case for the Lorentz and Coulomb forces on a particle with
charge q moving with velocity
v:
F = q(
E + +
v ×
B) .
(9.1)
Since the charge q is invariant under P, and the force
F and the velocity
v change
sign, the electric field strength
E must change sign and the magnetic field strength
B must remain unchanged.
For the electric potential
A, we obtain from the relations
E = −grad V − ∂
A/∂t,
(9.2)
B = rot
A,
(9.3)
the result that
A changes sign and V remains invariant, since the spatial differential
operator changes sign under the parity operation.
We therefore have four classes of quantities with different transformation behavior under P: axial vectors or pseudovectors such as
B and the angular momentum
J = =
r × ×
p, and scalars such as V , which remain invariant under P; and polar vectors
such as
r,
p,
F ,
E and
A, and pseudoscalars such as
E·
B, which change sign under
P.
521
[4]. The second discrete transformation is particle–antiparticle conjugation C [5].
This transforms each particle into its antiparticle, whereby all additive quantum
numbers change their sign. A third transformation of this kind is time reversal T ,
which reverses momenta and angular momenta [6]. This corresponds formally to an
inversion of the direction of time. According to the CPT theorem of Lüders and
Pauli [7–9], there is a connection between these three transformations such that,
under rather weak assumptions, in a local field theory all processes are invariant
under the combined operation C · P · T .
9.2.1 Discrete Symmetries in Classical Physics
9.2.1.1 Parity P
The parity operation consists in reversing the direction of the position vector
r =
(X, Y, Z) in Cartesian coordinates. This corresponds to reflection in a plane mirror,
followed by a rotation by 180 ◦ . Symmetry under parity operation is therefore also
called mirror symmetry.
The parity operation reverses the direction of all polar vectors derived from the
position vector; in particular, this is the case for the momentum
p = m v = md r/dt
and the acceleration
a = d 2
r/dt 2 . Therefore, the Newtonian force
F = d
p/dt is also
reversed under the parity operation.
This must be also the case for the Lorentz and Coulomb forces on a particle with
charge q moving with velocity
v:
F = q(
E + +
v ×
B) .
(9.1)
Since the charge q is invariant under P, and the force
F and the velocity
v change
sign, the electric field strength
E must change sign and the magnetic field strength
B must remain unchanged.
For the electric potential
A, we obtain from the relations
E = −grad V − ∂
A/∂t,
(9.2)
B = rot
A,
(9.3)
the result that
A changes sign and V remains invariant, since the spatial differential
operator changes sign under the parity operation.
We therefore have four classes of quantities with different transformation behavior under P: axial vectors or pseudovectors such as
B and the angular momentum
J = =
r × ×
p, and scalars such as V , which remain invariant under P; and polar vectors
such as
r,
p,
F ,
E and
A, and pseudoscalars such as
E·
B, which change sign under
P.