4
Lorentz Transformations and Discrete
Symmetries
In this chapter we shall review various covariances (see appendix D) of the KG
and Dirac equations, concentrating mainly on the latter. First, we consider
Lorentz transformations (rotations and velocity transformations) and show
how the scalar KG wavefunction and the 4-component Dirac spinor must
transform in order that the respective equations be covariant under these
transformations. Then we perform a similar task for the discrete transformations of parity, charge conjugation and time reversal. The results enable us
to construct ‘bilinear covariants’ having well-defined behaviour (scalar, pseudoscalar, vector, etc.) under these transformations. This is essential for later
work, for two reasons: first, we shall be able to do dynamical calculations in a
way that is manifestly covariant under Lorentz transformations; and secondly
we shall be ready to study physical problems in which the discrete transformations are, or are not, actual symmetries of the real world, a topic to which
we shall return in the second volume.
4.1 Lorentz transformations
4.1.1 The KG equation
In order to ensure that the laws of physics are the same in all inertial frames,
we require our relativistic wave equations to be covariant under Lorentz transformations – that is, they must have the same form in the two different frames
(see appendix D). In the case of the KG equation
(❗ + m
2 )φ(x) = −iq[∂ μ A
μ (x) + A
μ (x)∂ μ ]φ(x) + q
2 A
2 (x)φ(x)
(4.1)
for a particle of charge q in the field A
μ , this requirement is taken care of,
almost automatically, by the notation. Consider a Lorentz transformation
′
such that x → x . A
μ will transform by the usual 4-vector transformation
law (i.e. like x
μ ), which we write as A
μ (x) → A
′μ (x
′ ). Similarly we write
the transform of φ as φ(x) → φ
′ (x
′ ). Then in the primed coordinate frame
physics must be described by the equation
(❗
′ + m
2 )φ
′ (x
′ ) = −iq[∂ μ
′ A
′μ (x
′ ) + A
′μ (x
′ )∂ μ
′ ]φ
′ (x
′ ) + q
2 A
′2 (x
′ )φ
′ (x
′ ). (4.2)
87
Lorentz Transformations and Discrete
Symmetries
In this chapter we shall review various covariances (see appendix D) of the KG
and Dirac equations, concentrating mainly on the latter. First, we consider
Lorentz transformations (rotations and velocity transformations) and show
how the scalar KG wavefunction and the 4-component Dirac spinor must
transform in order that the respective equations be covariant under these
transformations. Then we perform a similar task for the discrete transformations of parity, charge conjugation and time reversal. The results enable us
to construct ‘bilinear covariants’ having well-defined behaviour (scalar, pseudoscalar, vector, etc.) under these transformations. This is essential for later
work, for two reasons: first, we shall be able to do dynamical calculations in a
way that is manifestly covariant under Lorentz transformations; and secondly
we shall be ready to study physical problems in which the discrete transformations are, or are not, actual symmetries of the real world, a topic to which
we shall return in the second volume.
4.1 Lorentz transformations
4.1.1 The KG equation
In order to ensure that the laws of physics are the same in all inertial frames,
we require our relativistic wave equations to be covariant under Lorentz transformations – that is, they must have the same form in the two different frames
(see appendix D). In the case of the KG equation
(❗ + m
2 )φ(x) = −iq[∂ μ A
μ (x) + A
μ (x)∂ μ ]φ(x) + q
2 A
2 (x)φ(x)
(4.1)
for a particle of charge q in the field A
μ , this requirement is taken care of,
almost automatically, by the notation. Consider a Lorentz transformation
′
such that x → x . A
μ will transform by the usual 4-vector transformation
law (i.e. like x
μ ), which we write as A
μ (x) → A
′μ (x
′ ). Similarly we write
the transform of φ as φ(x) → φ
′ (x
′ ). Then in the primed coordinate frame
physics must be described by the equation
(❗
′ + m
2 )φ
′ (x
′ ) = −iq[∂ μ
′ A
′μ (x
′ ) + A
′μ (x
′ )∂ μ
′ ]φ
′ (x
′ ) + q
2 A
′2 (x
′ )φ
′ (x
′ ). (4.2)
87