89
4.1. Lorentz transformations
The rule for finite rotations may be obtained from the infinitesimal form
by using the result
A
e = lim (1 + A/n)
n
(4.12)
n→∞
generalized to differential operators (the exponential of a matrix being un1
A
2
derstood as the infinite series expA = 1 + A +
+ . . . ). Let ∈ = α/n,
2
where α = (α x , α y , α z ) are three real finite parameters; we may think of the
direction of α as representing the axis of the rotation, and the magnitude of
α as representing the angle of rotation. Then applying the transformation
(4.11) n times, and letting n tend to infinity, we obtain for the finite rotation
ˆ
iα·
φ
′ (x) = e L φ(x) ≡ U ˆ R (α)φ(x).
(4.13)
†
Note that U ˆ R (α) is a unitary operator, since U ˆ is the inverse rotation.
R
Equation (4.13) is, of course, the familiar rule for rotations of scalar wavefunctions, exhibiting the intimate connection between rotations and angular
momentum in quantum mechanics. We recall that if a Hamiltonian is invariant under rotations, then the operators L ˆ commute with the Hamiltonian and
angular momentum is conserved.
A similar calculation may be done for velocity transformations (‘boosts’),
leading to corresponding operators K ˆ – see problem 4.1.
4.1.2 The Dirac equation
The case of the Dirac equation is more complicated, because (unlike the KG φ)
the wavefunction has more than one component, corresponding to the fact that
it describes a spin-1/2 particle. There is, however, a direct connection between
the angular momentum associated with a wavefunction, and the way that the
wavefunction transforms under rotations of the coordinate system. To take a
simple case, the 2p wavefunctions mentioned in section 3.2 correspond to l = 1
on the one hand and, on the other, to the components of a vector – indeed the
most basic vector of all, the position vector x = (x, y, z) itself. If we rotate
the coordinate system in the way represented by (4.5), the components in the
primed system transform into simple linear combinations of the components
in the original system.
Very much the same thing happens in the case of spinor wavefunctions,
except that they transform in a way different from – though closely related to
– that of vectors. In the present section we shall discuss how this works for
three-dimensional rotations of the spatial coordinate system, and explain how
it generalizes to boosts, which include transformations of the time coordinate
as well. It will be convenient to use the alternative representation (3.40) for
the Dirac matrices. In this representation, the components φ, χ of the freeparticle 4-spinor ω of (3.43) satisfy
Eφ = σ · pφ + mχ
(4.14)
Eχ = −σ · pχ + mφ
(4.15)
rather than (3.45) and (3.46).
4.1. Lorentz transformations
The rule for finite rotations may be obtained from the infinitesimal form
by using the result
A
e = lim (1 + A/n)
n
(4.12)
n→∞
generalized to differential operators (the exponential of a matrix being un1
A
2
derstood as the infinite series expA = 1 + A +
+ . . . ). Let ∈ = α/n,
2
where α = (α x , α y , α z ) are three real finite parameters; we may think of the
direction of α as representing the axis of the rotation, and the magnitude of
α as representing the angle of rotation. Then applying the transformation
(4.11) n times, and letting n tend to infinity, we obtain for the finite rotation
ˆ
iα·
φ
′ (x) = e L φ(x) ≡ U ˆ R (α)φ(x).
(4.13)
†
Note that U ˆ R (α) is a unitary operator, since U ˆ is the inverse rotation.
R
Equation (4.13) is, of course, the familiar rule for rotations of scalar wavefunctions, exhibiting the intimate connection between rotations and angular
momentum in quantum mechanics. We recall that if a Hamiltonian is invariant under rotations, then the operators L ˆ commute with the Hamiltonian and
angular momentum is conserved.
A similar calculation may be done for velocity transformations (‘boosts’),
leading to corresponding operators K ˆ – see problem 4.1.
4.1.2 The Dirac equation
The case of the Dirac equation is more complicated, because (unlike the KG φ)
the wavefunction has more than one component, corresponding to the fact that
it describes a spin-1/2 particle. There is, however, a direct connection between
the angular momentum associated with a wavefunction, and the way that the
wavefunction transforms under rotations of the coordinate system. To take a
simple case, the 2p wavefunctions mentioned in section 3.2 correspond to l = 1
on the one hand and, on the other, to the components of a vector – indeed the
most basic vector of all, the position vector x = (x, y, z) itself. If we rotate
the coordinate system in the way represented by (4.5), the components in the
primed system transform into simple linear combinations of the components
in the original system.
Very much the same thing happens in the case of spinor wavefunctions,
except that they transform in a way different from – though closely related to
– that of vectors. In the present section we shall discuss how this works for
three-dimensional rotations of the spatial coordinate system, and explain how
it generalizes to boosts, which include transformations of the time coordinate
as well. It will be convenient to use the alternative representation (3.40) for
the Dirac matrices. In this representation, the components φ, χ of the freeparticle 4-spinor ω of (3.43) satisfy
Eφ = σ · pφ + mχ
(4.14)
Eχ = −σ · pχ + mφ
(4.15)
rather than (3.45) and (3.46).