95
4.2. Discrete transformations: P, C and T
4.2 Discrete transformations: P, C and T
The transformations we considered in section 4.1 are known as ‘continuous’,
because the parameters involved (angles, speeds) vary continuously. This is
essentially the reason we were able to build up finite transformations from
infinitesimal ones, which differ only slightly from the identity transformation:
finite transformations could be reached continuously from the identity. But
there is another class of transformations, called ‘discrete’, which cannot be
reached continuously from the identity. Examples of discrete transformations
are parity (or space inversion), charge conjugation, and time reversal, and
their combinations. Although these discrete transformations are important
primarily in weak interactions, which we shall not cover until the second volume, it is useful to discuss the behaviour of Dirac wavefunctions under discrete
transformations at this stage. Among other things, more light will be cast on
antiparticles.
4.2.1 Parity
The parity (or space inversion) transformation P is defined by
′
P : x → x = −x, t → t;
(4.61)
that is, P inverts the spatial coordinates. It follows that P also inverts momenta (p → −p) but does not change angular momenta (x × p → x × p) or
spin (σ → σ). We already see that there are two kinds of 3-vectors: polar
3-vectors which change sign under P and axial vectors which do not. For example, the electric field E and the vector potential A are polar vectors, while
the magnetic field B is an axial vector. There are also scalar quantities (such
as x · p) which do not change sign under P, and pseudoscalar quantities (such
as σ · p) which do.
Consider first the KG equation (4.1). Since A is a polar vector, it changes
sign under parity, as does ∇, while both ∂/∂t and A
0 remain the same. The
scalar products ∂ μ A
μ and A
μ ∂ μ are therefore invariant under parity, as are ❗
and A
2 . Hence we may identify φ P (x
′ ) = φ(x), or equivalently
φ P (x) = φ(−x) ≡ P ˆ 0 φ(x),
(4.62)
where P ˆ 0 is the coordinate inversion operator. Note that we are calling the
transformed wavefunction φ P rather than yet another φ
′ since we need to
keep track of what transformation we are considering. If we take φ(x) to be
a positive-energy free particle solution with energy E and momentum p, φ P
will describe a positive energy particle with momentum −p, as we expect.
Now let us study the covariance of the free particle Dirac equation
∂ψ(x, t)
i
= −iα · ∇ψ(x, t) + βmψ(x, t)
(4.63)
∂t
4.2. Discrete transformations: P, C and T
4.2 Discrete transformations: P, C and T
The transformations we considered in section 4.1 are known as ‘continuous’,
because the parameters involved (angles, speeds) vary continuously. This is
essentially the reason we were able to build up finite transformations from
infinitesimal ones, which differ only slightly from the identity transformation:
finite transformations could be reached continuously from the identity. But
there is another class of transformations, called ‘discrete’, which cannot be
reached continuously from the identity. Examples of discrete transformations
are parity (or space inversion), charge conjugation, and time reversal, and
their combinations. Although these discrete transformations are important
primarily in weak interactions, which we shall not cover until the second volume, it is useful to discuss the behaviour of Dirac wavefunctions under discrete
transformations at this stage. Among other things, more light will be cast on
antiparticles.
4.2.1 Parity
The parity (or space inversion) transformation P is defined by
′
P : x → x = −x, t → t;
(4.61)
that is, P inverts the spatial coordinates. It follows that P also inverts momenta (p → −p) but does not change angular momenta (x × p → x × p) or
spin (σ → σ). We already see that there are two kinds of 3-vectors: polar
3-vectors which change sign under P and axial vectors which do not. For example, the electric field E and the vector potential A are polar vectors, while
the magnetic field B is an axial vector. There are also scalar quantities (such
as x · p) which do not change sign under P, and pseudoscalar quantities (such
as σ · p) which do.
Consider first the KG equation (4.1). Since A is a polar vector, it changes
sign under parity, as does ∇, while both ∂/∂t and A
0 remain the same. The
scalar products ∂ μ A
μ and A
μ ∂ μ are therefore invariant under parity, as are ❗
and A
2 . Hence we may identify φ P (x
′ ) = φ(x), or equivalently
φ P (x) = φ(−x) ≡ P ˆ 0 φ(x),
(4.62)
where P ˆ 0 is the coordinate inversion operator. Note that we are calling the
transformed wavefunction φ P rather than yet another φ
′ since we need to
keep track of what transformation we are considering. If we take φ(x) to be
a positive-energy free particle solution with energy E and momentum p, φ P
will describe a positive energy particle with momentum −p, as we expect.
Now let us study the covariance of the free particle Dirac equation
∂ψ(x, t)
i
= −iα · ∇ψ(x, t) + βmψ(x, t)
(4.63)
∂t