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3.1. The Klein–Gordon equation
3.1.2 Probability current for the KG equation
In exactly the same way as for the non-relativistic Schr¨ odinger equation, it
is possible to derive a conservation law for a ‘probability current’ of the KG
equation. We have
∂
2 φ − ∇
2 φ + m
2 φ = 0
(3.15)
∂t 2
and by multiplying this equation by φ
∗ , and subtracting φ times the complex conjugate of equation (3.15), one obtains, after some manipulation (see
problem 3.1), the result
∂ρ
∂t
+ ∇ · j = 0
(3.16)
where
ρ = i
[
φ
∗ ∂φ
∂t
−
(
∂φ
∗
∂t
)
φ
]
(3.17)
and
j = i
−1 [φ
∗
∇φ − (∇φ
∗ )φ]
(3.18)
(the derivatives (∂ μ φ
∗ ) act only within the bracket). In explicit 4-vector notation this conservation condition reads (cf problem 2.1 and equation (D.4)
in appendix D)
∂ μ j
μ = 0
(3.19)
with
j
μ
≡ (ρ, j) = i[φ
∗ ∂
μ φ − (∂
μ φ
∗ )φ].
(3.20)
Since φ of (3.11) is Lorentz invariant and ∂
μ is a contravariant 4-vector, equation (3.20) shows explicitly that j
μ is a contravariant 4-vector, as anticipated
in the notation.
The spatial current j is identical in form to the Schr¨ odinger current, but
for the KG case the ‘probability density’ now contains time derivatives since
the KG equation is second order in ∂/∂t. This means that ρ is not constrained
to be positive definite – so how can ρ represent a probability density? We can
see this problem explicitly for the plane-wave solutions
−iEt+ip·x
φ = N e
(3.21)
which give (problem 3.1)
ρ = 2|N |
2 E
(3.22)
and E can be positive or negative: that is, the sign of ρ is the sign of energy.
Historically, this problem of negative probabilities coupled with that of
negative energies led to the abandonment of the KG equation. For the moment we will follow history, and turn to the Dirac equation. We shall see in
section 3.4, however, how the negative-energy solutions of the KG equation
do after all have a role to play, following Feynman’s interpretation, in processes involving antiparticles. Later, in chapters 5–7, we shall see how this
interpretation arises naturally within the formalism of quantum field theory.
3.1. The Klein–Gordon equation
3.1.2 Probability current for the KG equation
In exactly the same way as for the non-relativistic Schr¨ odinger equation, it
is possible to derive a conservation law for a ‘probability current’ of the KG
equation. We have
∂
2 φ − ∇
2 φ + m
2 φ = 0
(3.15)
∂t 2
and by multiplying this equation by φ
∗ , and subtracting φ times the complex conjugate of equation (3.15), one obtains, after some manipulation (see
problem 3.1), the result
∂ρ
∂t
+ ∇ · j = 0
(3.16)
where
ρ = i
[
φ
∗ ∂φ
∂t
−
(
∂φ
∗
∂t
)
φ
]
(3.17)
and
j = i
−1 [φ
∗
∇φ − (∇φ
∗ )φ]
(3.18)
(the derivatives (∂ μ φ
∗ ) act only within the bracket). In explicit 4-vector notation this conservation condition reads (cf problem 2.1 and equation (D.4)
in appendix D)
∂ μ j
μ = 0
(3.19)
with
j
μ
≡ (ρ, j) = i[φ
∗ ∂
μ φ − (∂
μ φ
∗ )φ].
(3.20)
Since φ of (3.11) is Lorentz invariant and ∂
μ is a contravariant 4-vector, equation (3.20) shows explicitly that j
μ is a contravariant 4-vector, as anticipated
in the notation.
The spatial current j is identical in form to the Schr¨ odinger current, but
for the KG case the ‘probability density’ now contains time derivatives since
the KG equation is second order in ∂/∂t. This means that ρ is not constrained
to be positive definite – so how can ρ represent a probability density? We can
see this problem explicitly for the plane-wave solutions
−iEt+ip·x
φ = N e
(3.21)
which give (problem 3.1)
ρ = 2|N |
2 E
(3.22)
and E can be positive or negative: that is, the sign of ρ is the sign of energy.
Historically, this problem of negative probabilities coupled with that of
negative energies led to the abandonment of the KG equation. For the moment we will follow history, and turn to the Dirac equation. We shall see in
section 3.4, however, how the negative-energy solutions of the KG equation
do after all have a role to play, following Feynman’s interpretation, in processes involving antiparticles. Later, in chapters 5–7, we shall see how this
interpretation arises naturally within the formalism of quantum field theory.