78
3. Relativistic Quantum Mechanics
to Feynman: in essence, the idea is that the negative 4-momentum solutions
will be used to describe antiparticles, for both bosons and fermions.
We begin with bosons – for example pions, which for the present purposes
we take to be simple spin-0 particles whose wavefunctions obey the KG equation. We decide by convention that the π
+ is the ‘particle’. We will then
have
−ip·x
positive 4-momentum π
+ solutions: N e
(3.85)
negative 4-momentum π
+ solutions: N e
ip·x
(3.86)
2
2 )
1/2
where p
μ = [(m + p
, p]. The electromagnetic current for a free physical
(positive-energy) π
+ is given by the probability current for a positive-energy
solution multiplied by the charge Q(= +e):
j
μ (π
+ ) = (+e) × (probability current for positive energy π
+ )(3.87)
em
= (+e)2|N |
2 [(m
2 + p
2 )
1/2 , p]
(3.88)
using (3.20) and (3.85) (see problem 3.1). What about the current for the π
− ?
2
For free physical π
− particles of positive energy (m + p
2 )
1/2 and momentum
p we expect
2
j
μ (π
− ) = (−e)2|N |
2 [(m + p
2 )
1/2 , p]
(3.89)
em
by simply changing the sign of the charge in (3.88). But it is evident that
(3.89) may be written as
2
j
μ (π
− ) = (+e)2|N |
2 [−(m + p
2 )
1/2 , −p]
(3.90)
em
which is just j
μ (π
+ ) with negative 4-momentum. This suggests some equivem
alence between antiparticle solutions with positive 4-momentum and particle
solutions with negative 4-momentum.
Can we push this equivalence further? Consider what happens when a
system A absorbs a π
+ with positive 4-momentum p: its charge increases by
+e, and its 4-momentum increases by p. Now suppose that A emits a physical
π
− with 4-momentum k, where the energy k
0 is positive. Then the charge
of A will increase by +e, and its 4-momentum will decrease by k. Now this
increase in the charge of A could equally well be caused by the absorption
of a π
+ – and indeed we can make the effect (as far as A is concerned) of
the π
− emission process fully equivalent to a π
+ absorption process if we say
that the equivalent absorbed π
+ has negative 4-momentum, −k; in particular
the equivalent absorbed π
+ has negative energy −k
0 . In this way, we view
the emission of a physical ‘antiparticle’ π
− with positive 4-momentum k as
equivalent to the absorption of a ‘particle’ π
+ with (unphysical) negative 4momentum −k. Similar reasoning will apply to the absorption of a π
− of
positive 4-momentum, which is equivalent to the emission of a π
+ of negative
4-momentum. Thus we are led to the following hypothesis (due to Feynman):
μ
The emission (absorption) of an antiparticle of 4-momentum p is physically equivalent to the absorption (emission) of a particle of 4-momentum
μ
−p .
3. Relativistic Quantum Mechanics
to Feynman: in essence, the idea is that the negative 4-momentum solutions
will be used to describe antiparticles, for both bosons and fermions.
We begin with bosons – for example pions, which for the present purposes
we take to be simple spin-0 particles whose wavefunctions obey the KG equation. We decide by convention that the π
+ is the ‘particle’. We will then
have
−ip·x
positive 4-momentum π
+ solutions: N e
(3.85)
negative 4-momentum π
+ solutions: N e
ip·x
(3.86)
2
2 )
1/2
where p
μ = [(m + p
, p]. The electromagnetic current for a free physical
(positive-energy) π
+ is given by the probability current for a positive-energy
solution multiplied by the charge Q(= +e):
j
μ (π
+ ) = (+e) × (probability current for positive energy π
+ )(3.87)
em
= (+e)2|N |
2 [(m
2 + p
2 )
1/2 , p]
(3.88)
using (3.20) and (3.85) (see problem 3.1). What about the current for the π
− ?
2
For free physical π
− particles of positive energy (m + p
2 )
1/2 and momentum
p we expect
2
j
μ (π
− ) = (−e)2|N |
2 [(m + p
2 )
1/2 , p]
(3.89)
em
by simply changing the sign of the charge in (3.88). But it is evident that
(3.89) may be written as
2
j
μ (π
− ) = (+e)2|N |
2 [−(m + p
2 )
1/2 , −p]
(3.90)
em
which is just j
μ (π
+ ) with negative 4-momentum. This suggests some equivem
alence between antiparticle solutions with positive 4-momentum and particle
solutions with negative 4-momentum.
Can we push this equivalence further? Consider what happens when a
system A absorbs a π
+ with positive 4-momentum p: its charge increases by
+e, and its 4-momentum increases by p. Now suppose that A emits a physical
π
− with 4-momentum k, where the energy k
0 is positive. Then the charge
of A will increase by +e, and its 4-momentum will decrease by k. Now this
increase in the charge of A could equally well be caused by the absorption
of a π
+ – and indeed we can make the effect (as far as A is concerned) of
the π
− emission process fully equivalent to a π
+ absorption process if we say
that the equivalent absorbed π
+ has negative 4-momentum, −k; in particular
the equivalent absorbed π
+ has negative energy −k
0 . In this way, we view
the emission of a physical ‘antiparticle’ π
− with positive 4-momentum k as
equivalent to the absorption of a ‘particle’ π
+ with (unphysical) negative 4momentum −k. Similar reasoning will apply to the absorption of a π
− of
positive 4-momentum, which is equivalent to the emission of a π
+ of negative
4-momentum. Thus we are led to the following hypothesis (due to Feynman):
μ
The emission (absorption) of an antiparticle of 4-momentum p is physically equivalent to the absorption (emission) of a particle of 4-momentum
μ
−p .