Feynman’s interpretation of the negative-energy solutions of the KG and Dirac equations solutions of the KG and Dirac equations

It is clear that despite its brilliant success for spin-₂ ¹ particles, Dirac’s inter­

pretation cannot be applied to spin-0 particles, since bosons are not subject to the exclusion principle. Besides, spin-0 particles also have their corresponding antiparticles (e.g. π⁺ and π⁻), and so do spin-1 particles (W⁺ and W⁻, for instance). A consistent picture for both bosons and fermions does emerge from quantum field theory, as we shall see in chapters 5–7, which is perhaps one of the strongest reasons for mastering it. Nevertheless, it is useful to have an alternative, non-field-theoretic, interpretation of the negative-energy solu­

tions which works for both bosons and fermions. Such an interpretation is due

2At that time, this was not universally recognized. For example, Pauli (1933) wrote:

‘Dirac has tried to identify holes with antielectrons. . . we do not believe that this explanation canbeseriouslyconsidered.’

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 equa­

tion. We decide by convention that the π⁺ is the ‘particle’. We will then have

−ip·x

positive 4-momentum π⁺ solutions: Ne (3.85) negative 4-momentum π⁺ solutions: Ne ^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_em^μ (π⁺) = (+e)×(probability current for positive energy π⁺)(3.87)

= (+e)2|N|²[(m ²+p²)^1/2,p] (3.88) using (3.20) and (3.85) (see problem 3.1). What about the current for the π⁻? For free physical π⁻ particles of positive energy (m2+p²)^1/2and momentum pwe expect

j^μ _em(π⁻) = (−e)2|N|²[(m 2+p²)^1/2 ,p] (3.89) by simply changing the sign of the charge in (3.88). But it is evident that (3.89) may be written as

j_em^μ (π⁻) = (+e)2|N|²[−(m 2+p²)^1/2 ,−p] (3.90) which is just j_em^μ (π⁺) with negative 4-momentum. This suggests some equiv­

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 k⁰ 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 −k⁰. 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 4­

momentum −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):

The emission (absorption)of anantiparticle of 4-momentumpμ isphysi­

cally equivalenttothe absorption (emission)ofaparticleof4-momentum

−pμ.

79 3.4. The negative-energysolutions

FIGURE 3.2

Coulomb scattering of a π⁻ by a static charge Ze illustrating the Feynman interpretation of negative 4-momentum states.

In other words the unphysical negative 4-momentum solutions of the ‘particle’

equation do have a role to play: they can be used to describe physical processes involving positive 4-momentum antiparticles, if we reverse the role of ‘entry’

and ‘exit’ states.

The idea is illustrated in figure 3.2, for the case of Coulomb scattering of a π⁻ particle by a static charge Ze, which will be discussed later in section 8.1.3.

By convention we are taking π⁻ to be the antiparticle. In the physical process of figure 3.2(a) the incoming physical antiparticle π⁻ has 4-momentum pi, and the final π⁻ has 4-momentum pf: both Ei and Ef are, of course, positive.

Figure 3.2(b) shows how the amplitude for the process can be calculated using π⁺ solutions with negative 4-momentum. The initial state π⁻ of 4-momentum pibecomes a final state π⁺ with 4-momentum −pi, and similarly the final state π⁻ of 4-momentum pf becomes an initial state π⁺ of 4-momentum −pf. Note that in this and similar figures, the sense of the arrows always indicates the

‘flow’ of 4-momentum, positive 4-momentum corresponding to forward flow.

It is clear that the basic physical idea here is not limited to bosons. But there is a difference between the KG and Dirac cases in that the Dirac equation was explicitly designed to yield a probability density (and probability current density) which was independent of the sign of the energy:

ρ= ψ^†ψ j= ψ^†αψ. (3.91) Thus for any solutions of the form

ψ = ωφ(x, t) (3.92)

we have

ρ= ω^†ω|φ(x, t)|² (3.93) and

j= ω^†αω|φ(x, t)|² (3.94)

and ρ ≥ 0 always. We nevertheless want to set up a correspondence so that positive-energy solutions describe electrons (taken to be the ‘particle’, by con­

vention, in this case) and negative-energy solutions describe positrons, if we reverse the sense of incoming and outgoing waves. For the KG case this was straightforward, since the probability current was proportional to the 4-momentum:

j^μ(KG) ∼ p . μ (3.95)

We were therefore able to set up the correspondence for the electromagnetic current of π⁺ and π⁻:

π⁺: j_em^μ ∼ ep μ positive energy π⁺ (3.96) π⁻: j_em^μ ∼ (−e)p μ positive energy π⁻ (3.97)

≡ (+e)(−p ^μ) negative energy π⁺. (3.98) This simple connection does not hold for the Dirac case since ρ ≥ 0 for both signs of the energy. It is still possible to set up the correspondence, but now an extra minus sign must be inserted ‘by hand’ whenever we have a negative-energy fermion in the final state. We shall make use of this rule in section 8.2.4. We therefore state the Feynman hypothesis for fermions:

The invariant amplitude for the emission (absorption) of an antifermion of 4-momentum pμ and spin projection s_z in the rest frame is equal to the amplitude (minus the amplitude) for the absorption (emission) of a fermion of 4-momentum−pμ andspinprojection −s_z in the restframe.

As we shall see in chapters 5–7, the Feynman interpretation of the negative-energy solutions is naturally embodied in the field theory formalism.

3.5 Inclusion of electromagnetic interactions via the