16
1. The Particles and Forces of the Standard Model
netism, postulating a new field of force with an associated new field quantum,
analogous to the photon. In doing so, he showed with particular clarity how,
in quantum field theory, particles interact by exchanging virtual quanta, which
mediate the force.
Before proceeding, we should emphasize that we are not presenting Yukawa’s
ideas as a viable candidate theory of strong and weak interactions. Crucially,
Yukawa assumed that the nucleons and his quantum (later identified with the
pion) were point-like, but in fact both nucleons and pions are quark composites with spatial extension. The true ‘strong’ interaction relates to the quarks,
as we shall see in section 1.3.6. There are also other details of his theory which
were (we now know) mistaken, as we shall discuss. Yet his approach was profound, and – as happens often in physics – even though the initial application
was ultimately superseded, the ideas have broad and lasting validity.
Yukawa began by considering what kind of static potential might describe
the n–p interaction. It was known that this interaction decreased rapidly
for interparticle separation r ≥ 2 fm. Hence, the potential could not be of
coulombic type ∝ 1/r. Instead, Yukawa postulated an n–p potential energy
of the form
2 −r/a
−g e
N
U (r) =
(1.13)
4π
r
where ‘g N ’ is a constant analogous to the electric charge e, r = |r| and ‘a’ is
a range parameter (∼ 2 fm). This static potential satisfies the equation
(
)
1
2
∇
2
−
U (r) = g N δ(r)
(1.14)
a 2
(see appendix G) showing that it may be interpreted as the mutual potential
energy of one point-like test nucleon of ‘strong charge’ g N due to the presence
of another point-like nucleon of equal charge g N at the origin, a distance r
away. Equation (1.14) should be thought of as a finite range analogue of
Poisson’s equation in electrostatics (equation (G.3))
∇
2 V (r) = −ρ(r)/∈ 0 ,
(1.15)
the delta function in (1.14) (see appendix E) expressing the fact that the
‘strong charge density’ acting as the source of the field is all concentrated into
a single point, at the origin.
Yukawa now sought to generalize (1.14) to the non-static case, so as to
obtain a field equation for U (r, t). For r = 0, he proposed the free-space
/
equation (we shall keep factors of c and ħ explicit for the moment)
(
)
∇
2
−
∂
2
−
1 U (r, t) = 0
(1.16)
c 2 ∂t 2
a 2
which is certainly relativistically invariant (see appendix D). Thus far, U is
still a classical field. Now Yukawa took the decisive step of treating U quantum
1. The Particles and Forces of the Standard Model
netism, postulating a new field of force with an associated new field quantum,
analogous to the photon. In doing so, he showed with particular clarity how,
in quantum field theory, particles interact by exchanging virtual quanta, which
mediate the force.
Before proceeding, we should emphasize that we are not presenting Yukawa’s
ideas as a viable candidate theory of strong and weak interactions. Crucially,
Yukawa assumed that the nucleons and his quantum (later identified with the
pion) were point-like, but in fact both nucleons and pions are quark composites with spatial extension. The true ‘strong’ interaction relates to the quarks,
as we shall see in section 1.3.6. There are also other details of his theory which
were (we now know) mistaken, as we shall discuss. Yet his approach was profound, and – as happens often in physics – even though the initial application
was ultimately superseded, the ideas have broad and lasting validity.
Yukawa began by considering what kind of static potential might describe
the n–p interaction. It was known that this interaction decreased rapidly
for interparticle separation r ≥ 2 fm. Hence, the potential could not be of
coulombic type ∝ 1/r. Instead, Yukawa postulated an n–p potential energy
of the form
2 −r/a
−g e
N
U (r) =
(1.13)
4π
r
where ‘g N ’ is a constant analogous to the electric charge e, r = |r| and ‘a’ is
a range parameter (∼ 2 fm). This static potential satisfies the equation
(
)
1
2
∇
2
−
U (r) = g N δ(r)
(1.14)
a 2
(see appendix G) showing that it may be interpreted as the mutual potential
energy of one point-like test nucleon of ‘strong charge’ g N due to the presence
of another point-like nucleon of equal charge g N at the origin, a distance r
away. Equation (1.14) should be thought of as a finite range analogue of
Poisson’s equation in electrostatics (equation (G.3))
∇
2 V (r) = −ρ(r)/∈ 0 ,
(1.15)
the delta function in (1.14) (see appendix E) expressing the fact that the
‘strong charge density’ acting as the source of the field is all concentrated into
a single point, at the origin.
Yukawa now sought to generalize (1.14) to the non-static case, so as to
obtain a field equation for U (r, t). For r = 0, he proposed the free-space
/
equation (we shall keep factors of c and ħ explicit for the moment)
(
)
∇
2
−
∂
2
−
1 U (r, t) = 0
(1.16)
c 2 ∂t 2
a 2
which is certainly relativistically invariant (see appendix D). Thus far, U is
still a classical field. Now Yukawa took the decisive step of treating U quantum