130_notes.dvi

k = k′

−

∫∞

0

k

k^2 +μ^2

e−ikrdk+

∫∞

0

k

k^2 +μ^2

eikrdk = πie−μr

∫∞

0

k

k^2 +μ^2

eikrdk−

∫∞

0

k

k^2 +μ^2

e−ikrdk = πie−μr

∫∞

0

k

k^2 +μ^2

(

eikr−e−ikr

)

dk = πie−μr

This is exactly the integral we wanted to do.

Plug the integral into the Fourier transform we were computing.

φ(~x) =

−G

(2π)^2 ir

∫∞

0

k

k^2 +μ^2

(

eikr−e−ikr

)

dk

φ(~x) =

−G

(2π)^2 ir

πie−μr

φ(~x) =

−Ge−μr

4 πr

In this case, it is simple to compute theinteraction Hamiltonianfrom the interaction Lagrangian
and thepotential between two particles. Lets assume we have two particles, each with the same
interaction with the field.
ρ(~x,t) =ρ(x) =Gδ^3 (~x)

Now compute the Hamiltonian.

Lint = −φρ

Hint = φ ̇

∂Lint

∂φ ̇

−Lint=−Lint=φρ

Hint =

∫

Hintd^3 x 2

Hint(1,2) =

∫

φ 1 ρ 2 d^3 x 2 =

∫

−Ge−μr

4 πr

Gδ^3 (~x 2 )d^3 x 2 =

−G^2 e−μr^12

4 πr 12

We see that this is a short-range, attractive potential.

This wasproposed by Yukawa as the nuclear force. He predicted a scalar particle with a mass
close to that of the pion before the pion was discovered. His prediction of the mass was based on the
range of the nuclear force, on the order of one Fermi. In some sense, his prediction is approximately
correct. Pion exchange can explain much of the nuclear force but does not explain all the details.
Pions and nucleons have since been show to be composite particles with internal structure. Other
composites with masses larger than the pion also play a role in the force between nucleons.

Pions were also found to come in three charges:π+,π−, andπ^0. This would lead us to develop a
complex scalar field as done in the text. Its not our goal right now sowe will skip this. Its interesting
to note that the Higgs Boson is also represented by a complex scalarfield.