This is a field that falls off much faster than^1 r.A massive scalar field falls off exponentially
and the larger the mass, the faster the fall off. This fits the form of the force between nucleons fairly
well although the actual nuclear force needs a much more detailed study.
1.38 The Classical Electromagnetic Field
For the study of the Maxwell field, (See section 20) it is most convenient to make a small modification
to the system of units that are used. InRationalized Heaviside-Lorentz Unitsthe fields are
all reduced by a factor of
√
4 πand the charges are increased by the same factor. With this change
Maxwell’s equations, as well as the Lagrangians we use, are simplified. It would have simplified
many things if Maxwell had started off with this set of units.
As is well known from classical electricity and magnetism, the electricand magnetic field components
are actually elements of a rank 2 Lorentz tensor.
Fμν=
0 Bz −By −iEx
−Bz 0 Bx −iEy
By −Bx 0 −iEz
iEx iEy iEz 0
This field tensor can simply be written in terms of thevector potential, (which is a Lorentz vector).
Aμ = (A,iφ~ )Fμν =∂Aν
∂xμ−
∂Aμ
∂xνNote thatFμνis automatically antisymmetric under the interchange of the indices.
With the fields so derived from the vector potential, two of Maxwell’s equations are automatically
satisfied. The remaining two equations can be written as one 4-vector equation.
∂Fμν
∂xν=
jμ
cWe now wish to pick a scalar Lagrangian. Since E&M is a well understoodtheory, the Lagrangian
that is known to give the right equations is also known.
L=−
1
4
FμνFμν+1
cjμAμNote that (apart from the speed of light not being set to 1) the Lagrangian does not contain needless
constants in this set of units. The last term is a source term which provides the interaction between
the EM field and charged particles. In working with this Lagrangian, we will treat each component
ofAas an independent field. In this case, the Euler-Lagrange equationis Maxwell’s equation as
written above.
The free field Hamiltonian density can be computed according to the standard prescription yielding
∂L
∂(∂Aμ/∂x 4 )= Fμ 4