Assume we have a field defined everywhere in space and time.
For simplicity we will start with a scalar field(instead of the vector... fields of E&M).
The Euler-Lagrange equation derived from the principle of least action is
Since we are aiming for a description of relativistic quantum mechanics,
it will benefit us to write our equations in a covariant way.
I think this also simplifies the equations.
We will follow the notation of Sakurai.
(The convention does not really matter and one should not get hung up on it.)
As usual the Latin indices like
will run from 1 to 3 and represent the space coordinates.
The Greek indices like
will run from 1 to 4.
Sakurai would give the spacetime coordinate vector either as
We will not use the so called covariant and contravariant indices.
Instead we will put an
on the fourth component of a vector which give that component a
in a dot product.
The spacetime coordinate
is a Lorentz vector transforming under rotations and boosts as follows.
The Lorentz transformation matrix to a coordinate system boosted along the x direction is.
With this, lets work on the Euler-Lagrange equation to get it into covariant shape. Remember that the field is a Lorentz scalar.
This is the Euler-Lagrange equation for a single scalar field. Each term in this equation is a Lorentz scalar, if is a scalar.
Now we want to find a reasonable Lagrangian for a scalar field.
The Lagrangian depends on the
and its derivatives.
It should not depend explicitly on the coordinates
, since that would violate translation
and/or rotation invariance.
We also want to come out with a linear wave equation so that high powers of the field should not appear.
The only Lagrangian we can choose (up to unimportant constants) is
With this Lagrangian, the Euler-Lagrange equation is.
This is the known as the Klein-Gordon equation.
It is a good relativistic equation for a massive scalar field.
It was also an early candidate for the relativistic equivalent of the Schrödinger equation for electrons
because it basically has the relativistic analog of the energy relation inherent in the Schrödinger equation.
Writing that relation in the order terms appear in the Klein-Gordon equation above we get (letting
So far we have the Lagrangian and wave equation for a ``free'' scalar field.
There are no sources of the field (the equivalent of charges and currents in electromagnetism.)
Lets assume the source density is
The source term must be a scalar function so,
we add the term
to the Lagrangian.
Any source density can be built up from point sources so it is useful to understand the
field generated by a point source as we do for electromagnetism.
Now we solve for the scalar field from a point source by Fourier transforming the wave equation. Define the Fourier transforms to be.
We now have the Fourier transform of the field of a point source. If we can transform back to position space, we will have the field. This is a fairly standard type of problem in quantum mechanics.
If the integrand is a function of the complex variable , a pole is of the form where is called the residue at the pole. The integrand above has two poles, one at and the other at . The integral we are interested in is just along the real axis so we want the integral along the rest of the contour to give zero. That's easy to do since the integrand goes to zero at infinity on the real axis and the exponentials go to zero either at positive or negative infinity for the imaginary part of . Examine the integral around a contour in the upper half plane as shown below.
The integrand goes to zero exponentially on the semicircle at infinity so only the real axis contributes to the integral along the contour. The integral along the real axis can be manipulated to do the whole problem for us.
Plug the integral into the Fourier transform we were computing.
In this case, it is simple to compute the interaction Hamiltonian from the interaction Lagrangian
and the potential between two particles.
Lets assume we have two particles, each with the same interaction with the field.
This was proposed 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: , , and . This would lead us to develop a complex scalar field as done in the text. Its not our goal right now so we will skip this. Its interesting to note that the Higgs Boson is also represented by a complex scalar field.
We have developed a covariant classical theory for a scalar field. The Lagrangian density is a Lorentz scalar function. We have included an interaction term to provide a source for the field. Now we will attempt to do the same for classical electromagnetism.
Jim Branson 2013-04-22