The energy-momentum tensor of a point particle is give as:
There will be 4 conserved quantities:
Plugging it in, we get:
In EM, we have only one conserved quantity , which comes from:
Since GR is about the energy momentum tensor, in order for it to be covariant, we are essentially looking for a equation that looks like this:
Einstein didn't realize that this must be an equation of a 10-component tensor until 1912. This is the gravitational field. (In contrary, in Newtonian gravity, you can define everything with a one-component gravitational potential ).
Angular momentum and spin
Starting from energy-momentum conservation:
we define an object (angular momentum density):
Then we take the partial derivative of it:
This implies a conserved quantity (angular momentum) . The spatial components of the angular momentum tensor gives the classical angular momentum:
The time components of it gives mass moment:
A special note is that this quantity contains both orbital part and spin part of angular momentum, the latter does not depend on the origin. (I should make another note on this issue). This is called the Pauli–Lubanski pseudovector.
(This is essentially talking about the Lie algebra of the Lorentz Group as we discussed before in QFT notes.)
In QM, we discussed rotation group in 3 dimensional Euclidian space:
where is a 3x3 matrix.
it corresponds to the transformation of an indexed wave function:
and is the rotation matrix on the state vectors. This mean while we need to change the coordinate to index into a field under rotation, we also need to change the components of the field at the same time given the field has more than one components.
The rotation matrix has a simple form:
where is the angular momentum matrices. It satisfies certain algebra (in terms of commutation relations):
one of the outcome of this algebra is that only certain eigenvalues of the spin is allowed:
Considering special relativity, the true transformation of the world is not only rotation, but also boost.
In SR, , therefore, we need to redo the exercise to make sure we can still only get
spin but not and such.
which is the transformation associated with . They form a group:
To find , consider an infinitesimal Lorentz transformation looks like:
The product of two Lorentz transformation is:
So the product of two infinitesimal Lorentz transformation is the infinitesimal Lorentz transformation with the displacement equals to the
sum of the two transformations.
where is infinitesimal. From its being rotation:
Therefore is anti-symmetric.
For , we will pick an anti-symmetric to make a group:
These s, which we pick to be anti-symmetric., are called the "generators of the Lorentz group".
Lorentz group has 6 different parameters, therefore we need 6 of these 4x4 matrices, denoted as .
If is a 4-vector, then the , and then:
If all components of are independently arbitrary, then we can equalize the coefficients of the them (thus removing from the equation). Unfortunately is anti-symmetric. So we need to expand the summation:
Now we can cancel and get:
(Last time, I derived this from . This time its much simpler.)
The next step is to figure out the commutation relations. The result is:
So this breaks up into two rotation groups. So the eigenvalues of and should be:
Therefore all the representations of the Lorentz group should be classified by the a pair of numbers .
(The relationship between dimension of the field/particle and the pair requires some further explanation).