References: W. Greiner & J. Reinhardt, Field Quantization, Springer-Verlag (1996), Chapter 2, Section 2.4.
Now that we’ve seen that a general Lorentz transformation can be represented as a product of a pure boost and a pure 3-d rotation, we can return to Noether’s theorem and see what conserved property it predicts when we require a physical system to be invariant under a Lorentz transformation. As usual, we consider an infinitesimal transformation, which we can write as
where is the infinitesimal rotation in 4-dimensional spacetime. Here, we are treating a pure boost as a rotation; for example, a boost in the direction is given by the Lorentz transformation
for some ‘angle’ . This reduces to the more familiar form found in introductory relativity courses if we set
Returning to 1, we require, to first order in , that the Minkowski length of the 4-vector is the same before and after the transformation. That is,
In the last line, we renamed the dummy indices and to . To first order, we require the last term in the last line to be zero for all , which means we must impose a condition on . We can write this term as
From this, we see that we must have
so must be antisymmetric.
Incidentally, if this condition seems to be violated in the pure boost matrix 3, remember that 2 is an ordinary matrix product, while the last term in 1 is the product of a tensor and 4-vector, and thus includes the effect of the metric tensor
To first order in , 3 is
while for the infinitesimal rotation, we have
In matrix notation, 1 becomes
from which we can see that this gives the same result as 2.
In order to apply Noether’s theorem, we also need to know how the fields transform under a Lorentz transformation. The assumption is that, for infinitesimal transformations, the transformed field depends linearly on both the original fields and on the rotation . That is, we assume that
where are the infinitesimal generators of the Lorentz transformation. G & R don’t really explain this, apart from giving a reference to another book, but we won’t need to delve into the details to get the result needed in this post, so we’ll leave it for now.
From here, it’s a matter of plugging 1 and 16 into the equations for Noether’s theorem and seeing what comes out. Noether’s theorem says that
In the last line, we renamed the indices and to and respectively to avoid confusing the in the first term of 19 with any of the indices in .
and from 1
We also had the energy-momentum tensor
Putting all this together, we have
In the second line, we swapped the dummy indices and in the second term, which is allowed because both indices are summed. Therefore
In the second line, we swapped the positions of the indices on the terms . This is OK provided they are both summed over. The last line defines the term .
Because the infinitesimal rotations are arbitrary (subject to the condition that is antisymmetric), we can choose all of them to be zero except for one. For each such choice, we have a different , which leads to a conservation law for each choice. From Noether’s theorem, the quantity that is conserved is the integral of over 3-space, so we have the conserved quantities
From 25, we see that the first two terms are
where is the conjugate momentum density, defined by
Using the physical momentum density
we find that
This is one component of the angular momentum density , so the integral
is one component of the total angular momentum.
The other term in 35 is
depends on the generators , and thus on the specific way in which the fields transform. G & R tell us that this term describes the spin angular momentum, but at this stage, we just have to accept this on faith.
In any case, the overall conservation rule 35 shows that the sum of the ‘traditional’ angular momentum from the first two terms in the integrand, together with this mysterious other term, is a conserved quantity, so interpreting it as some other form of angular momentum seems reasonable. We’ll just have to wait and see how this plays out.