References: Anthony Zee, Einstein Gravity in a Nutshell, (Princeton University Press, 2013) – Chapter I.3, Problem 4.
We can generalize Lie’s method of generating rotations to any number of dimensions. [This post follows Appendix 2 in Zee’s chapter I.3, which I believe contains a few typos. I’ll try to correct them here.]
To review the last post, a rotation should leave the dot product of two vectors unchanged. If we consider an infinitesimal rotation given by the matrix , where is the identity matrix and is a matrix containing infinitesimal quantities, then the invariance of the dot product leads to the conclusion that which requires (to first order in ) that , that is, that is antisymmetric. In dimensions, there are independent antisymmetric matrices, which can be written down by choosing a row and a column (since diagonal elements are all zero in an antisymmetric matrix), setting the element and the element . Then any antisymmetric matrix can be decomposed into a linear combination of the matrices. To distinguish the s, we’ll given them a subscript to label which row and column are non-zero. For example in 3-d, we have
[Note that these matrices aren’t quite the same as those we used in the last post, since and .] The subscript labels the entire matrix and not just a single element within a matrix. To pick out an individual element, we’ll use superscript indices, so that, for example, is the element of in row 2 and column 3.
Because these matrices are related to the angular momentum operators in quantum mechanics, it’s customary to make them into hermitian operators, which, for a matrix, means that the complex conjugate of the transpose (the hermitian conjuage) is the same as the original matrix. That is, for a matrix we have . Since the are antisymmetric and real, their hermitian conjugates are antisymmetric, so they aren’t hermitian matrices. We can convert them into hermitian matrices by multiplying them by a multiple of ; by convention this multiple is taken to be . Thus we have the hermitian matrices
We can write out the in a general formula using the Kronecker delta:
To verify this formula, remember that with all other elements being zero. The first term in 5 is non-zero only if and , while the second term is non-zero only if and , so the formula works.
[In the paragraph following Zee’s equation 19, he says “there are only real antisymmetric -by- matrices “. The last should be since the contain purely imaginary elements, not real ones.]
To generate a rotation in dimensions, we can use Lie’s method of considering infinitesimal rotations , where is an infinitesimal linear combination of the . Since is real, we have
for some real values .
The next stage in the argument isn’t entirely clear to me, probably because I haven’t seen the use to which rotations are put in the rest of Zee’s book. However, let’s plow on for the moment.
We saw in the previous post that in 3-d, rotations about different axes do not commute; a rotation about and then will leave you in a different orientation than a rotation about and then . Generalizing to dimensions, suppose we have two infinitesimal rotations and , where and are infinitesimal, antisymmetric matrices as before. Then if we apply first, then , the overall rotation is given by
Switching the order [note that Zee has a typo here: he says ; it should be ] we get
Taking the difference, we get
where is the commutator of and (the same commutators that show up in quantum mechanics). This derivation seems to be a bit of a fudge, since the commutator is, by definition, of second order in and , so by separating it out from the general term, it seems we’re implicitly assuming that the is the same in both and , so it cancels out when taking the difference.
Zee gives a second argument for measuring the difference between the two compound rotations and . If the two orders of rotation commuted, then the inverse of one rotation should also be the inverse of the other. That is, we should have . A measure of how different the two orders of rotation are from each other can then be found by seeing how much differs from .
For an infinitesimal rotation , then to first order since . Therefore
Again, the term is neglected along with the terms, even though it contains the terms which are the same terms that appear in . It’s not clear to me how we can justify ignoring but not .
In any case, we can observe that the transpose of a commutator gives
so a commutator is always an antisymmetric matrix. In particular, for the matrices in 5, we have
with an implied sum over on the RHS. This follows because any antisymmetric matrix can be written as a linear combination of the s.
Since the s are purely imaginary, a product of two of them is always real. Therefore the coefficients must be real, since is a real matrix.
We can find the coefficients by a brute force calculation starting with 5. We get (with an implied sum over the index ):
So the commutator is
For general infinitesimal rotations from 6
The commutator is therefore
Thus if we know the commutators of the generator matrices we can work out the commutators of any antisymmetric matrix pair.