2
$\begingroup$

I'm reading through some notes (about the use of Lie groups/algebras in physics) obtained from a friend from a class that took a while back, and I can't quite figure out where one thing regarding some cosets spaces came from.

The setup is as follows:

Consider the group $ISO(m,n)$ of isometries of $\mathbb{R}^{m,n}$, i.e. maps $\varphi:\mathbb{R}^{m+n}\rightarrow\mathbb{R}^{m+n}$ that preserve $-x_1^2+\cdots-x_m^2+x_{m+1}^2+\cdots+x_{m+n}^2$.

The Lie algebra $\mathfrak{iso}(m,n)$ of this group can be written in terms of generators $J_{\mu\nu}$ and $K_{\mu}$ (where $0\leq\mu<\nu< m+n$) satisfying:

$$ [J_{\mu\nu},J_{\rho\sigma}]=i\hbar(\eta_{\mu\sigma}J_{\nu\rho}-\eta_{\mu\rho}J_{\nu\sigma}+\eta_{\nu\sigma}J_{\mu\rho}-\eta_{\nu\rho}J_{\mu\sigma}), $$

$$ [K_{\mu},J_{\nu\rho}]=i\hbar(\eta_{\mu\rho}K_{\nu}-\eta_{\mu\nu}K_{\rho}), \qquad [K_{\mu},K_{\nu}]=0. $$

Then the $J_{\mu\nu}$'s can be thought of as the infinitesimal generators for the subgroup $SO(m,n)<ISO(m,n)$. We write an arbitrary element of the Lie algebra as

$$X=-\frac{i}{\hbar}\left(\frac{1}{2}\omega^{\mu\nu}J_{\mu\nu}+c^{\mu}K_{\mu}\right).$$

(The factors of $\hbar$, etc., aren't terribly important here -- they are simply to get the units correct in whatever physical story one wants to tell using this language.)

Restricting to the case of $ISO(1,3)$ relevant to special relativity and such, the claim is that the coset space $M=ISO(1,3)/SO(1,3)$ is a pseudo-Euclidean space with metric $\eta_{\mu\nu}$ (i.e. Minkowski space), and moreover that this coset space can be written as a representation $\rho$ of the algebra $\mathfrak{iso}(1,3)$ in the following way:

$$ \rho(X)= \left( \begin{array}{cc} \omega^{\mu}_{\nu} & c^{\mu} \\ 0 & 0 \\ \end{array} \right), $$

(where $\omega^{\mu\nu}=-\omega^{\nu\mu}$) or in other words, $X$ represents an infinitesimal transformation on $M$ that does the following:

$$ \left( \begin{array}{c} dx^{\mu} \\ 0 \\ \end{array} \right)=\left( \begin{array}{cc} \omega^{\mu}_{\nu} & c^{\mu} \\ 0 & 0 \\ \end{array} \right)\left( \begin{array}{c} x^{\mu} \\ 1 \\ \end{array} \right). $$

I have absolutely no clue (and there is zero explanation in the notes as to) how one obtains these matrices. Any help to this end would be tremendously appreciated.

$\endgroup$

1 Answer 1

1
$\begingroup$

It turns out that this is fairly easy, so I will simply record the answer here in case it is useful for anyone else later.

All that is going on here is the natural action of the Lie algebra $\mathfrak{g}$ on itself (given by the Lie bracket) being projected onto the coset space. More specifically, writing a general element of $\mathfrak{iso}(1,3)$ as

$$ X=-\frac{i}{\hbar}\left(\frac{1}{2}\omega^{\mu\nu}J_{\mu\nu}+c^{\mu}K_{\mu}\right),$$

and an element of $\mathfrak{iso}(1,3)/\mathfrak{so}(1,3)$ as

$$ Y=-\frac{i}{\hbar}x^{\mu}K_{\mu}, $$

Then a quick computation yields

$$ [X,Y]\propto (\omega_{\mu}^{\nu}x^{\mu}+c^{\mu})K_{\mu} + (\text{some stuff})J_{\mu\nu}.$$

The matrices given in the question then amount to writing the parameter space of the coset space as $(x^{\mu},1)$ (where here the 1 just represents the $\mathfrak{so}(1,3)$ stuff that has been collapsed together in the quotient), and so we can write the above commutator in matrix form as

$$ [Y,X]=\left( \begin{array}{cc} \omega^{\nu}_{\mu} & c^{\mu} \\ 0 & 0 \\ \end{array} \right)\left( \begin{array}{c} x^{\mu} \\ 1 \\ \end{array} \right), $$

which, of course, gives us a representation of $\mathfrak{iso}(1,3)$ on the parameter space of $\mathfrak{iso}(1,3)/\mathfrak{so}(1,3)$ as

$$ \rho(X)=\left( \begin{array}{cc} \omega^{\nu}_{\mu} & c^{\mu} \\ 0 & 0 \\ \end{array} \right).$$

If anyone is ever reading this and needs some clarification, just let me know!

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .