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The $10$ generators of the Poincare group $P(1;3)$ are $M^{\mu\nu}$ and $P^\mu$. These generators can be determined explicitly in the matrix form. However, I have found that $M^{\mu\nu}$ and $P^\mu$ are often written in terms of position $x^\mu$ and momentum $p^\mu$ as

$$ M^{\mu\nu} = x^\mu p^\nu - x^\nu p^\mu $$ and $$ P^\mu = p^\mu$$

How do we get these relations?

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Related: physics.stackexchange.com/q/12559/2451 and links therein. –  Qmechanic Jul 15 at 19:17

2 Answers 2

up vote 4 down vote accepted

One obtains those expressions by considering a particular action of the Poincare group on fields.

Consider, for example, a single real scalar field $\phi:\mathbb R^{3,1}\to\mathbb R$. Let $\mathcal F$ denote the space of such fields. Define an action $\rho_\mathcal F$ of $P(3,1)$ acting on $\mathcal F$ as follows \begin{align} \rho_\mathcal F(\Lambda,a)(\phi)(x) = \phi(\Lambda^{-1} (x-a)) \end{align} Sometimes people will write this as $\phi'(x) = \phi(\Lambda^{-1} x)$ for brevity. Now let $G$ denote a generator of the Lie algebra of the Poincare group (namely an element of a chosen basis for this Lie algebra). We can use this generator to define a corresponding infinitesimal generator for group action $\rho_\mathcal F$ as follows: \begin{align} G_\mathcal F(\phi) = i\frac{\partial}{\partial\epsilon}\rho_\mathcal F(e^{-i\epsilon G})(\phi)\bigg|_{\epsilon = 0} \end{align}

Example - translations. Consider the translation generators $P^\mu$ which have the property \begin{align} e^{-ia_\mu P^\mu}x = x+a \end{align} The generator of $\rho_\mathcal F$ corresponding to $P^0$, for instance, is \begin{align} (P^0)_\mathcal F(\phi)(x) &= i\frac{\partial}{\partial\epsilon}\rho_\mathcal F(e^{-i\epsilon P^0})(\phi)\bigg|_{\epsilon = 0} \\ &= i\frac{\partial}{\partial\epsilon}\phi(x + \epsilon e_0)\bigg|_{\epsilon = 0} \\ &= i\partial^0\phi(x) \end{align} where $e_0 = (1,0,0,0)$, and similarly for the other $P^\mu$, which gives \begin{align} (P^\mu)_\mathcal F = i\partial^\mu. \end{align}

Example - Lorentz boosts.

If you use this same procedure for Lorentz boost generators, you will find that \begin{align} (M^{\mu\nu})_\mathcal F = i(x^\mu\partial^\nu - x^\nu\partial^\mu) = x^\mu p^\nu - x^\nu p^\mu \end{align}

Disclaimer about signs etc. There are a lot of conventional factors of $i$ and negative signs floating around which I wasn't super careful to keep track of, if you notice an error in this regard, please let me know and I'll fix it.

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Can you please tell me what the commutation relation is: $[x^\mu, P^\nu] = ?$ –  Ome Jul 15 at 20:52
    
@Ome Since that's a distinct question and would be useful for other users, you should ask it in another post! –  joshphysics Jul 15 at 20:54

The generators of isometry, also generators of the Poincare group, are the Killing vectors, hence we need the Killing vectors of Minkowski spacetime, $ds^2 = -dt^2 + dx^2 + dy^2+ dz^2$. The defining equation of the Killing vectors in terms of their components ($\xi = \xi^\mu \partial_\mu$ is a Killing vector with components $\xi^\mu$) is $$\partial_{(\mu} \xi_{\nu)}=0,$$ where the parentheses denote symmetrization over the enclosed indices. We differentiate once, then cyclically permute the indices, $$\begin{split}\partial_c\partial_a \xi_b + \partial_b\partial_c \xi_a &=0,\\ \partial_a\partial_b\xi_c + \partial_c\partial_a\xi_b&= 0,\\ \partial_b\partial_c\xi_a + \partial_a\partial_b\xi_c &=0,\end{split}$$ which is a linear system with unknowns $\partial_a\partial_b\xi_c$ and its permutations. The only solution of the system is the trivial $\partial_a\partial_b\xi_c=0$, from which we obtain $\xi_a = a_{ab} x^b + b_a$ with $a_{ab}$ and $b_a$ constants. From the defining equation we obtain $a_{ab} = -a_{ba}$ and, since the Killing vectors are unique up to multiplication with a constant and translation, we obtain $$ \begin{aligned} \partial_{t} && \partial_{x} && \partial_{y} && \partial_{z},\\ -x\partial_{t} - t\partial_{x}, && -y\partial_{t} - t \partial_{y} && - z\partial_{y} - t\partial_{z},\\ x\partial_{y} - y\partial_{x} && y\partial_{z} - z\partial_{x} && z\partial_{x} - x \partial_{z}.&& \end{aligned}$$ On the first line are the translation generators, on the second the boost generators and on the third the rotation generators.

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