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I am currently working through this paper on relativistic field theories in three dimensions. I have come to terms with the classification of Unitary irreducible representations (UIR's) given. The problem I am now trying to solve is the explicit realisation of the physical UIR's (corresponding to steps 4 and 5 in Binegar's program). I would be very happy with an answer/guidance for either of my questions, you need not answer both.

First of all, I have an issue with step 5 of his program:

"Finally, from each UIR $\mathcal{D}[~~]$ of each stability subgroup $S_{\hat{O}}$ we form the induced UIR of $\pi_+^{~~\uparrow}$ (connected component of the Poincare group) given by $$U_{D}^{~\hat{O}}(a,\Lambda)\psi(p)=\exp(ip_{\mu}a^{\mu})\mathcal{D}[\Omega^{-1}(p)\Omega_{\Lambda}\Omega(\Lambda^{-1}p)]\psi(p) \tag{1}$$ where $\Omega(p)\in \text{SL}(2,\mathbb{R})$ corresponds to a Lorentz transformation which takes the momentum $p$ (on mass shell with mass m) to the stability point $\hat{p}$ (both momentum 3-vectors), $\Omega_{\Lambda}\in\text{SL}(2,\mathbb{R})$ corresponds to the Lorentz transformation $\Lambda$ and $\Omega^{-1}(p)\Omega_{\Lambda}\Omega(\Lambda^{-1}p)\in\text{SL}(2,\mathbb{R})$ is to be interpreted as its ($\Lambda$'s) projection onto the stability subgroup $S_{\hat{O}}$."

Now shouldn't $\Omega(p)$ in $\Omega^{-1}(p)\Omega_{\Lambda}\Omega(\Lambda^{-1}p)\equiv W(\Lambda,p)$ correspond to a Lorentz transformation which takes $\hat{p}$ to $p$ (not the other way around)? Otherwise I do not see how $W(\Lambda,p)$ can be an element of the stability group of $\hat{p}$. Observe that if this were the case then $$\begin{align}W(\Lambda,p)\cdot \hat{p}&=\bigg(\Omega^{-1}(p)\Omega_{\Lambda}\Omega(\Lambda^{-1}p)\bigg)[\hat{p}^{\mu}\tau_{\mu}]\bigg(\Omega^{-1}(p)\Omega_{\Lambda}\Omega(\Lambda^{-1}p)\bigg)^T\\ &=\Omega^{-1}(p)\Omega_{\Lambda}\bigg(\Omega(\Lambda^{-1}p)[\hat{p}^{\mu}\tau_{\mu}]\Omega^T(\Lambda^{-1}p)\bigg)\Omega^T_{\Lambda}\Omega^{-1T}(p)\\ &=\Omega^{-1}(p)\bigg(\Omega_{\Lambda}[(\Lambda^{-1}p)^{\mu}\tau_{\mu}]\Omega^T_{\Lambda}\bigg)\Omega^{-1T}(p)\\ &=\Omega^{-1}(p)[p^{\mu}\tau_{\mu}]\Omega^{-1T}(p)\\ &=\hat{p}^{\mu}\tau_{\mu}\\ &=\text{Id}\cdot\hat{p} \end{align}$$ Where $p^{\mu}\tau_{\mu}$ is a symmetric 2x2 matrix given in the introduction and $\cdot$ denotes the action of $\text{SL}(2,\mathbb{R})$ on $\mathbb{R}^3$. It is unlikely that he wrote it wrong because he writes it in the same way multiple times throughout the paper (not to mention that this paper has ~ 350 citations). So how should I be interpreting $W(\Lambda,p)$?

Secondly, at the end of page 3/beginning of page 4 the author derives some properties for these Wigner rotations, $W(\Lambda,p)$, for some specific orbits, viz., the (physical) massive and massless cases. For example, he claims that in the massive case for an infinitesimal rotation $R(\theta)$ we have $$W(R(\theta),p)=\Omega^{-1}(p)\Omega_{R(\theta)}\Omega(R^{-1}(\theta)p)=R(\theta)$$ and that for an infinitesimal boost $L(\vec{\theta})$ in the $\vec{\theta}$ direction, $$W(L(\vec{\theta}),p)=\Omega^{-1}(p)\Omega_{L(\vec{\theta})}\Omega(L^{-1}(\vec{\theta})p)=R\bigg(\frac{\theta_1p_2-\theta_2p_1}{E+m}\bigg).$$ Given that the author does not supply a 'standard boost', or an $\text{SL}(2,\mathbb{R})$ matrix which corresponds to one (a.k.a an explicit $\Omega(p)$) for that matter, I would assume that there is a simple way to derive these relations. However I couldn't find such a simple method, and so I constructed a standard Lorentz transformation (SLT) which does the trick and was able to show the first and almost the second of these relations (after alot of cumbersome bookkeeping). The problem is that I think I should be using an $\text{SL}(2,\mathbb{R})$ standard boost instead of a Lorentz standard boost, but it has proved extremely computationally expensive (and ugly) to obtain the $\text{SL}(2,\mathbb{R})$ matrix to which my SLT corresponds (not to mention there is that pesky $\pm$ uncertainty due to the double valued-ness of the double cover $\text{SL}(2,\mathbb{R})$). Furthermore, when trying to prove a similar relation (given just after these) for the massless case I think it is imperative to use an $\text{SL}(2,\mathbb{R})$ boost rather than a Lorentz one, so I must conclude that my method is crap. Do you know of a better approach which I can take to proves these relations?

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Answer to questions:

  1. The way I interpreted it is correct; $\Omega (p)$ should be an $\text{SL}(2,\mathbb{R})$ matrix corresponding to a Lorentz transformation which takes the standard momentum $\hat{p}$ to $p$. In terms of $\text{SL}(2,\mathbb{R})$ transformations, this should be; $\Omega (p)\cdot\hat{p}=\Omega (p)\hat{p}^{\mu}\tau_{\mu}\Omega^T (p)=p^{\mu}\tau_{\mu}=\text{Id}\cdot p$.

  2. Indeed, one has to use $\text{SL}(2,\mathbb{R})$ transformations instead of Lorentz transformations since we are interested in projective (multi-valued) representations (one can go even farther and use the universal cover of the Lorentz group, which can be found here). The relations stated at the end of the OP for the massive case can be proven using the following 'standard $\text{SL}(2,\mathbb{R})$ boost':$$\Omega (p)=\frac{1}{\sqrt{2m^2+2mE(\mathbf{p})}} \begin{pmatrix}m+E(\mathbf{p})-p^1&&p^2\\p^2&&m+E(\mathbf{p})+p^1\end{pmatrix}$$$$\text{ where }E(\mathbf{p})=\sqrt{m^2+(p^1)^2+(p^2)^2}.$$ Both proof's are very tedious, the second more so than the first. I used the universal cover of the Lorentz group to prove both, but it must be possible to do it by just considering $\text{SL}(2,\mathbb{R})$ since it seems Binegar did it that way. I am still working on the proof for the massless case.

In short, don't take Binegar's notation too seriously because it is ambiguous, inconsistent and incorrect on multiple accounts throughout the paper.

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