I know from theory that the adjoint representation $\tau_a$ of group $\mathcal G$ is faithful iff $\mathcal G$ is centerless. From basic theory it turns out that parametrizing a rotation in $\mathbb R^3$ with direction $\hat{n}$ and angle $\alpha$ ($\boldsymbol{\alpha}=\alpha\hat{n}$) the adjoint representation is not faithful because if $\boldsymbol{\alpha}'=-\hat{n}\frac{2\pi-\alpha}{\alpha}$ we have $\tau_a(\boldsymbol{\alpha}')=\tau_a(\boldsymbol{\alpha})$. On the other hand we can set a correspondence with the elements of the fundamental representation of SU(2) (in which we choose a parametrization consistent with SO(3)'s one) $\tau_f$ and we find that $\tau_f$ is faithful. On the lecture notes i'm reading of QFT (http://www.robertosoldati.com/archivio/news/107/Campi1.pdf) on page 31 it says that with euler angles the situation is reversed with $\tau_a$ faithful and $\tau_f$ not. If a representation (for a Lie group) is a map from the group manifold $\mathcal G$ to $Aut(V)$, with V linear space, how should a property of the representation depend on the parametrization of the manifold? I'm a bit confused, am I missing something that this appears to me so strange or i have just misunderstood something?


There is some confusion about what these notes claim. It is better stated that there are two distinct groups,$ \mathrm{SU}(2)$ and $\mathrm{SO}(3)$, two representation spaces, and two possible points of view.

The first group ($\mathrm{SU}(2)$) acts faithfully on $\mathbb{C}^2$ which is called the spin or fundamental representation (this is $\tau_f$), while the second group ($\mathrm{SO}(3)$) acts faithfully by rotations on $\mathbb{R}^3$ called the adjoint representation (this is $\tau_a$).

There is a surjective $2$-to-$1$ homomorphism $\mathrm{SU}(2)\to \mathrm{SO}(3)$, which allows us to consider the action by rotations of $\mathrm{SU}(2)$ on the adjoint representation. This cannot be faithful, since it acts though a non-injective homomorphism. Regardless, working with the group $\mathrm{SU}(2)$ allows us to think about either of these representations at the same time, with the small caveat that it does not act faithfully on the adjoint representation $\tau_a$.

If we were uncomfortable with the physical meaning of $\mathrm{SU}(2)$ as a group, and wished to understand spin in terms of rotations, then we could try to phrase the fundamental representation $\mathbb{C}^2$ in terms of elements of $\mathrm{SO}(3)$. This is what the author wishes to do, with Euler angles a nice way of expressing elements of the rotation group.

There is a problem with this: there is no action of $\mathrm{SO}(3)$ on $\mathbb{C}^2$! That is, there is no map $\tau_f:\mathrm{SO}(3)\to Aut(\mathbb{C}^2)$ giving an analogue of the fundamental representation. All we can do instead is write down a multi-valued function $\mathrm{SO}(3)\to Aut(\mathbb{C}^2)$, and this is what is described on page 31. This is not a true representation, but a multivalued function.

So there isn't a contradiction, just two different points of view depending on whether you wish to work with $\mathrm{SU}(2)$ or $\mathrm{SO}(3)$.

  • $\begingroup$ I couldn't have said it better! $\endgroup$ – Vincent Apr 8 at 19:37

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