====== Cocycles ======

<tabbox Why is it interesting?> 

Every particle transforms under spatial rotations according to a [[advanced_tools:group_theory:representation_theory:projective_representation|projective representation]] of the rotation group $SO(3)$. Cocycles appear in the definition of a projective representation.


<tabbox Layman> 

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
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<tabbox Student> 

<blockquote>
Because state vectors differing only by a phase are the same for certain purposes, it turns out that symmetries need not correspond to unitary representations; they can correspond to [[advanced_tools:group_theory:representation_theory:projective_representation|projective representations,]] in which the rules of a representation hold only 'up to phases': 

$$ \rho(1) = e^{i\theta}, $$
$$ \rho(g)\rho(h)= e^{i\theta(g,h)} \rho(gh) .$$

Here $\theta$ is a fixed real number, while the **cocycle** $e^{i\theta(g,h)}$ is any function of $g$ and $h$. [...] Now for the same reason, one can change a projective representation $\rho$ to another $\rho'$ by throwing in an extra phase without changing the physics:

$$ \rho'(g)= e^{i\varphi(g)} \rho(g).$$
[...]
If nochoice of $\varphi$ makes $\theta'(g,h)=0$ for all $g,h$, we say the cocycle $e^{i\theta(g,h)}$ is **essential**. What this means is that it is impossible to 'straighten out' the projective representation $\rho$ into an actual representation.

<cite>page 179 in Gauge fields, knots, and gravity by John Baez</cite>
</blockquote>
 
<tabbox Researcher> 

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

  
<tabbox Examples> 

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<tabbox FAQ> 
  
<tabbox History> 

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