The universe and the flowers

Peeking the infinite increases the space, the breath, the brain of whoever is watching it.
Erri De Luca, italian writer

The universe is a perilous but also a beautiful place. But the beauty of the universe it's not only in galactic shots, but also in mathematics. For example the maps of the E8 group seems flowers, and if we follow Garrett Lisi and his preprint An exceptionally simple theory of everything⁽¹⁾, these maps are also a sort of universe's flowers!
The E8 maps in this post are extracted from Lisi's preprint and they represent the structure of the group (the first image is F4. E8 is a Lie group, most important in physics because all symmetries group of the physical systems are Lie groups. The Lie group is an analytical group: all functions that we can define in the group are continuous. A physical example is the Galilei's group, and studying it we can argue information about free particles, described by Schroedinger equation.

Ray representations and Galilei group

For my PhD thesis I performed a work in group teory, precisely in the theory of representations, applied to quantum mechanics. So, in order to describe my work, recently published by the Journal of Mathematical Physics, I need to introduce some concepts. The group theory was founded indipendetly by Niels Abel and Evariste Galois and it is focused on group, a set $G$ of elements with a multiplication operation $\cdot$ and such that the following properties are true⁽⁹⁾:

1. $\forall a, b \in G, a \cdot b \in G$
2. $\forall a, b, c \in G, a(bc) = (ab) c$
3. $\forall a \in G \, \exists e \in G \, \text{:} \, ae = ea = a$
4. $\forall a \in G \, \exists b \in G \, \text{:} \, ab = ba = e$ and $b = a^{-1}$

A group is called abelian group if

1. $\forall a, b \in G, ab = ba$

For every group, we can write a representation, that is a set of operators or functions that act in a mathematical space and change it in a same way at the elements of the group change their own space⁽¹⁾. In other (trivial) words we have a given world with a group (for example our real world with a group of symmetry, for example the rotations), and in order to find properties of this world we must use a mathematical representation.
If the world is a given physical system (for example a free particle), we have a symmetry group, that is a set of all symmetry transformation⁽²⁾ of our physical system, and his representation acts in a so called Hilbert space. In this space, following Wigner's theorem⁽⁴⁾, the most general representation is a ray (unitary) representation. In order to understand the ray (or projective) representations, we must enunciate the theorem:

For every transformation of symmetry $T: \mathcal R \rightarrow \mathcal R$ between the rays of a Hilbert space $\mathcal H$ and such that conserve the transition probabilities, we can define an operator $U$ on the Hilbert space $\mathcal H$ such that, if $|\psi> \in {\mathcal R}_\psi$, then $U |\psi> \in {\mathcal R}'_\psi$, where ${\mathcal R}_\psi$ is the radius of the state $|\psi>$, ${\mathcal R}'_\psi = T {\mathcal R}_\psi$, and $U$ uniform and linear \[< U \psi | U \varphi> = <\psi | \varphi>, \qquad U |\alpha \psi + \beta \varphi> = \alpha U |\psi> + \beta U |\varphi>\] or with $U$ antiunitario and antilinear: \[< U \psi | U \varphi> = <\varphi | \psi>, \qquad U |\alpha \psi + \beta \varphi> = \alpha^* U |\psi> + \beta^* U |\varphi>\] Further, $U$ is uniquely determined except for a phase factor.

So a ray representation is the association between an element of the symmetry group $G$ to a set of unitary (or antiunitary) operators which differ only for a phase: in other worlds a ray of operators⁽³⁾.