How do i navigate in Hilbert space?
How do i find de wey?
Ill ellaborate with a simple example: Consider the set of vectors that point to the surface of the common sphere of radius 1. I can take any vector and rotate it into any other position by making 2 rotations, on the polar and the azimuthal angle. I can also reach any point by making a rotation about a specific axis trace some convoluted path. But in general i need two rotations on "standard" axes by some angle. Its equivalent to give the latitud and longitud. These transformations for a group the "rotation group" because two rotations are also a rotation.
In Hilbert space theres transformations from one ket to another via a type of rotation called unitary rotations. They also form a group because two "unitary transformations" acting on a ket are also a unitary transformation.
What i want to know is which group is defined by these transformations. Which unitary transformations form the basis of this group, in the same sense that rotations of the azimuthal and polar angles are what creates the rotation group.
To know this group is essentially the same as defining any ket in Hilbert space, the information of Hilbert space must be encoded in this group, just like the concept of a sphere is encoded in the definitions of the rotation group.