Transformations

Consider transformations of the state of a system to state that conserve probabilities .

Wigner’s Theorem

In QM, these transformations are represented by a linear and unitary operator with (also, by definition )
(Or an anti linear anti unitary operator, but we’ll not consider this possibility in PQM)

These transformations form a Group
is a homomorphism from to linear operators on ie for we write for the corresponding operator.
In mathematical terms we study the unitary representations of on .

We can equivalently transform operators instead of states:
Let with . After transform we have so we could have just sent whilst leaving the states unchanged.
This is called a similarity transformation
Similarity transformations preserve the spectrum
if then is an eigenvector of with eigenvalue .

Continuous transformations

Let depend smoothly on a parameter with
For we expand
is called the generator of the transformation ("" factor is convention)
is Hermetian

Generators can be observables!

From we can derive:

Translations

is a translation by vector
For infinitesimal translation :

and therefore
By definition, are generators of translations