Introduction to Quantum Symmetries

This chapter continues in the description of the mathematical structure of Quantum Mechanics, by introducing fundamental notions and tools of great relevance.

1. 1.

   There is a certain ambiguity in defining A _ψ and B _ψ , because the subscripts n of the possible real coefficients a _n can be chosen either in A _n or in B _n indifferently.

2. 2.

   The map (g, x) ↦ gx is customarily taken so that g′(gx) = (g′g)x and ex = x for every g, g′ ∈ G, x ∈ X, where e ∈ G is the neutral element.

3. 3.

   The word special, for matrix groups, indicates determinant equal 1, and is often denoted by putting an S before the group’s name.

4. 4.

   NB: some authors require the last condition in the definition of regular Borel measure.

5. 5.

   In fact h(e) = h(e · e) = h(e) ○ h(e), so applying h(e)^–1 we get e′ = h(e).

6. 6.

   The structure constants are the components of a tensor, called the structure tensor of the Lie group.

7. 7.

   Statements (a), (b) from theorem 2 in [BaRa86, Chapter 11.3]. Statement (c) follows from lemma 7 [BaRa86, Chapter 11.2] and from Preposition 9.21(c) of this book.

8. 8.

   Back when the author was an undergraduate, the procedure was impertinently known among students by the cheeky name of computation of “Flash Gordon coefficients”.

Authors and Affiliations

1. Department of Mathematics, University of Trento, Italy

   Valter Moretti

Reprints and permissions

© 2013 Springer-Verlag Italia

Policies and ethics