This chapter is devoted to providing a physical motivation for group cohomology and group extensions by analysing some specific features of classical and quantum mechanics of non-relativistic particles. In particular the need of considering projective representations of the Galilei group and Bargmann's superselection rule are discussed. The projective representations lead to the introduction of two-cocycles; the three-cocycle appears as a result of a consistent breach of associativity. The consistency conditions that have to be fulfilled in both cases are analysed, and illustrated in the case of two-cocycles with the examples of the Weyl-Heisenberg group and the Galilei group. All these concepts will be mathematically defined in chapters 4, 5, and 6, and addressed again in a more elaborate way from the point of view of classical physics in chapter 8.

The relation between cohomology and quantization and in particular the topics of symplectic cohomology, dynamical groups and geometric quantization are briefly treated in this chapter. Again, the Galilei and the Weyl-Heisenberg groups will be used as illustrations of the theory.

Finally, it is shown how the group contraction procedure can generate non-trivial group cohomology.

Some known facts of ‘non-relativistic’ mechanics: two-cocycles

A theory is called ‘non-relativistic’ if its formulation is covariant under the Galilei group G, which is the relativity group of a ‘non-relativistic’ theory. It should be noticed, however, that theories covariant under the Poincare group and the Galilei group are both, strictly speaking, equally relativistic; only their relativity group is different.