a double cover of , meaning that there is a two-to-one group homomorphism from to , see relationship between SO(3) and SU(2). Quaternions The real linear span of is isomorphic to the real algebra of quaternions, , represented by the span of the basis vectors The isomorphism from to this set is given by the following map (notice the reversed signs for the Pauli matrices): Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order, As the set of versors forms a group isomorphic to , gives yet another way of describing . The two-to-one homomorphism from to may be given in terms of the Pauli matrices in this formulation. Physics Classical mechanics In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein parameters. The matrix corresponding to the position of a point in space is defined in terms of the above Pauli vector matrix, Consequently, the transformation matrix for rotations about the -axis through an angle may be written in terms of Pauli matrices and the unit matrix as Similar expressions follow for general Pauli vector rotations as detailed above. Quantum mechanics In quantum mechanics, each Pauli matrix is related to an angular momentum operator that corresponds to an observable describing the spin of a spin particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin . The states of the particles are represented as two-component spinors. In the same way, the Pauli matrices are related to the isospin operator. An interesting property of spin particles is that they must be rotated by an angle of 4 in order to return to their original configuration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north/south pole on the 2-sphere they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space. For a spin particle, the spin operator is given by , the fundamental representation of SU(2). By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated