7.3: Internal and Space-Time Symmetries

Above I have mentioned angular momentum, the vector product of position and momentum. This is defined in terms of properties of space (or to be more generous, of space-time). But we know that many particles carry the spin of the particle to form the total angular momentum, \[{\vec{J}} = {\vec{L}} + {\vec{S}}.\] The invariance of the dynamics is such that \({\vec{J}}\) is the conserved quantity, which means that we should not just rotate in ordinary space, but in the abstract “intrinsic space” where \({\vec{S}}\) is defined. This is something that will occur several times again, where a symmetry has a combination of a space-time and intrinsic part.