1.1.11 The Commutator

An es­sen­tial math­e­mat­i­cal re­la­tion­ship be­tween matrix groups and their el­e­ments is whether they com­mute. Ma­tri­ces that com­mute are called abelian, which de­fines that the com­mu­ta­tor op­er­a­tion be­tween by two ma­tri­ces $A,\ B$ will equal zero

Non-com­mu­ta­tive ma­tri­ces are re­ferred to as non-abelian, where the prod­uct of their com­mu­ta­tor op­er­a­tion can take on var­i­ous values. An ex­am­ple might be

In higher-di­men­sion­al matrix groups, we will en­counter many non-com­mu­ta­tive re­la­tion­ships be­tween the el­e­ment ma­tri­ces of these groups. We will use the fol­low­ing Lie brack­et syntax to define the com­mu­ta­tor op­er­a­tion, where the ex­am­ple above, is de­not­ed

The or­der­ing of the brack­et­ed ma­tri­ces is im­por­tant, since $\lbrack B,A\rbrack$ may not be equal to $\lbrack A,B\rbrack$!

Ma­tri­ces are typ­i­cal­ly de­fined in math­e­mat­ics as a group­ing of linear op­er­a­tions. But for our pur­pos­es, it is best to view the ones we dis­cuss as multi-di­men­sion­al num­bers. These multi-di­men­sion­al num­bers do not nec­es­sar­i­ly com­mute. This is an im­por­tant con­cept to re­mem­ber, as non-com­mu­ta­tive/com­mu­ta­tive op­er­a­tions play a cen­tral role in quan­tum me­chan­ics, the un­cer­tain­ty prin­ci­ple, matter, and forces. We will con­tin­ue to con­sid­er per­mu­ta­tion ma­tri­ces as the al­ge­bra of a group, which de­fines the sym­me­try trans­for­ma­tions pos­si­ble for an object in a par­tic­u­lar space. By matrix ex­po­nen­ti­a­tion, we can take a set number of per­mu­ta­tions and make them con­tin­u­ous by the process of in­fin­i­tes­i­mal trans­for­ma­tions.