### D.70 Emer­gence of spin from rel­a­tiv­ity

This note will give a (rel­a­tively) sim­ple de­riva­tion of the Dirac equa­tion to show how rel­a­tiv­ity nat­u­rally gives rise to spin. The equa­tion will be de­rived with­out ever men­tion­ing the word spin while do­ing it, just to prove it can be done. Only Dirac’s as­sump­tion that Ein­stein's square root dis­ap­pears,

will be used and a few other as­sump­tions that have noth­ing to do with spin.

The con­di­tions on the co­ef­fi­cient ma­tri­ces for the lin­ear com­bi­na­tion to equal the square root can be found by squar­ing both sides in the equa­tion above and then com­par­ing sides. They turn out to be:

 (D.45)

Now as­sume that the ma­tri­ces are Her­mit­ian, as ap­pro­pri­ate for mea­sur­able en­er­gies, and choose to de­scribe the wave func­tion vec­tor in terms of the eigen­vec­tors of ma­trix . Un­der those con­di­tions will be a di­ag­o­nal ma­trix, and its di­ag­o­nal el­e­ments must be for its square to be the unit ma­trix. So, choos­ing the or­der of the eigen­vec­tors suit­ably,

where the sizes of the pos­i­tive and neg­a­tive unit ma­tri­ces in are still un­de­cided; one of the two could in prin­ci­ple be of zero size.

How­ever, since must be zero for the three other Her­mit­ian ma­tri­ces, it is seen from mul­ti­ply­ing that out that they must be of the form

The ma­tri­ces, what­ever they are, must be square in size or the ma­tri­ces would be sin­gu­lar and could not square to one. This then im­plies that the pos­i­tive and neg­a­tive unit ma­tri­ces in must be the same size.

Now try to sat­isfy the re­main­ing con­di­tions on , , and us­ing just com­plex num­bers, rather than ma­tri­ces, for the . By mul­ti­ply­ing out the con­di­tions (D.45), you see that

The first con­di­tion above would re­quire each to be a num­ber of mag­ni­tude one, in other words, a num­ber that can be writ­ten as for some real an­gle . The sec­ond con­di­tion is then ac­cord­ing to the Euler for­mula (2.5) equiv­a­lent to the re­quire­ment that

this im­plies that all three an­gles would have to be 90 de­grees apart. That is im­pos­si­ble: if and are each 90 de­grees apart from , then and are ei­ther the same or apart by 180 de­grees; not by 90 de­grees.

It fol­lows that the com­po­nents can­not be num­bers, and must be ma­tri­ces too. As­sume, rea­son­ably, that they cor­re­spond to some mea­sur­able quan­tity and are Her­mit­ian. In that case the con­di­tions above on the are the same as those on the , with one crit­i­cal dif­fer­ence: there are only three ma­tri­ces, not four. And so the analy­sis re­peats.

Choose to de­scribe the wave func­tion in terms of the eigen­vec­tors of the ma­trix; this does not con­flict with the ear­lier choice since all half wave func­tion vec­tors are eigen­vec­tors of the pos­i­tive and neg­a­tive unit ma­tri­ces in . So you have

and the other two ma­tri­ces must then be of the form

But now the com­po­nents and can in­deed be just com­plex num­bers, since there are only two, and two an­gles can be apart by 90 de­grees. You can take and then or . The ex­is­tence of two pos­si­bil­i­ties for im­plies that on the wave func­tion level, na­ture is not mir­ror sym­met­ric; mo­men­tum in the pos­i­tive -​di­rec­tion in­ter­acts dif­fer­ently with the and mo­menta than in the op­po­site di­rec­tion. Since the ob­serv­able ef­fects are mir­ror sym­met­ric, do not worry about it and just take the first pos­si­bil­ity.

So, the goal of find­ing a for­mu­la­tion in which Ein­stein's square root falls apart has been achieved. How­ever, you can clean up some more, by re­defin­ing the value of away. If the four-di­men­sion­al wave func­tion vec­tor takes the form , de­fine , and sim­i­lar for and .

In that case, the fi­nal cleaned-up ma­tri­ces are

 (D.46)

The s word has not been men­tioned even once in this de­riva­tion. So, now please ex­press au­di­ble sur­prise that the ma­tri­ces turn out to be the Pauli (it can now be said) spin ma­tri­ces of chap­ter 12.10.

But there is more. Sup­pose you de­fine a new co­or­di­nate sys­tem ro­tated 90 de­grees around the -​axis. This turns the old -​axis into a new -​axis. Since has an ad­di­tional fac­tor , to get the nor­mal­ized co­ef­fi­cients, you must in­clude an ad­di­tional fac­tor in , which by the fun­da­men­tal de­f­i­n­i­tion of an­gu­lar mo­men­tum dis­cussed in ad­den­dum {A.19} means that it de­scribes a state with an­gu­lar mo­men­tum . Sim­i­larly cor­re­sponds to a state with an­gu­lar mo­men­tum and and to ones with .

For nonzero mo­men­tum, the rel­a­tivis­tic evo­lu­tion of spin and mo­men­tum be­comes cou­pled. But still, if you look at the eigen­states of pos­i­tive en­ergy, they take the form:

where is a small num­ber in the non­rel­a­tivis­tic limit and is the two-com­po­nent vec­tor . The op­er­a­tor cor­re­spond­ing to ro­ta­tion of the co­or­di­nate sys­tem around the mo­men­tum vec­tor com­mutes with , hence the en­tire four-di­men­sion­al vec­tor trans­forms as a com­bi­na­tion of a spin state and a spin state for ro­ta­tion around the mo­men­tum vec­tor.