8.3 Global Sym­metriza­tion

When com­put­ing, say a hy­dro­gen mol­e­cule, it is all nice and well to say that the wave func­tion must be an­ti­sym­met­ric with re­spect to ex­change of the two elec­trons 1 and 2, so the spin state of the mol­e­cule must be the sin­glet one. But what about, say, elec­tron 3 in fig­ure 8.1, which can with 50% chance be found on Mars and oth­er­wise on Venus? Should not the wave func­tion also be an­ti­sym­met­ric, for ex­am­ple, with re­spect to ex­change of this elec­tron 3 in one of two places in space with elec­tron 1 on the hy­dro­gen mol­e­cule on Earth? And would this not lo­cate elec­tron 3 in space also in part on the hy­dro­gen mol­e­cule, and elec­tron 1 also partly in space?

The an­swer is: ab­solutely. Na­ture treats all elec­trons as one big con­nected bunch. The given so­lu­tion for the hy­dro­gen mol­e­cule is not cor­rect; it should have in­cluded every elec­tron in the uni­verse, not just two of them. Every elec­tron in the uni­verse is just as much present on this sin­gle hy­dro­gen mol­e­cule as the as­sumed two.

From the dif­fi­culty in de­scrib­ing the 33 elec­trons of the ar­senic atom, imag­ine hav­ing to de­scribe all elec­trons in the uni­verse at the same time! If the uni­verse is truly flat, this num­ber would not even be fi­nite. For­tu­nately, it turns out that the ob­served quan­ti­ties can be cor­rectly pre­dicted pre­tend­ing there are only two elec­trons in­volved. An­ti­sym­metriza­tion with far-away elec­trons does not change the prop­er­ties of the lo­cal so­lu­tion.

If you are think­ing that more ad­vanced quan­tum the­o­ries will even­tu­ally do away with the pre­pos­ter­ous no­tion that all elec­trons are present every­where, do not be too con­fi­dent. As men­tioned in ad­den­dum {A.15.1}, the idea has be­come a fun­da­men­tal tenet in quan­tum field the­ory.