English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We represent vector rotation operators in terms of bras or kets of half-angleexponentials in Clifford (geometric) algebra Cl_{3,0}. We show that SO_3 is arotation group and we define the dihedral group D_4 as its finite subgroup. Weuse the Euler-Rodrigues formulas to compute the multiplication table of D_4 andderive its group algebra identities. We take the linear combination of rotationoperators in D_4 to represent the four Fermion matrices in Sakurai, which inturn we use to decompose any 2x2 matrix. We show that bra and ket operatorsgenerate left- and right-acting matrices, respectively. We also show that thePauli spin matrices are not vectors but vector rotation operators, except for\sigma_2 which requires a subsequent multiplication by the imaginary number igeometrically interpreted as the unit oriented volume.

A geometric algebra reformulation of 2x2 matrices: the dihedral group D_4 in bra-ket notation