be constructed by multiplying these for different angles in different coordinate
planes. The Lie algebraspin(n) can be identified with the Lie algebraso(n) by
jk↔−1
2
γjγkYet another way to see this would be to compute the commutators of the−^12 γjγk
for different values ofj,kand show that they satisfy the same commutation
relations as the corresponding matricesjk.
Recall that in the bosonic case we found that quadratic combinations of the
Qj,Pk(or of theaBj,aB†j) gave operators satisfying the commutation relations
of the Lie algebrasp(2n,R). This is the Lie algebra of the groupSp(2n,R),
the group preserving the non-degenerate antisymmetric bilinear form Ω(·,·) on
the phase spaceR^2 n. The fermionic case is precisely analogous, with the role of
the antisymmetric bilinear form Ω(·,·) replaced by the symmetric bilinear form
(·,·) and the Lie algebrasp(2n,R) replaced byso(n) =spin(n).
In the bosonic case the linear functions of theQj,Pjsatisfied the commuta-
tion relations of another Lie algebra, the Heisenberg algebra, but in the fermionic
case this is not true for theγj. In chapter 30 we will see that a notion of a “Lie
superalgebra” can be defined that restores the parallelism.
29.3 For further reading
Some more detail about spin groups and the relationship between geometry and
Clifford algebras can be found in [55], and an exhaustive reference is [68].