and
Spin(2,2) =SL(2,R)×SL(2,R)
correspond to different so-called “real forms” of a fact about complex groups
that one can get by complexifying any of the examples (considering elements
(x 0 ,x 1 ,x 2 ,x 3 )∈C^4 , not just inR^4 ). For instance, in theSpin(4)case, taking
thex 0 ,x 1 ,x 2 ,x 3 in the matrix
(
x 0 −ix 3 −x 2 −ix 1
x 2 −ix 1 x 0 +ix 3
)
to have arbitrary complex valuesz 0 ,z 1 ,z 2 ,z 3 one gets arbitrary 2 by 2 complex
matrices, and the transformation
(
z 0 −iz 3 −z 2 −iz 1
z 2 −iz 1 z 0 +iz 3)
→Ω 1
(
z 0 −iz 3 −z 2 −iz 1
z 2 −iz 1 z 0 +iz 3)
Ω 2
preserves this space as well as the determinant (z 02 +z 12 +z^22 +z^23 ) forΩ 1 andΩ 2
not just inSU(2), but in the larger groupSL(2,C). So we find that the group
SO(4,C)of complex orthogonal transformations ofC^4 has spin double cover
Spin(4,C) =SL(2,C)×SL(2,C)Sincespin(4,C) =so(3,1)⊗C, this relation between complex Lie groups corre-
sponds to the Lie algebra relation
so(3,1)⊗C=sl(2,C)⊕sl(2,C)we found explicitly earlier when we showed that by taking complex coefficients
of generatorsljandkj ofso(3,1)we could find generatorsAjandBjof two
differentsl(2,C)sub-algebras.
40.5 For further reading
Those not familiar with special relativity should consult a textbook on the
subject for the physics background necessary to appreciate the significance of
Minkowski space and its Lorentz group of invariances. An example of a suitable
such book aimed at mathematics students is Woodhouse’sSpecial Relativity
[104].
Most quantum field theory textbooks have some sort of discussion of the
Lorentz group and its Lie algebra, although the issue of how complexification
works in this case is routinely ignored (recall the comments in section 5.5).
Typical examples are Peskin-Schroeder [67], see the beginning of their chapter
3, or chapter II.3 [105] of Zee.