Quantum Mechanics for Mathematicians

Spin(4) elements that act by unitary (for the Hermitian inner product 31.3)
transformations on the spinor state space, but change| 0 〉and correspond to
a change in complex structure, are given by exponentiating the Lie algebra
representation operators

i(aF† 1 aF† 2 +aF 2 aF 1 ), aF† 1 aF† 2 −aF 2 aF 1

The possible choices of complex structure are parametrized bySO(4)/U(2),
which can be identified with the complex projective sphereCP^1 =S^2.
The construction in terms of matrices is well-suited to calculations, but
it is inherently dependent on a choice of coordinates. The fermionic version
of Bargmann-Fock is given here in terms of a choice of basis, but, like the
closely analogous bosonic construction, only actually depends on a choice of
inner product and a choice of compatible complex structureJ, producing a
representation on the coordinate-independent objectFd+= Λ∗VJ+.
In chapter 41 we will consider explicit matrix representations of the Clifford
algebra for the case ofSpin(3,1). The fermionic oscillator construction could
also be used, complexifying to get a representation of

so(4)⊗C=sl(2,C)⊕sl(2,C)

and then restricting to the subalgebra

so(3,1)⊂so(3,1)⊗C=so(4)⊗C

This will give a representation ofSpin(3,1) in terms of quadratic combinations
of Clifford algebra generators, but unlike the case ofSpin(4), it will not be
unitary. The lack of positivity for the inner product causes the same sort of
wrong-sign problems with the CAR that were found in the bosonic case for
the CCR whenJand Ω gave a non-positive symmetric bilinear form. In the
fermion case the wrong-sign problem does not stop one from constructing a
representation, but it will not be a unitary representation.

31.6 For further reading

For more about pseudo-classical mechanics and quantization, see [90] chapter

7. The Clifford algebra and fermionic quantization are discussed in chapter
   20.3 of [46]. The fermionic quantization map, Clifford algebras, and the spinor
   representation are discussed in detail in [59]. For another discussion of the
   spinor representation from a similar point of view to the one here, see chapter
   12 of [94]. Chapter 12 of [70] contains an extensive discussion of the role of
   different complex structures in the construction of the spinor representation.