corresponding to the action of an even or odd number of creation operators on
| 0 〉F. This is because quadratic combinations of theaFj,aF†jpreserve the parity
of the number of creation operators used to get an element ofSby action on
| 0 〉F.
31.4 Complex structures,U(d)⊂SO(2d) and the spinor representation
The construction of the spinor representation given here has used a specific
choice of theθj,θj(see equations 31.2) and the fermionic annihilation and cre-
ation operators. This corresponds to a standard choice of complex structure
J 0 , which appears in a manner closely parallel to that of the Bargmann-Fock
case of section 26.1. The difference here is that, for the analogous construction
of spinors, the complex structureJmust be chosen so as to preserve not an
antisymmetric bilinear form Ω, but the inner product, and one has
(J(·),J(·)) = (·,·)We will here restrict to the case of (·,·) positive definite, and unlike in the
bosonic case, no additional positivity condition onJwill then be required.
Jsplits the complexification of the real dual phase spaceV∗=V=R^2 dwith
its coordinatesξjinto addimensional complex vector space on whichJ= +i
and a conjugate complex vector space on whichJ=−i. As in the bosonic case
one has
V ⊗C=VJ+⊕VJ−
and quantization of vectors inVJ+ gives linear combinations of creation op-
erators, while vectors inVJ−are taken to linear combinations of annihilation
operators. The choice ofJis reflected in the existence of a distinguished di-
rection| 0 〉Fin the spinor spaceS=Fd+which is determined (up to phase) by
the condition that it is annihilated by all linear combinations of annihilation
operators.
The choice ofJalso picks out a subgroupU(d)⊂SO(2d) of those orthogonal
transformations that commute withJ. Just as in the bosonic case, two different
representations of the Lie algebrau(d) ofU(d) are used:
- The restriction tou(d)⊂so(2d) of the spinor representation described
above. This exponentiates to give a representation not ofU(d), but of a
double cover ofU(d) that is a subgroup ofSpin(2d). - By normal ordering operators, one shifts the spinor representation ofu(d)
by a constant and gets a representation that exponentiates to a true rep-
resentation ofU(d). This representation is reducible, with irreducible
components the Λk(Cd) fork= 0, 1 ,...,d.
In both cases the representation ofu(d) is constructed using quadratic combina-
tions of annihilation and creation operators involving one annihilation operator