176
7 CFT Systems
Structure of the Hilbert Space (Grassmann Odd)
Since the zero-modes commute with the Hamiltonian and with all other negativeand positive-frequency modes, the Hilbert space is decomposed in several subspaces, each associated to a zero-mode. 11
Starting with the holomorphic sector only, the Hilbert space H gh is
H gh = H gh,0 ⊕ c 0 H gh,0 ,
H gh,0 := H gh ∩ ker b 0 ,
(7.169)
which follows from the 2-state algebra (7.148). Obviously, one has c 0 H gh,0 = H gh ∩
ker c 0 . The oscillator basis of the Hilbert space H gh,0 is generated by applying the
negative-frequency modes and has the structure of a fermionic Fock space without
zero-modes:
H gh,0 = Span
↓; {N
b
n }; {N
c
n }
,
(7.170a)
↓; {N
b
n }; {N
c
n }
=
n≥1
(b −n )
N b
n (c −n )
N c
n | ↓↓ ,
N
b
n , N
c
n ∈ N
∗
(7.170b)
(again, number operators and their eigenvalues are not distinguished). This means
that H gh,0 can also be regarded as a Fock space built on the vacuum | ↓↓, for which
c 0 and b 0 are, respectively, creation and annihilation operators. Conversely, c 0 and
b 0 are, respectively, annihilation and creation operators for c 0 H gh,0 .
In particular, this means that any state can be written as the sum of two states
ψ = ψ ↓ + ψ ↑ ,
ψ ↓ ∈ H gh,0 ,
ψ ↑ ∈ c 0 H gh,0 ,
(7.171)
with ψ ↓ and ψ ↑ built, respectively, on top of the | ↓↓ and | ↑↑ vacua.
This pattern generalizes when considering both the holomorphic and antiholomorphic sectors. In that case, the Hilbert space is decomposed in four subspaces: 12
H gh = H gh,0 ⊕ c 0 H gh,0 ⊕ ¯
c 0 H gh,0 ⊕ c 0 ¯
c 0 H gh,0 ,
H gh,0 := H gh ∩ ker b 0 ∩ ker ¯
b 0 .
(7.172)
11 Due to the specific structure of the inner product defined below, these subspaces are not
orthonormal to each other.
12 The reader should not get confused by the same symbol H gh,0 as in the case of the holomorphic
sector.