7.1 Free Scalar
157
7.1.6 Hilbert Space
The Hilbert space of the free scalar has the structure of a Fock space.
From (7.76), the momentum p commutes with the Hamiltonian L
+
0 such that
it is a good quantum number to label the states: 7 this translates the fact that the
action (7.1) does not depend on the conjugate variable x. As a consequence, there
exists a family of vacua |k.
The vacua |k are the states related to the vertex operators (7.33) through the
state–operator correspondence:
|k := lim
z,¯ z→0
V k (z, ¯
z) |0 = e
ikx
|0 ,
(7.78)
where |0 is the SL(2, C) vacuum and x is the zero-mode of X(z, ¯
z). That this
identification is correct follows by applying the operator p:
p |k = k |k .
(7.79)
The notation is consistent with the one of the SL(2, C) vacuum since p |0 = 0.
The vacuum is annihilated by the action of the positive-frequency modes:
∀n > 0 :
α n |k = 0,
(7.80)
which is equivalent to
N n |k = 0.
(7.81)
The different vacua are each ground state of a Fock space (they are all equivalent),
but they are not ground states of the Hamiltonian since they have different energies:
L
+
0 |k = 2
2 k
2
|k ,
L
−
0 |k = 0,
(7.82)
using (7.70). The SL(2, C) vacuum is the lowest (highest) energy state if = 1
( = −1).
The Fock space F(k) built from the vacuum at momentum k is found by acting
repetitively with the negative-frequency modes. A convenient basis, the oscillator
basis, is given by the states:
F(k) = Span
|k; {N n }
,
(7.83a)
|k; {N n } :=
n≥1
(α −n ) N n
n N n N n !
|k ,
N n ∈ N
∗
(7.83b)
7 To simplify the discussion, we do not consider winding but only vertex operators of the
form (7.33).