6.4 Operator Formalism and Radial Quantization
131
For this to make sense, the modes which diverge as z → 0 must annihilate the
vacuum. In particular, for a weight h field φ(z), one finds
∀n ≥ −h + 1 : φ n |0 = 0.
(6.125)
Thus, the φ n for n ≥ −h + 1 are annihilation operators for the vacuum |0, and
conversely the states φ n with n < −h + 1 are creation operators. As a consequence,
the state |φ is found by applying the mode n = −h to the vacuum:
|φ = φ −h |0 =
dz
2π i
φ(z)
z
|0.
(6.126)
Since L −1 is the generator of translations on the plane, one finds
φ(z)|0 = e
zL −1 φ(0)e
−zL −1 |0 = e
zL −1 |φ.
(6.127)
The vacuum |0 is the state associated with the identity 1. Translating the conditions
(6.125) to the energy–momentum tensor gives
∀n ≥ −1 : L n |0 = 0.
(6.128)
This is consistent with the definition (6.121) since it includes the sl(2, C) subalgebra.
If h < 0, some of the modes with n > 0 do not annihilate the vacuum: (6.117)
implies that some states have an energy lower than the one of |0. The state |
(possibly degenerate) with the lowest energy is called the energy vacuum
∀|φ ∈ H :
:|L 0 | ≤≤φ|L 0 |φ.
(6.129)
It is obtained by acting repetitively with the modes φ n>0 . This vacuum defines a new
partition of the non-zero-modes operators into annihilation and creation operators.
If there are zero-modes, i.e. n = 0 modes, then the vacuum is degenerate since they
commute with the Hamiltonian, [L 0 , φ 0 ] = 0 according to (6.117). The partition
of the zero-modes into creation and annihilation operators depends on the specific
state chosen among the degenerate vacua.
The energy a of |, which is also its L 0 eigenvalue
L 0 | := a |,
(6.130)
is called zero-point energy. Bosonic operators with negative h are dangerous
because they lead to an infinite negative energy together with an infinite degeneracy
(from the zero-mode).
The conjugate vacuum is defined by BPZ or Hermitian conjugation
0| = |0
‡
= |0
t
(6.131)
131
For this to make sense, the modes which diverge as z → 0 must annihilate the
vacuum. In particular, for a weight h field φ(z), one finds
∀n ≥ −h + 1 : φ n |0 = 0.
(6.125)
Thus, the φ n for n ≥ −h + 1 are annihilation operators for the vacuum |0, and
conversely the states φ n with n < −h + 1 are creation operators. As a consequence,
the state |φ is found by applying the mode n = −h to the vacuum:
|φ = φ −h |0 =
dz
2π i
φ(z)
z
|0.
(6.126)
Since L −1 is the generator of translations on the plane, one finds
φ(z)|0 = e
zL −1 φ(0)e
−zL −1 |0 = e
zL −1 |φ.
(6.127)
The vacuum |0 is the state associated with the identity 1. Translating the conditions
(6.125) to the energy–momentum tensor gives
∀n ≥ −1 : L n |0 = 0.
(6.128)
This is consistent with the definition (6.121) since it includes the sl(2, C) subalgebra.
If h < 0, some of the modes with n > 0 do not annihilate the vacuum: (6.117)
implies that some states have an energy lower than the one of |0. The state |
(possibly degenerate) with the lowest energy is called the energy vacuum
∀|φ ∈ H :
:|L 0 | ≤≤φ|L 0 |φ.
(6.129)
It is obtained by acting repetitively with the modes φ n>0 . This vacuum defines a new
partition of the non-zero-modes operators into annihilation and creation operators.
If there are zero-modes, i.e. n = 0 modes, then the vacuum is degenerate since they
commute with the Hamiltonian, [L 0 , φ 0 ] = 0 according to (6.117). The partition
of the zero-modes into creation and annihilation operators depends on the specific
state chosen among the degenerate vacua.
The energy a of |, which is also its L 0 eigenvalue
L 0 | := a |,
(6.130)
is called zero-point energy. Bosonic operators with negative h are dangerous
because they lead to an infinite negative energy together with an infinite degeneracy
(from the zero-mode).
The conjugate vacuum is defined by BPZ or Hermitian conjugation
0| = |0
‡
= |0
t
(6.131)