6.4 Operator Formalism and Radial Quantization
125
where the derivatives of w are with respect to z. This vanishes if the transformation
is in SL(2, C), and it transforms as
S(u, z) = S(w, z) +
dw
dz
2
S(u, w)
(6.92)
under successive changes of coordinates.
Computation: Equation (6.89)
δT (z) =
C z
dw
2π i
v(w)T (w)T (z) ∼
C z
dw
2π i
v(w)
c/2
(z − w) 4 +
2T (w)
(z − w) 2 +
∂T (w)
z − w
=
c
2 × 3!
∂
3 v(z) + 2∂v(z) T (z) + v(z)∂T (z).
6.4.3 Hermitian and BPZ Conjugation
In this section, we introduce two different notions of conjugations: one is adapted for
amplitudes because it defines a unitary Euclidean time evolution, while the second is
more natural as an inner product of CFT states. Both can be interpreted as providing
a map from in-states to out-states on the cylinder.
Given an operator O, we need to define an operation O ‡ —called Euclidean
adjoint (or simply adjoint)—which, after Wick rotation from Euclidean to
Lorentzian signature, can be interpreted as the Hermitian adjoint. 9 This is necessary
in order to define a Hermitian inner product and to impose reality conditions.
To motivate the definition, consider first the cylinder in Lorentzian signature. Since Hermitian conjugation does not affect the Lorentzian coordinates, the
Euclidean time must reverse its sign:
t
†
= −iτ
†
= t ⇒ τ
†
= −τ.
(6.93)
Hence, an appropriate definition of the Euclidean adjoint is a Hermitian conjugation
together with time reversal. 10 Another point of view is that the time evolution
operator U(τ ) := e −τ H is not unitary when H is Hermitian H † = H : the solution
is to define a new Euclidean adjoint U(τ ) ‡ := U(−τ ) † such that U(τ ) is unitary for
it.
9 In [28], it is denoted by a bar on top of the operator: we avoid this notation since the bar already
denotes the anti-holomorphic sector. In [40], it is indicated by a subscript hc. Otherwise, in most
of the literature, it has no specific symbol since one directly works with the modes.
10 The Euclidean adjoint can be used to define an inner product: positive-definiteness of the latter
is called reflection positivity or OS-positive and is a central axiom of constructive QFT.
125
where the derivatives of w are with respect to z. This vanishes if the transformation
is in SL(2, C), and it transforms as
S(u, z) = S(w, z) +
dw
dz
2
S(u, w)
(6.92)
under successive changes of coordinates.
Computation: Equation (6.89)
δT (z) =
C z
dw
2π i
v(w)T (w)T (z) ∼
C z
dw
2π i
v(w)
c/2
(z − w) 4 +
2T (w)
(z − w) 2 +
∂T (w)
z − w
=
c
2 × 3!
∂
3 v(z) + 2∂v(z) T (z) + v(z)∂T (z).
6.4.3 Hermitian and BPZ Conjugation
In this section, we introduce two different notions of conjugations: one is adapted for
amplitudes because it defines a unitary Euclidean time evolution, while the second is
more natural as an inner product of CFT states. Both can be interpreted as providing
a map from in-states to out-states on the cylinder.
Given an operator O, we need to define an operation O ‡ —called Euclidean
adjoint (or simply adjoint)—which, after Wick rotation from Euclidean to
Lorentzian signature, can be interpreted as the Hermitian adjoint. 9 This is necessary
in order to define a Hermitian inner product and to impose reality conditions.
To motivate the definition, consider first the cylinder in Lorentzian signature. Since Hermitian conjugation does not affect the Lorentzian coordinates, the
Euclidean time must reverse its sign:
t
†
= −iτ
†
= t ⇒ τ
†
= −τ.
(6.93)
Hence, an appropriate definition of the Euclidean adjoint is a Hermitian conjugation
together with time reversal. 10 Another point of view is that the time evolution
operator U(τ ) := e −τ H is not unitary when H is Hermitian H † = H : the solution
is to define a new Euclidean adjoint U(τ ) ‡ := U(−τ ) † such that U(τ ) is unitary for
it.
9 In [28], it is denoted by a bar on top of the operator: we avoid this notation since the bar already
denotes the anti-holomorphic sector. In [40], it is indicated by a subscript hc. Otherwise, in most
of the literature, it has no specific symbol since one directly works with the modes.
10 The Euclidean adjoint can be used to define an inner product: positive-definiteness of the latter
is called reflection positivity or OS-positive and is a central axiom of constructive QFT.