12.2 Curved Spacetime Is a Neural Network
181
The relation between the boundary value of the scalar field in the asymptotic AdS
region and the one-point function of the quantum field theory on the boundary side
is specified by the general principle defined in the literature [139]. Suppose that the
operator of quantum field theory on the boundary side is O, and the term gO exists
in the Lagrangian of the quantum field theory as its source term. Given the source g,
the one-point function means that the vacuum expectation value is given as
a function of g. For example, when an external magnetic field is given as a source,
is a magnetization operator, that is, the response to the external field. In QCD,
for example, if the source g is the mass of the quark, the mass term of the QCD is
m q ψψ, so the chiral condensate ψψ corresponds to the response. In this manner,
when g and are obtained by the quantum field theory on the boundary side, the
AdS/CFT correspondence tells us that there is the following relation [139] 6 on the
gravity side, under the normalization condition that the AdS radius R is 1,
φ(η ini ) = g exp[− − η ini ] + +O
exp[− + η ini ]
+ − −
.
(12.9)
Here is determined from the mass of the bulk, as ± ≡ (d/2) ±
d 2 /4 + m 2 R 2 ,
and + is the conformal dimension 7 of the operator O. In addition, a cutoff was
made in the region η → ∞: the boundary is given at η = η ini . Differentiating this
relation by η gives
π(η ini ) = −gg − exp[− − η ini ] − −O
+ exp[− + η ini ]
+ − −
.
(12.10)
These are the formulas that give the values of φ and π at η → ∞ for a given
one-point function of the quantum field theory.
On the other hand, what about the output data? At the even horizon of the
black hole, only waves that enter the horizon are allowed. Therefore, the solution
of the equation of motion only allows wave solutions in the negative direction of
η. Specifically, we can find out (by restoring the time derivative term as well) the
following condition:
0 = F ≡
2
η
π − m
2 φ −
δV (φ)
δφ
η=η fin
.
(12.11)
6 When the field is evaluated by its integration in the bulk coordinates, the first term is generally a
non-normalizable mode, and the second term is a normalizable mode.
7 The conformal dimension means that is in quantum field theory which is invariant under
conformal symmetry, the operator O transforms, when the coordinate scales as x → αx, like
O → α − O. In a free field theory, is the mass dimension of the operator (or the field).
181
The relation between the boundary value of the scalar field in the asymptotic AdS
region and the one-point function of the quantum field theory on the boundary side
is specified by the general principle defined in the literature [139]. Suppose that the
operator of quantum field theory on the boundary side is O, and the term gO exists
in the Lagrangian of the quantum field theory as its source term. Given the source g,
the one-point function means that the vacuum expectation value is given as
a function of g. For example, when an external magnetic field is given as a source,
is a magnetization operator, that is, the response to the external field. In QCD,
for example, if the source g is the mass of the quark, the mass term of the QCD is
m q ψψ, so the chiral condensate ψψ corresponds to the response. In this manner,
when g and are obtained by the quantum field theory on the boundary side, the
AdS/CFT correspondence tells us that there is the following relation [139] 6 on the
gravity side, under the normalization condition that the AdS radius R is 1,
φ(η ini ) = g exp[− − η ini ] + +O
exp[− + η ini ]
+ − −
.
(12.9)
Here is determined from the mass of the bulk, as ± ≡ (d/2) ±
d 2 /4 + m 2 R 2 ,
and + is the conformal dimension 7 of the operator O. In addition, a cutoff was
made in the region η → ∞: the boundary is given at η = η ini . Differentiating this
relation by η gives
π(η ini ) = −gg − exp[− − η ini ] − −O
+ exp[− + η ini ]
+ − −
.
(12.10)
These are the formulas that give the values of φ and π at η → ∞ for a given
one-point function of the quantum field theory.
On the other hand, what about the output data? At the even horizon of the
black hole, only waves that enter the horizon are allowed. Therefore, the solution
of the equation of motion only allows wave solutions in the negative direction of
η. Specifically, we can find out (by restoring the time derivative term as well) the
following condition:
0 = F ≡
2
η
π − m
2 φ −
δV (φ)
δφ
η=η fin
.
(12.11)
6 When the field is evaluated by its integration in the bulk coordinates, the first term is generally a
non-normalizable mode, and the second term is a normalizable mode.
7 The conformal dimension means that is in quantum field theory which is invariant under
conformal symmetry, the operator O transforms, when the coordinate scales as x → αx, like
O → α − O. In a free field theory, is the mass dimension of the operator (or the field).