12.2 Curved Spacetime Is a Neural Network
177
Researchers are experimenting with various gravity models because no gravityside theory equivalent to QCD has been found. On the gravity side, one can write
a spacetime metric in a spacetime with one dimension higher, and a Lagrangian of
various matter fields in it. One can perform various classical calculations based on
that, and can use the dictionary for the gauge/gravity duality to calculate the QCD
physical quantities which live on the boundary side. So what is the gravity theory
equivalent to QCD?
As you can see, this is one of the inverse problems. QCD is a well-defined
quantum field theory whose Lagrangian is given, and its physical quantities have
been widely studied. Because of the strong coupling, it is not easy to calculate,
but it has been established that it is possible to evaluate various physical quantities
by numerical calculation using a supercomputer: lattice QCD. 5 In particular, as for
the spectrum (mass distribution) of hadrons, which are bound states of elementary
quarks in QCD, the calculation results of the lattice QCD match very well with the
measurement results of accelerator experiments, thus our world is confirmed to be
described by QCD. The problem of finding a gravitational description of such a
given quantum field theory is an inverse problem. This is because, until now, almost
all holographic QCD calculations have been performed for a given gravitational-side
Lagrangian by hand and are interpreted as QCD-like quantities.
In the next section, we will introduce a method using deep learning [118, 130]
to solve this inverse problem. In this method, the data of the lattice QCD simulation
can be used to determine the metric on the gravity side so that it reproduces the data,
by learning.
One of the fundamental problems of the gauge/gravity correspondence is how to
construct the gravity side for a given quantum field theory on the boundary side,
and how to judge whether there is a description of the gravity side in the first place.
This kind of problem is not limited to QCD. In recent years, wider gauge/gravity
correspondences based on entanglement entropy and quantum information theory
have been studied. The concept of “emergent gravity” is gaining widespread
acceptance and there is hope for how the inverse problem will be solved in the
future.
12.2 Curved Spacetime Is a Neural Network
In order to partially solve the inverse problem of finding the gravitational theory
emerging from the boundary quantum field theory such as QCD, the following
simple settings are made. We shall regard the gravitational spacetime itself as a
neural network.
5 This is the field of research where the inside of the nucleus can be numerically calculated at
the quark/gluon level from the first principle. Due to the large amount of numerical calculation,
calculations are performed by the Markov chain Monte Carlo method using a supercomputer. The
formulation itself is analogous to statistical mechanics. For details, see [104].
177
Researchers are experimenting with various gravity models because no gravityside theory equivalent to QCD has been found. On the gravity side, one can write
a spacetime metric in a spacetime with one dimension higher, and a Lagrangian of
various matter fields in it. One can perform various classical calculations based on
that, and can use the dictionary for the gauge/gravity duality to calculate the QCD
physical quantities which live on the boundary side. So what is the gravity theory
equivalent to QCD?
As you can see, this is one of the inverse problems. QCD is a well-defined
quantum field theory whose Lagrangian is given, and its physical quantities have
been widely studied. Because of the strong coupling, it is not easy to calculate,
but it has been established that it is possible to evaluate various physical quantities
by numerical calculation using a supercomputer: lattice QCD. 5 In particular, as for
the spectrum (mass distribution) of hadrons, which are bound states of elementary
quarks in QCD, the calculation results of the lattice QCD match very well with the
measurement results of accelerator experiments, thus our world is confirmed to be
described by QCD. The problem of finding a gravitational description of such a
given quantum field theory is an inverse problem. This is because, until now, almost
all holographic QCD calculations have been performed for a given gravitational-side
Lagrangian by hand and are interpreted as QCD-like quantities.
In the next section, we will introduce a method using deep learning [118, 130]
to solve this inverse problem. In this method, the data of the lattice QCD simulation
can be used to determine the metric on the gravity side so that it reproduces the data,
by learning.
One of the fundamental problems of the gauge/gravity correspondence is how to
construct the gravity side for a given quantum field theory on the boundary side,
and how to judge whether there is a description of the gravity side in the first place.
This kind of problem is not limited to QCD. In recent years, wider gauge/gravity
correspondences based on entanglement entropy and quantum information theory
have been studied. The concept of “emergent gravity” is gaining widespread
acceptance and there is hope for how the inverse problem will be solved in the
future.
12.2 Curved Spacetime Is a Neural Network
In order to partially solve the inverse problem of finding the gravitational theory
emerging from the boundary quantum field theory such as QCD, the following
simple settings are made. We shall regard the gravitational spacetime itself as a
neural network.
5 This is the field of research where the inside of the nucleus can be numerically calculated at
the quark/gluon level from the first principle. Due to the large amount of numerical calculation,
calculations are performed by the Markov chain Monte Carlo method using a supercomputer. The
formulation itself is analogous to statistical mechanics. For details, see [104].