Comments by Giotis on my previous post, about emergent gravity, reminded me about one of the tricky points in gauge-gravity dualities such as the AdS/CFT correspondence. This is the understanding of them as dualities between one theory that lives in the “bulk” spacetime (say quantum gravity on five dimensional AdS space) and another that lives on the boundary of that spacetime (four dimensional supersymmetric gauge theory in that case). This is kind of correct, if you know precisely what you mean, but it can also lead you to scratching your head if you don’t.
The tricky part in thinking about dualities is internalizing the idea that the two sides are actually two descriptions of one and the same object. So, the gauge theory has the same properties as quantum gravity in five dimensional space, it is a five dimensional quantum gravity theory. Like any other description of this quantum theory, it has a bulk and a boundary, it has gravitational forces and propagating gravitons, it has black holes forming and evaporating, etc., etc. … All of those are realized in a somewhat unfamiliar language, more suited for the quantum rather than the classical theory, but all the same they are still there. It’s a good mental exercise to phrase gravitational properties of the theory in terms of the gauge theory variables, it forces you to distinguish physics from language. So, I’ll do that here by discussing bulk and boundary of spacetime in the gauge theory language.
To start, what distinguishes bulk and boundary in the gravity language? We’d have to think about physical properties that don’t tie us to particular description or limit of the theory. The answer is that the physical distinction between bulk and boundary is really the distinction between fluctuating and dynamical properties of the theory and non-fluctuating or background parameters thereof. This distinction exists in any physical theory, and is the key to understanding the issue of background independence, so let me elaborate just a little a bit.
Consider for example an harmonic oscillator, just an idealized pendulum. You start it of at some position and watch it oscillate back and forth. Now, depending on the initial conditions, position and velocity, the amplitude of oscillation will vary. This is an example of a dynamical mode, it’s value depends on the state of the system, it is a property of a specific classical solution (or for the quantum theory it would be a property of a particular wavefunction). Similarly for quantum gravity in AdS space you can have different states of the system with different properties. Some of them will have gravitons propagating and colliding, some of them will have galaxies with gigantic black holes in their centers. Some will have carbon-based humans living on planets orbiting a star. You get the picture. All of those are specific states of the same theory, described in either set of variables.
On the other hand, the harmonic oscillator has some properties which are common to all its states. For an idealized oscillator an example is the period of oscillation, which depends only on external parameters of the system: the length of the pendulum and the strength of the gravitational force driving the oscillations. Those parameters are fixed for all states of the system, and specify which system we are discussing. They are unchanged by any physical process in the theory.
The gauge theory, like any theory in physics, has both dynamical modes and non-dynamical parameters. For example you can have different states where electromagnetic waves propagate, or hadrons collide emitting high energy jets, or gold ions collide and form, for a split second, a quark-gluon fluid. On the other hand it also has parameters that are the same for all states. The simplest are the rank of the gauge group, or the manifold on which the gauge theory lives. Those do not change in any dynamical process, they are fixed once and for all.
In gravity in AdS space, this universal distinction holds as well, and very beautifully it becomes a geometrical one. Dynamical modes are local processes, ones taking place somewhere in the bulk of spacetime. For example we can follow the lifetime of a star, from the initial gas to the final stages of its death (which depending on its mass can be a white dwarf, a neutron star, a black hole, or more exotic objects). We don’t have to think about the boundary of spacetime for that, in fact if it were not for these esoteric semi-philosophical discussions we are having , that thought would never cross our minds.
But, there are also those background parameters, ones that specify the theory and are given once and for all, fixed for all states. We saw examples of them in the gauge theory variables, and in the gravitational language they are simply boundary conditions. Every differential equation needs boundary conditions to be well defined, and those boundary conditions, by definition, are the same for all solutions. In the classical gravity limit, our theory is described by a set of differential equations, which have boundary conditions for all fields, and those are fixed for all states of the theory. In a more technical language, they determine superselection sectors. Needless to say, the set of background parameters, like the set of all states, is independent on which set of variables we choose to utilize.
So, where is the gauge theory? it is everywhere. As the theory of AdS quantum gravity that it is, it has both bulk and boundary. It’s “bulk” consists of all dynamical processes that can occur in the theory, it’s “boundary” consists of all fixed parameters of the theory.