When Leonard Susskind recently visited UCSD and gave a talk here, he strongly advocated the holographic principle. According to this principle, roughly, the physics in an (n+1)-dimensional space can be mapped, without loss, onto an n-dimensional space. The idea seems to be that this principle asserts a sort of equivalence between the physics in an (n+1)-dimensional space and that of an n-dimensional one. Let’s call a map of the physics of an (n+1)-dimensional space onto the physics of an n-dimensional space, which satisfies the holographic principle, a holographic map.
Now, interestingly, Susskind also said that such a map was necessarily discontinuous. This seemed intuitive enough (unlike other things he said), but when I tried to find a mathematical theorem that would establish a precisified version of the statement that all holographic maps are discontinuous, I quickly realized that this is less straightforward than I thought. I asked a local mathematician at UCSD, Justin Roberts, who explained some of what follows to me.
Suppose that the spaces are differentiable manifolds of the relevant dimension and that the “physics” is captured by scalar, vector, or tensor fields defined on these manifolds.
The qualification that the physics ought to be mapped from one space to the other without loss, or “preserving the physical information” is important. Suppose that was not a requirement and you’d simply want to know whether there exist continuous maps from an (n+1)-dimensional to an n-dimensional space. Now a simple restriction of the functions defined on R^2 to an axis is a continuous map. Conversely, we can pull back a scalar function on R^1 to R^2 using a projection map. This will give us a continuous mapping. But it’s obvious that in the former case, we haven’t preserved the original information, so the map isn’t holographic.
Justin pointed out to me that one can of course inject a ring of functions of one space into that of another of equal or higher dimension, which would satisfy one sense of “preserving the information”. But what you want is to preserve the information contained in the higher-dimensional space and map it into the lower-dimensional space holographically; injecting a ring of functions of a space can only be done continuously into that of another if the dimension of the latter is equal or higher. In particular, if one demands that the holographic map be bijective–which seems a fairly simple yet intuitive rendering of “preserving the information”–, then the dimension of the two spaces must be equal.
Does this mean that a mapping of the “physics” of an (n+1)-dimensional space onto that of an n-dimensional space cannot be both continuous and bijective? Or are there any other mathematical theorems that would help in clarifying the situation?