In this month’s Scientific American there’s an article entitled The Illusion of Gravity. On page 63 the author states, “ So far no example of the holographic correspondence has been rigorously proved—the mathematics is too difficult.”
What does he mean by this? Obviously he doesn’t mean the algebra, calculus, linear algebra etc. is too difficult; he must be talking about something else altogether. Like the development of a whole new mathematics? If something entirely new is required how do they know there is something new?
Holographic correspondence might mean two people exchanging ideas in hand-written letters. Without more information, the maths they are discussing in those letters might be anything.
Actually, in a sense he does mean calculus. If your sense of “math” is “(polynomial) algebra, calculus, and linear algebra”, then I really don’t know how I can respond at all coherently. Roughly, there are certain quantities that should be given by certain mathematical constructions, but the constructions don’t make rigorous sense in the naive way one would think they did. Finding a better way to understand them is the “new mathematics”.
He may mean something more basic than that. In physics, it’s quite common to write down an equation for some quantity of interest which can’t be solved. For example, I might find that the function y(x) that has particular significance in the problem I’m looking at satisfies the differential equation x[sup]2[/sup] y’’ + x y’ + (x[sup]2[/sup] - n[sup]2[/sup]) y = 0. For general values of n, you can’t solve this equation in terms of “nice” functions (trigonometric functions, exponential functions, powers of x, etc.), so you have to make certain approximations: find out how the function behaves near x = 0, figure out where its zeroes are, etc. But you can’t properly “solve” this equation other than saying something like “the solution to this equation is the solution to this equation.”
Compared to some of the other equations that end up making their way into physics, the above equation is a very simple one. So he may just mean that “nobody’s been able to solve these equations yet.”
(Aside: As it happens, the above equation is a very well-known one; it’s called Bessel’s equations, and the so-called “Bessel functions” are defined as the solutions of it. A lot of work has been done over hundreds of years on them, and a lot is known about them, but except for special values of n they’re not expressible in terms of simpler functions.)
Actually (not having read the article since my subscription to SA has lapsed), the most common type of thing that comes up in trying to come up with an extension of gravity is sums that make heuristic – but not rigorous – sense.
As a toy example: conformal field theories. Closed strings trace out surfaces. If there are no particles at the beginning and end of the interaction, the surfaces are themselves closed, and thus are parametrized by their genus. Closed surfaces of genus g have a finite-dimensional moduli space of conformal structures, and to each one we want to assign a complex number. We do this by integrating some form over the moduli space for each g, which gives the contribution for g to the partition function of vacuum-vacuum interactions.
But we want particles at the beginning and end, which means considering surfaces with boundaries. We can sew together surfaces along boundary components (that we can do this when there are conformal structures around is a theorem), but even the simplest surfaces now have infinite-dimensional moduli spaces, and the notion of “integration” is no longer well-defined in the simple straightforward way we’d like. It’s possible that a new way of understanding what we mean by “integration” over this space will give meaning to this heuristic, but… well, the math is too hard as yet.
Maybe you’re thinking that it should be “no example. . .has been. . .proven,” which might be a common usage but does not render “proved” incorrect.
Then again maybe you think “mathematics” should take the plural verb, but I would say that “mathematics” is a singular noun for a branch of study, rather than more than one “mathematic.” Merriam-Webster sez:
Not to mention that I have commited the strawman fallacy.