Talks

A single classical system is characterized by its manifold of states; and to combine several systems, we take the product of manifolds. A single quantum system is characterized by its Hilbert space of states; and to combine several systems, we take the tensor product of Hilbert spaces. But what if we choose to combine an infinite number of systems? A naive attempt to describe such combinations fails, for there is apparently no natural notion of an infinite product of manifolds; nor of an infinite tensor product of Hilbert spaces. But, at least in the quantum case, the situation is not as hopeless as it might appear. We argue that there does indeed exist a natural mathematical framework for combinations of infinite numbers of quantum systems.

There are two notions that play a central role in the mathematical theory of computation. One is that of a computable problem, i.e., of a problem that can, in principle, be solved by an (idealized) computer. It is known that there exist problems that \'have answers\', but for which those answers are not computable. The other is that of the difficulty of a computation, i.e. of the number of (idealized) steps required actually to carry out that computation. It is known that, given any appropriate \'degree of difficulty\', there exists a problem that, while computable, is at least that difficult. These two notions, while purely mathematical, are designed to reflect, in some broad sense, the physics of the computation process. But there are indications that physics may have something further to say about them. Indeed, it has been suggested that, by using general relativity, some problems that are (mathematically) non-computable may become computable; and that, by using quantum mechanics, some problems that are (mathematically) difficult may become less so. Are there, in principle, any limitations on what physics can do for us in this area?