Is Wave Propagation Computable or Can Wave Computers Beat the Turing Machine?

By the Church±Turing Thesis a numerical function is computable by a physical device if and only if it is computable by a Turing machine. The `if'-part is plausible since every (suf®ciently small) Turing machine can be simulated by a computer program which operates correctly as long as suf®cient time and storage are available and no errors occur. On the other hand, every program for a modern digital computer can be simulated by a Turing machine. What is more, most physicists believe that for processes which can be described by well-established theories (®nitely many point masses interacting gravitationally, electromagnetic waves, quantum systems etc.) the future behavior can be computed with arbitrary precision, at least in principle, from suf®ciently precisely given initial conditions, where the computations can be performed on digital computers, and hence on Turing machines. Nevertheless, there might exist physical processes which are not Turing computable in this way. For discussions and further references see [4, 13]. Is Turing's computability concept suf®ciently powerful to model all kinds of physical processes? If the answer is `yes', then perhaps the Church±Turing Thesis could be derivable from the laws of physics, or perhaps it should even be considered as a fundamental law of physics itself. Otherwise, the Church±Turing Thesis should possibly be corrected. In this paper we shall concentrate on a special type of physical process, namely one on scalar waves in Euclidean space. We start from remarkable results by Pour-El and Richards [11] and Pour-El and Zhong [9], who constructed computable initial conditions f for the threedimensional wave equation utt ˆ Du; u…0; x† ˆ f …x†; ut…0; x† ˆ 0; t 2 R; x 2 R; …1:1†