Number theory as the ultimate physical theory. (English) Zbl 1258.81074
Summary: At the Planck scale doubt is cast on the usual notion of space-time and one cannot think about elementary particles. Thus, the fundamental entities of which we consider our Universe to be composed cannot be particles, fields or strings. In this paper the numbers are considered as the fundamental entities. We discuss the construction of the corresponding physical theory. A hypothesis on the quantum fluctuations of the number field is advanced for discussion. If these fluctuations actually take place then instead of the usual quantum mechanics over the complex number field a new quantum mechanics over an arbitrary field must be developed. Moreover, it is tempting to speculate that a principle of invariance of the fundamental physical laws under a change of the number field does hold. The fluctuations of the number field could appear on the Planck length, in particular in the gravitational collapse or near the cosmological singularity. These fluctuations can lead to the appearance of domains with non-Archimedean \(p\)-adic or finite geometry. We present a short review of the \(p\)-adic mathematics necessary, in this context.
MSC:
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 11E95 | \(p\)-adic theory |
| 81Q65 | Alternative quantum mechanics (including hidden variables, etc.) |
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