#### Abstract

It is known that the entire geometry of many relativistic space-times can be summed up in two concepts, a space-time measure $\mu $ and a space-time causal or chronological order relation $C$, defining a causal measure space. On grounds of finiteness, unity, and symmetry, we argue that macroscopic space-time may be the classical-geometrical limit of a causal quantum space. A tentative conceptual framework is provided. Mathematical individuals that naturally form causal spaces are symbol sets or words, taken in the order of their generation. The natural extension of this purely logical concept to quantum symbols is formulated. The problem is posed of giving finite quantum rules for the generation of quantum symbol sets such that the order of generation becomes, in the classical limit, the causal order of space-time—as it were, to break the space-time code. The causal quantum spaces of three simple codes are generated for comparison with reality. The singulary code (repetitions of one digit) gives a linearly ordered external world of one time dimension and a circular internal space. The binary code gives the future null cone of special relativity and a circular internal space. The causal quantum space of word pairs in the binary code gives the solid light cone $t>\mathrm{}{({x}^{2}+{y}^{2}+{z}^{2})}^{\frac{1}{2}}$ of special relativity and an internal space $U(2,C)\mathrm{}$ suitable for the description of charge and isospin. In the classical limit, there is full translational and proper Lorentz invariance except at the boundary of the light cone, where the classical-geometrical limit fails. Plausible consequences of this model for cosmology and elementary particles are discussed. There is a quantum of time $\tau \lesssim \frac{\hslash}{{m}_{\mu}{c}^{3}}$ and a space-time complementarity relation $\Delta t\Delta x\Delta y\Delta z\gtrsim {\tau}^{4}$.

DOI: http://dx.doi.org/10.1103/PhysRev.184.1261

- Received 25 March 1968
- Published in the issue dated August 1969

© 1969 The American Physical Society