Non-commutative space-time and the uncertainty principle

The full algebra of relativistic quantum mechanics (Lorentz plus Heisenberg) is unstable. Stabilization by deformation leads to a new deformation parameter εl, l being a length and ε a ± sign. The implications of the deformed algebras for the uncertainty principle and the density of states are worked out and compared with the results of past analysis following from gravity and string theory. PACS: 03.65.Bz Physical theories are approximations to Nature and physical constants may never be known with absolute precision. Therefore, a wider range of validity is expected to hold for theories that do not change in a qualitative manner under a small change of parameters. Such theories are called stable or rigid. The stable-model point of view originated in the field of non-linear dynamics, where it led to the notion of structural stability[1] [2]. However, as emphasized by Flato[3] and Faddeev[4], the same pattern seems to occur in the fundamental theories of Nature. Indeed, the most important physical revolutions of this century, the transition from non-relativistic to relativistic and from classical to quantum mechanics, may be interpreted as the replacement of two unstable theories by two stable ones. Mathematically this corresponds to stabilizing deformations leading, in the first case, from the Galilean to the School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332 USA, [email protected]; Work partially supported by NSF grant DMS 00-70589 Grupo de F́isica Matemática, Universidade de Lisboa, Av. Gama Pinto 2, 1699 Lisboa, Portugal, [email protected]