Physical Equivalence on Non-Standard Spaces and Symmetries on Infinitesimal-Lattice Spaces

Equivalence in physics is discussed on the basis of experimental data accompanied by experimental errors. The introduction of the equivalence being consistent with the mathematical definition is possible only in theories constructed on nonstandard number spaces by taking the experimental errors as infinitesimal numbers of the non-standard spaces. Following the idea for the equivalence (the physical equivalence), a new description of space-time in terms of infinitesimal-lattice points on non-standard real number space R is proposed. The infinitesimal-lattice space, L, is represented by the set of points on R which are written by ln = n ε, where the infinitesimal lattice-spacing ε is determined by a non-standard natural number N such that ε ≡ ∗N−1. By using infinitesimal neighborhoos (Mon(r| L)) of real number r on L we can make a space M which is isomorphic to R as additive group. Therefore, every point on ( ∗M)N automatically has the internal confinedsubspace Mon(r| L). A field theory on L is proposed. To determine a projection from L to M, a fundamental principle based on the physical equivalence is introduced. The physical equivalence is expressed by the totally equal treatment for indistinguishable quantities in our observations. Following the principle, we show that U(1) and SU(N) symmetries on the space ( ∗M)N are induced from the internal substructure (Mon(r| ∗L))N . Quantized state describing configuration space is constructed on ( ∗M)N . By providing that the subspace (Mon(r| ∗L))N is local inertial system of general relativity, infinitesimal distance operators are consistently introduced. We see that Lorentz and general relativistic transformations are also represented by operators which involve the U(1) and SU(N) internal symmetries. (To appear in Proceedings “Symmetries in Science XI“ (Plenum, New york). 2000) 1