Field Theory on Infinitesimal-Lattice Spaces

Equivalence in physics is discussed on the basis of experimental data accompanied by experimental errors. It is pointed out that the introduction of the equivalence being consistent with the mathematical definition is possible only in theories constructed on non-standard number spaces by taking the experimental errors as infinitesimal numbers. Following the idea for the equivalence, a new description of space-time $\SL$ in terms of infinitesimal-lattice points on non-standard real number space $\SR$ is proposed. By using infinitesimal neighborhoos ($\MON$) of real number r on $\SL$ we can make a space $\SM$ which is isomorphic to $\RE$ as additive group. Therefore, every point on $(\SM)^N$ automatically has the internal confined-subspace $\MON$. A field theory on $\SL$ is proposed. It is shown that U(1) and SU(N) symmetries on the space $(\SM)^N$ are induced from the internal substructure $(\MON)^N$. Quantized state describing configuration space is constructed on $(\SM)^N$. We see that Lorentz and general relativistic transformations are also represented by operators which involve the U(1) and SU(N) internal symmetries.