Field Theory on Infinitesimal-Lattice Spaces

TL;DR

This paper introduces a novel space-time framework based on non-standard infinitesimal lattice points, enabling a new field theory that naturally incorporates internal symmetries and aligns with experimental error considerations.

Contribution

It proposes a space-time model using non-standard analysis with infinitesimal lattice points, leading to a field theory with embedded internal symmetries and compatible with relativistic transformations.

Findings

Space-time modeled on infinitesimal-lattice points in non-standard real numbers.

U(1) and SU(N) symmetries emerge from internal substructure.

Lorentz and general relativistic transformations involve internal symmetries.

Abstract

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.