[Paper Review] How Far Should the Principle of Relativity Go ?

Elemér E Rosinger|ArXiv.org|Oct 1, 2007

Mathematical and Theoretical Analysis16 citations

TL;DR

This paper proposes extending the Principle of Relativity beyond reference frame covariance to include covariance with respect to a broad class of non-Archimedean algebras of scalars—such as reduced power algebras—instead of restricting physics to real or complex numbers. By replacing fields with more general algebras, the framework enables a deeper, wider formulation of physical laws that can naturally handle infinities and divergences, offering a novel path to resolving long-standing issues in theoretical physics related to infinities.

ABSTRACT

The Principle of Relativity has so far been understood as the {\it covariance} of laws of Physics with respect to a general class of reference frame transformations. That relativity, however, has only been expressed with the help of {\it one single} type of mathematical entities, namely, the scalars given by the usual continuum of the field $\mathbb{R}$ of real numbers, or by the usual mathematical structures built upon $\mathbb{R}$, such as the scalars given by the complex numbers $\mathbb{C}$, or the vectors in finite dimensional Euclidean spaces $\mathbb{R}^n$, infinite dimensional Hilbert spaces, etc. This paper argues for {\it progressing deeper and wider} in furthering the Principle of Relativity, not by mere covariance with respect to reference frames, but by studying the possible covariance with respect to a {\it large variety} of algebras of scalars which extend significantly $\mathbb{R}$ or $\mathbb{C}$, variety of scalars in terms of which various theories of Physics can equally well be formulated. First directions in this regard can be found naturally in the simple Mathematics of Special Relativity, the Bell Inequalities in Quantum Mechanics, or in the considerable amount of elementary Mathematics in finite dimensional vector spaces which occurs in Quantum Computation. The large classes of algebras of scalars suggested, which contain $\mathbb{R}$ and $\mathbb{C}$ as particular cases, have the important feature of typically {\it no longer} being Archimedean, see Appendix, a feature which can prove to be valuable when dealing with the so called "infinities" in Physics. The paper has a Comment on the so called "end of time".

Motivation & Objective

• To challenge the assumption that physical theories must be formulated using only real or complex scalars, which are Archimedean and incapable of distinguishing infinite magnitudes.
• To argue that theoretical physics is confined by an unexamined reliance on Archimedean structures, leading to persistent issues with infinities in quantum field theory and general relativity.
• To propose that the Principle of Relativity should be generalized from covariance under reference frame transformations to covariance under changes in the algebraic structure of scalars used in physical theories.
• To demonstrate that non-Archimedean algebras—such as matrix algebras and reduced power algebras—offer a viable, mathematically rich, and physically meaningful alternative to traditional scalar fields.
• To show that abandoning the requirement of a field structure (as restricted by Pontrjagin’s theorem) allows access to a vastly larger class of algebras that are non-Archimedean and better suited for handling infinities in physics.

Proposed method

• Utilizes reduced power algebras constructed from sequences of real or complex numbers, quotiented by ultrafilters, to form non-Archimedean algebras that include R and C as substructures.
• Applies the concept of algebraic structures (algebras) that are less restrictive than fields—allowing non-invertible elements—thereby enabling the use of matrix algebras as viable scalar systems.
• Introduces a partial order ≼ on sequence spaces like R^N to define non-Archimedean behavior, where nx ≼ y for all n ∈ N even when x ≠ 0.
• Demonstrates that standard mathematical structures like R^n with lexicographic order or R^N with coordinate-wise order fail to satisfy the Archimedean property (ARCH), thus being non-Archimedean.
• Uses Pontrjagin’s theorem to argue that only R, C, and H are non-discrete fields, and since H is also Archimedean, any attempt to go beyond R and C in the field framework is mathematically impossible.
• Proposes that physical theories can be reformulated in these non-Archimedean algebras, preserving physical content while allowing for a more nuanced treatment of infinities and divergences.

Experimental results

Research questions

• RQ1 Can the Principle of Relativity be extended beyond covariance under reference frame transformations to include covariance under changes in the algebraic structure of the scalars used in physical theories?
• RQ2 Why do standard physical theories consistently encounter infinities, and can this be traced to the use of Archimedean scalar fields like R and C?
• RQ3 Are there viable mathematical alternatives to R and C that are non-Archimedean and still rich enough to support the formulation of physical laws?
• RQ4 What is the mathematical and physical significance of abandoning the field structure in favor of more general algebras in physical theories?
• RQ5 Can non-Archimedean algebras such as reduced power algebras provide a consistent and useful framework for re-expressing quantum mechanics, relativity, and quantum field theory?

Key findings

• The class of non-Archimedean algebras, such as reduced power algebras and matrix algebras, is vastly larger than the class of Archimedean fields, making the latter a negligible subset in mathematical space.
• Matrix algebras of order n ≥ 2 are not fields due to non-invertible elements, but they are algebras that support addition, subtraction, multiplication, and division except for zero-determinant elements, making them suitable for physical formulations.
• The Archimedean property (ARCH) fails in infinite-dimensional spaces like R^N under standard partial orders, and even in finite-dimensional spaces like R^2 under lexicographic order, demonstrating that Archimedean structures are rare.
• The use of a partial order ≼ on R^N allows for sequences x and y such that nx ≼ y for all n ∈ N with x ≠ 0, violating (ARCH+), proving non-Archimedean behavior.
• Pontrjagin’s theorem implies that any non-discrete field must be R, C, or H—none of which are non-Archimedean—thus proving that non-Archimedean structures cannot be fields, necessitating a move to algebras.
• The paper concludes that the restriction to Archimedean fields like R and C is not a natural or necessary choice, but an arbitrary confinement that limits the scope of theoretical physics and leads to persistent issues with infinities.