Non-Standard Numbers and TGD

I had opportunity to read articles of Elemer Rosinger about possible physical applications of non-standard numbers and it was natural to compare these numbers with the generalization of real numbers inspired by the notion of infinite primes. This lead to the idea of writing a commentary about the articles.

• How Far Should the Principle of Relativity Go?
• Quantum Foundations: Is Probability Ontological?
• Group Invariant Entanglements in Generalized Tensor Products
• Heisenberg Uncertainty in Reduced Power Algebras
• Surprising Properties of Non-Archimedean Field Extensions of the Real Numbers
• No-Cloning in Reduced Power Algebras

I have a rather rudimentary knowledge about non-standard numbers and my comments are very subjective and TGD centered. I however hope that they might tell also something about Rosinger's work. My interpretation of the message of articles relies on associations with my own physics inspired ideas related to the notion of number. I divide the articles to physics related and purely mathematical ones. About the latter aspects I am not able to say much.

The construction of ultrapower fields (generalized scalars) is explained using concepts familar to physicist using the close analogies with gauge theories, gauge invariance, and with the singularities of classical fields. Some questions related to the physical applications of non-standard numbers are discussed including interpretational problems and the problems related to the notion of definite integral. The non-Archimedean character of generalized scalars is discussed and compared with that of p-adic numbers.

Rosinger considers several physical ideas inspired by ultrapower fields including the generalization of general covariance to include the independence of the formulation of physics on the choice of generalized scalars, the question whether generalized scalars might allow to understand the infinities of quantum field theories, and the question whether the notion of measurement precision could realized in terms of scale hierarchy with levels related by infinite scalings. These ideas are commented in the article by comparison to p-adic variants of these ideas.

Non-standard numbers are compared with the numbers generated by infinite primes. It is found that the construction of infinite primes, integers, and rationals has a close similarity with construction of the generalized scalars. The construction replaces at the lowest level the index set Λ=N of natural numbers with algebraic numbers A, Frechet filter of N with that of A, and R with unit circle S¹ represented as complex numbers of unit magnitude. At higher levels of the hierarchy generalized -possibly infinite and infinitesimal- algebraic numbers emerge. This correspondence maps a given set in the dual of Frechet filter of A to a phase factor characterizing infinite rational algebraically so that correspondence is like representation of algebra. The basic difference between two approaches to infinite numbers is that the counterpart of infinitesimals is infinitude of real units with complex number theoretic anatomy: one might loosely say that these real units are exponentials of infinitesimals.

With motivations coming from quantum computation, Rosinger discusses also a possible generalization of the notion of entanglement allowing to define it also for what could be regarded as classical systems. Entanglement is also number theoretically very interesting notion. For instance, for infinite primes and integers the notion of number theoretical entanglement emerges and relates to the physical interpretation of infinite primes as many particles states of second quantized super-symmetry arithmetic QFT. What is intriguing that the algebraic extension of rationals induces de-entanglement. The de-entanglement corresponds directly to the replacement of a polynomial with rational coefficients with a product of the monomials with algebraic roots in general.

For details see the new chapter Non-Standard Numbers and TGD of "Physics as a Generalized Number Theory".