# Symmetries in Foundation of Quantum Theory and Mathematics

## Abstract

**:**

## 1. Introduction

## 2. Comparison of Different Physical and Mathematical Theories

**Definition**

**1.**

**Definition**not only from physical but also from mathematical considerations.

**Statement:**

**Proof.**

**Definition**are satisfied. ☐

**Statement**, classical theory is a special degenerated case of quantum one in the formal limit $\hslash \to 0$. Since in classical theory all operators commute with each other then in this theory operators are not needed and one can work only with physical quantities. A question why ℏ is as is does not arise since the answer is: because people’s choice is to measure angular momenta in $\mathrm{kg}\xb7{\mathrm{m}}^{2}/\mathrm{s}$.

**Statement**, NT is a special degenerated case of RT in the formal limit $c\to \infty $. The question of why $c=3\xb7{10}^{8}$ m/s and not, say $c=7\xb7{10}^{9}$ m/s does not arise since the answer is: because people’s choice is to measure velocities in m/s.

**Statement**, RT is indeed a special degenerated case of dS and AdS theories in the formal limit $R\to \infty $. By analogy with the abovementioned fact that c must be finite, R must be finite too: the formal case $R=\infty $ corresponds to the situation when the dS and AdS algebras do not exist because they become the Poincare algebra.

**Definition**above. Namely, the more general theory contains a finite parameter which is introduced for having a possibility to compare this theory with a less general one. Then the less general theory can be treated as a special degenerated case of the former in the formal limit when the parameter goes to zero or infinity. The more general theory can reproduce all results of the less general one by choosing some value of the parameter. On the contrary, when the limit is already taken one cannot return back from the less general theory to the more general one.

**must be finite**. Indeed, ℏ is the contraction parameter from quantum Lie algebra to the classical one, c is the contraction parameter from Poincare invariant quantum theory to Galilei invariant quantum theory, and R is the contraction parameter from dS and AdS quantum theories to Poincare invariant quantum theory.

## 3. Cosmological Acceleration as a Consequence of Quantum dS Symmetry

#### 3.1. Brief Overview of the Cosmological Constant and Dark Energy Problems

#### 3.2. A System of Two Bodies in Quantum dS Theory

## 4. What Mathematics Is More Pertinent for Describing Nature?

**Definition**that describes when theory A is more general than theory B and the latter is a special degenerated case of the former in the formal limit when a finite parameter in the former goes to zero or infinity. In the subsequent sections we prove that the same

**Definition**applies for the relation between finite quantum theory and finite mathematics on one hand, and standard quantum theory and classical mathematics on the other. Namely, the former theories are based on a ring or field with a finite characteristic p and the latter theories are special degenerated cases of the former ones in the formal limit $p\to \infty $.

## 5. The Problem of Potential vs. Actual Infinity

**Statement**

**1:**

**Definition**in Section 2, the ring ${R}_{p}$ is more general than Z, and Z is a special degenerated case of ${R}_{p}$.

**Definition**in Section 2 describes conditions when one theory is more general than the other. A question arises whether

**Definition**can be used for proving that finite mathematics is more general than classical one. As shown in Section 9, as a consequence of

**Statement 1**, quantum theory based on finite mathematics is more general than standard quantum theory based on classical mathematics. Since quantum theory is the most general physical theory (all other physical theories are special cases of quantum one), this implies that in applications finite mathematics is more pertinent than classical one and that

**Main Statement: even classical mathematics itself is a special degenerated case of finite mathematics in the formal limit when the characteristic of the field or ring in the latter goes to infinity**.

## 6. Remarks on Arithmetic

## 7. Remarks on Statement 1

**Statement 1**is the first stage in proving that finite mathematics is more general than classical one. Therefore this statement should not be based on results of classical mathematics. In particular, it should not be based on properties of the ring Z derived in classical mathematics. The statement should be proved by analogy with standard proof that a sequence of natural numbers $\left({a}_{n}\right)$ goes to infinity if $\forall M>0$$\exists {n}_{0}$ such that ${a}_{n}\ge M\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall n\ge {n}_{0}$. In particular, the proof should involve only potential infinity but not actual one.

**Statement 1**in the literature. However, the fact that Z can be treated as a limit of ${R}_{p}$ when $p\to \infty $ follows from a sophisticated construction called ultraproducts. As shown, for example, in References [39,40], infinite fields of zero characteristic (and Z) can be embedded in ultraproducts of finite fields. This fact can also be proved by using only rings (see e.g., Theorem 3.1 in Reference [41]). This is in the spirit of mentality of majority of mathematicians that sets with characteristic 0 are general, and for investigating those sets it is convenient to use properties of simpler sets of positive characteristics.

## 8. Proof of Statement 1

**Proposition**

**1.**

**Proof.**

**Statement 1**are satisfied. ☐

**Proposition**and

**Definition**,

**Statement 1**is valid, that is, the ring Z is the limit of the ring ${R}_{p}$ when $p\to \infty $ and Z is a special degenerated case of ${R}_{p}$.

## 9. Why Finite Mathematics Is More General Than Classical One

**Statement**

**2:**

**Statement**

**3:**

**Statement 3**.

**Definition**in Section 2 and

**Statements 1–3**,

- Standard quantum theory based on classical mathematics is a special degenerated case of FQT in the formal limit $p\to \infty $.
**Main Statement**formulated in Section 5 is valid.

## 10. Discussion

**Statement 1**that Z is a special degenerated case of the ring ${R}_{p}=(0,1,\dots p-1)$ in the formal limit $p\to \infty $, and the proof does not involve actual infinity. We did not succeed in finding such a proof in the literature and, as noted above, even in standard textbooks on classical mathematics, it is not even posed a problem whether Z can be treated as a limit of finite sets. As noted in Section 7, the fact that Z can be treated as a limit of ${R}_{p}$ in the formal limit $p\to \infty $ can be proved proceeding from ultraproducts and other sophisticated approaches. However, those approaches involve actual infinity and therefore they cannot be used in the proof that finite mathematics is more general than classical one.

**Main Statement**, finite mathematics is more general than classical one.

## Funding

## Acknowledgments

## Conflicts of Interest

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Lev, F.M.
Symmetries in Foundation of Quantum Theory and Mathematics. *Symmetry* **2020**, *12*, 409.
https://doi.org/10.3390/sym12030409

**AMA Style**

Lev FM.
Symmetries in Foundation of Quantum Theory and Mathematics. *Symmetry*. 2020; 12(3):409.
https://doi.org/10.3390/sym12030409

**Chicago/Turabian Style**

Lev, Felix M.
2020. "Symmetries in Foundation of Quantum Theory and Mathematics" *Symmetry* 12, no. 3: 409.
https://doi.org/10.3390/sym12030409