# Symmetry at the Foundation of Science and Nature

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## Abstract

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## 1. Introduction

- reproducibility,
- predictability,
- reduction,
- symmetry of evolution,
- symmetry of states,
- gauge symmetry,
- symmetry at the foundation of quantum theory.

- Possibility of change. It must be possible to perform a change, although the change does not actually have to be performed.
- Immunity. Some aspect of the situation would remain unchanged, if the change were performed.

## 2. Reproducibility

- Possibility of change. Changes belonging to the set associated with reproducibility can be performed on the experiment and its result.
- Immunity. The changed result remains the actual result of the changed experiment.

## 3. Predictability

- Possibility of change. The input to the experiment can be changed.
- Immunity. The experimental result maintains the same relation R with the experimental input.

## 4. Reduction

- Possibility of change. It is possible to make changes in the rest of Nature.
- Immunity. These changes do not affect important aspects of the part of Nature that can be individually understood.

- Possibility of change. We can change our observational activities.
- Immunity. The behavior of the rest of Nature is unaffected by and independent of such changes.

- Possibility of change. Observers can be changed.
- Immunity. The validity of the data is independent of the observer.

- Possibility of change. Changes can be made in the environment of a quasi-isolated system.
- Immunity. A quasi-isolated system possesses aspects that are not affected by certain such changes.

- Possibility of change. The state of a system, as an initial state, can be changed.
- Immunity. The law of evolution for the system is the same no matter what the system’s initial state.

## 5. Symmetry of Evolution

- Possibility of change. A process in a quasi-isolated system can be transformed to another process.
- Immunity. A physical process retains its physicality, while a hypothetical unphysical process remains unphysical.

## 6. Symmetry of States

- Possibility of change. A state of a quasi-isolated system can be transformed to another state of the system.
- Immunity. The object and image states are indistinguishable within the system.

## 7. Gauge Symmetry

- Possibility of change. A process in a quasi-isolated system can be transformed to another process by any gauge transformation of certain kinds.
- Immunity. A physical process retains its physicality, while a hypothetical unphysical process remains unphysical. In particular, a dynamic process related to a physical inertial process by such a gauge transformation is also physical.

## 8. Symmetry at the Foundation of Quantum Theory

#### 8.1. Association of a Hilbert space with a physical system

#### 8.2. Correspondence of observables to Hermitian operators

#### 8.3. Complete set of compatible observables

#### 8.4. Heisenberg commutation relations

#### 8.5. Operators for canonical variables

#### 8.6. A measurement result is an eigenvalue

#### 8.7. Expectation values and probabilities

#### 8.8. The Hamiltonian operator

#### 8.9. Planck’s constant as a parameter

#### 8.10. The correspondence principle

## 9. Science and Nature

## References

- Rosen, J. Symmetry Rules: How Science and Nature Are Founded on Symmetry; Springer-Verlag: Berlin, Germany, 2008. [Google Scholar]
- Rosen, J. The Symmetry Principle. Entropy
**2005**, 7, 308–313. [Google Scholar] [CrossRef]

© 2009 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Rosen, J.
Symmetry at the Foundation of Science and Nature. *Symmetry* **2009**, *1*, 3-9.
https://doi.org/10.3390/sym1010003

**AMA Style**

Rosen J.
Symmetry at the Foundation of Science and Nature. *Symmetry*. 2009; 1(1):3-9.
https://doi.org/10.3390/sym1010003

**Chicago/Turabian Style**

Rosen, Joe.
2009. "Symmetry at the Foundation of Science and Nature" *Symmetry* 1, no. 1: 3-9.
https://doi.org/10.3390/sym1010003