MATHEMATICAL STRUCTURE OF IDENTITY

Introduction.

A way of speaking about something can be inspirational. Metaphor is a powerful tool of understanding. We explore a way of speaking about the application of group theory to describe the physical world.

The mathematical theory of groups is what gives precision and substance to the word symmetry - itself a metaphor. To link this to identity we examine another metaphor - sameness under altered scrutiny. This suggestive verbal structure conforms to all the elemental rules by which the mathematics is used in examining physical systems.

Both symmetry and identity derive from examining things. From its role in physics the empirical meaning of symmetry emerges. It relates quite directly to examining the physical world. The meaning of identity is then found to be operationally the same as that for symmetry.

Particle physics has long recognized that particles are identified by their symmetry properties. It is the universality of the connection between symmetry and identity that is explored here.^{1}

In what follows Sections 1 to 6 lay the conceptual basis. In sections 6 and 7 the equivalence of symmetry and identity emerges. The remaining sections show, in detail, how the metaphors match the mathematical structure. The nature of non-visual symmetry is explored. Degeneracy is discussed. The quest for hidden symmetries emerges as a metaphor for understanding.

1. The System and the Observer

In the orthodox quantum view of nature the two key elements to consider are the system and the observer. The system is what is studied by the observer. It is something accessible to measurement. The observer makes measurements on the system.

The difficulty about the concept lies in the matter of isolation; that the system not be coupled to the rest of the universe. Evidently such systems are not to be found in nature. It is an idealization. Coupling is a matter of degree; never equal to zero. We follow tradition on the matter assuming coupling weak enough to qualify as zero.^{2}

An isolated system is always to be found in a 'state'. The state of a system is defined by the ultimate set of measurement results on it. These are eigenvalues of each of a complete set of commuting operators^{3}; those representing compatible measurement results. It is a set of numbers. We label that set of numbers by a single integer, n. A state of the system is designated by |n›, using standard Dirac notation .^{4}

The observer makes the measurements. He is equipped with a battery of instruments. The observer's eyes are one such instrument but he will usually have others - like clocks, meter sticks, electrometers, particle detectors, etc. In accordance with von Neumann's principle of Psychophysical Parallelism^{5}, it doesn't make any difference whether the instrument is within the observer's body or not. He scrutinizes the system by recording the measurement results his instruments read. Making sure that all his measurements are compatible he assigns to a particular collection of his measurement results a particular value of n. He concludes that the system is in the state |n›.

Continue to 2. Altered Scrutiny

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© m chester, August 1999, Occidental, CA

chester@physics.ucla.edu