## Jordan algebras

In Kevin McCrimmon's excellent book *A Taste of Jordan Algebras* (2004) he describes how Pascual Jordan's study of quantum mechanics led to the definition of Jordan algebras. These have been of great importance in mathematics but it was eventually shown by Efim Zelmanov in 1983 that they they could never provide a setting for quantum mechanics. McCrimmon writes: |

In 1932 the physicist Pascual Jordan proposed a program to discover a new algebraic setting for quantum mechanics, which would be freed from dependence on an invisible all-determining metaphysical matrix structure, yet would enjoy all the same algebraic benefits as the highly successful Copenhagen model. He wished to study the intrinsic algebraic properties of hermitian matrices, to capture these properties in formal algebraic properties, and then to see what other possible non-matrix systems satisfied these axioms. The first step in analyzing the algebraic properties of hermitian matrices or operators was to decide what the basic observable operations were. There are many possible ways of combining hermitian matrices to get another hermitian matrix, but after some empirical experimentation Jordan decided that they could all be expressed in terms of quasi-multiplication

*x* · *y* := ^{1}/_{2} (*xy* + *yx*)

(we now call this symmetric bilinear product the Jordan product). Thus in addition to its observable linear structure as a real vector space, the model carried a basic observable product, quasi-multiplication. The next step in the empirical investigation of the algebraic properties enjoyed by the model was to decide what crucial formal axioms or laws the operations on hermitian matrices obey. Jordan thought the key law governing quasi-multiplication, besides its obvious commutativity, was

*x*^{2} · (*y* · *x*) = (*x*^{2} · *y*) · *x*

(we now call this equation of degree four in two variables the Jordan identity, in the sense of identical relation satisfied by all elements). Quasi-multiplication satisfied the additional "positivity" condition that a sum of squares never vanishes, which (in analogy with the recently-invented formally real fields) was called formal reality. The outcome of all this experimentation was a distillation of the algebraic essence of quantum mechanics into an axiomatically defined algebraic system. ... In a fundamental 1934 paper, Jordan, John von Neumann, and Eugene Wigner showed that every finite-dimensional formally real Jordan algebra is a direct sum of a finite number of simple ideals, and that there are only five basic types of simple building blocks ...

JOC/EFR July 2015
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