Jordan Algebra - Formally Real Jordan Algebras

Formally Real Jordan Algebras

A (possibly nonassociative) algebra over the real numbers is said to be formally real if it satisfies the property that a sum of n squares can only vanish if each one vanishes individually. In 1932, Pascual Jordan attempted to axiomatize quantum theory by saying that the algebra of observables of any quantum system should be a formally real algebra which is commutative (xy = yx) and power-associative (the associative law holds for products involving only x, so that powers of any element x are unambiguously defined). He proved that any such algebra is a Jordan algebra.

Not every Jordan algebra is formally real, but Jordan, Neumann & Wigner (1934) classified the finite dimensional formally real Jordan algebras. Every formally real Jordan algebra can be written as a direct sum of so-called simple ones, which are not themselves direct sums in a nontrivial way. In finite dimensions, the simple formally real Jordan algebras come in four infinite families, together with one exceptional case:

• The Jordan algebra of n×n self-adjoint real matrices, as above.
• The Jordan algebra of n×n self-adjoint complex matrices, as above.
• The Jordan algebra of n×n self-adjoint quaternionic matrices. as above.
• The Jordan algebra freely generated by Rn with the relations

where the right-hand side is defined using the usual inner product on Rn. This is sometimes called a spin factor or a Jordan algebra of Clifford type.

• The Jordan algebra of 3×3 self-adjoint octonionic matrices, as above (an exceptional Jordan algebra called the Albert algebra).

Of these possibilities, so far it appears that nature makes use only of the n×n complex matrices as algebras of observables. However, the spin factors play a role in special relativity, and all the formally real Jordan algebras are related to projective geometry.