The wave equation on space time admits 3 species of waves: those that can be represented by real functions, those that can be represented by complex functions, and those that can be represented by quaternions. It is remarkable that the 3 associative division algebras can be put into correspondence with the three sets of characteristic solutions admitted by the wave equation. Remember, the characteristic solutions are those special solutions built on point sets that do not demand uniqueness. In other words, such characteristic solutions can represent propagating discontinuities.

In space time, the characteristic solutions satisfy the eikonal expression, which is a quadratic form with a specific signature. Fock showed that the relativistic signature is preserved by TWO types of transformations. The linear Lorentz group, and the non-linear projective fractional transformations. (The idea is that with respect to these groups, a discontinuity -e.g. a signal - to one observer looks like a discontinuity to another observer, where the second observer is related to the first by means of transformations of either of the two groups mentioned above). It is remarkable, that the second non-linear projective group does not require a finite limiting speed for the propagation of the discontinuity, or defect. Why this last concept has not been exploited in applied physics is beyond me.)

The solutions to to eikonal equation can be put into three equivalence classes defined by a certain group structures, but essentially related to the three types of waves: longitudinal, transverse, torsion. Consider the three groups:

The Orthogonal group O(2n) -- which preserves a euclidean structure.

The Complex group GL(n,C) -- which preserves a complex structure.

The Symplectic group Sp(2n) -- which preserves a symplectic structure.

The common intersection of these groups is the Unitary group U(n) -- which preserves the Hermitian scalar product of quantum mechanics.

Choose n=2,
then dim O(4) is 6 and is a compact Lie group.

then dim GL(2,C) is 8

then dim Sp(4) is 10 and is a non-compact Lie group.

and then dim U(2) is 4 and is a compact Lie group.

There has been a recent and remarkable interest in the application of the theory of random matrices and their effects on energy level statistics. The applications range from trying to understand phase transitions, such as the Metal to Insulator Transition, elementary particle structure, to behavior of nanometer circuitry. The idea is to study probability distributions built from random matrices that belong to the three groups mentioned above. There is a quantity called the Dyson index which is 1, 2, and 4 respectively for the Gaussian Orthogonal Ensemble (GOE), the Gaussian Complex Ensemble (GCE) and the Gaussian Symplectic ensemble (GSE).

This ideas are still in the formulation stage, but note the correspondence to real (longitudinal), complex (transverse), and symplectic (quaternionic) waves fields.

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