Orthogonal, Complex, Symplectic to Unitary

Arnold in his book on "Mathematical Methods of Classical Mechanics" brings to attention the fact that on the space R^2N, the linear transformations that preserve the euclidean structure form the Orthogonal group O(2n), that preserve the symplectic structure form the Symplectic group Sp(2n), and that preserve the complex structure form the Complex linear group GL(n,C).

These three groups have a common intersection (equal to the intersection of any pair) which is the Unitary group U(n).

The extraordinary prediliction of Quantum mechanics in Unitary evolution hides this common feature. That is quantum mechanics could be interpreted as a special subset of three distinct pairs. There seems to be some evidence (due to the quaternion representations) that the Symplectic group has the germ of Fermion Spin. Is it possible that the Complex group has the germ of Boson spin? It is still a mystery to me how this all fits together.

However, recent interest in nano-circuit stadium oscillators and random matrices of the three group species given above appear to demonstrate that the Symplectic system is to be associated with chaotic trajectories. That is, the measured resonance frequencies for electrons in ellipsoidal stadia have differences that appear to be explained by random symplectic matrix analysis. !!

How to Dirac and Weyl spinors fit into these ideas? More Later.


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Last update 12/01/2000
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