Redundancy of the Unitary|Hermitian Postulate

Summary

Chapter 8 shows the imaginary unit is a logically independent scalar. Insight into the mechanisms of indeterminacy reduces to understanding how and where Quantum Mathematics drives the necessity for this number's presence in the theory. This chapter eliminates fundamental symmetry as a source; and shows that (complex) unitarity arises logically independently, due to complementarity --- when required consistent.

Examination of the homogeneity of space shows that the Canonical Commutation Relation cannot be derived purely from homogeneity alone, but is a composite requiring extra information. That extra information is an imaginary unit, logically independent of the homogeneity symmetry, which arises from the demand that position and momentum spaces be complementary, as well as mutually consistent. The significance is that this single example contradicts the textbook doctine that symmetries underlying quantum systems are ontologically unitary, and instead, unitarity originates in the complementarity of mixed states.

The distinction is important because the a posteriori unitarity has logical consequences for Quantum Mathematics and Quantum Theory, which the a priori unitarity does not. Textbook Quantum Theory is unaware of this logic.

The effect of this wrong is to cover and obscure a pre-existing mathematical system of information, lying beneath, possessing intricate structure --- with an axiomatic homogeneous structure that hides all below. Historically, this a priori condition, falsely imposed on Quantum Mathematics, has been the obstacle blocking progress in the advancement of Indeterminacy Theory. If indeterminacy is to reveal itself in the mathematics, the a posteriori behaviour must be allowed to propagate its effects.