Quantum mechanics promoted position vector into an operator expressed as the generalized ��-dimensional tuple (��) in a multidimensional configuration mathematical phase space. Consequently, together with the generalized momentum operator (��) they become the observables as conjugate variables of one of the fundamental equations of quantum mechanics. However, the basic assumption needed is that both �� and �� must be in the same direction in order for this given equation to be true: ����-����=��ℎ/2�� which is intrinsically non-commutative. In 1925, it was Werner Heisenberg then Max Born and Pascual Jordan whose triple collaboration and the independently alternative formulation of Paul Dirac that ushered in the new language of quantum operators. This promotion of position vectors into quantum operators extended the concept of three dimensional vectors into real multidimensional tensors. Moreover, in the complex domain these were transformed into the multidimensional spinors.

All these are properly described by using the algebras of real or complex matrices. Their applications allowed the creations of Pauli’s 2 by 2 spin matrices and the 4 by 4 Dirac matrices. Further extensions into the mathematics of quantum field theories allowed the creations of orthogonal matrices whose transposes equal their inverses such that special orthogonal matrices ����(��) are the ones with positive unit determinants. More abstract extensions made possible the algebras of unitary operators and matrices wherein special unitary matrices ����(��) having positive unit determinants become the symmetry groups of the standard model of elementary particles of high energy physics. Unfortunately, all these become possible if and only if position vector is demoted from being an operator back down into a parameter since parameterized time could never be theoretically promoted to an operator without destroying the reality of the space-time parameter. The hidden variable attributes of all parameters are the properties of directional invariance in contrast to properties of gauge invariance or phase invariance.