Natural geometric representation for electron local observables

An existence of the quartic identities for the electron local observables that define orthogonality relations for the 3D quantities quadratic in the electron observables is found. It is shown that the joint solution of the quartic and bilinear identities for the electron observables defines a unique natural representation of the observables. In the natural representation the vector type electron local observables have well-defined fixed positions with respect to a local 3D orthogonal reference frame. It is shown that the natural representation of the electron local observables can be defined in six different forms depending on a choice of the orthogonal unit vectors. The natural representation is used to determine the functional dependence of the electron wave functions on the local observables valid for any shape of the electron wave packet.