Warnock's Dilemma as an Eigenvector and/or Eigenvalue
As follows is a very-small-tweak to an idea in Consciousness: A Hyperspace View By Saul-Paul Sirag
First consider this: A quantum system is represented by a vector (called the state vector) which rotates in an abstract space (which may very well be infinite-dimensional). Note: a vector is an arrow-like entity with both length and direction. The rotation of the state vector is deterministic in the sense that if its postion is known at one time, its position at another time can be calculated. (It is just like a clock hand except that a clock hand is rotating in a 2-d space, whereas the quantum state vector might be rotating in a hyperspace.) As long as no measurement is made on the system, the state vector keeps rotating smoothly in the state space according to a deterministic equation called Schroedinger's equation; but, as soon as a measurement is made, the state vector immediately jumps to a vector (an eigenvector) corresponding to an allowed value (an eigenvalue) of the particular measurement that is being made. This jump is called the collagse of the wave function (another name for the state vector), but it is more appropirately described as the projection of the state vector onto an eigenvector belonging to a measurement. Most important: It is completely undetermined which eigenvector is projected out of the state vector by the measurement; however, we can calculate the probability of this projection, and we can verify this probability by repeating the measurement over and over or by making simultaneous measurements on a multitude of similarly prepared systems. Each type of measurement (e.g., a position measurement or a momentum measurement has its own set of allowed states (eigenvectors) which belong to allowed values (eigenvalues). The eigenvectors corresponding to a type of measurement provide a coordinate system for the space in which the state vector rotates (state space). Each type of measurement provides such a coordinate system.
Since a coordinate system is imposed on a vector space arbitrarily (by choosing a measurement), let's consider a special case, whereby the coordinate system is equivalent to point of indeterminacy (where the unanswered question exists), prior to the collapse of the wave function, within our space(s) containing Warnock's Dilemma.*
"A crucial question is: Under what conditions do different types of measurement provide identical coordinate systems for state space? And what is the consequence of this coincidence?" | Read more...
*The emphases were added above. Whereas, Saul-Paul suggests that we "consider a coordinate system as equivalent to a pont of view", if we may, we extend the concept to include a coordinate system as equivalent to a point of indeterminacy.