The Superposition Continuum
We’ll start with the idea, already well established in current quantum theory, that the state of a particle is represented by a complex vector. The dimension of this vector is equal to the number of possible states provided by the quantum system and can be infinite. As a special case, we’ll typically consider only four dimensions. In quantum theory, the magnitude of the vector is considered irrelevant, only the direction it points has physical meaning. It is a mathematically convenient way to indicate a “ray.” For this reason, quantum physicists are typically sloppy about normalization, for it can always be corrected for after the fact. While this is convenient mathematically, it is possible that this short cut is masking real physics. Therefore, we shall be anal about normalization.
Given a normalized vector to represent the state of a quantum particle, we will use a taxonomy to characterize the extent of an object’s superposition. This taxonomy will be called the superposition continuum, even though superposition is non denumerable. On one end will be a particle in a classical state, no superposition at all, and on the other end, a full equally weighted superposition amongst all the possible states. The first case will be called “classical” the second “flat.” In between will be the state where the particle is equally weighted between only two possibilities, we’ll call this one “spooky.” States in between classical and spooky, will be called “localized” and between spooky and flat, “diffuse.” Summarizing, the taxonomy is: classical, localized, spooky, diffuse, and flat. This is a loose and not entirely consistent taxonomy, but it serves a useful purpose; it allows us to concentrate on the essentials by suppressing some of the mathematical noise. A quantum object in a classical state in basis A will be in a flat state in any basis conjugate to A. If QTP is correct, any quantum object entangled with a cyclic entanglement will be in a spooky state in the collapse basis. In general, localized or diffused states indicate an uninteresting basis.
If we want to use the power of a spreadsheet to help explore the QTP hypothesis, we’ll need to be able to represent vectors in either rectilinear or polar coordinates, and then use cells to determine if the vector is normalized and if so, where it lies on the superposition continuum.
Tags: entanglement, qtp, quantum mechanics