hat exactly is the experimental difference in applying the projection operator P to a spin 1/2 x system from that of its ortho-complement (i think that is the word) 1-P (1 being the identity operator) which itself is a projector of that same quantum state. I know that one converts the yes outcome, 'spin up' say, to a no outcome and vice versa (the no outcome in P corresponds to the yes outcome in 1-P of the same spin 1/2 particle which will be found as either spin or spin down but whether its designated as a yes or a no outcome depends on which of the two projectors are applied.
But is there an ontological difference, are they two distinct experimental contexts, or at least corresponding to two physically distinct ways of measuring the exact same quantum state?
And secondly are there any theories (probably speculative) out there which indicate whether there are possible counterfactual correlations between what result you get if you apply P, get a yes outcome (spin up), and whether in the 'morgenbesser sense' you still would have gotten a yes, had we used 1-P instead (and thus the opposite result, spin down) or a no (the same result, spin up)?
Moreover, if this is true or speculated to be true, is there any speculation or literature which theorizes as to whether this has anything to do with the particular probability values involved? (ie in this case the chance of yes for both P and 1-P are the same as the chance values for spin up and spin down here lets say are both 1/2 and thus equal)
This problem is connected with the life of otho- and parapositronium. It is shown experimentally that orthopositronium (the spins of both positrons are parallel) lives more long that parapositronium (spins are anti parallel). The lifetime of orthopositronium and rapositronium are 1,4x10-7 and 1.25x10-10 correspondingly. It means that the spin of orthopositronium posses the additional energy of rotation (summary spin = 1). The spin-energy of parapositronium equals zero. These experiments were realized my Michigan Group of researchers, repeated by Levin. See:, Borissova L.B., Rabounski D.D. Fields, Vacuum and the Mirror Universe. Editorial URSS, 2001.
In general a measurement on a quantum system can be represented as a POVM, i.e. a positive operator valued measure. A special case is a test where the measurement has two outcomes like 0 and 1. In this case the measurement can be represented by two positive operators P_0 and P_1 such that P_0 + P_1 = 1. Knowing one of the operators is the same as knowing the other operator because the sum is the identity. Therefore a test can be represented by a single positive operator satisfying that one minus the operator is positive. There is no difference between choosing P_0 and P_1 to make such representation. In the set of tests the extreme points are the simple measurements where P_0 and P_1 are projections.
If a measurement has three or more possible outcomes there exists extreme points in the set of measurements that cannot be represented by orthogonal projections.
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