Probability, geometry, and irreversibility in quantum mechanics
Quantum mechanics is in very good accordance with observations. A natural question -- started by Einstein -- is whether unusual features of quantum mechanics (such as the impossibility to measure a state exactly) follow from these observations, or whether another classical (``hidden variables'') theory is possible that can explain all these observations. This question was answered when J. Bell discovered experimentally testable inequalities that are valid for quantum mechanics (but not for any classical theory), and when Bell's inequalities were experimentally confirmed. From the mathematical viewpoint, the situation with the 1964 Bell's inequality remained somewhat unsatisfactory: while quantum physics in general is closely related with operator theory, this physically important inequality is a stand-our result, with no visible relation to any relevant math. The authors shows that Bell-style inequalities can be naturally reformulated in terms of a relatively new area of operator theory -- operator trigonometry. Its main notion -- the cosine \(\cos(A)\) of an operator \(A\) -- is defined as the largest angle through which \(A\) can turn a vector (i.e., the largest angle between \(x\) and \(Ax\)). Since an eigenvector is defined as the vector for which the angle between \(x\) and \(Ax\) is the smallest (namely, 0), the vector \(x\) corresponding to \(\cos(A)\) is also called an antieigenvector (there is also a related notion of an antieigenvalue). It turns out that Bell's inequality is a particular case of more general inequalities between cosines of different operators and related characteristics from the operator trigonometry. It also turns out that Bell's inequality is not the only physical application of operator trigonometry; its formulas can also clarify the physical description of CP symmetry violation in kaon decay, in which the (somewhat heuristically) proposed combinations of kaons turn out to be antieigenvectors of the corresponding operator.
scientific article; zbMATH DE number 1755371
scientific article