Bracket conditions were formalized as quantum conditions for describing the non-commutative products of dynamical observables, known as Poisson brackets of quantum mechanics (QM). Before their appearances, continuous parenthetical conditions appeared throughout classical physics including general relativity.
In the real numbers domain, extreme bracket conditions are often indicated by the following: [-¥,¥], [0,¥], and [-¥,0] or parenthetically (-¥,¥), (0,¥), and (-¥,0). Unfortunately none of these could set realistic lower or upper bound of dynamical variables. When probability was introduced into QM, the bracket condition would have been [0,1]. However, the parenthetical one, (0,1) is more plausible since random variables of computational probability rarely take on values of exactly zeros or exactly ones. On the other hand, no one could hardly say that they have seen bracket conditions for negative probability as [-1,0] or (-1,0). They describe products of antiparticles as well as Dirac equation with its spinors. When these negative and positive conditions are combined together, the resulting Abelian conditions [-1,1] and (-1,1) could represent squares of energy.


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