Unitarity—which stipulates that the probabilities of all possible events add up to one—is one of the pillars of quantum mechanics. Yet over the last few years, significant advances have been achieved in "non-unitary quantum mechanics." The loss of unitarity is a necessary sacrifice to use a local field theoretic description for a problem that is initially non-local (such as percolation) or disordered (such as the whole Hall effect).
Even though the sum of the probabilities still equals one, some probabilities can now be negative. Moreover, events with a zero probability can no longer be discarded. Luckily, these difficulties are compensated by the appearance of symmetries known as "super-group symmetries." For instance, for processes involving as many bosons as fermions (i.e., particles with a zero or integral spin vs. particles with a half-integral spin), probabilities can vanish by compensation, but the events associated with these zero probabilities can still occur during intermediary steps.
As part of his Advanced ERC Grant, Hubert Saleur and his collaborators were able to achieve significant advances that broadened the understanding of corresponding field theories (especially in the conformal invariant case).
These advances have relied on and contributed to recent mathematical results in the theory of non-semi-simple associative algebras. A direct consequence of non-unitarity is the emergence of logarithmic terms inside correlation functions instead of the pure power laws in the unitary case.