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It is usually said that unitarity is necessary for the consistent probabilistic interpretation. But is that really so? Suppose that ##|\psi(t)\rangle## does not evolve unitarily with time, so that ##\langle\psi(t)|\psi(t)\rangle## changes with time. Even then one can propose that probability ##p_k## is given by the formula
$$p_k(t)=\frac{\langle\psi(t)|\hat{\pi}_k|\psi(t)\rangle}{\langle\psi(t)|\psi(t)\rangle}$$
where ##\hat{\pi}_k## is a projector. Indeed, if the projectors obey ##\sum_k \hat{\pi}_k =1## (which does not depend on unitarity of the time evolution), then the sum of probabilities obeys
$$\sum_k p_k(t) =1$$
despite nonunitarity. So is unitarity really necessary, and if it is, why exactly is it so?
$$p_k(t)=\frac{\langle\psi(t)|\hat{\pi}_k|\psi(t)\rangle}{\langle\psi(t)|\psi(t)\rangle}$$
where ##\hat{\pi}_k## is a projector. Indeed, if the projectors obey ##\sum_k \hat{\pi}_k =1## (which does not depend on unitarity of the time evolution), then the sum of probabilities obeys
$$\sum_k p_k(t) =1$$
despite nonunitarity. So is unitarity really necessary, and if it is, why exactly is it so?