Quantum mechanics and unitarity (part 1 of 4)
I will start a sequence of posts showing why quantum mechanics demands only unitary time evolution despite the collapse postulate and how to solve the problem. For reference, this is based on http://arxiv.org/abs/1305.3594
The quantum mechanics reconstruction project presented in http://arxiv.org/abs/1303.3935 shows that in the algebraic approach, the Leibniz identity plays a central and early role. But what is the Leibniz identity? It is the chain derivation rule: D(fg) = D(f) g + f D(g).
All standard calculus follows from this rule. For example using recursion one proves D(X^n) = n X^(n-1) and form this and the
series, the derivation rules for all usual functions follow.
In the algebraic formalism of quantum mechanics, the Leibniz identity corresponds in the state space to unitarity. Any breaking of unitarity means that Leibnitz identity is violated as well. This is the case for example in the epistemological interpretation of the wavefunction where the collapse postulate is understood as simply information update. However (and here is the big problem), breaking the Leibniz identity destroys the entire quantum mechanics formalism. In other words, any non-unitary time evolution is fatal for quantum mechanics.
So how can we understand the collapse postulate? Is quantum mechanics inconsistent? Should quantum mechanics be augmented by classical physics to describe the system and the measurement apparatus? From http://arxiv.org/abs/1303.3935 we know that there cannot be any consistent classical-quantum description of nature. Also the formalism which highlighted the problem shows the way out of the conundrum. Part 2 of the series will present preliminary mathematical structures which will be used to show how quantum mechanics can be fully unitary even during measurement.