I am reading Quantum Theory: Concepts and Methods by Asher Peres. Terminology used in the book: $P_{\mu m}$ are &quot;transition probabilities&quot;. They are the squares of &quot;transition amplitudes&quot;. That is, \begin{equation}| C_{\mu m}|^2 = P_{\mu m}\end{equation} He says on page 40,  Up to this point , nothing was assumed that had any physical consequence. The phases of the transition amplitudes $C_{\mu m}$ still are irrelevant, and we could as well have stayed with the probabilities $P_{\mu m}$. It is only now that we introduce a new physical hypothesis (borrowed from classical wave theory): Law of Composition of Transition amplitudes. The phase of the transition amplitudes can be chosen in such a way that, if several paths are available from the initial state to the final outcome, and if the dynamical process leaves no trace allowing to distinguish which path was taken, the complete amplitude for the final outcome is the sum of the amplitudes for the various paths.  Then on page 43 he says  The amplitudes $C_{\mu m}$ should be considered as the primary, fundamental object, and the probabilities $P_{\mu m}$ should be derived from them, in spite of the fact that the $P_{\mu m}$ are the only quantities that are directly observable.  My Question I don't understand why we work with transition amplitudes instead of probabilities if the probabilities are the only direct obserables. Can someone explain why transition amplitudes are regarded as more fundamental?
Why is the application of probability in  (QM) fundamentally different from its application in other areas? QM applies probability according to the same  as in other areas of physics, engineering, etc.  Why is there a difference?  Naively one would assume one of these possibilities:   It is not the same probability (theory?) It is a matter of interpretation (of the formalism?) Something else?   Many answers (which I am still studying) focus on the fact that the combined probability of two mutually-exclusive events in QM is not equal to the sum of the probabilities of each event (which holds classically by definition). This fact (appears to) makes the formulation of another probability (a quantum one) a necessity.   Yet this again breaks down to assumed independent, if this is not so, the "classic probability" is applicable (as indeed in other areas). This is one of the main points of the question.
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