If such experimental verification would have not been successful, Born's proposal would have not been retained, and the Born rule would not be part today of the physics' manuals. Of course, Schrödinger's equation would also have, consequently, less empirical value, but nevertheless would still be considered to be physically relevant, for instance for the determination of the atomic energy levels, hence without necessarily relying on a Born rule (Schrödinger did not derive his equation as an equation for probability amplitudes). This reasoning proves that it makes no sense to speak about the Born rule without considering its validity in the description of the experimental data obtained from repeated measurements.

Now, when we state that in our hidden-measurement approach we 'derive' the Born rule, we mean by this that we add something that is fundamentally new to the standard quantum formalism (and consequently also to the PTI interpretation). More specifically, considering the hypothesis that (i) quantum indeterminism would be rooted in the presence of fluctuations at the level of the interaction between the measurement apparatus and the measured entity, and (ii) exploiting a representation of the states that is a natural generalization of that of the Bloch sphere, allowing for the 'calculation of the probabilities involved in these fluctuations', one obtains in a direct, unique a non-circular way the Born rule.

Concerning point (i), note that this hypothesis is already in part present in most interpretations of quantum mechanics, as it is generally assumed that a measurement can change the state of the measured entity, hence we only add to this standard view that fluctuations would be also present and play a role in what happens. Concerning point (ii), note that the 'extended Bloch representation' is deeply linked, at the mathematical level, to the complex Hilbert space geometry, which indicates that our model is most probably not ad hoc, but one with a content that, when further explored, might well reveal new aspects of the quantum reality; and we think it did so already, for instance when used to understand entanglement and interference effects. In that respect, see our recent article in the J. Math. Phys. 57, 122110, 2016)

The above does not presuppose anything about 'how quantum things truly are’. For example, it does not need to assume that the measurement induced change of state should necessarily be as in 'von Neumann’s second stage'. Also, it does not presuppose any of the deterministic and classical views that are typical of hidden-variables theories where the hidden variables are associated with the state of the entity. Indeed, although our hidden-measurements approach remains compatible with the view of a 'deterministic reality as a whole', it is certainly not a return to classical physics. This not only because the hidden-measurement interactions are non-spatial in nature (as explained in the article), but also because one cannot restore, not even in principle, determinism at the level of the measurement process.

Finally, we emphasize that our analysis does not weaken or contradict in whatsoever way the PTI. Quite on the contrary, by providing a general description of the weighted symmetry breaking process (only using mathematical structures that are already part of the quantum formalism), it shows that the weights appearing in the description of an incipient transaction can be consistently interpreted as outcome probabilities. In other words, our analysis, in a sense, confirms the validity of the PTI narrative, explaining how one can go from 'weights' to 'probabilities'. But this is not a trivial passage, as it requires the full power of the extended Bloch representation and the associated hidden-measurements interpretation.

Diederik Aerts & Massimiliano Sassoli de Bianchi

]]>Similarly, the author seems to have expectations for a derivation of the Born Rule that misconstrues the Rule as applying to a measurement process that leads deterministically to one outcome. But that is not what it is. The Born Rule is no more and no less than a rule for obtaining the probability of each outcome as a square of the probability amplitude for that outcome; as noted in the Stanford Encyclopedia of Philosophy, it is a “definition of the probabilities for measurement results.” (http://plato.stanford.edu/entries/qt-nvd/)

The author seems to be confusing the Born Rule with the second stage of von Neumann’s “Process 1,” in which the density operator manifesting the Born Rule (as weights of a sum of projection operators) collapses to a single projection operator. But that is not a part of the Rule.

To derive something means to trace the origin of that thing from a source (e.g., Merriam-Webster). The Transactional Interpretation obtains the quantities appearing in the Born Rule from a specific theoretical source--the direct action theory of fields. The weights of outcomes (that latter represented by projection operators) derived from the direct-action theory are the absolute squares of the Schrödinger probability amplitude. That is a derivation of the Born Rule. To say that it is not is to say that Schrödinger failed to ‘derive’ that his wave equation was an equation for a probability amplitude, which is to confuse physics with mathematics. All physical theories must have physical correlates, and those are a matter of empirical correspondence, not mathematical deduction.

Thus the author’s allegation that TI is incomplete is just a statement that it does not satisfy an optional metaphysical preference for a deterministic account that the author apparently does not recognize as optional. If the wave function is a probability amplitude (the square root of a probability), then an interpretation showing that each outcome is associated with the square of the probability amplitude is to show that that outcome is characterized by that probability—which is the entire content of the Born Rule. That is, TI clearly derives von Neumann’s “Process 1,” but the author apparently does not recognize von Neumann’s Process 1 transition as an expression of the Born Rule—at variance with all precedent in the peer-reviewed literature on this topic, as far as I know.

If the author wants to insist that TI does not derive the Born Rule in the sense that the square of the wave function corresponds to a probability, then he must also criticize Schrödinger for not ‘deriving’ that his equation was an equation for a probability amplitude. As noted above, the wave function has as its basic physical correlate a probability amplitude. That is not subject to mathematical derivation from any more basic principle, unless additional assumptions are imposed that might not even be correct, such as an epistemic interpretation of quantum probability--which is what the author appears to be assuming without recognizing it.

]]>