I wonder if you would apply the same criticism to so-called “derivations” of quantum theory from information theoretic principles, specifically those which work within the environment of general probabilistic theories. For example:
The above links, despite having perhaps overly strong titles, are fairly clear about what assumptions are made, and what is derived. These assumptions are more than simply uncertainty and robust reproducibility: e.g. one assumption that is made by all the above links is that any two pure states are linked by a reversible transformation (in the first link, a slightly modified version of this is assumed). Of course, “pure state” and “reversible transformation” are well-defined concepts within the general probabilistic framework which generalize the meaning of the terms in quantum theory.
Since this research is closely related to my PhD, I feel compelled to give an answer your questions about uncertainty relations and complex numbers in this context. General probabilistic theories provide an abstracted formalism for discussing experiments in terms of measurement choices and outcomes. Essentially any physical theory that predicts probabilities for experimental outcomes (a “prediction calculus” if you like) occupies a place within that formalism, including the complex Hilbert space paradigm of quantum theory. The idea is to whittle down, by means of minimal reasonabe assumptions, the full class of general probabilistic theories until one ends up with the theory that corresponds to quantum theory. What you then have is a prediction calculus equivalent to that of complex Hilbert space quantum theory. In short, complex numbers aren’t directly derived from the assumptions; rather, they can be seen simply as part of a less intuitive representation of the same prediction calculus. Uncertainty relations can of course be deduced from the general probabilistic theory if desired, but since they are not part of the actual postulates of quantum theory, there hasn’t been much point in doing so. It bears mentioning that this “whittling down process” has so far been achieved only for finite-dimensional quantum theory, as far as I’m aware, although there is work being done on the infinite-dimensional case.
I have no problem with alternative derivations of quantum theory—if they are correct! But the framework in this paper is too weak to qualify. Look at their definition of ‘category 3a’ models. They are sort of suggesting that quantum mechanics is the appropriate prediction calculus or framework for reasoning, for anything matching that description.
But in fact category 3a also includes scenarios which are completely classical. At best, they have defined a class of prediction calculi which includes quantum mechanics as a special case, but then go on to claim that this definition is the whole story about QM.
Different formalisms may be more or less convenient for reasoning about certain concepts. Of course there is a reason physicists keep using complex numbers in QM.
I wonder if you would apply the same criticism to so-called “derivations” of quantum theory from information theoretic principles, specifically those which work within the environment of general probabilistic theories. For example:
http://arxiv.org/abs/1011.6451 ; http://arxiv.org/abs/1004.1483 ; http://arxiv.org/abs/quantph/0101012
The above links, despite having perhaps overly strong titles, are fairly clear about what assumptions are made, and what is derived. These assumptions are more than simply uncertainty and robust reproducibility: e.g. one assumption that is made by all the above links is that any two pure states are linked by a reversible transformation (in the first link, a slightly modified version of this is assumed). Of course, “pure state” and “reversible transformation” are well-defined concepts within the general probabilistic framework which generalize the meaning of the terms in quantum theory.
Since this research is closely related to my PhD, I feel compelled to give an answer your questions about uncertainty relations and complex numbers in this context. General probabilistic theories provide an abstracted formalism for discussing experiments in terms of measurement choices and outcomes. Essentially any physical theory that predicts probabilities for experimental outcomes (a “prediction calculus” if you like) occupies a place within that formalism, including the complex Hilbert space paradigm of quantum theory. The idea is to whittle down, by means of minimal reasonabe assumptions, the full class of general probabilistic theories until one ends up with the theory that corresponds to quantum theory. What you then have is a prediction calculus equivalent to that of complex Hilbert space quantum theory. In short, complex numbers aren’t directly derived from the assumptions; rather, they can be seen simply as part of a less intuitive representation of the same prediction calculus. Uncertainty relations can of course be deduced from the general probabilistic theory if desired, but since they are not part of the actual postulates of quantum theory, there hasn’t been much point in doing so. It bears mentioning that this “whittling down process” has so far been achieved only for finite-dimensional quantum theory, as far as I’m aware, although there is work being done on the infinite-dimensional case.
I have no problem with alternative derivations of quantum theory—if they are correct! But the framework in this paper is too weak to qualify. Look at their definition of ‘category 3a’ models. They are sort of suggesting that quantum mechanics is the appropriate prediction calculus or framework for reasoning, for anything matching that description.
But in fact category 3a also includes scenarios which are completely classical. At best, they have defined a class of prediction calculi which includes quantum mechanics as a special case, but then go on to claim that this definition is the whole story about QM.
There is nothing special about complex numbers in quantum mechanics. You can get rid of them by adding an extra dimension to the Hilbert space.
That doesn’t seem true without causing SU(n) to lose its privileged status as the transformation group on quantum states.
Different formalisms may be more or less convenient for reasoning about certain concepts. Of course there is a reason physicists keep using complex numbers in QM.