*Volume 4, Issue 1, pages 1-116*

Per Östborn [Show Biography]

We derive the Hilbert space formalism of quantum mechanics from epistemic principles. A key assumption is that a physical theory that relies on entities or distinctions that are unknowable in principle gives rise to wrong predictions. An epistemic formalism is developed, where concepts like individual and collective knowledge are used, and knowledge may be actual or potential. The physical state S corresponds to the collective potential knowledge. The state S is a subset of a state space S = {Z}, such that S always contains several elements Z, which correspond to unattainable states of complete potential knowledge of the world. The evolution of S cannot be determined in terms of the individual evolution of the elements Z, unlike the evolution of an ensemble in classical phase space. The evolution of S is described in terms of sequential time n belonging to N, which is updated according to n -> n+1 each time potential knowledge changes. In certain experimental contexts C, there is knowledge at the start of the experiment at time n that a given series of properties P, P’,… will be observed within a given time frame, meaning that a series of values p, p’,… of these properties will become known. At time n, it is just known that these values belong to predefined, finite sets {p},{p’},… In such a context C, it is possible to define a complex Hilbert space HC on top of S, in which the elements are contextual state vectors Sc. Born’s rule to calculate the probabilities to find the values p,p’,… is derived as the only generally applicable such rule. Also, we can associate a self-adjoint operator P with eigenvalues {p} to each property P observed within C. These operators obey [P, P'] = 0 if and only if the precise values of P and P’ are simultaneoulsy knowable. The existence of properties whose precise values are not simultaneously knowable follows from the hypothesis that collective potential knowledge is always incomplete, corresponding to the above-mentioned statement that S always contains several elements Z.

The author noticed an error in Section 3.3 in the discussion about a universal ordering of events, as described by sequential time n. It is written in the paper that two events A and B, perceived by two subjects k and k’, should be regarded as simultaneous whenever the separation between A and B is space-like. This is not always appropriate, as the following simple example shows.

Consider another event C, perceived by another subject k’’, and suppose that the separation between A and C is space-like just like that between A and B, but that the separation between B and C is time-like. If A and B are considered simultaneous, as well as A and C, then so should B and C. This is inappropriate, of course.

Even though space-like separation isn’t a sufficient condition for simultaneity, we assume that the question whether two events are simultaneous or not always has a definite answer. This is necessary in order to construct the universal sequential time n, which is at the core of the present reconstruction of quantum mechanics.

(It should be noted that simultaneity with respect to n, which is considered here, is not the same thing as simultaneity with respect to the time variable t that appears in relativity. Even though I argue that simultaneity with respect to n can have a universal meaning, simultaneity with respect to t cannot, of course. The relation between n and t is discussed in a follow-up paper: arXiv:1801.03396v2)