2016 International Workshop on Quantum Observers

The absolute and relative truth in quantum theory: the superposition principle

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    Jiri Soucek
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    Using concepts of the absolute and relative truth in quantum mechanics (QM) we obtain that the individual superposition principle is scientifically unfounded and, as a consequence, the solution to the basic quantum observer`s problem.

    We shall define the concept of a truth in the situation where different theories may have the same empirical content: (i) Two theories are empirically equivalent if their empirical predictions are identical. (ii) The QMversion is each theory which is empirically equivalent to QM.

    A recent example is the existence of the modified QM (introduced in J. Soucek, The Restoration of Locality: the axiomatic formulation of the Modified Quantum Mechanics, available at http://vixra.org/abs/1503.0109). The modified QM is theoretically different from QM but it gives the same empirical predictions as QM (this is proved in this paper).

    QMversions are empirically indistinguishable among them and thus they are equally verified by experiments. With respect to this fact we can define the concept of the absolute and relative truth: (i) The statement S is absolutely true in QM if S is true in each QMversion. (ii) The statement S is relatively true in QM if S is true in some QMversion but S is false in some other QMversion.

    It is clear that every empirical prediction of QM is absolutely true.

    There are following consequences of these concepts: (i) The absolutely true statement is scientifically well-founded. (ii) The relatively true statement is not scientifically well-founded and should be considered as the scientifically unfounded statement.
    Proof: (i) In this case the statement is true in all QMversions so that the choice of the QMversion is irrelevant. (ii) In this case the truth of the statement depends on the QMversion and since no QMversion is empirically preferred we have to conclude that this statement is unfounded.

    This means that only absolutely true statements can be considered as the scientifically well-founded statements.

    The empirical success of QM does not imply that the standard QM is true – it implies only that some QMversion may be true.
    Proof: All QMvesions are equally supported by the empirical success of QM. Then we cannot say that some of them is better supported than the other QMversion. Assuming that at least two QMversions are different (e.g. the standard QM and the modified QM) we cannot state that certain QMversion is true since there is no empirical criterion to discriminate among QMversions.

    This means that we are not able to choose the “right” QMversion (i.e. the choice based on the empirical evidence is impossible) since all QMversions are empirically equivalent.

    It is clear that the superposition principle is the main ingredient in the basic problem of quantum observers: it implies the possibility of the superposition of quantum observersstates. Based on the previous considerations of the absolute and relative truth in QM we shall argue that the individual superposition principle is scientifically unfounded (see the analyses described below). This implies that the problem of the superposition of observers states is a false problem.

    It is possible to define two forms of the superposition principle: the collective form and the individual form. By the collective state in QM we consider any possible state of an ensemble described by the density operator in the system`s Hilbert space.

    The pure state is a state generated by the vector from the Hilbert space. It is clear that the superposition of two pure states is generated by the linear combination of two vectors, so that the result is again the pure state. This implies that the collective superposition principle (i.e. for states of ensembles) is a trivial consequence of basic axioms of QM.

    The individual superposition principle requires the concept of an individual state. This is usually defined using the concept of a homogeneous ensemble. The ensemble is in a homogeneous state if all elements of this ensemble are in the same individual state (von Neumann ). This is clearly the circular definition, but we can suppose that there is a set of individual states (a subset of the set of pure states) such that each individual state can be considered as the state of some individual system.

    In the standard QM it is supposed that each pure state is an individual state. This means that each wave function describes a possible state of an individual system.

    Then the individual superposition principle says that the superposition of any two individual states is an individual state. It is clear that the individual superposition principle is satisfied in the standard QM.

    On the other hand, in the modified QM it is assumed that any two different individual states must be orthogonal (i.e. the set of all individual states is a subset of some orthogonal base). This implies that in the modified QM the individual superposition principle does not hold: any non-trivial superposition of two different individual states is not an individual state.

    Using the fact (proved in the paper cited above) that the modified QM is empirically equivalent to the standard QM we obtain the following theorem: (i) The individual superposition principle is the relatively true statement. (ii) The empirical success of QM does not imply the validity of the individual superposition principle.
    Proof: The individual superposition principle holds in the standard QM, but it does not hold in the modified QM which is the QMversion. The second statement is a consequence of the arguments given above.

    This means that the individual superposition principle is not a scientifically well-founded statement and it should be considered as an unfounded statement.

    The role of the individual superposition principle is essential in the considerations of quantum observers. The main problem for quantum observers is a consequence of the application of the individual superposition principle. The nullity of the individual superposition principle can solve the basic quantum observer`s problem.

    More details (and the discussion of the role of Occam`s razor) can be found in an attached paper.

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