Abstract
Brukner and Dakić (Quantum theory and beyond: is entanglement special? 2009. https://arxiv.org/abs/0911.0695) proposed a very simple axiom system as a foundation for quantum theory. It implies the qubit and quantum entanglement. Because this axiom system aims at the core of our understanding of nature, it must be brought to the forum of the philosophy of nature. For philosophical reasons, a completely denied champion of quantum theory, imaginarity i, is added into this axiom system. In relation to Bell’s inequality, this leads to a deeper ‘philosophical’ understanding of quantum nature based on qubits and entanglement. Both opens a way as well as one can get to the fundamental Schrödinger equation of quantum mechanics with the help of a complex valued Brownian motion.
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Notes
Schrödinger, Discussion of Probability Relations Between Separated Systems, Proceedings of the Cambridge Philosophical Society 31 (1935), pp. 555–563; 32 (1936), pp. 446–451.
With kind permission of CERN. Email from the Press Office from 9.12.2014.
The formula on the blackboard is a different form of representation.
I apologize for the triviality, but natural philosophy must be based on the most elementary premise of mathematical calculus, in the way how many mathematical entities can be derived from axiom systems.
For a more in-depth philosophical discussion of the qubit, I refer to the book of quantum physicist Thomas Görnitz (2016), especially chapters 9 and 10.
Classical minds could argue that the complex numbers would not be a unique characteristic of quantum physics, one can also work very well with them in classical terms. I suppose that is true. If the real numbers are a subset of the complexes number one can work with them naturally also in the complex domain. However, the representatives of the thesis of an equivalence with the real numbers should pay attention to what they do when they have lavished themselves on the honey of complex numbers. After the work is done, they only declare the real cosine to be ‘real’ and dispose of the imaginary sinus portion as ‘unreal’ in the Orcus. They are right to do so, because they confirm that only the real numbers are decisive for classical physics. They understand nothing of the quantum reality. Her wild claims that even the complex numbers are irrelevant are bordering on hubris or stupidity.
I don’t want to get any further work on the geometric algebra. This algebra, conceived by Hermann Grassmann, is beautiful and fits to a good Philosophy of Nature, but unfortunately it has been completely ‘messed up’ by the successors. One really ‘hates’ the imaginarity of this algebra and subsumes it under ‘pseudo-scalars’; moreover, one tries with its help to reinterpret quantum physics in a classical way. The physicist and mathematician David Hestenes is outstanding in this way.
For people with esoteric attitudes, I would like to draw attention to the fact that Brahman operates in a square of four square unitary bricks, which also form the foundation for every Hindu temple. Far more esoteric for physicists and mathematicians could appear my following considerations. The complex number \(z = a + ib\) can also be defined as the product of two ordered pairs \((a,b)\) or real numbers a and b. The product is defined as: \((a,b) \cdot (c,d) = (ac - bd,ad + bc) \Rightarrow z = a + ib\). What is not usually done can be written in a different way: \((a,b) \cdot (c,d) = (ac - bd,ad + bc) \Rightarrow ac + ad + bc - bd\). Now replace the two ordered pairs of numbers with \((a,a^{{\prime }} ) \cdot (b,b^{{\prime }} )\). Analogous to the previous development this results in \((a,a^{{\prime }} ) \cdot (b,b^{{\prime }} ) = ab + ab^{{\prime }} + a^{{\prime }} b - a^{{\prime }} b^{{\prime }}\). However, this leads to quantum entanglement as a violation of Bell's inequality \(ab + ab^{{\prime }} + a^{{\prime }} b - a^{{\prime }} b^{{\prime }} = 2\sqrt 2 = 2.828 \ldots\) Conclusion: The Qubit and the quantum entanglement are already contained in the complex number \(z = a + ib\) ! But it’s very, very hidden.
See for example Jean-François Le Gall (2016).
See for example Klaus Schulten from the University of Illinois at Urbana-Champaign: http://www.ks.uiuc.edu/~kschulte/.
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Paul Drechsel Retired: Priv. Keltenstr. 18a, 55130, Mainz, Germany.
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Drechsel, P. Foundation of Quantum Mechanics: Once Again. Found Sci 24, 375–389 (2019). https://doi.org/10.1007/s10699-018-9555-1
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DOI: https://doi.org/10.1007/s10699-018-9555-1