Abstract
The dynamics-from-permutations of classical Ising spins is generalized here for an arbitrarily long chain. This serves as an ontological model with discrete dynamics generated by pairwise exchange interactions defining the unitary update operator. The model incorporates a finite signal velocity and resembles in many aspects a discrete free field theory. We deduce the corresponding Hamiltonian operator and show that it generates an exact terminating Baker–Campbell–Hausdorff formula. Motivation for this study is provided by the Cellular Automaton Interpretation of Quantum Mechanics. We find that our ontological model, which is classical and deterministic, appears as if of quantum mechanical kind in an appropriate formal description. However, it is striking that (in principle arbitrarily) small deformations of the model turn it into a genuine quantum theory. This supports the view that quantum mechanics stems from an epistemic approach handling physical phenomena.
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Notes
Denoting \(\mathcal OS\) by \(|A\rangle ,\; |B\rangle ,\; |C\rangle ,\; |D\rangle , \dots \), for example, such a dynamics could be simply this:
$$\begin{aligned} |A\rangle \rightarrow |D\rangle \rightarrow |B\rangle \rightarrow \dots \;. \end{aligned}$$(1)This kind of evolution is the only possible one, unless the set of states itself changes, i.e., grows or shrinks. We do not consider a changing set of states here, however, this could be interesting when it comes to the evolving Universe.
Diagonal operators on this basis are beables and their eigenvalues characterize physical properties of the states, corresponding to the labels \(A,\; B,\; C, \dots \) used in the previous footnote.
The amplitudes that specify a \(\mathcal QS\) need interpretation, when describing experiments. By experience, relating amplitudes to probabilities has been extraordinarily useful. In this way, the Born rule is built in by definition! The Born rule can also be understood as a counting procedure related to a conserved two-time function of Hamiltonian cellular automata, which generalizes the norm of \(\mathcal QS\) [10]. It is is not forbidden by any element of quantum theory to abandon the proportionality between absolute values squared of complex amplitudes and probabilities, however, at the price of unnecessarily complicating its mathematical tools [1].
Usually, they are thought to describe limiting situations of quantum mechanics, e.g. in presence of environment induced decoherence. They constitute the realm of classical physics.
Using quantum superpositions of \(\mathcal OS\) to describe the initial state approximately, we obtain for an evolving \(\mathcal QS\) \(|Q\rangle \):
$$\begin{aligned}&|Q\rangle :=\alpha |A\rangle +\delta |D\rangle +\dots,\;|\alpha |^2+|\delta |^2 +\dots \; =1, \end{aligned}$$(2)$$\begin{aligned}&\quad \text{ then }, |Q\rangle \;\longrightarrow \; \alpha |D\rangle +\delta |B\rangle +\dots . \end{aligned}$$(3)We see that the amplitudes remain the initial ones, while the \(\mathcal OS\) evolve by permutations, in the chosen example according to (1) of footnote 1.
This does not imply that quantum mechanical superposition states are to be avoided. On the contrary, part of the motivation for CAI is to better understand, why they are so extremely effective in describing experiments probing nature.
On a historical note, we add here that the discussion of ontological vs. epistemological approaches to the building of theories of physics has an intense precursor in times when Newtonian physics was superseded by field theories of forces in the sequel of Maxwell’s electrodynamics [15, 16]. The fine distinction between qualitatively different types of theories or theory building seems to have been lost during the rapid developments leading to Quantum Mechanics in the sequel. Attention to this has been drawn by Khrennikov, indicating possible consequences for the persisting interpretational problems of quantum theory [17].
The following considerations work with other boundary conditions as well. However, periodic boundary conditions turn out to yield the most transparent picture of the dynamics.
Shifting by an odd integer, instead, converts leftmovers and rightmovers into each other.
Overall, beginning with \({\hat{U}}\) itself on the first line, the relations produce also an interesting functional equation for \({\hat{U}}\).
Which is one motivation for the search of physics beyond the Standard Model, namely to yield stronger constraints on the large set of its parameters.
...who might suffer with us from the all-too-often blurred terminology when it comes to the foundations of quantum theory.
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It is a great pleasure to thank Andrei Khrennikov for instigating the writing of this paper.
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Elze, HT. Are Quantum Spins but Small Perturbations of Ontological Ising Spins?. Found Phys 50, 1875–1893 (2020). https://doi.org/10.1007/s10701-020-00370-4
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DOI: https://doi.org/10.1007/s10701-020-00370-4