A Baker–Campbell–Hausdorff formula for the logarithm of permutations

Elze, Hans‐Thomas

doi:10.1142/s0219887820500528

Cited by 6 publications

(18 citation statements)

References 13 publications

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“…Another particular case is the permutation operator P23 P12 P34 applied to a system of four Ising spins: in [13] a similar procedure was followed and another BCH formula has been obtained.…”

Section: A Bch Formula With a Qubit Hamiltonian For Bitsmentioning

confidence: 99%

How perturbing a classical 3-spin chain can lead to quantum features

Rizzo

2020

Preprint

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It is well known that Quantum Mechanics (QM) describes a world that follows completely counter intuitive rules, that has famous paradoxes as postulates (e.g. superposition of states, wave/particle dualism, the measurement problem...), but that at the same time it is the foundation for the description of the physical world that we experience. While in the Copenhagen Interpretation these paradoxes are fully accepted as a reality, there are other interpretations that try to find a compromise, between the classic macroscopic world and the QM results. A quite interesting one is the Cellular Automata Interpretation of QM, by Gerard 't Hooft [1], according to whom particles evolve following the rules of Cellular Automata (CA), a mathematical model consisting of discrete units that evolve following deterministic laws in discrete space and time. The states of a Cellular Automaton are, by definition, classical and thus deterministic and do not form superpositions: they are identified, in this interpretation, as Ontological States. These states are the ones in which microscopic particles find themselves in reality, whether they are observed or not. Unfortunately, it is not always possible to determine exactly the Ontological States. To overcome this obstacle, the formalism of QM works as a "template" with which a scientist can work and get reliable results even without fully knowing what is happening at the microscopic level. The problem of measuring these states is solved by considering the observer and his/her instruments as a separate physical system which is equally deterministic, so the act of measurement is interpreted as the interaction between the two systems.Finally, we will illustrate how this classical system of three Ising spins that evolves exclusively through Ontological States behaves like a quantum system, evolving through superposition states, in particular. This is an example of the potential implications of the CA Interpretation of Quantum Mechanics. Consequences of perturbations on a classical chain of Ising spins3.

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Section: A Bch Formula With a Qubit Hamiltonian For Bitsmentioning

confidence: 99%

How perturbing a classical 3-spin chain can lead to quantum features

Rizzo

2020

Preprint

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“…In this way, the Born rule is built in by definition! -The Born rule can also be understood as a counting It will often be difficult to relate unitary evolution of OS by permutations to a familiar looking Hamiltonian operator, in particular in presence of interactions [1,2,3,4,8,9,10]. Which makes the accessible spin chain model of Section 3 interesting.…”

Section: A Synopsis Of the Cellular Automaton Interpretationmentioning

confidence: 99%

“…Also classical states need to be defined in relation to OS. 4 For CAI, classical states belong to deterministic macroscopic systems, including billiard balls, pointers of apparatus, planets, etc., i.e., situations where large numbers of ontological states must be involved.…”

Section: A Synopsis Of the Cellular Automaton Interpretationmentioning

confidence: 99%

“…-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]. 4 Usually, they are thought to describe limiting situations of quantum mechanics, e.g. in presence of environment induced decoherence.…”

Section: A Synopsis Of the Cellular Automaton Interpretationmentioning

confidence: 99%

“…We have earlier studied simple three-and four-body models consisting of interacting two-state Ising spins, in order to illustrate the approach of CAI in some detail [3,4]. Presently, our aim is to generalize this for the case of arbitrarily long Ising spin chains.…”

Section: Introductionmentioning

confidence: 99%

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Are quantum spins but small perturbations of ontological Ising spins?

Elze

2020

Preprint

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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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Are Quantum Spins but Small Perturbations of Ontological Ising Spins?

Elze

2020

Found Phys

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A Baker–Campbell–Hausdorff formula for the logarithm of permutations

Cited by 6 publications

References 13 publications

How perturbing a classical 3-spin chain can lead to quantum features

How perturbing a classical 3-spin chain can lead to quantum features

Are quantum spins but small perturbations of ontological Ising spins?

Are Quantum Spins but Small Perturbations of Ontological Ising Spins?

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