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Unveiling Quantum Mechanics through Set-Level Mathematics

In a recent article published in the journal AppliedMath, a new approach to understanding quantum mechanics (QM) was introduced, using a toy model over ℤ to illustrate the concepts.

Unveiling Quantum Mechanics through Set-Level Mathematics
Study: Unveiling Quantum Mechanics through Set-Level Mathematics. Image Credit: Robsonphoto/Shutterstock.com

The New Approach

In this study, the author proposed a novel approach to QM by demonstrating that QM's distinctive mathematical formalism can be seen as the linearization of the mathematics of partitions on a set. This mathematical framework is used to represent distinctions/inequivalences and indistinctions/equivalences at the set level.

The paper elaborates on this new approach by using the vector space over the mathematics of partitions in its ℤ form. The result is a non-relativistic, finite-dimensional toy model referred to as "quantum mechanics over sets" (QM/Sets). The main goal of this model is to provide pedagogical insights into some of QM's complex aspects using the simplest possible calculations (modulo 2) where 1 + 1 = 0.

This model aims to intuitively illustrate the typical oddities and paradoxes of QM, such as the double-slit experiment, without relying on the wave-interpreted mathematics over complex numbers (ℂ). In the model, integers modulo 2 are represented as ℤ = {0, 1}, where vectors denoted by 0 and 1 are interpreted as sets, and the rules for addition and multiplication are uniquely defined so that 1 + 1 = 0.

In the QM/Sets toy model, Dirac brackets take on natural values, representing the cardinality of set overlaps. When probabilities are introduced via density matrices, real numbers are used, creating a more intricate model for depicting quantum phenomena.

The key concepts of partitions on a set include logical-level notions for modeling indistinctions versus distinctions, indefiniteness versus definiteness, or indistinguishability versus distinguishability. These concepts are critical for comprehending the QM's non-classical 'weirdness'. In QM, the primary non-classical notion is superposition, which is the notion of a state that is indefinite between two or more eigen- or definite states.

Vector Spaces over Z2

A vector space was formed using ℤ by employing columns of 1s and 0s as the vectors. For instance, the column vectors are added component-wise, with each of the third, second, or first components adding to the other vector modulo 2's corresponding component in the three-dimensional (3D) vector space of column vectors like Z23.

Every component is viewed as the absence or presence of an element of a three-element set like U = {a, b, c} for interpreting these 3D column vectors in a meaningful way. Thus, the above addition operation would be {a, b} + {b, c} = {a, c}. Such addition on sets is known as the symmetric difference. The author used this set interpretation of Z23/Z2n in general for the n-dimensional case of QM/Sets.

In quantum interpretation, the multiple-element subsets and single-element/singleton subsets represent superposition states/indefinite states of the quantum particle and eigenstates or definite states of a quantum particle, respectively. No state is represented by the empty set/zero vector. Definite states like {c}, {b}, or {a} form the basis for the vector space, as all other states/subsets can be derived by sums of them.

Double-Slit Experiment in QM/Sets

The author considered a setup where the three states in U = {a, b, c} primarily stand for the vertical positions for modeling the necessary aspects. A particle was sent from {b} to a screen having two slits at positions {c} and {a}. One time period took the particle to the screen, and the next time period took it to the wall.

In the first case, the superposition state {a, c} was reduced to {a} or to {c} with 1/2 probability upon detection at the slits. Subsequently, {a} evolved to {a'} = {a, b} and hit the detection wall at {b} or {a} with 1/2 probability, or {c} evolved to {b, c} and hit the wall at {c} or {b} with 1/2 probability in the next time period.

In the second case, the superposition state {a, c} evolved as a superposition/indefinite state as no state reduction occurred at the slits with no detection at the slits. The interference pattern's stripes characteristic was {a, b} + {b, c} = {a, c} without detection at the slits.

In this case, the evolution happened at a lower level/a level of indefiniteness, where the states {a, c} remained indistinguishable. Classical evolution takes definite states to definite states, as every state is distinguished in classical physics. 

Overall, the simplified pedagogical model could allow the use of a lattice of partitions to assign an intuitive image to the classical world of entirely distinguished states and the quantum ‘underworld’ of indefinite states.

Journal Reference

Ellerman, D. (2024). A New Approach to Understanding Quantum Mechanics: Illustrated Using a Pedagogical Model over ℤ2. AppliedMath, 4(2), 468-494. https://doi.org/10.3390/appliedmath4020025, https://www.mdpi.com/2673-9909/4/2/25

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Samudrapom Dam

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Samudrapom Dam

Samudrapom Dam is a freelance scientific and business writer based in Kolkata, India. He has been writing articles related to business and scientific topics for more than one and a half years. He has extensive experience in writing about advanced technologies, information technology, machinery, metals and metal products, clean technologies, finance and banking, automotive, household products, and the aerospace industry. He is passionate about the latest developments in advanced technologies, the ways these developments can be implemented in a real-world situation, and how these developments can positively impact common people.

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