In a recent article published in the journal AppliedMath, a new approach to understanding quantum mechanics (QM) was introduced, using a toy model over ℤ₂ to illustrate the concepts.
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 ℤ₂ 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 ℤ₂ = {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 ℤ₂ 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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