Cellular Automaton Interpretation of Quantum Mechanics using Beables — Part 1
WOMANIUM Global Quantum Project Initiative — Mega Project -Winner of Global Quantum Media Project
Table of Contents:
01) Quantum Mechanics and Hilbert Space Reconsidered
02) Ising Model: Quantum Mechanics’ Non-Foundational Role
03) Harmonic Oscillator: Quantum-Classical Link
04) Neutrino Sheets: Unique Quantum Perspective
05) Automata Models: Quantum Insights
06) Cellular Automaton
07) Cellular Automaton Interpretation
08) Ontological Basis: Bridging Quantum and Classical
Introduction
In the endeavor to unravel intricate queries, an approach grounded in pragmatism frequently unveils valuable insights. This vantage point leads us into the realm of quantum mechanics, assuming a pivotal role within the tapestry of contemporary physics. Quantum mechanics, a product of human ingenuity, equips the discerning observer to decipher the intricacies underlying the environment.
Quantum Mechanics and Hilbert Space Reconsidered
Yet, a cautious stance prompts contemplation regarding the Hilbert space’s relationship with quantum mechanics — does it serve merely as a technical scaffold, veering away from an accurate portrayal of natural phenomena? In this contemplation, a parallel emerges with methodologies employed in resolving complex problems, where intricate mathematical enigmas are transformed into quantum field theories to extract classical solutions. This juxtaposition prods us to ponder the plausibility of quantum mechanics aligning with this framework, potentially assuming a non-foundational stance within the bedrock of physics.
Quantum mechanics functions as an indispensable tool for solving intricate equations. Complex numbers, resident within this toolkit, may not directly correspond to tangible reality; rather, they operate as mathematical aids. Analogous to the application of complex numbers in solving integral equations, real numbers can simplify mathematical handling by creating an abstraction of reality. However, this does not render them fundamental in the schema of nature’s laws.
The essence of nature may well be rooted in simplicity, exemplified by the enduring notion of information. Quantum black holes serve as a lens through which information processing becomes apparent. At the minutest scales, finite integers, rather than infinite or real numbers, hold sway.
Central to comprehension is the deconstruction of complex inquiries. Quantum mechanics serves as a tool to fathom the nuances of our surroundings. It is crucial to recognize that the Hilbert space does not inherently form part of nature; rather, it can be likened to a technical implement.
Ising Model: Quantum Mechanics’ Non-Foundational Role
The approach of the Ising Model, which transforms problems into quantum field theories, suggests that quantum mechanics might not constitute a foundational pillar. Complex numbers, while effective tools in solving intricate equations, do not necessarily mirror tangible reality. Similarly, real numbers, despite their role in mathematical simplification, do not inherently possess foundational significance.
The crux of nature’s essence could potentially be rooted in simplicity and information processing, mirroring phenomena observed in quantum black holes. When operating at small scales, finite integers take precedence over infinite or real numbers.
Harmonic Oscillator: Quantum-Classical Link
Our exploration commences with an introduction to the quintessential quantum mechanical system: the harmonic oscillator. Despite being traditionally associated with particles, its ambit extends to encompass fields or any oscillating entity. The harmonic oscillator fascinatingly transcends its conventional domain to describe anything manifesting periodic behavior.
Entities characterized by vibration inherently possess a period ‘t,’ conceivably residing within a parabolic potential well. The energy levels tied to these systems find facile derivation through the solution of the Schrödinger equation pertinent to the potential well. Intriguingly, these energy levels align with quantized values, entailing the integer multiplied by a constant factor ‘h’ times ‘μ,’ where ‘μ’ signifies the oscillator’s frequency and ‘h’ stands emblematic of the Planck constant.
The intrigue intensifies as the quantum model’s transformation into a classical archetype takes center stage. It often surprises observers to learn that the quantum harmonic oscillator, despite its intrinsic quantum nature, lends itself to a classical interpretation. Ponder a scenario wherein attention narrows to a finite assortment of energy levels, contrasting with the infinite sequence of ascending energy states.
Through the application of a discrete or finite Fourier transform to these ’N’ states, a compelling pattern materializes. These states, numbering ‘N,’ can be envisioned as points distributed along a circular trajectory. The oscillator’s evolution signifies that, following a fraction of its period, it transitions from one state to another akin to a discrete Fourier transform, encompassing this state-to-state progression. The comprehensive period of this circular motion aligns harmoniously with the harmonic oscillator’s period. This circular path demarcates segments, the quantity of which corresponds to the energy levels encapsulated within the finite Fourier transform.
Embarking on the trajectory towards an infinite scope engenders contemplation. As discrete transitions around the circle approach infinitude, the discrete segments meld into an unbroken circular continuum. In essence, a particle embarks on a rotational journey along this circular path, adhering faithfully to the harmonic oscillator’s period. Within this context, the particle’s journey unfolds with uniform constancy, echoing a distinct classical behaviour.
Astonishingly, even in the realm of finite objects, their dynamics evolve with a markedly classical character, unfolding through a sequence of discrete, classical phases across temporal dimensions. These transitions transpire devoid of dispersion or quantum superposition, indicative of an inherently classical nature.
This concept remarkably lays the foundation for the quantum harmonic oscillator to bridge the chasm between the quantum and classical domains. Recognizing the importance of comprehending transformation rules remains pivotal, a characteristic often encountered in similar circumstances. When ’n’ remains finite across temporal dimensions, it signifies a discretized mesh of temporal values. This underpins a completely deterministic system. Delving into intermediate temporal instances reveals the complete tapestry of quantum dynamics, thereby reaffirming the intricate interplay governing the quantum and classical realms.
In harmonic oscillator:
Within the framework of the quantum harmonic oscillator, a set of familiar operators emerges: x(t) and p(t), where ‘x’ denotes the deviation and ‘p’ represents the momentum of the oscillator. However, the distinctive aspect lies in the presence of an angle termed ‘ϕ’ or the label assigned to discrete states. These labels exhibit a classical evolution, resembling classical objects.
Termed ‘beables’ in honor of John Bell’s profound insights, these states bear a significant role in this context.
Operators, often regarded as observables, occasionally entail subjective uncertainty relations. However, the essence of these ‘beables’ contrasts remarkably. The values attributed to them are not reflective of subjective uncertainties; instead, they maintain sharp definiteness across temporal instances.
Such values are unequivocally defined, denoted as ‘script B’ to signify their nature as ‘beables.’ Notably, multiple ‘beables’ can coexist, sharing the fundamental property of commutation throughout.
This interplay results in a transformation where the intricacies of quantum mechanics fade, yielding a semblance of triviality or non-quantum mechanical behavior. Fundamentally, the concept underlying the quantum harmonic oscillator’s dynamics is remarkably elementary: an entity tracing periodic trajectories along a circular path. Importantly, this analytical approach can be extended to any naturally occurring periodic phenomenon, thereby establishing a set of ‘beables.’ Leveraging these ‘beables’ enables the establishment of an ontological basis within the Hilbert space.
Upon reframing the fundamental tenets governing the harmonic oscillator, the ontological basis takes prominence. Within this perspective, the progression unfolds deterministically, devoid of the complexities intrinsic to quantum mechanics. This naturally prompts inquiries concerning quantum superposition, a subject that will be explored subsequently. Remarkably, the notion arises that quantum superposition could potentially be a contrived complexity, distanced from the authentic physical realities that underlie natural processes.
Neutrino Sheets: Unique Quantum Perspective
Introducing a model that aligns with a comparable paradigm, we venture into the domain of massless chiral neutrinos. While on the surface they may appear unremarkable, these neutrinos possess intricate attributes warranting exploration. Acknowledging their non-interacting nature, one might dismiss these neutrinos as possessing limited physical significance. However, their undeniable place within the quantum framework as neutrinos, and their attendant characteristics, warrant careful consideration. Neutrinos, as quantum particles, are quintessential quanta of the neutrino field. It’s noteworthy that neutrinos are emitted individually in processes such as radioactive decay.
Neutrinos share parallels with electrons and other fundamental particles across various aspects. Yet, the intrigue intensifies when we scrutinize the scenario wherein neutrinos are massless — a supposition at odds with current understanding, as neutrinos indeed bear mass. In the scenario where neutrinos are massless and chiral, rotating exclusively along a single dimension, the foundational Dirac equation for neutrinos simplifies from four components to two. This reduction emerges from the constraint that neutrino spin can only manifest within one of two dimensions. Consequently, neutrinos assume two states, analogous to the electron nucleus.
Second Quantized Theory:
First Quantized Theory:
Critical to this discourse is the distinction between the first-quantized and second-quantized theories. The latter introduces the familiar field ‘psi,’ while the former encapsulates a single neutrino and its antiparticle in entirety. Comparatively more intricate than the harmonic oscillator’s beables, the beables associated with massless chiral neutrinos yield particular fascination. Among these, ‘p’ delineates the orientation of the momentum vector, devoid of its magnitude. This magnitude has been factored out. Functioning as a unit vector, ‘p’ aligns with or opposes the vector direction. ‘s’ denotes spin in alignment with the momentum direction ‘p’ — being positive when ‘p-hat’ coincides with ‘p’ and negative when they oppose each other. ‘r’ signifies the component of the neutrino’s position aligned with its momentum. Remarkably, these attributes coalesce into a set of beables, a claim that can be substantiated without undue complexity.
Substantiation involves demonstrating the commutativity of these beables at any given temporal point. Notably, two independent components of ‘p-hat’ mutually commute, while the commutation between ‘s’ and ‘p’ is discernible. Although the commutation between ‘r’ and ‘p-hat’ might appear intricate, it can indeed be demonstrated that ‘r’ commutes with ‘p-hat.’ Moreover, the temporal constancy of the commutativity of these operators arises from the straightforward equations of motion governing their evolution. Consequently, this consistency in commutation pervades temporal sequences. When considering the separate eigenstates of these operators, a foundation for an ontological basis encompassing the complete Hilbert space for a single neutrino becomes apparent.
This ontological basis’s physical interpretation is profound — it encapsulates the concept of a neutrino sheet, akin to a two-dimensional membrane. The sheet’s orientation along ‘p-hat’ decisively determines its position, confined to the longitudinal coordinate with transverse coordinates left undefined. The motion of the sheet manifests as rotation, a perpetual and unvarying phenomenon over time. In essence, the neutrino can be analogized to a classical sheet in unceasing motion at the speed of light along a singular trajectory.
In undertaking a calculation that initially appeared straightforward, a complex process unfolded, necessitating several days to formulate the correct expression. This endeavor entailed mapping the original operators ‘x’ and ‘sigma’ in terms of the coordinates {r, s, p-hat}.
Despite its apparent simplicity, the task proved more intricate than anticipated. The resultant complex expression mandated the involvement of the inverse of the ‘p-r’ operator — an integral operator demanding meticulous definition. This procedure engendered intricate expressions involving operators such as ‘L,’ responsible for the sheet’s rotation, and ‘s1’ and ‘s2,’ spin flip operators integral to the Pauli matrices. Operators ‘theta’ and ‘phi,’ contingent on ‘p-hat,’ also came into play. Challenges arose concerning the zeros of the ‘p-r’ operator, complicating the definition of ‘x,’ particularly in scenarios where functions are independent of ‘r.’
Within this context, the Hamiltonian of the first-quantized theory lacks a ground state, mirroring the aspects of Dirac’s theory. Remarkably, the second-quantized theory defies this trend, presenting a substantive ground state. Addressing the predicament posed by quantum mechanics in the absence of a ground state, Dirac proffered a profound solution — second quantization.
By considering both positive and negative energy states and designating the latter as antiparticle annihilation states, an elegant framework emerges. This juxtaposition underscores the dual nature of neutrinos as both particles and antiparticles. Although this approach yields the Hamiltonian’s ground state, it introduces an intriguing trade-off — an innumerable proliferation of these sheets across the cosmos. Each neutrino corresponds to a distinct sheet. Consequently, we encounter a quantum fermionic field governed by the Fermionic Dirac equation. It’s captivating to note that this quantum fermionic field finds equivalence in classical sheets, each in dynamic motion against an encompassing backdrop.
Automata Models: Quantum Insights
The classical models — the classical harmonic oscillator and neutrino sheets — find guidance in the concept of automata. These systems can be effectively emulated within a computational framework — a notion analogous to programming within a classical laptop. These automata manifest as systems describable through finite sets of states. As these state sets evolve following permutation rules, they can be expressed through an evolution operator, akin to a kind of automaton. The allure lies in the fact that this entire system can be simulated within a classical laptop. A straightforward illustration involves ’N’ states, governed by an evolution operator corresponding to permutations of these ’N’ states. This transition to a quantum theory involves a mathematical technique — representation of permutations in vector spaces.
Although this facilitates mathematical treatment, it’s important to recognize that this maneuver replaces the automaton with a quantum theory featuring a Hilbert space. In cases where systems evolve through an infinite number of states, the Hilbert space transmutes from finite to infinite dimensions. In this expanded space, the permutation operator assumes the form of an operator within an infinite-dimensional Hilbert space.
At its core, a foundational endeavor involves defining an operator ‘U’ dictating system evolution over a discrete time increment ‘delta.’ This operator can be cast as the exponential of another operator ‘A,’ wherein ‘A’ corresponds to the product of the Hamiltonian and ‘delta t.’ This encapsulates the dynamical essence of the system. Because ‘U’ emerges as a unitary operator — permutations within this framework correspond to unitary operations — an arrangement of states along a circular trajectory becomes tenable, prompting inquiries into their evolution. To extract the Hamiltonian from ‘U,’ the logarithm function is applied. Nonetheless, a challenge emerges due to the logarithm’s inherent multivalued character, resulting in diverse solutions.
The ensuing solution, though seemingly correct, possesses limitations — it pertains accurately to a solitary state. This complication reaches its zenith at the endpoints of the circular trajectory, where mathematical intricacies surface. These endpoints coincide with the points where the Hamiltonian reaches its extremities, reflecting either the lowest or highest eigenvalue. Consequently, while the highest eigenvalue aligns with the lowest eigenvalue within the Hilbert space, inconsistencies manifest for the lowest eigenstates. To address this disparity, a correction term involving -π and a delta function when the energy of the function equals zero is introduced.
Cellular Automaton
Similarly, a cellular automaton represents an extension of the automaton concept, with the distinction that its states are distributed across a lattice. This configuration bears some resemblance to the Ising model, although in this instance, the lattice can possess three dimensions or even infinite dimensions. In the limit where the lattice attains continuity, an analogy with the universe or the standard model arises — albeit with a crucial caveat. While the standard model is quantum mechanical, the universe, as conceived within this context, unfolds in a classical manner.
Intriguingly, the evolution of a cellular automaton can be conceptually visualized with time progressing vertically and space unfolding vertically as well. It is customary to assume that interactions transpire among nearest neighbors within the automaton. A notable exemplar of this concept is the “game of life,” where a lattice-defined universe adheres to specific evolution laws, offering the opportunity to observe outcomes under varying initial conditions.
An intriguing subset emerges when the evolution law takes on a distinct form that enables seamless progression both forward and backward in time. Consider a point labeled as ‘A’ on the graph: the cell’s contents evolve from A0 to A1, along with a contribution from the contents of cell B0. By formulating the evolution law in this manner, temporal progression and regression become equally attainable. However, this subset does not encompass the game of life, as it adheres to time-reversible cellular automata — a set of systems whose evolution laws can be inverted, transforming them into permutators. Consequently, these systems possess evolution operators that, when represented, align with unitary operators. As a result, their eigenvalues populate the unit circle.
Within this framework, operator A forms a unit circle expressed as e^-iA. Remarkably, along a specific line, all A-type operators commute. This observation extends to operator B, which is the sum of local terms B(x). The evolution operator distinguishes between A and B type cells, and intriguingly, A and B operators do not commute when their respective positions, x and x’, are neighboring. Transforming this into a Hamiltonian is facilitated by the Baker-Campbell-Hausdorff formula. If the series can be terminated, a local Hamiltonian reminiscent of the standard model ensues.
Baker-Campbell-Hausdorff formula:
Cellular Automaton Interpretation
Implicit in this discourse is the assumption of the existence of “beables,” yet their proof remains elusive. While for the standard model, the identification of beables remains uncertain, various theories bearing remote similarities to the standard model abound. Consequently, the interpretation hinges upon the presupposition of beables’ existence. If the universe’s Hamiltonian aligns with that of an automaton, it becomes plausible to identify observables termed “beables.”
These beables, Bi(t), constitute ordinary quantum operators bound by an equation. The eigenstates of Bi(t) at a given time ‘t’ form a basis known as the ontological basis. Although constructing an ontic basis within a quantum theory remains a challenge, the Cellular Automaton Interpretation (CAI) posits its existence. It asserts that ontic states can be constructed from conventional quantum states, thereby converging closely with an actual basis. When beables can be largely fashioned from known states, a classical “hidden variable theory” ensues — termed “templates.”
Ontological Basis: Bridging Quantum and Classical
With a defined set of beables and an ontological basis, any other quantum state belonging to a different basis can be expressed via quantum superposition of all beable states. This superposition, while a human construct, proves remarkably utilitarian, leading to the characterization of such states as “template states.” These states, employed in quantum mechanics, constitute the foundation for quantum calculations and are juxtaposed with ontological states. The interrelation between these states involves a unitary transformation, as templates represent quantum superpositions of ontic states and vice versa. Both categories adhere to the Schrödinger equation.
P.S. To be continued!
References
[1] G. ‘t Hooft, “The Cellular Automaton Interpretation of Quantum Mechanics,” Presentation at Tohoku University, Jan 21, 2016. [Online]. Available: https://www.youtube.com/watch?v=F3hPvusB0ds
[2] Bell, J. S. (1976). The theory of local beables. Epistemological Letters, 9, 11–24.
[3] I. T. Durham, “Bell’s Theory of Beables and the Concept of ‘Universe’,”
arXiv:1805.02143 [physics.hist-ph], May 2018.
