Why an Interpretation Is Needed
The discovery of quantum mechanics stands as a pivotal scientific revolution of the 20th century, fundamentally shaping our understanding of the atomic and subatomic realms This branch of physics governs not only the behavior of particles but also provides essential insights into solid state physics, chemistry, thermodynamics, and various radiation phenomena, all of which are intricately linked to the principles of quantum theory.
The successes of quantum mechanics are phenomenal, and furthermore, the theory appears to be reigned by marvellous and impeccable internal mathematical logic.
The remarkable scientific breakthrough has intrigued not only scientists from various disciplines but also philosophers and the general public Despite nearly a century passing, physicists remain divided on the implications of the theory and its limitations regarding our understanding of reality.
5 At various places in this book, we explain what is wrong with those ‘few lines’.
Quantum mechanics excels in its applications because it focuses on the outcomes of experiments rather than the underlying nature of reality This theory provides precise expectations for experimental results, their statistical distributions, and insights into internal parameters, particularly concerning elementary particles The Standard Model, a key framework in quantum mechanics, identifies around 25 fundamental constants of Nature that current knowledge cannot predict While many of these parameters can be determined with varying degrees of accuracy from experimental data, quantum mechanics consistently delivers accurate predictions.
Quantum mechanics is considered one of the most significant breakthroughs in physics, fundamentally transforming our comprehension of the atomic and sub-atomic realms.
Physics remains an ongoing journey, as the elusive Theory of Everything has yet to be uncovered despite optimistic claims at the turn of the century Numerous unanswered questions highlight that physicists still have work to do Inspired by past achievements, scientists are actively designing new experiments and developing innovative theories, integrating knowledge from previous discoveries into their evolving ideas.
When considering our strategic approach to completing the final pieces of our puzzle, we must ask ourselves which paths to take and what we envision those pieces to resemble A critical question arises: should we anticipate that the ultimate theory of the future will be rooted in quantum mechanics?
In the realm of scientific inquiry, differing opinions among researchers are not only expected but welcomed, as they inspire deeper exploration into overlooked areas This book delves into the 'reality' of quantum mechanics, proposing that its true nature may diverge significantly from traditional textbook interpretations We suggest that this reality could be simpler than commonly perceived, potentially enhancing our pursuit of a clearer theoretical understanding.
This treatise presents fundamental ideas that, while not new, have been overlooked despite their simplicity and clarity Historically, these observations have been articulated numerous times, both in recent and ancient contexts, yet they have often been dismissed outright.
Quantum mechanics is often referred to as a 'theory,' but it may be more accurately described as a framework that aids in developing theories for particle interactions and sub-systems This distinction is important to avoid confusion regarding the term 'theory' in the context of quantum mechanics.
Classical deterministic models have been largely dismissed due to their failure to replicate the elegance and effectiveness of quantum mechanics Attempts to adapt these models resulted in complex theories that lacked the simplicity of quantum mechanics Quantum field theory, which describes relativistic sub-atomic particles, adheres to essential principles like causality, locality, and unitarity, forming the foundation of the successful Standard Model Efforts to derive quantum field theory from a deterministic framework often necessitate sacrificing one of these fundamental properties, undermining the inherent beauty of the accepted theory Consequently, the pursuit of a classical underlying theory is frequently abandoned in favor of the straightforwardness of quantum mechanics.
The assumption of a classical world beneath quantum mechanics appears both unnecessary and impossible Researchers often dismiss this notion, yet before moving on, it's worth exploring whether deterministic models designed to replicate quantum effects consistently lead to contradictions This investigation could ultimately demonstrate the impossibility of such models, potentially providing a definitive closure on the topic.
A way to do this was to address the famous Gedanken experiment designed by
Einstein, Podolsky, and Rosen proposed that quantum particles possess characteristics beyond a mere wave function, indicating that to accurately represent 'reality' in quantum mechanics, the existence of 'hidden variables' is necessary.
What could be done was to prove that such hidden variables are self-contradictory.
We call this a ‘no-go theorem’ The most notorious, and most basic, example was
Bell's theorem investigates the correlations between measurements of entangled particles, revealing that when the initial state of these particles is sufficiently generic, the correlations predicted by quantum mechanics cannot be replicated by classical information carriers This conclusion is articulated through Bell inequalities, later expanded into the CHSH inequality, which classical systems adhere to but which quantum mechanics significantly violates Consequently, this leads to the unavoidable conclusion that classical, local, realistic theories are not viable.
So why the present treatise? Almost every day, we receive mail from amateur physicists telling us why established science is all wrong, and what they think a
The concept of a "theory of everything" has evolved significantly, and I do not intend to undermine the century-long advancements in quantum mechanics Rather, I believe that the extensive research conducted over the past hundred years has yielded remarkable discoveries The challenge lies not in the accuracy of the equations or the technology employed, but in the need for a more radical and precise articulation of the findings, particularly regarding what is commonly known as the Copenhagen interpretation.
Interpretation, should be replaced Up to now, the theory of quantum mechanics
The motivation behind this work is to establish rigorous rules connecting wave function amplitudes to the probabilities of various experimental outcomes, emphasizing that these amplitudes do not reflect 'what is really happening.' Instead of seeking an underlying reality, one should focus on the predictions of experimental results The notion that no definitive 'reality' exists may seem enigmatic, but the goal is to eliminate any mysticism from quantum theory and derive concrete facts about reality in the process.
Outline of the Ideas Exposed in Part I
We will begin with a simple and straightforward approach, acknowledging that some may find it overly basic Our focus will be on a classical system that resembles our universe, allowing us to refine our model as necessary Key questions will arise, such as the necessity of non-local interactions, the potential for information loss, and whether to incorporate a version of gravitational force Ultimately, the success of our project hinges on exploration and experimentation.
The modest price we pay involves selecting a unique set of mutually orthogonal states in Hilbert space, designated as 'real.' These states represent the only configurations the universe can occupy, with one state chosen at any given moment, while all others have a probability of zero We refer to these as ontological states, which form the ontological basis of Hilbert space Although this may seem merely semantic, we posit that this special basis possesses unique properties, suggesting that the resulting quantum theories are a distinct subset of all possible theories This perspective could pave the way for new physics, as our goal extends beyond merely reinterpreting quantum mechanics; we aim to uncover innovative tools for model building.
Despite a seemingly precarious relationship with standard quantum mechanics and its interpretations, our approach surprisingly yields consistent results Remarkably, we can construct several models that accurately replicate quantum mechanics without any modifications, as detailed in Part II These straightforward models have garnered numerous positive responses, affirming their effectiveness.
Critics of our work often claim that our models are not genuine quantum mechanics, but this is a misunderstanding; they are indeed quantum mechanical However, it is valid to point out that our models may be overly simplistic, failing to capture the complexities of interacting particles, or they may have other subtle flaws Specifically, creating realistic quantum models for locally interacting particles has proven to be a significant challenge Achieving this requires not only a complete, renormalizable quantum field theoretical model but also potentially a perfectly quantized version of gravitational forces, which explains the difficulty we face in this endeavor.
Despite numerous attempts to challenge Bell's arguments, most falsification claims have been dismissed Now, we aim to investigate the source of the perceived disagreement between our models and quantum mechanics We will explore whether the issue lies within our classical models or in Bell's assertions Specifically, we will analyze what might go wrong in a Bell experiment involving entangled particles and whether any assumptions made in our models fail to hold true Through this inquiry, we hope to contribute to the ongoing discourse surrounding quantum mechanics and locality.
This study aims to explore essential physical principles that are closely aligned with the foundational theories of classical mechanics By incorporating discrete kinetic variables more often than traditional approaches, we reveal that these models share significant similarities with quantum mechanics.
In many cases, they are quantum mechanical, but also classical at the same time.
Some of our models occupy a domain in between classical and quantum mechanics, a domain often thought to be empty.
Will this lead to a revolutionary alternative view on what quantum mechanics is?
The challenges related to the energy sign and the locality of effective Hamiltonians in our theories remain unresolved In reality, a lower bound exists for total energy, indicating the presence of a vacuum state The complexities surrounding this issue will be addressed in Part II, as they necessitate thorough calculations In conclusion, we believe there are multiple potential solutions to navigate this difficulty.
10 1 Motivation for This Work still, that it can be used to explain some of the apparent contradictions in quantum mechanics.
This treatise does not provide definitive answers to all questions, but it highlights significant observations regarding Bell's theorem and its assumptions, which, while seemingly reasonable, may need to be reconsidered Our theory remains incomplete, and those who oppose our perspective might find flaws that challenge its validity.
I hope, will be inspired to continue along this path.
This article invites readers to explore new perspectives on the deeper meanings of quantum mechanics by developing models and performing calculations Unlike previous models that often rely on complex concepts like infinite universes or modified quantum equations, our approach maintains the coherence of the original equations We suggest that Einstein's skepticism towards Bohr and Heisenberg may have merit, proposing that nature might not be fundamentally random, but rather governed by consistent laws Every event in the universe occurs for a reason, dictated by physical laws rather than chance While Bell's inequalities challenge this view, particularly regarding locality, we aim to address these critical questions throughout the discussion.
While my arguments may appear lengthy, the core elements of my reasoning are straightforward and concise To ensure that each chapter of this book stands alone and remains easily comprehensible, some repetition of ideas is inevitable, for which I apologize Additionally, certain calculations are presented at a basic level to enhance accessibility for a broader audience of scientists and students.
Dirac's bra-ket formalism is the most elegant approach to understanding quantum mechanics in its entirety Hilbert space serves as a fundamental tool in physics, extending beyond quantum mechanics to apply in a variety of systems, including those that are not typically represented by standard quantum models like the hydrogen atom Additionally, Hilbert space can be utilized in completely deterministic frameworks, such as Newton's description of planetary systems.
In modeling within Hilbert space, the initial step involves selecting a basis To effectively describe the dynamics of the system, a Hamiltonian is essential A notable characteristic of Hilbert space is the flexibility to utilize any preferred basis for analysis.
A valuable lesson from life experiences is that lengthy arguments can often be less credible than concise ones Additionally, transitioning from one basis to another is a unitary transformation, which we will frequently utilize The information provided in Sections 1.6, 3.1, and 11.3 is entirely standard and widely accepted.
In Part Iof the book, we describe the philosophy of the Cellular Automaton
Interpretation (CAI) without too many technical calculations After the Introduction, we first demonstrate the most basic prototype of a model, the Cogwheel Model, in
Chapters 3 and 4 focus on the interpretation of quantum mechanics, specifically examining the Copenhagen Interpretation This section highlights key aspects, particularly the significance of the Bell and CHSH inequalities, providing insights into our analysis of these foundational concepts in quantum theory.
In this article, we aim to clarify our perspective on deterministic quantum mechanics The Cellular Automaton Interpretation of quantum mechanics, discussed in Chapters 4 and 5, may be perceived as controversial by some physicists, but this stems from our departure from commonly accepted assumptions in the field.
Most notably, it is the assumption that space-like correlations in the beables of this world cannot possibly generate the ‘conspiracy’ that seems to be required to violate
Bell’s inequality We derive the existence of such correlations.
A 19th Century Philosophy
In the 19th century, envision a scenario where mathematics is highly advanced, yet 20th-century physics remains unknown Imagine a hypothesis suggesting that the universe operates as a cellular automaton, defined by sequences of integers and governed by a classical algorithm for time evolution This automaton, while lacking any quantum mechanics, can function as a universal computer, capable of solving any mathematical equation at its smallest scales However, predicting its behavior over large time intervals proves too complex, leading mathematicians to consider alternative approaches, such as making statistical statements regarding its large-scale behavior.
Initially, one might perceive only white noise, yet a deeper analysis reveals significant correlations within the integer series Some correlation functions can be computed similarly to those in a Van der Waals gas, where individual molecular trajectories remain unpredictable Instead, we can determine properties like free energy, pressure, and viscosity based on density and temperature This approach mirrors the work of 19th-century mathematicians applying cellular automaton models In this book, we will explore how 20th and 21st-century physicists and mathematicians can leverage quantum mechanics to uncover more complex details, although they too must acknowledge the limitations of exact calculations Ultimately, the laws they derive will predominantly be statistical, allowing for predictions of average experimental outcomes rather than precise results.
E Fredkin, a specialist in numerical computation techniques, engaged in extensive discussions about the concept, which has roots that date back much further However, accurately interpreting the results of individual experiments remains challenging due to the complexity of the evolution equations involved.
In our imagined 19th-century world, effective laws appear to govern reality while incorporating a significant element of randomness This suggests that alongside deterministic laws, unpredictable random number generators may be influencing outcomes Consequently, these combined effective laws bear a striking resemblance to the quantum mechanical principles we currently understand for sub-atomic particles.
The Van der Waals gas, while it adheres to general equations of state and can explain the behavior of sound waves, does not exhibit quantum mechanical properties This discrepancy may stem from the fundamental microscopic laws underlying a Van der Waals gas, which differ significantly from those of a cellular automaton However, it remains unclear if these differences are enough to account for the absence of quantum mechanics in the behavior of the Van der Waals gas.
One clear takeaway from this reasoning is that the presence of effective laws requiring a stochastic element, akin to a perfect random number generator, is not surprising 19th-century physicists would have welcomed the mathematical insights of their time and would have been well-prepared for the 20th-century discoveries, which revealed that the laws governing hydrogen atoms indeed incorporate a stochastic component, such as the precise moment an excited atom emits a photon.
The fundamental philosophical reason behind quantum mechanics may lie in the limitations of extrapolating the features of a cellular automaton that underpins our universe to larger scales While this chapter presents a non-technical overview and may not fully capture the complexities of quantum theory, it highlights key elements of the narrative we aim to convey If 19th-century physicists had access to contemporary mathematics, they might have derived an effective quantum theory based on their automaton model The intriguing question of whether these physicists could have conducted experiments with entangled photons will be addressed in Section 3.6 and beyond.
In the 19th century, while the theory of atoms emerged as a significant breakthrough in understanding matter, the concepts of energy, momentum, and angular momentum remained perceived as continuous This raises the question: could these quantities also be discrete? The advent of quantum mechanics revealed insights into the discrete nature of the physical world, yet space and time are still treated as continuous This prompts speculation about a future discovery that could reveal a fundamentally discrete and deterministic framework underlying current physical theories, a notion supported by thinkers like Fredkin.
14 1 Motivation for This Work mechanics as we know it today, is the imperfect logic resulting from an incomplete discretization 9
Brief History of the Cellular Automaton
A cellular automaton is a mathematical model that simplifies physical systems by representing variables as discrete integers on a one- or multi-dimensional grid Each position on this grid, known as a 'cell', is identified by a series of integer coordinates At regular intervals, or "beats of a clock," the values within these cells are updated based on specific rules that reflect physical laws Typically, the new values for each cell depend on its previous state and the states of its neighboring cells, a characteristic known as 'locality'.
The concept of self-replicating structures in a physical world was first introduced by John von Neumann and Stanislaw Ulam in the 1940s, focusing on the emergence of life from simple evolutionary laws This topic gained significant attention in the 1970s when John Conway created Conway's Game of Life, an automaton operating on a two-dimensional grid In this system, each rectangular grid cell interacts with eight neighbors, and can exist in one of two states: 0 or 1, representing the basic laws of evolution in this model.
‘dead’ and ‘alive’ At each beat of the clock every cell was renewed as follows:
– Any live cell with fewer than 2 of its 8 neighbours alive, will die, “as if caused by loneliness”;
– Any live cell with 2 or 3 live neighbours lives on to the next generation;
– Any live cell with more than 3 live neighbours will die, “as if by over-population”; – Any dead cell with exactly 3 live neighbours becomes alive, “as if by reproduc- tion”.
The initial state of the system can vary, and with each tick of the clock, every cell is renewed based on specific rules This process allows for the continuous evolution of the entire system Although the grid is theoretically infinite, it is also possible to explore various boundary conditions.
The game gained popularity following Martin Gardner's description in the October 1970 issue of Scientific American During this period, physicists observed the evolution of these automata on computers, recognizing that the "game of life" could act as a basic model for an evolving universe inhabited by living creatures.
As I compose this, I anticipate receiving many letters from amateurs; however, it's crucial to recognize that while suggesting a simplistic or arbitrary theory may be easy, identifying a robust theory that accurately interprets the complexities of our world through rigorous mathematics is significantly more challenging.
Research has revealed that certain structures can maintain stability or exhibit periodic behavior when surrounded by empty cells Additionally, some configurations, known as "gliders" or "space ships," are capable of moving in horizontal, vertical, or diagonal directions.
Research indicates that simple fundamental laws of physics can give rise to complex systems, potentially leading to the emergence of consciousness and free will Cellular automata are categorized into different classes based on their global properties; some quickly stabilize or become chaotic, while the most intriguing class 4 automata evolve into recognizable structures with increasing complexity These class 4 automata are believed to be capable of executing complex calculations.
Many intriguing instances, like the 'game of life,' demonstrate that certain processes are not time-reversible, as multiple initial configurations can result in the same end state.
At first glance, many physical laws at the atomic level appear time-reversible, making them seem less intriguing for physics However, our study reveals the significance of time non-reversibility in cellular automata, particularly through models like the "game of life." Notably, many members of the interesting class 4 are not time-reversible, suggesting that time non-reversibility may contribute an intriguing form of stability to these systems, enhancing their relevance in physics Further exploration of time non-reversibility will be discussed in Chapter 7.
Cellular automata serve as effective models for physical systems, including liquids and complex particle mixtures There is growing interest in exploring cellular automata as foundational theories of physics, raising the question of whether the fundamental principles of physics are rooted in discrete laws This concept was first introduced by Konrad Zuse in 1967, marking a significant step in understanding the relationship between cellular automata and the laws governing physical phenomena.
In "Rechnender Raum" (Calculating Space), the author proposes that the universe operates as the result of deterministic computational laws within an automaton This concept gains credibility when observing that fundamental particles, particularly fermions, can be likened to bits of information in motion, especially when represented in coordinate form.
John Archibald Wheeler coined the phrase “It from Bit” to express the notion that particles of matter (“it”) are intrinsically linked to the information they convey, which is essential for their characterization (“bit”).
An extensive study of the role of cellular automata as models for addressing sci- entific questions was made by Stephen Wolfram in his book A New Kind of Science
Wolfram's approach is grounded in a unique philosophy that highlights the complexity and computational universality of cellular automata, suggesting that they can provide insights into physical systems While one might view our book as a continuation of Wolfram's groundbreaking research, we aim to explore broader questions, particularly focusing on the origins of quantum mechanical phenomena, which may extend beyond the restrictive models he considers.
Both Zuse and Wolfram already speculated that quantum mechanical behaviour should be explained in terms of cellular automata, but did not really attempt to get
This article explores the motivation behind explaining quantum mechanics through the lens of cellular automata It raises the question of whether a specific type of automaton is required or if any automaton can eventually exhibit quantum mechanical behavior.
Computational scientists have explored various aspects of cellular automata, but this study will focus on generic states rather than specific initial conditions influenced by quantum mechanics.
Modern Thoughts About Quantum Mechanics
The advent of quantum mechanics has profoundly transformed our understanding of reality, leaving even esteemed physicists like Richard Feynman perplexed, as he noted, “I think I can safely say that nobody today understands quantum mechanics.” Despite this confusion, one undeniable aspect is that the theory is remarkably coherent and aligns exceptionally well with experimental results.
Finding a cellular automaton evolution law that generates the particles of the Standard Model and their interaction characteristics seems unlikely in the near future Current understanding can be viewed as a dictionary of information, where particles symbolize information that is transmitted and processed This processing manifests as quantum mechanical information, characterized by superpositions of eigenstates of observables If one system of information carriers could be perfectly transformed into another with different processing rules, it would be impossible to determine which system is superior.
In the context of cellular automaton systems, we may encounter various classes where identifying a specific element that accurately represents our reality becomes challenging This concept is articulated by David Deutsch in his constructor theory, emphasizing the importance of distinguishing between different physical systems.
This book proposes that at least one element within certain classes should be identified as a classical automaton, although this step is often overlooked Instead, a 'many world' interpretation frequently appears to be unavoidable Furthermore, the notion that even minor non-linear modifications to the Schrödinger equation are necessary to account for the wave function's collapse remains prevalent The density matrix derived from the Schrödinger equation includes off-diagonal terms that, regardless of their rapid oscillation or unstable phases, require a mechanism for their complete elimination However, our theory demonstrates that such a mechanism is unnecessary.
A poll conducted by A Zeilinger et al revealed significant insights from conference participants on the foundations of quantum mechanics While the questions may have been biased, the majority agreed that quantum information fundamentally differs from classical information and rejected the idea of an underlying deterministic theory Most participants viewed Einstein's criticism of Bohr's quantum mechanics as misguided This book aims to demonstrate that a deterministic underpinning theory could be feasible and suggests necessary amendments to the Copenhagen doctrine, acknowledging Bohr's pragmatic correctness A summary of these views is provided in Reference [109], along with preliminary explorations by the author.
Most researchers support the concept of freedom of choice, which allows observers to select which properties of a system to measure at any time Zeilinger argues that this freedom can be assured in experiments; however, we contend that strong spacelike correlations may significantly limit this freedom By precisely defining freedom of choice, we replace the notion of 'free will' with a more mathematically rigorous concept While observers can choose their settings, correlation functions dictate the ontological states of observed entities, such as elementary photons, in a non-local manner Consequently, observer choices must align with the correlation functions defined by physical laws, which are local, whereas the correlations themselves are not This interplay between correlation functions and observer choices may illuminate some of the perplexities surrounding quantum mechanics.
Notation
In this book, quantum mechanics serves primarily as a toolkit rather than a standalone theory We will utilize various mathematical tools, including Hilbert space, to facilitate our exploration While we expect readers to have a foundational understanding of these concepts, we will provide a brief overview of Hilbert space for clarity.
Hilbert space is a complex vector space that typically has an infinite number of dimensions, although it can also be finite The elements within this space are referred to as states, commonly represented by symbols such as |ψ, |ϕ, or other "kets."
We have linearity: whenever|ψ 1andψ 2are states in our Hilbert space, then
Some critical readers question the origin of complex numbers in quantum mechanics, given their classical roots The reality is that complex numbers, like real numbers, are human inventions In Hilbert space, they serve as valuable tools for discussing conserved quantities, such as baryon number, and for diagonalizing Hamiltonians While quantum mechanics can be expressed without complex numbers by considering the Hamiltonian as an anti-symmetric matrix, this leads to imaginary eigenvalues Ultimately, imaginary numbers are essential for mathematical operations, making them indispensable in the realm of physics.
In this work, we explore the relationship between complex numbers λ and μ within a Hilbert space, where each ket-state |ψ has a corresponding conjugated bra-state ψ| This relationship indicates that if the equation (1.1) is satisfied, then the conjugated state ϕ| can be expressed as a linear combination of the states ψ1| and ψ2|, specifically ϕ| = λ*ψ1| + μ*ψ2|.
Furthermore, we have an inner product, or inproduct: if we have a bra,χ|, and a ket,|ψ, then a complex number is defined, the inner product denoted by χ|ψ, obeying χ| λ|ψ 1 +μ|ψ 2
=λχ|ψ 1 +μχ|ψ 2; χ|ψ = ψ|χ ∗ (1.3) The inner product of a ket state|ψwith its own bra is real and positive: ψ 2 ≡ ψ|ψ =real and≥0, (1.4) while ψ|ψ =0 ↔ |ψ =0 (1.5)
Therefore, the inner product can be used to define a norm A state|ψis called a physical state, or normalized state, if ψ 2 = ψ|ψ =1 (1.6)
In this article, we will refer to the concept of "template" to describe a specific state, as using the term "physical state" may lead to confusion The comprehensive application of Dirac's notation will be further explored in Part II.
In Hilbert space, variables can represent either numerical values or operators When clarity is necessary, we will explicitly denote operators by using superscripts or subscripts.
“op” to the symbol in question 11
The Pauli matrices,σ=(σ x , σ y , σ z )are defined to be the 2×2 matrices σ x op 0 1
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It can be excessive to apply the same notation universally for all operators When an operator merely represents multiplication by a function, we often choose to omit the "op" superscript or subscript In other instances, we clearly specify that we are addressing an operator.
Deterministic Models in Quantum Notation
The Basic Structure of Deterministic Models
Operators: Beables, Changeables and Superimposables 21
We plan to distinguish three types of operators:
(I) beables: these denote a property of the ontological states, so that beables are diagonal in the ontological basis{|A,|B, }of Hilbert space:
(II) changeables: operators that replace an ontological state by another ontological state, such as a permutation operator:
O op|A = |B, (changeable); (2.11) These operators act as pure permutations.
3 In Part II, we shall see the importance of having one state for which our identities fail, the so-called edge state
22 2 Deterministic Models in Quantum Notation
(III) superimposables: these map ontological states onto superpositions of ontolog- ical states:
Now, we will construct a number of examples In PartII, we shall see more examples of constructions of beable operators (e.g Sect.15.2).
The Cogwheel Model
Generalizations of the Cogwheel Model: Cogwheels
The cogwheel model's initial generalization involves a system that permutes N 'ontological' states, denoted as |n ont, where n ranges from 0 to N−1 and N is a positive integer greater than 1 The evolution of this system is governed by a clock-driven law.
The equation |n ont→ |n+1 modN ont (2.20) serves as a universal model for systems that exhibit periodic behavior with a cycle of N time steps This evolution equation describes the various states within the system.
The 24 2 deterministic models in quantum notation are identified as 'ontological' states, focusing solely on integer time steps This approach does not address the ontological states that may exist between these discrete time intervals We refer to this framework as the simple periodic cogwheel model, characterized by a period of N.
As a generalization of what was done in the previous section, we perform a dis- crete Fourier transformation on these states:
Normalizing the time stepδt to one, we have
N − 1 n = 0 e 2π ikn/N |n+1 modN ont=e − 2π ik/N |k H , (2.23) and we can conclude
The Hamiltonian under consideration has eigenvalues confined to the interval [0, 2π), indicating that while 0 is included, 2π is not This periodicity suggests that the Hamiltonian repeats every 2π, although it is typically treated as restricted to this interval Notably, the most significant physical scenarios arise when examining very small time intervals, such as those near the Planck time, where the Hamiltonian's highest eigenvalues become exceedingly large, rendering the corresponding eigenstates practically negligible.
In the original, ontological basis, the matrix elements of the Hamiltonian are ont m|H op |n ont= N 2π 2
This sum can we worked out further to yield
Note that, unlike Eq (2.8), this equation includes the corrections needed for the ground state For the other energy eigenstates, one can check that Eq (2.26) agrees with Eq (2.8).
For later use, Eqs (2.26) and (2.8), without the ground state correction for the caseU (t )|ψ = |ψ, can be generalised to the form
In equation (2.27), C represents a large constant, T denotes the period, and t_n = nδt indicates the specific times at which the operator U(t_n) must have a defined value It is important to note that this formulation is a sum rather than an integral, leading to a significant increase in the Hamiltonian when time values are closely spaced While a straightforward continuum limit appears elusive, Part II of this article will explore the construction of a continuum limit and examine the resulting implications.
By applying the Schrödinger equation \( \frac{d}{dt} |\psi(t)\rangle = -iH_{\text{op}}|\psi(t)\rangle \) with the boundary condition \( |\psi(0)\rangle = |n_0\rangle \), the state follows a deterministic evolution law at integer time steps Furthermore, when considering superpositions of the states \( |n\rangle \) and interpreting the complex coefficients through the Born rule, the Schrödinger equation accurately describes the evolution of these Born probabilities.
The energy spectrum of a Zeeman atom, characterized by total angular momentum J = 1/2 (N−1) and a magnetic moment μ in a weak magnetic field, is a significant concept in physics Following the discrete Fourier transformation, this atom can be viewed as a fundamental deterministic system that transitions between N distinct states at discrete time intervals.
In the context of the Zeeman atom, we can introduce a finite, universal quantity δE to the Hamiltonian, which results in a rotation of all states by the complex amplitude e^{-iδE} after each time step While this modification may initially appear harmless, similar to a simple cogwheel, its implications can become significantly impactful in later analyses.
Introducing perturbations to the Zeeman atom leads to unequal energy level splits, resulting in a loss of the cogwheel-like structure Consequently, these systems become significantly more challenging to describe within a deterministic framework, requiring an understanding of their complexity in a broader context.
The Most General Deterministic, Time Reversible, Finite
In this article, we explore finite models characterized by a limited number of states and an arbitrary time evolution law Starting from an initial state |n₀⟩, we observe its evolution over a finite number of time steps, denoted as N₀, after which the system returns to |n₀⟩ However, not all possible states |n⟩ may be visited during this process If we initiate the evolution from a different state, |n₁⟩, a new sequence of states is generated, potentially resulting in a different periodicity, N₁.
To achieve a comprehensive understanding of the model, we must explore all its existing states Ultimately, the most generalized model can be represented by a collection of simple periodic cogwheel models, each exhibiting different periodicities while operating under a consistent universal time step, δt, which can be normalized to one.
26 2 Deterministic Models in Quantum Notation
Fig 2.2 Example of a more generic finite, deterministic, time reversible model
Figure 2.3 illustrates the energy levels of a simple periodic cogwheel model, a combination of these models, and the most general deterministic, time-reversible finite model The energy levels of the cogwheels have been adjusted by varying amounts (δE i), which is permissible since the index (i) indicating the specific cogwheel is conserved, thus these shifts do not impact physical outcomes However, significant effects are observed when the spectra are combined, as shown in Fig 2.3c.
The energy spectrum of a finite discrete deterministic model can quickly become complex, prompting the question of whether a deterministic model can replicate the energy spectrum of any quantum system This raises important considerations about the trade-offs in locality when attempting such a mimicry Additionally, it invites exploration into the existence of deterministic theories that can be mapped onto quantum models, and which of these theories may offer intriguing possibilities for further research.
The energy spectrum of the simple periodic cogwheel model is illustrated in Fig 2.3a, where δE represents an arbitrary energy shift Fig 2.3b shows the energy spectrum of a more complex model, combining multiple simple cogwheel models, with each cogwheel i potentially shifted by its own arbitrary amount δE i By integrating these energy levels, we derive the spectrum of a generic finite model, as depicted in Fig 2.3c.
The models presented in the figures are merely simplified examples, as the actual universe is far more complex A critic questioned the significance of using a model with 31 states, but this number is arbitrary and solely serves to demonstrate the mathematical principles involved.
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This book does not aim to provide a comprehensive analysis of all interpretations of quantum mechanics, as previous literature has extensively covered these approaches However, we believe that each interpretation has its shortcomings The most traditional and pragmatic viewpoint is known as the Copenhagen Interpretation, which some mainstream researchers argue encompasses all essential knowledge regarding quantum mechanics.
The unexplained aspects of the Copenhagen interpretation often intrigue us, prompting a closer examination In this article, we will explore how the cellular Automaton interpretation can provide clarity on these unanswered questions.
The Copenhagen Doctrine
The late 1920s marked an exhilarating era in early modern science as researchers began to grasp the complexities of quantum mechanics, leading to the establishment of what is now known as the Copenhagen Doctrine Initially, physicists faced significant challenges with the equations and technical aspects of quantum mechanics However, advancements have since enabled a clearer understanding of these concepts, allowing us to articulate the foundational principles with greater precision Quantum mechanics was originally expressed through wave functions, which described the states of electrons, represented as ψ (x, t) = x|ψ (t) Today, while we still refer to 'wave functions,' we often use the term to encompass ket states in a more generalized context.
Leaving aside who said exactly what in the 1920s, here are the main points of what one might call the Copenhagen Doctrine Somewhat anachronistically, 1 we employ Dirac’s notation:
1 Let me stress here again that from our use of terms such as “Copenhagen Interpretation”, or
The "Copenhagen doctrine" should not be interpreted as an attempt to rewrite history; rather, it reflects the philosophical discussions that occurred within the "Copenhagen group."
G ’t Hooft, The Cellular Automaton Interpretation of Quantum Mechanics,
Fundamental Theories of Physics 185, DOI 10.1007/978-3-319-41285-6_3
A system is fully characterized by its wave function |ψ(t), which belongs to Hilbert space, allowing for various bases to describe it This wave function follows Schrödinger’s equation, a linear first-order differential equation in time, the precise form of which can be established through repeated experimentation.
In quantum mechanics, any observable can be measured using Hermitian operators within Hilbert space The theory predicts that after conducting multiple repetitions of an experiment, the average value obtained for the observable will converge to a specific result.
Upon measurement, the system's wave function collapses to a state within the Hilbert space that corresponds to an eigenstate of the observable O, or it may represent a probabilistic distribution of these eigenstates.
When two observables, O1 and O2, do not commute, accurate simultaneous measurement is impossible The commutator [O1, O2] reveals the expected magnitude of the uncertainties δO1 and δO2 Additionally, the measuring device is treated as a classical object, and in large systems, quantum mechanical measurements increasingly align with classical descriptions.
The concept of probability is inherently present in the wave function equation When we decompose the wave function |ψ into its eigenstates |ϕ corresponding to an observable O, we can determine the likelihood of measuring the eigenvalue associated with the state |ϕ This probability is mathematically expressed as P = |ϕ|ψ|², a principle known as Born's probability rule.
We note that the wave function may not be given any ontological significance.
The concept of a 'pilot wave' is not a necessity in quantum mechanics, as the wave function, specifically the amplitudes, are considered psi-epistemic rather than psi-ontic This means that we cannot directly measure the wave function itself; instead, we can only determine probabilities through repeated experiments, which inherently come with margins of error.
The Copenhagen interpretation emphasizes that the focus should solely be on predicting the outcomes of experiments, rather than questioning the underlying reality of what is occurring This perspective discourages inquiries into the actual processes at play, suggesting they are unnecessary According to the Copenhagen group, possessing knowledge of the Schrödinger equation provides all the information required to anticipate experimental results, rendering further questions irrelevant.
The Copenhagen doctrine highlights a significant strength in quantum mechanics, yet it presents notable limitations While knowledge of the Schrödinger equation provides a comprehensive understanding of quantum systems, the challenge arises when this equation is not yet known This complexity reflects the diverse interpretations and attitudes toward quantum mechanics, emphasizing the need for a deeper exploration of its foundational principles.
The Einsteinian View
one arrive at the correct equation? In particular, how do we arrive at the correct Hamiltonian if the gravitational force is involved?
For over 30 years, gravity has been a central topic in elementary particle theory and the study of space and time, leading to significant developments in (super)string theory Despite these advancements, no convincing model has successfully unified gravity with other fundamental forces or accurately explained the values of key constants in nature, such as the masses of fundamental particles, the fine structure constant, and the cosmological constant This raises critical questions about the underlying mechanisms at play in our understanding of gravity and the universe.
One strong feature of the Copenhagen approach to quantum theory was that it was also clearly shown how a Schrửdinger equation can be obtained if the classical limit is known:
In a classical system governed by Hamilton equations, classical variables \( p_i \) and \( q_i \) allow for the definition of Poisson brackets By substituting these classical variables with commutators, a quantum model is derived, which aligns with the original classical system in its classical limit as \( \hbar \) approaches zero.
While there are powerful techniques to approach gravitational force, they fall short of achieving 'quantum gravity.' The challenges extend beyond the non-renormalizability of gravity and the complexities in defining quantum space-time coordinates and non-trivial topologies Some researchers view these issues as mere technical hurdles, but the fundamental concern lies in the uncontrolled curvature of space-time at the Planck scale.
We will be forced to turn to a different book keeping system for Nature’s physical degrees of freedom there.
A promising approach involves utilizing local conformal symmetry as a fundamental principle, allowing for the relativity of distance and time scales This perspective challenges the absolute nature of 'small distances.' The theory is summarized in Appendix B and requires further refinement, potentially necessitating a Cellular Automaton interpretation to incorporate its quantum features.
This section is called ‘The Einsteinian view’, rather than ‘Einstein’s view’, because we do not want to go into a discussion of what it actually was that Einstein thought.
Einstein's discomfort with the Copenhagen Doctrine is widely recognized, as he believed in an Einsteinian view where all phenomena in the universe are governed by deterministic equations, leaving nothing to chance This perspective raises important questions, including whether the quantum-mechanical description of physical reality can be deemed complete.
53], or, does the theory tell us everything we might want to know about what is going on?
In the Einstein–Podolsky–Rosen discussion of a Gedanken experiment, two par- ticles (photons, for instance), are created in a state x 1−x 2=0, p 1+p 2=0 (3.2)
Since[x 1 −x 2 , p 1 +p 2 ] =0, both equations in (3.2) can be simultaneously sharply imposed.
Einstein, Podolsky, and Rosen were troubled by the phenomenon where, despite the cessation of interaction between two particles, measuring the momentum of particle #2 would reveal the momentum of particle #1, and measuring its position would similarly determine the position of particle #1.
How can such a particle be described by a quantum mechanical wave function at all? Apparently, the measurement at particle # 2 affected the state of particle #1, but how could that have happened?
In contemporary quantum terminology, the proposed measurements in this Gedanken experiment would disturb the wave function of the entangled particles The measurements conducted on particle #2 influence the probability distributions of particle #1, but this interaction should not be interpreted as a mysterious signal transmitted between the two systems.
Despite the challenges posed by their arguments, Einstein, Podolsky, and Rosen were able to accurately calculate the quantum mechanical probabilities for measurement outcomes, demonstrating that quantum mechanics remained intact throughout this discourse.
It is much more difficult to describe the two EPR photons in a classical model.
Such questions will be the topic of Sect.3.6.
Einstein had difficulties with the relativistic invariance of quantum mechanics
(“does the spooky information transmitted by these particles go faster than light?”).
Recent advancements have addressed the technical challenges previously associated with information transmission over distances This concept aligns with Copenhagen's Doctrine, which posits that effective communication occurs only when we can identify non-commuting operators A at space-time point x1 and operators B at space-time point x2, satisfying the condition [A, B] = 0.
In elementary particle theory, it is established that all space-like separated observables commute, preventing any faster-than-light signaling This fundamental principle is a key aspect of the Standard Model, contributing significantly to its success.
With the technical challenges addressed, we can focus on the fundamental objections raised by Einstein against the Copenhagen interpretation of quantum mechanics He argued that this theory is inherently probabilistic and fails to provide a clear understanding of reality Additionally, it is often implied that one must suspend conventional logical reasoning to accept its principles.
The ongoing Einstein–Bohr debate highlights the necessity for a theory that preserves classical logic without requiring a redefinition of its principles Some argue that by focusing on what not to ask and adjusting our logical framework, we can achieve clarity and resolution in this scientific discourse.