When investigating theories at the smallest imaginable scales in nature, almost all researchers today return to quantum language and accept the verdict that we must nickname the "Copenhagen doctrine", that the only way to describe what is going on will always involve states in Hilbert space, governed by operator equations. Meanwhile, we develop mathematical notions that appear to be coherent and beautiful.
Why an Interpretation Is Needed
Everything this theory6 says, and it needs to be said, is about the reality of the results of an experiment. Indeed, it is in quite a few competing theories of the interpretation of quantum mechanics where authors are led to introduce nonlinearities in the Schrödinger equation or violations of the Born rule that will be impermissible in this work.
Outline of the Ideas Exposed in Part I
10 1 The motivation for this work is still that it can be used to explain some of the apparent contradictions in quantum mechanics. In Chapters 3 and 4, we begin to deal with the real subject of this research: the question of the interpretation of quantum mechanics.
A 19th Century Philosophy
Well, you don't have to take everything for granted; there are still unresolved issues and more avenues to explore. In short, our imaginary 19th century world will seem to be governed by effective laws with a large stochastic element in them.
Brief History of the Cellular Automaton
Cellular automata are often used as models for physical systems such as fluids or other complex mixtures of particles. An extensive study of the role of cellular automata as models for addressing scientific questions was made by Stephen Wolfram in his book A New Kind of Science [92].
Modern Thoughts About Quantum Mechanics
We then note that although all observers at a given time are indeed free to choose their settings, the correlation functions then dictate, non-locally, what the ontological states of observed objects, such as elementary photons, may be. We will see how these correlation functions can affect our conclusions about the mysteries of quantum mechanics.
Notation
Virtually all investigators [22, 23] adhere to the concept called freedom of choice, which means that an observer should enjoy the freedom at any time to choose which observable property of a system should be measured. 18 1 Motivation for this work whereλandμ are complex numbers is also a condition in this Hilbert space.
The Basic Structure of Deterministic Models
Operators: Beables, Changeables and Superimposables . 21
Imagine that δ is not as small as the Planck time, 10−43 seconds (see Chapter 6), then if noticeable changes only occur on much larger time scales, deviations from the ontological model will not be noticeable. The fact that the ontological and quantum models coincide for all integer multiples of time dt is physically significant.
Generalizations of the Cogwheel Model: Cogwheels
The fact that the equivalence holds only for integer multiples of δt is not a restriction. In fact, its definition implies that the Hamiltonian is periodic with period 2π, but in most cases we will treat it as defined, which is restricted to an interval.
The Most General Deterministic, Time Reversible, Finite
As in the Zeeman atom, we can consider the possibility of adding a finite, universal quantity δE to the Hamiltonian. Yet it is the things that are not explained in the Copenhagen picture that often catch our attention.
The Copenhagen Doctrine
If we expand the wave function|ψ to the eigenstates|ϕ of an observableO, then we find that the probability that the experiment on|ψ actually yields that the eigenvalue of the state|ϕ is found will be given byP = |ϕ|ψ|2. This is a strong point of the Copenhagen doctrine, but it also brings serious limitations.
The Einsteinian View
In any case, Einstein, Podolsky, and Rosen also had no problem calculating quantum mechanical probabilities for the outcomes of measurements, so quantum mechanics emerged from this sequence of arguments in principle unscathed. It is a built-in feature of the standard model, to which it actually owes much of its success.
Notions Not Admitted in the CAI
In an ontological description of our universe, in terms of its ontological basis, there are only two values that a wave function can take: 1 and 0. It is only for mathematical reasons that one later wants to equip this wave function with a phase,eiϕ.
The Collapsing Wave Function and Schrödinger’s Cat
The overlay cat mathematically solves the equations in a perfectly acceptable way, but it does not describe a situation that may occur in the real world. The most likely answer to this will be that the transformation need not always be entirely local, but in practice may involve many viewer states in the environment.
Decoherence and Born’s Probability Axiom
We can ask what initial conditions could have led to the condition we see now and how likely they were. Our question is therefore whether we can explain whether and how expression (3.4) is related to these numbers.
Bell’s Theorem, Bell’s Inequalities and the CHSH Inequality
On either side of the filter there is a detector, so both Alice and Bob observe that one of their two detectors gives a signal. All one needs to assume is that the atom sends a signal to Alice and one to Bob, regarding the polarization of the emitted photons.
The Mouse Dropping Function
Ontology Conservation and Hidden Information
It's still true that the effect of the mouse droppings seems mysterious, as we deny, in a sense, that Alice, Bob, and their mice have "free will." They do not tell us the settings themselves, but they do contain information about the correlation function of these settings.
Free Will and Time Inversion
If a change is made in the given values of the kinetic variables, they may have a much greater or lesser probability than the original configuration. So when an amplitude at time=t3 turns out to be low and the Born probability is small, this can actually be attributed to a smaller probability of the chosen initial state, att=t1.
Introduction
What really matters is that an ontological basis allows a meaningful subset of observers to be defined as operators that are diagonal to this basis. Most likely, the model will first need to be extended to include gravity in some way.
The Classical Limit Revisited
In terms of the basis we would normally use in quantum mechanics, these states will be complicated quantum superpositions. In terms of the original, ontological basis, the beables will only describe the elementary basic elements.
Born’s Probability Rule
The Use of Templates
Reversing this transformation, one finds that the ontological states will in turn be complicated superpositions of the template states. What was argued in the previous section was that the classical, or macroscopic, states are diagonal in terms of the sub-microscopic states, so they are all ontic states.
Probabilities
The final ontological states are the ontological states that lead to a particular outcome of the experiment. If we now assume that the template states used always reflect the relative probabilities of the ontic states of the theory, the Born rule seems to be an inevitable consequence [113].
Time Reversible Cellular Automata
The simple gear model allows for both classical and quantum mechanical description without any changes to the physics. Section 2.2.2 of Section 2.2 shows how quite complex energy spectra can be generated in this way in relatively simple generalizations of the cogwheel model.
The CAT and the CAI
The origin of the continuous symmetries of the Standard Model will remain beyond what we can deal with in this book, but we can discuss the question to what extent cellular automata can be used to approach and understand the quantum nature of this world. One would then be tempted to argue that any computationally universal cellular automaton could be used to simulate systems as complicated as the Standard Model of subatomic particles.
Motivation
The Wave Function of the Universe
Standard quantum mechanics can confront us with a question that seems difficult to answer: Does the universe as a whole have a wave function. If the universe is in an ontological state, its wave function will automatically collapse whenever necessary.
The Rules
We will never change its mathematical form; only the absolute squares of the inner products can be used. However, they are on a basis which is a rather complicated unit transformation of the ontological basis.
Features of the Cellular Automaton Interpretation (CAI)
- Beables, Changeables and Superimposables
- Observers and the Observed
- Inner Products of Template States
- Density Matrices
All wave functions can be used to describe a physical process, but then we have to tolerate the axiom of collapse if we are not working in the ontological basis. Thus, it can be assumed that one of our template states represents a probability distribution of the ontological states of the universe, the other is a model of an ontological state.
The Hamiltonian
Locality
The problem is that the cellular automaton defines only the local evolution operator at finite time steps, not the local Hamiltonian that defines the infinitesimal time evolution. Note that perturbation expansions have been used very successfully in most quantum field theories (think, for example, of calculating the irregular magnetic moment g-2 of an electron, which could be calculated and compared to experiment with extreme precision), while it is still suspected that the expansions do not ultimately converge.
The Double Role of the Hamiltonian
How does this work in discrete, deterministic systems as studied here. Our problem is that in a discrete classical system the energy must also be discrete, but the generator of the evolution must be an operator with continuous eigenvalues.
The Energy Basis
76 5 Concise Description of the CA Interpretation sufficiently large time steps (much larger than the Planck time). If we were to consider the basis of energy as our ontological basis, we would consider all energy to be classical, but then the part describing evolution disappears; this is not good physics.
Miscellaneous
- The Earth–Mars Interchange Operator
- Rejecting Local Counterfactual Definiteness and Free Will 78
- The Superposition Principle in Quantum Mechanics
- The Vacuum State
- A Remark About Scales
- Exponential Decay
- A Single Photon Passing Through a Sequence of
- The Double Slit Experiment
This is a feature due to the somewhat artificial nature of the model used for detection. Given the ontological state of the filter, we have the probability cos2ψ that the photon is transmitted but rotated in the direction ϕn and the probability sin2ψ that it is absorbed (or reflected and rotated).
The Quantum Computer
It is not unreasonable to suspect that the Planck length is the smallest length scale meaningful in physics, and the Planck time is the smallest time scale on which things can happen, but there is more. General Relativity is known to cause curvature of space and time, so if one can speak of some "grid" in space and time, curvature can be expected to cause defects in this grid.
Cogwheels with Information Loss
Information equivalence classes will show some similarity to measure equivalence classes, and they can actually be quite large. Therefore, the equivalence classes are all very large, but the total number of equivalence classes is also quite large, and the model is physically non-trivial.
Time Reversibility of Theories with Information Loss
Intuitively, one might think that information loss would make our models non-invariant under time reversal. A "quantum observer" in a model with information loss can determine a perfect validity of symmetries such as CP T invariance.
The Arrow of Time
Then, in Section.7.2, we showed that, even if the classical equations contain information loss on a large scale – so that only small fractions of information are preserved – the emerging quantum mechanical laws are still precisely time-reversible, so that, as long as we adhere to a description of things in terms of Hilbert space, we cannot understand the symmetry of time. Our introduction of information loss can have another advantage: two states can be in the same equivalence class, even if we cannot follow the evolution very far back in time.
Information Loss and Thermodynamics
This is because the math will be much more difficult; we have simply not yet been able to use information loss in our more physically relevant examples. It could take years, decades, perhaps centuries to arrive at a comprehensive theory of quantum gravity combined with a theory of quantum matter that would be an extensive extension of the Standard Model.
What Will Be the CA for the SM?
This could be done for free, massless particles, which is not a bad start, but it is also not good enough to continue. Then we have some rudimentary ideas about bosonic force fields, as well as a suggestive string-like excitation (further elaborated in Part II), but again, it is not yet known how to combine these into something that can compete with the Standard Model known today.
The Hierarchy Problem
Presumably, a procedure similar to second quantization must be used, but as yet it is only clear how to do this for fermionic fields. In short, the most natural way to incorporate scale hierarchies into our theory is not clear.
Positivity of the Hamiltonian
All solutions of the Schrödinger equation would have such instabilities, which would describe a world quite different from the quantum world we are used to. Having approximate solutions that only slightly violate our requirements will not be very useful, because of the hierarchy problem, see Section 8.2: The Standard Model Hamiltonian (or what we usually substitute for it, its Lagrangian), requires fine-tuning.
Second Quantization in a Deterministic Theory
If we diagonalize the U operator for each 'particle', we find that half the states have positive energy, half negative. This means that we are working with the energy levels close to the center of the spectrum, where we see that the Fourier expansions of Sect.14 still converge rapidly.
Information Loss and Time Inversion
Then the local speed of light, expressed in the coordinates used, is given by c2= |λ|/λ0. We can impose an inequality onc: assume that the values of the metric tensor are constrained to obey|c| ≤1.
Holography and Hawking Radiation
Not only in deterministic theories, but also in pure quantum theories, the arrangement of the outgoing particles must depend to some extent on the particles that entered to form a black hole. This means that our identification of the basic elements of the Hilbert space with the information equivalence classes appears to be not completely local.
The CAI
Our theory for the interpretation of what we observe is now clear: humanity discovered that phenomena on the distance and energy scales of the Standard Model (which includes distances far greater and energies far less than the Planck scale) can be captured by postulating the effectiveness of templates. Of course, if we want to know how our units of measurement work, we use our templates, and this is the origin of the usual "measurement problem".
Counterfactual Definiteness
Superdeterminism and Conspiracy
The Role of Entanglement
Resetting Alice's experiment without making changes near Bob would result in a state that, in quantum mechanical terms, is not orthogonal to the original one, and therefore not ontological. The fact that the new state is not orthogonal to the previous one is completely in line with the standard quantum mechanical descriptions;.
Choosing a Basis
In reality, these templates reflected the relative correlations of the ontological variables underlying these photons. We point out that also all classical observables, describing stars and planets, cars, people, and eventually pointers on detectors, will be diagonal to the same ontological basis.
Correlations and Hidden Information
There could exist a basis in which all supermicroscopic physical observables are beables; these are the observable features on the Planck scale, they must all be diagonal on this basis. This particular basis, called 'ontological basis', is hidden from us today, but it should in principle be possible to identify such a basis.
The Importance of Second Quantization
No one can build such classical computers, so quantum computers can indeed work miracles; but there are clear limits: the quantum computer will never outperform classical computers when scaled up to Planckian dimensions. We predict that there will be fundamental limitations in our ability to avoid decoherence of the qubits.
Calculation Techniques
Outline of Part II
This is the lowest energy state and solving the equation "energy = lowest" actually creates non-localities. In Chapter 20, we explain some of the most important features of quantum field theory.
Notation
We try a number of alternative approaches with some modest successes, but not all problems will be solved, and the conjecture is raised about the source of our difficulties: quantum gravity effects may be crucial, while it is precisely these effects that are not yet understood as well as they need to be here. Here, the square bracket indicates a limit whose value itself may be included, a round bracket excludes that value.
More on Dirac’s Notation for Quantum Mechanics
They are considered as vectors in the Hilbert space where the set of peak wave functions delta|is chosen as basis:. 11.5) Many special functions, such as the Hermite, Laguerre, Legendre, and Bessel functions, can be viewed as generators of various sets of Hilbert space basis elements.
The Group SU(2), and the Harmonic Rotator
Thus we see that the lowest energy states of the gear approach the lowest energy states of the harmonic oscillator. Note that the spectrum of the Hamiltonian of the harmonic rotator is exactly that of the harmonic oscillator, except that there is an upper bound, Hop<2π.
Infinite, Discrete Cogwheels
By construction, the period T =(2+1)δ of the harmonic rotator, as well as that of the harmonic oscillator, is exactly that of the periodic gear. The importance of this approach is that the representation is a unit and that there is a natural ground state, the ground state of the harmonic oscillator.
Automata that Are Continuous in Time
12.22) Thus the spectrum of eigenvalues for the energy eigenstates|nH runs over all integers from −∞ to∞. The significance of the Hamiltonian's ground state was discussed in Chapter 9 of Part I.
The Operator ϕ op in the Harmonic Rotator
One now sees that the operator. increases the value by one unit, except for the state|m=, which goes to|m= −. Care was taken to correctly represent the square roots in the definitions of L±: since|H| ≤, division by 0 is never encountered. The eigenstates |ϕ of the operator ϕop are closely related to Glauber's coherent states in the harmonic oscillator [42], but the Hermitian operator ϕopis and its eigenstates are orthonormal; Glauber's states are eigenstates of creation or annihilation operators, which were introduced by him following a different philosophy.
The Harmonic Rotator in the x Frame
It is not difficult to assess the extent of the deformation close to the origin. Here MPl is the "Planck mass" or whatever is the inverse of the elementary time scale in the model.
The Jordan–Wigner Transformation
- Algebra of the Beable ‘Neutrino’ Operators
- Orthonormality and Transformations of the ‘Neutrino’
- Second Quantization of the ‘Neutrinos’
Since J= L+12σ, one can also write using Eq. 15.60), the following expression for the operators nexi in the neutrino wave function in the form of beablesq, rˆ ands3 and changeables4Lontk , pr, s1ands2:. This derivative originates from factorpr in Eq. 15.39), which is necessary for a proper normalization of the states.
The ‘Neutrino’ Vacuum Correlations
They probably play an important role in the mysterious behavior of the beabel models when Bell's inequalities are taken into account, see Part I, Chapter 3.6 et seq. In the chapters on Bell inequalities and the cellular automaton interpretation (Section 5.2 and Chapter 3 of Part I), it is argued that the ontological theories proposed in this book must contain strong, space-like correlations throughout the universe.
The Algebra of Finite Displacements
- From the One-Dimensional Infinite Line
- The States | Q, P in the q Basis
As should be clear from Eq. 16.5), we can consider the angleκ as the generator of a shift in the integer X, and the angleξaxis generating a shift inK:. The wave function (16.24), which is equal to its own Fourier transform, is special because it is close to a square wave, with only very small tails outside the domain.
Transformations in the PQ Theory
The sum over N and the integral over κ from 0 to 2π combine into an integral over all real values of κ, to obtain the remarkably simple expression.
Resume of the Quasi-periodic Phase Function φ (ξ, κ)
Note, in all these expressions we have the symmetry under the combined exchange. the latter expression is found by contour integration).
The Wave Function of the State | 0, 0
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Two Dimensional Model of Massless Bosons
- Second-Quantized Massless Bosons in Two Dimensions . 182
- An Alternative Ontological Basis: The Compactified
- The Quantum Ground State
However, we claim that there is an exact mapping between the basic elements of the quantized field theory of Section 17.1.1 and the states of a cellular automaton (again with the possible exception of edge states). The states of the cellular automaton can be used as a basis for describing quantum field theory.
Bosonic Theories in Higher Dimensions?
- Instability
- Abstract Formalism for the Multidimensional Harmonic
- String Basics
- Strings on a Lattice
- The Lowest String Excitations
- The Superstring
- Deterministic Strings and the Longitudinal Modes
- Some Brief Remarks on (Super)string Interactions
Surprisingly, one finds that these operators obey the same equations (17.67) and (17.68): the operation. has the same effect as κop. where the sum is the same as in Eq. 17.71), and noting that QandP are integers, while κopandξopare is restricted to the interval [0,2π ), we conclude that the same equations are again obeyed by the real number operators. Up to a factor sink/k, this is the Hamiltonian (17.25) (since the equations of motion at different k values are independent, conservation of one of these Hamiltonians implies conservation of the other).
Classical and Quantum Symmetries
Continuous Transformations on a Lattice
- Continuous Translations
- Continuous Rotations 1: Covering the Brillouin Zone
- Continuous Rotations 2: Using Noether Charges
- Continuous Rotations 3: Using the Real Number
- Quantum Symmetries and Classical Evolution
- Quantum Symmetries and Classical Evolution 2
Large Symmetry Groups in the CAI
The Vacuum State, and the Double Role of the Hamiltonian
The Hamilton Problem for Discrete Deterministic Systems
Conserved Classical Energy in PQ Theory
- Multi-dimensional Harmonic Oscillator
More General, Integer-Valued Hamiltonian Models
- One-Dimensional System: A Single Q, P Pair
- The Multi-dimensional Case
- The Lagrangian
- Discrete Field Theories
- From the Integer Valued to the Quantum Hamiltonian
General Continuum Theories—The Bosonic Case
Fermionic Field Theories
Standard Second Quantization
Perturbation Theory
- Non-convergence of the Coupling Constant Expansion
The Algebraic Structure of the General, Renormalizable,
Vacuum Fluctuations, Correlations and Commutators
Commutators and Signals
The Renormalization Group
Local Time Reversibility by Switching from Even to Odd Sites
- The Time Reversible Cellular Automaton
- The Discrete Classical Hamiltonian Model
The Baker Campbell Hausdorff Expansion
Conjugacy Classes
Second Quantization in Cellular Automata
More About Edge States
Invisible Hidden Variables
How Essential Is the Role of Gravity?
Discreteness of Time
