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Cambridge University Press eBooks

Discrete time dynamics architecture

George Jaroszkiewicz

Published Apr 17, 2014• 0 citations
Publisher

Abstract

Mappings, functions In this book, the term architecture refers to a non-mathematical description of the sequence of events which collectively describes an experiment. Continuous time (CT) classical mechanics (CM) is based on some explicit and implicit assumptions that generate architectures radically different from those describing quantum processes. In the following list, which is not complete by any means, most of these assumptions may appear obvious and overstated, but one or two are subtle and should be brought to light, since they have an enormous impact on the theories which are based on them. Our comments are in square brackets at the end of each item in the list. At any given time, a system under observation (SUO) exists in a unique physical state with absolute properties independent of any observer [a standard classical metaphysical assumption]. The physical state of an SUO at any given time t may be represented by a single element Ψ( t ) called a state of some fixed state space U (or universe) [a reasonable mathematical assumption]. The state Ψ( t ') of an SUO at any time t ' later than t is also an element of U , not necessarily the same state Ψ( t ) at the earlier time [a reasonable classical mathematical assumption]. Observers are exophysical, meaning that they stand outside of U [a nearly universal metaphysical assumption; observers are not discussed in detail in CM or QM simply because there is no comprehensive theory of what constitutes an observer].

DOI
10.1017/cbo9781139525381.006

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Structure of Physical Theories

K. Kong Wan

This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).

•Aug 27, 2020•6 citations

Discrete time quantum mechanics

George Jaroszkiewicz

Quantization We now discuss the quantization of discrete time (DT) classical mechanics (CM). We restrict our attention in this chapter to a system under observation (SUO) consisting of a single point particle of mass m moving in one spatial dimension with Cartesian coordinate x . The generalization to two or more degrees of freedom is as straightforward here as in CT quantum mechanics (QM). The Dirac bra-ket notation will be used for convenience. The standard principles of QM have proven remarkably successful and consistent over the years (Peres, 1993) and we have no reason to alter them apart from changing from CT to DT. This is a significant change. Quantum mechanics became dominant in physics and chemistry because of the success of the Schrödinger equation, which is a first-order-in-time differential equation. Therefore, we should take care to ensure that discretizing time does not undermine the successes of CT QM. There are two reasons why discretizing time in QM might be considered. First, the Schrödinger equation is hard, or even impossible, to solve exactly in many situations and temporal discretization might be seen as a step towards numerical simulation by computer. This motivated the work of Bender (Bender et al. , 1985a, 1985b, 1993) and others on the DT Schrödinger equation. The second reason is one of principle: we may want to explore the properties of DT QM as a self-consistent theory in its own right rather than as an approximation to CT QM.

Cambridge University Press eBooks•Apr 17, 2014•0 citations

A strict epistemic approach to physics

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The general view is that all fundamental physical laws should be formulated within the framework given by quantum mechanics (QM). In a sense, QM therefore has the character of a metaphysical theory. Consequently, if it is possible to derive QM from more basic principles, these principles should be of general, philosophical nature. Here, we derive the formalism of QM from well-motivated epistemic principles. A key assumption is that a physical theory that relies on entities or distinctions that are unknowable in principle gives rise to wrong predictions. First, an epistemic formalism is developed, using concepts like knowledge and potential knowledge, identifying a physical state $S$ with the potential knowledge of the physical world. It is demonstrated that QM emerges from this formalism. However, Hilbert spaces, wave functions and probabilities are defined in certain well-defined observational contexts only. This means that the epistemic formalism is broader than QM. In the fundamental layer of description, the physical state $S$ is a subset of a state space $\mathcal{S}=\{Z\}$, such that $S$ always contains many elements $Z$. These elements correspond to unattainable states of complete knowledge of the world. The evolution of $S$ cannot be determined in terms of the individual evolution of the elements $Z$, unlike the evolution of an ensemble in classical phase space. The evolution of $S$ is described in terms of sequential time $n\in \mathbf{\mathbb{N}}$, which is updated according to $n\rightarrow n+1$ each time an event occurs, each time potential knowledge changes. Sequential time $n$ can be separated from relational time $t$, which describes distances between events in space-time. There is an entire space-time associated with each $n$, in which $t$ represents the knowledge at sequential time $n$ about the temporal relations between present and past events.

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World Theory

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Quantum Mechanics Based on the Density of Quantum Space (Part V)

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DESCRIPTION This work presents a complete, deterministic framework for quantum mechanics derived entirely from a single, testable postulate: the density of quantum space. The Core Problem For over a century, quantum mechanics has been interpreted as a theory of fundamental randomness, wave-particle duality, instantaneous jumps, and irreducible uncertainty. The standard Copenhagen interpretation treats the wave function as a probability amplitude, measurement as a "collapse" of that probability, and the Heisenberg uncertainty principle as a fundamental limit of nature. This work demonstrates that all these "mysteries" are not properties of nature—they are artifacts of our limited perception. We observe quantum phenomena through slow, macroscopic instruments that cannot resolve the actual continuous dynamics occurring in a space that is 6.54 trillion times denser than what we perceive as "empty space." 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The "wave" is the charge density distribution—a standing wave in the dense quantum medium. From the standing wave condition 2πr = nλ, the Poynting vector in the far field is exactly zero (S = 0), meaning the electron does not radiate. The apparent "wave behavior" is an optical illusion caused by the density ρ smearing the point-like trajectory in our low-resolution projection. Wave Function = Energy Density, Not Probability The squared modulus of the wave function |Ψ|² is the energy density fraction, not a probability density. The particle itself follows a deterministic trajectory y(t) = v₀t - ½g₀t². Probability appears only because we cannot resolve the trajectory (due to t_total = 118 as ≪ Δt_instr) and we do not know the initial phase φ of the oscillation. Schrödinger Equation = Effective Approximation The Schrödinger equation is an effective, averaged description of how the energy density evolves. The fundamental law is classical motion in the dense quantum medium. 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The uncertainty is a practical limitation of invasive measurement—we disturb the particle's trajectory when we hit it with a photon. Quantum Randomness = Ignorance Residual "quantum randomness" is simply unmodeled environmental force noise. The apparent randomness arises from three sources: temporal resolution (t_total = 118 as ≪ Δt_instr—we cannot resolve the trajectory), initial phase ignorance (we do not know the phase φ of sin(2πt/T)), and measurement perturbation (we do not know the exact collision parameters). If we could measure the outgoing photon completely and know the initial phase, the outcome would be fully deterministic. Randomness is in our ignorance, not in nature. Five Testable Predictions The theory makes five clear, quantitative predictions that distinguish it from standard quantum mechanics: Prediction 1 - Electron Transit Time: The electron transit time during the n=1 → n=2 transition in hydrogen is 118 ± 5 attoseconds. Standard quantum mechanics predicts zero duration (instantaneous jump). Testable with attosecond pump-probe spectroscopy. Prediction 2 - Heisenberg Product Can Approach Zero: With better detectors, the uncertainty product Δx·Δp can approach zero, not ħ/2. Standard quantum mechanics predicts Δx·Δp ≥ ħ/2 always. Testable with weak measurement experiments. Prediction 3 - Energy Oscillation at 116 kHz: The energy of gold-197 oscillates at 116 kHz with amplitude 1.252 × 10⁻²⁶ J. Standard quantum mechanics predicts constant energy for a given state. Testable with precision spectroscopy. Prediction 4 - Bell Inequality Violations Decrease with Measurement Strength: As measurement impulse strength decreases, Bell inequality violations decrease. Standard quantum mechanics predicts violations are independent of measurement strength. Testable with weak measurement entanglement experiments. 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Mathematical Consistency The work provides rigorous derivations of: The density ρ from first principles (Planck length, electron mass, fine structure constant) The quantum coordinate norm ‖n‖ from the discrete metric of quantum space The oscillation period T from the de Broglie relation The coupling constant ε from fundamental constants The regularized metric eliminating all singularities The thermodynamic properties of quantum space (entropy, temperature, heat capacity) The covariant Einstein equation in quantum space The connection to quantum anomalies and topological invariants The Lamb shift from quantum space density oscillation Dark energy from quantum space density variation Comparison with Other Approaches String theory requires approximately 10⁵⁰⁰ free parameters and offers no testable predictions with current technology, while introducing new physics such as strings, extra dimensions, and supersymmetry. Loop quantum gravity requires approximately 10 parameters and also offers no testable predictions, introducing spin networks and loop quantization. Both approaches have difficulty connecting to the Standard Model. In contrast, this theory has only one parameter (ρ, which is derived from fundamental constants rather than being free), offers five testable predictions with current or near-future technology, requires no new physics beyond standard physics applied in a dense medium, and connects directly to the Standard Model through the energy-momentum tensor. The Core Message Quantum mechanics is not magic. It is not a new kind of physics. It is the same old physics—Newton, Maxwell, Coulomb—operating in a space that is 6.54 trillion times denser than we perceive. Three things to remember: Space is dense, not empty. What we call "empty space" is actually a dense, discrete medium. We are the ones who are sparse—not the universe. The particle has a trajectory. Every particle follows a precise, continuous path. We don't see it because our instruments are too slow. The path exists—we just can't resolve it. Energy breathes. Energy is not a static number. It oscillates in time, like a heartbeat. The universe is alive with resonance. Conclusion This work demonstrates that quantum mechanics is not a theory of fundamental randomness and irreducible uncertainty. It is the low-resolution, static-space approximation of a deeper, deterministic, dynamic reality. The deeper reality is quantum space—dense, discrete, resonant, and absolute. The final statement of this

•Jul 1, 2026•0 citations

Classical Systems

K. Kong Wan

From our daily contact with the physical world, we gradually acquire a strong intuition as to how objects around us would behave. These are objects which we can examine directly through our senses and some simple instruments are known as classical systems. Typically a classical system consists of a number of particles. This chapter discusses a number of guiding principles of the behaviour of classical systems. A system may consist of a single particle. More complex systems can be built up from a finite number of particles. The studies of such systems, known as discrete systems, lead to a large body of theories known as classical mechanics. The properties of a discrete classical system are intuitively meaningful and their existence is taken for granted. These properties can be given a mathematical description. There is an underling objective reality and our task is to describe and to reveal such reality.

•Jul 9, 2019•0 citations
Mappings, functions In this book, the term architecture refers to a non-mathematical description of the sequence of events which collectively describes an experiment. Continuous time (CT) classical mechanics (CM) is based on some explicit and implicit assumptions that generate architectures radically different from those describing quantum processes. In the following list, which is not complete by any means, most of these assumptions may appear obvious and overstated, but one or two are subtle and should be brought to light, since they have an enormous impact on the theories which are based on them. Our comments are in square brackets at the end of each item in the list. At any given time, a system under observation (SUO) exists in a unique physical state with absolute properties independent of any observer [a standard classical metaphysical assumption]. The physical state of an SUO at any given time t may be represented by a single element Ψ( t ) called a state of some fixed state space U (or universe) [a reasonable mathematical assumption]. The state Ψ( t ') of an SUO at any time t ' later than t is also an element of U , not necessarily the same state Ψ( t ) at the earlier time [a reasonable classical mathematical assumption]. Observers are exophysical, meaning that they stand outside of U [a nearly universal metaphysical assumption; observers are not discussed in detail in CM or QM simply because there is no comprehensive theory of what constitutes an observer].
This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).
Quantization We now discuss the quantization of discrete time (DT) classical mechanics (CM). We restrict our attention in this chapter to a system under observation (SUO) consisting of a single point particle of mass m moving in one spatial dimension with Cartesian coordinate x . The generalization to two or more degrees of freedom is as straightforward here as in CT quantum mechanics (QM). The Dirac bra-ket notation will be used for convenience. The standard principles of QM have proven remarkably successful and consistent over the years (Peres, 1993) and we have no reason to alter them apart from changing from CT to DT. This is a significant change. Quantum mechanics became dominant in physics and chemistry because of the success of the Schrödinger equation, which is a first-order-in-time differential equation. Therefore, we should take care to ensure that discretizing time does not undermine the successes of CT QM. There are two reasons why discretizing time in QM might be considered. First, the Schrödinger equation is hard, or even impossible, to solve exactly in many situations and temporal discretization might be seen as a step towards numerical simulation by computer. This motivated the work of Bender (Bender et al. , 1985a, 1985b, 1993) and others on the DT Schrödinger equation. The second reason is one of principle: we may want to explore the properties of DT QM as a self-consistent theory in its own right rather than as an approximation to CT QM.
The general view is that all fundamental physical laws should be formulated within the framework given by quantum mechanics (QM). In a sense, QM therefore has the character of a metaphysical theory. Consequently, if it is possible to derive QM from more basic principles, these principles should be of general, philosophical nature. Here, we derive the formalism of QM from well-motivated epistemic principles. A key assumption is that a physical theory that relies on entities or distinctions that are unknowable in principle gives rise to wrong predictions. First, an epistemic formalism is developed, using concepts like knowledge and potential knowledge, identifying a physical state SSS with the potential knowledge of the physical world. It is demonstrated that QM emerges from this formalism. However, Hilbert spaces, wave functions and probabilities are defined in certain well-defined observational contexts only. This means that the epistemic formalism is broader than QM. In the fundamental layer of description, the physical state SSS is a subset of a state space S={Z}\mathcal{S}=\{Z\}S={Z}, such that SSS always contains many elements ZZZ. These elements correspond to unattainable states of complete knowledge of the world. The evolution of SSS cannot be determined in terms of the individual evolution of the elements ZZZ, unlike the evolution of an ensemble in classical phase space. The evolution of SSS is described in terms of sequential time n∈Nn\in \mathbf{\mathbb{N}}n∈N, which is updated according to n→n+1n\rightarrow n+1n→n+1 each time an event occurs, each time potential knowledge changes. Sequential time nnn can be separated from relational time ttt, which describes distances between events in space-time. There is an entire space-time associated with each nnn, in which ttt represents the knowledge at sequential time nnn about the temporal relations between present and past events.
In general, a physical theory is based on some fundamental principles concerning space, time, matter, and the laws prescribing how matter is distributed in space and how it evolves in time, and even how spacetime itself changes in response to the changes of matter. The spectrum of the known theories is very diverse, since they are based on different assumptions. Some theories assume that space and time are totally independent, some that they are warped in a continuum named "spacetime", and some even that there are more dimensions than the usual 3+1. Most theories consider that space and time are continuous, but many recent theories consider them to be discrete. Most dynamical laws of fundamental Physics are deterministic, but apparently Quantum Mechanics shows that they may have an indeterministic component. Despite these distinctions between various theories, our intuition and experience tells us that they have a lot in common. In this paper I will try to capture the common features in a single mathematical structure. This is not a Theory of Everything, but only a mathematical structure which is common to most physical theories known so far. We will see that these theories can be obtained as particular cases of this structure, which I will call "world". In particular, the worlds described by the Classical Mechanics, the Theory of Relativity and the Quantum Mechanics are examples of worlds according to this definition, but also some theories attempting to unify gravity and QM, like String Theory. The purpose of this distillation is to provide a mathematical common background to both physical and metaphysical discussions about the various theories of the World.
DESCRIPTION This work presents a complete, deterministic framework for quantum mechanics derived entirely from a single, testable postulate: the density of quantum space. The Core Problem For over a century, quantum mechanics has been interpreted as a theory of fundamental randomness, wave-particle duality, instantaneous jumps, and irreducible uncertainty. The standard Copenhagen interpretation treats the wave function as a probability amplitude, measurement as a "collapse" of that probability, and the Heisenberg uncertainty principle as a fundamental limit of nature. This work demonstrates that all these "mysteries" are not properties of nature—they are artifacts of our limited perception. We observe quantum phenomena through slow, macroscopic instruments that cannot resolve the actual continuous dynamics occurring in a space that is 6.54 trillion times denser than what we perceive as "empty space." The Fundamental Postulate The foundation of this work is a single, geometrically derived constant: ρ = (number of Planck lengths in 1 mm) / (number of millimeters in 1 light-year) = 6.54 × 10¹² Physical Meaning: One millimeter of physical space contains 6.54 trillion times more quantum spatial units than one light-year contains millimeters. What we perceive as "empty space" is actually a dense, discrete, information-rich medium. The entire observable universe (93 billion light-years in diameter) maps to only 0.014 light-years in quantum space—well within our Solar System. Consequence: The density inversion explains why quantum phenomena appear strange: we are observing a dense, fast, discrete reality through slow, coarse, macroscopic instruments. The apparent "weirdness" is not in nature—it is in our perception. The Comprehensive Law From the density postulate and classical physics (Newton, Coulomb, Maxwell, Einstein), a single comprehensive law is derived that contains all known physics as limit cases: E(t,v,n) = (m₀c²/ρ) · ‖n‖ · √(1 + (γv/c)²) · sin(2πt/T) This formula contains: Einstein's E = mc² as the limit ρ → 1, ‖n‖ → 1, sin → 1, v ≪ c Newtonian mechanics as the limit v ≪ c Special relativity as the limit ρ → 1, ‖n‖ → 1, sin → 1 Quantum mechanics as the discrete geometry n ∈ ℤ³ and resonance sin(2πt/T) Each term has a precise physical meaning: the rest mass energy (m₀c²) is scaled by the quantum space density (1/ρ), modified by the position relative to the absolute zero point Z₀ (‖n‖), adjusted for relativistic motion (√(1 + (γv/c)²)), and modulated by temporal resonance (sin(2πt/T)). Resolution of All Quantum Paradoxes Wave-Particle Duality = Optical Illusion The electron is always a localized particle with a definite trajectory. The "wave" is the charge density distribution—a standing wave in the dense quantum medium. From the standing wave condition 2πr = nλ, the Poynting vector in the far field is exactly zero (S = 0), meaning the electron does not radiate. The apparent "wave behavior" is an optical illusion caused by the density ρ smearing the point-like trajectory in our low-resolution projection. Wave Function = Energy Density, Not Probability The squared modulus of the wave function |Ψ|² is the energy density fraction, not a probability density. The particle itself follows a deterministic trajectory y(t) = v₀t - ½g₀t². Probability appears only because we cannot resolve the trajectory (due to t_total = 118 as ≪ Δt_instr) and we do not know the initial phase φ of the oscillation. Schrödinger Equation = Effective Approximation The Schrödinger equation is an effective, averaged description of how the energy density evolves. The fundamental law is classical motion in the dense quantum medium. The Schrödinger equation emerges when we average over the fast oscillation sin(2πt/T), ignore the density force F_ρ, and treat trajectories statistically. Quantization = Resonance, Not Arbitrary Jumps Energy levels are the resonant harmonics of the quantum space medium. The standing wave condition 2πr = nλ gives E_n = -13.6 eV/n²—exactly the same physics as a vibrating string. Only frequencies that satisfy the standing wave condition are stable. The transition between levels is not an instantaneous "jump" but a continuous trajectory y(t) = v₀t - ½g₀t² with total duration t_total = 118 as. The apparent "instantaneous jump" is an artifact of our slow instruments. Heisenberg Uncertainty = Collision, Not Fundamental Limit The uncertainty product depends on the measurement geometry: Δx·Δp = h·tan(θ/2), where θ is the scattering angle of the measurement photon. As θ → 0 (forward scattering with perfect detection), the product approaches zero. There is no fundamental lower bound. The uncertainty is a practical limitation of invasive measurement—we disturb the particle's trajectory when we hit it with a photon. Quantum Randomness = Ignorance Residual "quantum randomness" is simply unmodeled environmental force noise. The apparent randomness arises from three sources: temporal resolution (t_total = 118 as ≪ Δt_instr—we cannot resolve the trajectory), initial phase ignorance (we do not know the phase φ of sin(2πt/T)), and measurement perturbation (we do not know the exact collision parameters). If we could measure the outgoing photon completely and know the initial phase, the outcome would be fully deterministic. Randomness is in our ignorance, not in nature. Five Testable Predictions The theory makes five clear, quantitative predictions that distinguish it from standard quantum mechanics: Prediction 1 - Electron Transit Time: The electron transit time during the n=1 → n=2 transition in hydrogen is 118 ± 5 attoseconds. Standard quantum mechanics predicts zero duration (instantaneous jump). Testable with attosecond pump-probe spectroscopy. Prediction 2 - Heisenberg Product Can Approach Zero: With better detectors, the uncertainty product Δx·Δp can approach zero, not ħ/2. Standard quantum mechanics predicts Δx·Δp ≥ ħ/2 always. Testable with weak measurement experiments. Prediction 3 - Energy Oscillation at 116 kHz: The energy of gold-197 oscillates at 116 kHz with amplitude 1.252 × 10⁻²⁶ J. Standard quantum mechanics predicts constant energy for a given state. Testable with precision spectroscopy. Prediction 4 - Bell Inequality Violations Decrease with Measurement Strength: As measurement impulse strength decreases, Bell inequality violations decrease. Standard quantum mechanics predicts violations are independent of measurement strength. Testable with weak measurement entanglement experiments. Prediction 5 - Quantum Randomness is Environmental Noise: Residual "quantum randomness" equals unmodeled environmental force noise. Full force accounting restores classical determinism. Standard quantum mechanics predicts randomness is fundamental and irreducible. Testable with complete force reconstruction in closed quantum systems. The Absolute Zero Point Z₀ This work introduces Z₀ = (0,0,0) as a universal, absolute reference point in quantum space. Every point in space has a discrete address: n = (n_r, n_θ, n_φ), ‖n‖ = √(n_r² + n_θ² + n_φ²) Z₀ is physically grounded in the cosmic microwave background (CMB) rest frame, vacuum energy (the lowest possible energy state), and the geometric center of the quantum space lattice. This eliminates the relativity of reference frames—providing an absolute coordinate system that standard physics lacks. Mathematical Consistency The work provides rigorous derivations of: The density ρ from first principles (Planck length, electron mass, fine structure constant) The quantum coordinate norm ‖n‖ from the discrete metric of quantum space The oscillation period T from the de Broglie relation The coupling constant ε from fundamental constants The regularized metric eliminating all singularities The thermodynamic properties of quantum space (entropy, temperature, heat capacity) The covariant Einstein equation in quantum space The connection to quantum anomalies and topological invariants The Lamb shift from quantum space density oscillation Dark energy from quantum space density variation Comparison with Other Approaches String theory requires approximately 10⁵⁰⁰ free parameters and offers no testable predictions with current technology, while introducing new physics such as strings, extra dimensions, and supersymmetry. Loop quantum gravity requires approximately 10 parameters and also offers no testable predictions, introducing spin networks and loop quantization. Both approaches have difficulty connecting to the Standard Model. In contrast, this theory has only one parameter (ρ, which is derived from fundamental constants rather than being free), offers five testable predictions with current or near-future technology, requires no new physics beyond standard physics applied in a dense medium, and connects directly to the Standard Model through the energy-momentum tensor. The Core Message Quantum mechanics is not magic. It is not a new kind of physics. It is the same old physics—Newton, Maxwell, Coulomb—operating in a space that is 6.54 trillion times denser than we perceive. Three things to remember: Space is dense, not empty. What we call "empty space" is actually a dense, discrete medium. We are the ones who are sparse—not the universe. The particle has a trajectory. Every particle follows a precise, continuous path. We don't see it because our instruments are too slow. The path exists—we just can't resolve it. Energy breathes. Energy is not a static number. It oscillates in time, like a heartbeat. The universe is alive with resonance. Conclusion This work demonstrates that quantum mechanics is not a theory of fundamental randomness and irreducible uncertainty. It is the low-resolution, static-space approximation of a deeper, deterministic, dynamic reality. The deeper reality is quantum space—dense, discrete, resonant, and absolute. The final statement of this
From our daily contact with the physical world, we gradually acquire a strong intuition as to how objects around us would behave. These are objects which we can examine directly through our senses and some simple instruments are known as classical systems. Typically a classical system consists of a number of particles. This chapter discusses a number of guiding principles of the behaviour of classical systems. A system may consist of a single particle. More complex systems can be built up from a finite number of particles. The studies of such systems, known as discrete systems, lead to a large body of theories known as classical mechanics. The properties of a discrete classical system are intuitively meaningful and their existence is taken for granted. These properties can be given a mathematical description. There is an underling objective reality and our task is to describe and to reveal such reality.