The existence of the so-called ‘dark’ issues of mechanics implies that our present accounting for mass and energy is incorrect in terms of applicability on a cosmological scale, and the question arises as to where the difficulty might lie. The phenomenon of quantum entanglement indicates that systems of particles exist that individually display certain characteristics, while collectively the same characteristic is absent simply because it has cancelled out between individual particles. It may therefore be necessary to develop theoretical frameworks in which long-held conservation beliefs do not necessarily always apply. The present paper summarises the formulation described in earlier papers (Hill, JM. On the formal origin of dark energy. Z Angew Math Phys 2018; 69:133-145; Hill, JM. Some further comments on special relativity and dark energy. Z Angew Math Phys 2019; 70: 5–14; Hill, JM. Special relativity, de Broglie waves, dark energy and quantum mechanics. Z Angew Math Phys 2019; 70: 131–153.), which provides a framework that allows exceptions to the law that matter cannot be created or destroyed. In these papers, it is proposed that dark energy arises from conventional mechanical theory, neglecting the work done in the direction of time and consequently neglecting the de Broglie wave energy [Formula: see text]. These papers develop expressions for the de Broglie wave energy [Formula: see text] by making a distinction between particle energy [Formula: see text] and the total work done by the particle [Formula: see text], that which accumulates from both a spatial physical force [Formula: see text] and a force [Formula: see text] in the direction of time. In any experiment, either particles or de Broglie waves are reported, so that only one of [Formula: see text] or [Formula: see text] is physically measured, and particles appear for [Formula: see text] and de Broglie waves occur for [Formula: see text], but in either event both a measurable and an immeasurable energy exists. Conventional quantum mechanics operates under circumstances such that [Formula: see text] vanishes and [Formula: see text] becomes purely imaginary. If both [Formula: see text] and [Formula: see text] are generated as the gradient of a potential, the total particle energy is necessarily conserved in the conventional manner.
Those defile the purity of mathematical and philosophical truthy who confound real quantities with their relations and sensible measures. —ISAAC NEWTON, Principia Before we say anything about quantum mechanics, let us take a quick look at the Newtonian (or ‘classical’) view of the physical universe. According to that view, nature consists of an enormous number of particles. When Newtonian physics is combined with the theory of the electromagnetic field, it becomes convenient to think of these particles as dimensionless (even if there is a kind of conceptual strain involved in trying to think of something as having a mass but not any size ), and as possessing electrical properties – negative charge, or positive charge, or neutrality. This leads to the well-known ‘solar system’ view of the atom – with the electrons whirling around the nucleons (neutrons and protons) just as the planets whirl around the sun. Out of atoms are built molecules; out of molecules, macroscopic objects, scaling from dust motes to whole planets and stars. These latter also fall into larger groupings – solar systems and galaxies – but these larger structures differ from the ones previously mentioned in being held together exclusively by gravitational forces. At every level, however, one has trajectories (ultimately that means the possibility of continuously tracing the movements of the elementary particles) and one has causality (ultimately that means the possibility of extrapolating from the history of the universe up to a given time to its whole sequence of future states).
The proposed paper presents the unobserved inadequacies in de Broglie's concepts of wave-particle duality and matter waves in the year 1923. The commonly admitted quantum energy or frequency expression hν=γmc2 is shown to be inappropriate for matter waves and is acceptable only for photons. The superluminal phase velocity expression c2/\u{psion}, for matter waves, is investigated in detail and is also reported to be inadequate. The rectifications in the inadequate concepts of de Broglie's theory and refinements in the analogy implementation between light waves and matter waves are presented, which provides the modified frequency and phase velocity expression for matter waves. Mathematical proofs for the proposed modified frequency and phase velocity expression are also presented. In accordance with the proposed concepts, a wave-particle duality picture is presented which elucidates the questions coupled with the wave-particle duality, existing in the literature. Consequently, particle type nature is shown to be a characteristic of waves only, independent from the presence of matter. The modifications introduced in the frequency expression for matter waves leads to variation in the wave function expression for a freely moving particle and its energy operators, with appropriate justifications. A new relation between the Kinetic energy and Momentum of the moving body is proposed and is subsequently applied to introduce novel General and Relativistic Quantum Mechanical Wave Equations. Applications of these equations in bound state quantum mechanical systems provide the information regarding particle's general and relativistic behavior in such systems. Moreover, the proposed wave equations can also be transformed into Schrödinger's and Dirac's equations. The interrelation of Schrödinger's, Dirac's and proposed equations with the universal wave equation is also presented.
Now I shall try to give you an idea of the way in which physicists at present endeavour to overcome this failure. One might term it an ‘emergency exit’, though it was not intended as such, but as a new theory. I mean, of course, wave mechanics. (Edding-ton called it ‘not a physical theory but a dodge— and a very good dodge too’.) The situation is about as follows. The observed facts (about particles and light and all sorts of radiation and their mutual interaction) appear to be repugnant to the classical ideal of a continuous description in space and time. (Let me explain myself to the physicist by hinting at one example: Bohr's famous theory of spectral lines in 1913 had to assume that the atom makes a sudden transition from one state into another state, and that in doing so it emits a train of light waves several feet long, containing hundreds of thousands of waves and requiring for its formation a considerable time. No information about the atom during this transition can be offered.) So the facts of observation are irreconcilable with a continuous description in space and time; it just seems impossible, at least in many cases. On the other hand, from an incomplete description—from a picture with gaps in space and time—one cannot draw clear and unambiguous conclusions; it leads to hazy, arbitrary, unclear thinking—and that is the thing we must avoid at all costs! What is to be done?
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
Foundations of physics · For the general reader The Relational Architecture of Time Emergent temporal order, delayed-choice photon experiments, and the thermodynamic arrow — a complete mathematical account, explained from the ground up János Gábor Melegh The Relational Architecture of TimeEmergent temporal order, delayed-choice photon experiments, and the thermodynamic arrow. Copyright © 2026 János Gábor Melegh. All rights reserved. No part of this book may be reproduced in any form without written permission from the author, except for brief quotations in reviews and scholarly works. This book unifies and expands two research manuscripts by the author: Time as an Emergent Relational Structure (2026) and Relational Time Structure in Delayed-Choice Photon Experiments (axiomatically extended version, 2026). Every definition, theorem, proof, and derivation of those works is reproduced here in full and without abridgement; the surrounding prose has been substantially expanded so that a motivated reader without formal training in physics or mathematics can follow the entire argument. First English edition, 2026. Preface How to Read This Book This is a book about a deceptively simple question: what is time, physically? Not what a clock reads, and not how time feels, but what structure in the physical world earns the name "time" at all. The answer defended here can be stated in one sentence: time is not a fundamental ingredient of the universe but an emergent, relational structure — an ordering that arises when physical descriptions defined through interaction fit together consistently, with its familiar one-way direction supplied separately by thermodynamics. That single sentence carries a lot of machinery, and this book unpacks all of it. The book makes an unusual promise, and it is worth stating explicitly: The promise of this book Nothing has been simplified away. Every equation, every definition, every theorem, and every proof from the underlying research papers appears here in full. What has been added — greatly — is explanation: before and after each formal step you will find plain-language accounts of what the symbols say, why the step matters, and what would go wrong without it. If you skip the equations entirely, the prose still tells a complete story. If you read the equations, the prose tells you exactly how to read them. Throughout the book you will meet a small family of recurring boxes, each with a fixed job: In plain language (copper boxes) — the intuition. What a formal statement means in everyday terms, often with an analogy. Definitions (blue boxes) — precise introductions of new mathematical objects. These are the vocabulary of the theory. Theorems (framed boxes) — the claims that are actually proved, followed by their proofs in full. A proof ends with the traditional square symbol □. Warnings and scope limits (red boxes) — the places where the theory deliberately stops claiming things. These boxes are part of the scientific content, not decoration: a theory earns trust partly by declaring its own boundaries. A five-minute notation primer Readers who have never used the notation of quantum mechanics need only a handful of symbols, all introduced properly when they first appear. As a preview:
∣ψ⟩ — a "ket," the standard label for a quantum state. Think of it as the complete catalogue of what quantum theory says about a system.
ρ — a density operator, the more general bookkeeping device for quantum states, including states known only statistically.
Tr — the trace, a summing-up operation.
TrB ("partial trace over
B") means: average away everything about subsystem
B and keep the rest.
P(x∣b) — the probability of outcome
x given setting
b. Ordinary conditional probability.
e1≺e2 — event
e1 causally precedes event
e2; it can influence it.
∑ — a sum;
sup and
inf — the least upper bound and greatest lower bound of a set of numbers, close cousins of "maximum" and "minimum." Everything else — beam splitters, POVMs, quantum channels, entropy, modular flow — is built up step by step in the chapters where it is needed, and Appendix A collects every symbol in one table. The shape of the argument Part I poses the problem of time and develops the conceptual core: relational description, the principle of relational consistency, the removal of subjective experience from the foundations, and the division of labor between relational ordering and thermodynamic direction. Part II takes the framework into the laboratory. It builds the full mathematical toolkit from scratch and then works through the two most notorious "time-bending" experiments of quantum optics — Wheeler's delayed choice and the delayed-choice quantum eraser — with complete derivations, showing that nothing in them requires the future to reach back into the past. Part III formalizes relational consistency exactly, explains what a physical record is, de-mystifies the "observer" (including a careful treatment of the Wigner's-friend and Frauchiger–Renner thought experiments), and shows the correct use of causal event graphs. Part IV separates order from direction, develops the thermodynamic arrow, connects the framework to thermal time and to neighboring research programs. Part V asks the hardest questions of any theory — what does it mean, and how could it be wrong? — and closes with a complete empirical research program and conclusions. What this book does not claim This book does not announce a new physical force, a new dynamical law, or an experimental result that contradicts standard quantum mechanics. Its framework is deliberately, provably empirically equivalent to standard quantum theory in every laboratory situation treated here. Its contribution is a rigorously constrained conceptual and mathematical reconstruction: an account of what time is, and of what delayed-choice experiments do and do not show, that is causally clean, fully formalized, and — crucially — falsifiable in the precise, limited sense worked out in Chapter 22. Readers deserve to know which statements are established mathematics, which are structural proposals, and which are open questions; this book labels all three throughout. Contents Table of Contents Preface: How to Read This Book Part I — The Problem of Time 1. Why Time Is a Problem 2. Relational Descriptions in Contemporary Physics 3. Relational Consistency as a Structural Principle 4. Time Without Experience 5. Relational Ordering and Thermodynamic Asymmetry 6. Ontological Implications and Open Questions Part II — The Laboratory of Time 7. Aim and Scientific Status 8. The Experimental Foundation 9. The Mathematical Toolkit 10. Postulates of the Relational-Time Delayed-Choice Model 11. Wheeler's Delayed Choice: The Complete Two-Path Derivation 12. The Delayed-Choice Quantum Eraser: A Density-Matrix Derivation 13. The General No-Signalling Theorem Part III — Consistency, Records, and the Event Graph 14. Relational Consistency: Exact and Experimental Formalization 15. Physical Records, Decoherence, and the Observer De-Mystified 16. Relational Time and the Correct Use of the Event Graph Part IV — The Arrow and the Landscape 17. Relational Order and the Thermodynamic Direction of Time 18. Distinctions from Related Approaches 19. A Formal Bridge: Consistent Quantum Histories 20. Separating Temporal Consistency from Quantum Coherence Part V — Meaning and Testability 21. What Does "Determined Later" Mean, Mathematically? 22. Empirical Content, Falsifiability, and a Research Program 23. Conclusions Back Matter Appendix A — Notation and Conceptual Correspondence Appendix B — The Logical Core in a Single Chain of Formulas Appendix C — Claim–Assumption Matrix References Part I The Problem of Time In which time is neither denied nor assumed, but asked to explain itself — and the concepts of relational description, relational consistency, and the thermodynamic arrow are assembled. Chapter 1 Why Time Is a Problem The conceptual status of time remains deeply problematic in contemporary physics. Relativity eliminates absolute simultaneity; quantum theory challenges the notion of observer-independent states; and neither framework explains how temporal structure itself arises. Ask a physicist of the nineteenth century what time is, and the answer would have been serene: time is a universal, externally given parameter. Every event in the universe happens at some moment of a single cosmic clock, and physics describes how things change as that clock ticks. Newton wrote it into the foundations: absolute time "flows equably without regard to anything external." This picture is intuitive, it matches everyday experience, and it is built into the very grammar of classical mechanics, where every equation of motion says how a quantity changes with respect to
t — a
t that is simply assumed to be there, the same for everyone, everywhere. Modern physics profoundly destabilized this picture — twice, from two independent directions. The first blow came from special relativity. Einstein's 1905 analysis showed that there is no such thing as absolute simultaneity: whether two distant events happen "at the same time" depends on the observer's state of motion. Time intervals themselves are frame-dependent. This is not a limitation of our measuring devices; it is a structural feature of spacetime. There is no privileged universal "now" slicing the universe into past and future — no preferred foliation, in the technical vocabulary — that all observers could agree on. What relativity leaves behind is a network of relations: which events can influence which, encoded in the invariant light-cone structure. The second blow came from quantum theory, which undermines something even more basic: the idea that a physical system simply has a defin