de totstandkordng van dit proefschriftThis thesis consists of four parts.Notes and references are given at the end of each part.In part I we will give a review of the various notions of "representation" and "equivalence" that are used in theoretical physics and we will give a precise definition of the concept of "gauge equivalence".There we will meet the notion of gauge matrix.A method will be developed for obtaining the gauge matrices up to equivalence, as this is necessary in order to find the gauge equivalence classes of representations.In this method we will use the concept of superequivalence of group exponents, generalised here to group multipliers.Part I has a rather abstract mathematical character.Its results will be applied to physically interesting symmetry groups in the subsequent parts, where the notation and terminology of part I will be used freely.In part II we will consider the Poincaré as well as the Galilei symmetry group of uniform parallel electromagnetic fields.We will see that there are representations of these groups that are projectively equivalent, but that describe particles which arc, indeed, not "equivalent in every physical aspect".Specifically, particles in different Landau levels are described by projectively equivalent representations.However, those particles are not at all equivalent from the physical point of view; a transition between different Landau levels is not a trivial transformation of the mathematical description but it is a physical process that involves emission or absorption of energy.This is one of the most striking examples showing that, in general, projective equivalence of representations does not correspond to equivalence of physical systems.On the other hand, the different -13 -Landau levels are inequivalent with respect to the gauge equivalence concept.Another example which shows that gauge equivalence corresponds better than projective equivalence to "equivalence in every physical aspect" is provided by particles with different masses in uniform parallel electromagnetic fields (with Ε φ 0) in the Poincaré theory.Surprisingly, these particles are described by projectively equivalent representations; these representations are, however, gauge inequivalent.As a by-product we will find in part II what might be called a group theoretical derivation of the minimal coupling of a spinless particle in uniform parallel electromagnetic fields.Traditionally, the minimal coupling in quantum mechanics is "derived" by correspondence with classical mechanics.A derivation has also been given by a pseudo-invariance argument using instantaneous transformations.Although the latter argument may be the best that is available for arbitrary electromagnetic fields (that break space-time symmetry) it is in fact a "waste" of invariance in those cases where a special field is present that preserves a part of the symmetry.Indeed, in part II we obtain the minimal coupling in the Schrödinger and Klein-Gordon equations from the (exact rather than pseudo) symmetry group of the uniform parallel fields, without the detour via the free particle equations and without the "trick" of the substitution ІЭ ->• ІЭ -eA.This may be seen as a contribution to the process of remodelling traditional quantum mechanics by modern (group) theoretical tools.In part III other uniform electromagnetic fields are considered which are particular, in the sense that their connected symmetry groups are not isomorphic to the connected symmetry group of the uniform parallel fields.These are the uniform crossed fields under the Poincaré group and the uniform electric fields under the Galilei group.The equations -14of motion, including the minimal coupling, will be derived from the symmetry, analogous to part II.We will find a "Galilean subdynamics" for a Poincaré particle in uniform crossed fields.The exponents of the symmetry groups will give rise to an unexpected parameter in the equations of motion, which can be interpreted in terms of an anisotropic mass, corresponding to the anisotropy of the physical system due to the external field.In part IV we apply the notion of gauge equivalence of locally operating representations to some cosmologica!symmetry groups, operating on curved universes.For these groups the notion of superequivalence of group exponents plays a non-trivial role in the determination of a complete set of inequivalent gauge matrices.We will find that in cosmologica!models with one space dimension a free particle and a particle in an external field are described by projective representations of the symmetry group that are projectively equivalent but gauge inequivalent.This shows once more that the concept of gauge equivalence of locally operating representations is a more natural concept for the description of physical systems than that of the simple projective equivalence.As a by-product we will find that the minimal coupling is not necessarily the correct description of the interaction of a charged particle in an external field in a (relativistic) cosmologica!space-time.
The first paper: We describe principles of new cybernetics and use these principles for resolution of basic physical paradoxes. It demonstrates universality of the principles of new cybernetics. The second paper: recent letter by Maccone presents a solution based on the existing laws of quantum mechanics to the arrow-of-time dilemma. He argues that all phenomena in which the entropy decreases must not leave any information (in the observer's memory) of their having occurred because the observer is a part of the whole system. Maccone concludes that quantum mechanics is necessary to his argument, which he believes does not otherwise work in classical mechanics. This Comment consists of four parts. We discuss the basic problems in the first part. This Comment and the previously published Comment by Jennings and Rudolph describes flaws in Maccone's arguments. However, the main argument (erasure of the observer's memory), which was previously formulated in our work and was repeated by Maccone, is correct under the conditions described in this Comment. Moreover, this argument can be used to resolve a reduction paradox (the Schrodinger's Cat paradox) in quantum mechanics. This use is demonstrated in the second part. In the third part, the synchronisation (decoherence) of time arrows is discussed. In the fourth part, the synchronisation (decoherence) of time arrows in quantum gravity is considered.
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