Quantum theory reads several of its quantities on the loops of configuration space rather than at its points: the inequivalent quantizations of a system are classified by the characters of the fundamental group (Schulman 1968; Laidlaw and DeWitt 1971), and the phase factors on loops give a complete description of a gauge field (Wu and Yang 1975). We propose a framework in which reading on loops is the rule. It has three elements: the site ℝℙ², realized in Euclidean space by the cross-cap; its covering S²↓ℝℙ², the only connected non-trivial covering of the site, hence its universal covering; and the reading, with values in a sheaf on the site. The obstruction of the site is the class a = w₁(ℝℙ²). Four known calculations are reread in this framework, and each reads the same class a: the Schrödinger equation on ℝℙ² and on SO(3) (sectors split by the parity of ℓ, or of 2j); the Dirac equation on ℝℙ² (a single type of Pin structure, in two copies, and an asymmetric spectrum); the two degrees of the reading (the holonomy −1 and the torsion class of H²(ℝℙ²; ℤ) are one element); and the 4π periodicity of spin ½ (the monodromy of SU(2)↓SO(3)). The thesis (Th) is that this framework offers a common perspective on scattered approaches. Numerical checks are supplied as scripts. This record contains the same paper in two versions: English (primary) and French (original). The French text prevails in case of discrepancy. — Ce dépôt contient le même texte en deux versions : anglaise (principale) et française (originale) ; le texte français fait foi.
Quantum theory reads several of its quantities on the loops of configuration space rather than at its points: the inequivalent quantizations of a system are classified by the characters of the fundamental group (Schulman 1968; Laidlaw and DeWitt 1971), and the phase factors on loops give a complete description of a gauge field (Wu and Yang 1975). We propose a framework in which reading on loops is the rule. It has three elements: the site ℝℙ², realized in Euclidean space by the cross-cap; its covering S²↓ℝℙ², the only connected non-trivial covering of the site, hence its universal covering; and the reading, with values in a sheaf on the site. The obstruction of the site is the class a = w₁(ℝℙ²). Four known calculations are reread in this framework, and each reads the same class a: the Schrödinger equation on ℝℙ² and on SO(3) (sectors split by the parity of ℓ, or of 2j); the Dirac equation on ℝℙ² (a single type of Pin structure, in two copies, and an asymmetric spectrum); the two degrees of the reading (the holonomy −1 and the torsion class of H²(ℝℙ²; ℤ) are one element); and the 4π periodicity of spin ½ (the monodromy of SU(2)↓SO(3)). The thesis (Th) is that this framework offers a common perspective on scattered approaches. Numerical checks are supplied as scripts. This record contains the same paper in two versions: English (primary) and French (original). The French text prevails in case of discrepancy. — Ce dépôt contient le même texte en deux versions : anglaise (principale) et française (originale) ; le texte français fait foi.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.
Cet article passe en revue les multiples applications de la théorie des groupes aux problèmes de symétrie en physique. En physique classique, il s’agit surtout de la relativité : euclidienne, galiléenne, einsteinienne (relativité restreinte). Passant à la mécanique quantique, on remarque d’abord que les principes de base impliquent que l’espace des états d’un système quantique a une structure intrinsèque d’espace préhilbertien, que l’on complète ensuite en un espace de Hilbert. Dans ce contexte, la description de l’invariance sous un groupe G se base sur une représentation unitaire de G. On parcourt ensuite les différents domaines d’application : physique atomique et moléculaire, matière condensée, optique quantique, ondelettes, symétries internes, symétries approchées. On discute ensuite l’extension aux théories de jauge, en particulier au Modèle Standard des interactions fondamentales. On conclut par quelques indications sur des développements récents. * * * The present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity : Euclidean, Galilean, Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the state space of a quantum system has an intrinsic structure of pre-Hilbert space, that one completes into a genuine Hilbert space. In this framework, the description of the invariance under a group G is based on a unitary representation of G. Next we survey the various domains of application : atomic and molecular physics, condensed matter physics, quantum optics, wavelets, internal symmetries, approximate symmetries. Next we discuss the extension to gauge theories, in particular to the Standard Model of fundamental interactions. We conclude with some remarks about recent developments.
A finitist, relational axiomatization of physics: quantum theory and a relativity sector derived from four axioms about group-valued relations and closure. Contributions to the foundations of quantum mechanics (Born rule derivation, quantum reconstruction, local tomography, complex field selection), the emergence of time and causal structure in discrete networks, discrete Noether theorems, lattice gauge structure, and the Schrödinger equation derived by two independent routes. Every claim is graded in a Claim Ledger; every numerical claim is machine-verified with archived scripts; negative results are reported at full strength. What this paper does The universe is taken to be constituted by relations rather than objects. Stable relations form closures (self-consistent cycles) whose holonomies carry the physical content. Four axioms — primacy of relations (group-valued, edge-borne), closure stability, chirality from order, and observation as bidirectional group exchange — yield, at explicitly graded and conditional levels: Quantum sector Exchange-completeness theorems for finite label groups: abelian at all cycle ranks; non-abelian on the single cycle (Wilson readouts plus class-sum convolutions generate the full matrix algebra); the multi-cycle non-abelian case on the operational-equivalence quotient, with the obstruction completely characterized via class-preserving outer automorphisms — and the quotient's necessity now stratified by the source of word blindness (see New in Version 24.0). The Born rule, proved for finite abelian networks from a strictly weaker relational readout premise (statistics of realizable effects depend only on relational content); non-contextuality has this single source; additivity is basis-relative and declared as such. The Schrödinger equation by two routes of unequal premise grade, maintained side by side: a spectral route — the action operator is a weighted Cayley-graph Laplacian in the Pontryagin dual basis, the derived Gibbs state's unique canonical (modular) flow yields the Schrödinger equation with Hamiltonian equal to the action operator, no continuum limit, zero conjectures consumed, the single non-theorem step one flagged identification; and a Madelung route — physical and mechanism-rich (probability density derived from constraint gaps), valid exactly on the domain of the pseudo-continuity conjecture, with its full premise bill stated (continuity equation, Wallstrom circulation condition, m, V). A dedicated comparison subsection gives the line-by-line premise accounting and the machine-measured agreement region. Complex field selection: ℂ selected over ℝ and ℍ within the GPT comparison class in the covered sectors, with a second independent route from chirality (Frobenius–Schur dichotomy, quaternionic double exclusion); "uniquely selected" expressly not claimed. Positivity as pairing: all five in-text positive-semidefiniteness theorems traced to one mechanism (M = B†B with native factor); the Kraus form and complete positivity derived rather than assumed. The Noether sector on Noether's two theorems exactly: Noether I recovered on the holonomy action (compact Lie realization); Noether II identified as present (Bianchi as horn-filling, Schläfli as its Regge form); selection rules recovered for finite groups; the GNS bridge proved non-vacuously on the full exchange algebra; the sole potential extension — conservation as the kernel of exchange — isolated as a registered conjecture, consumed nowhere. A local–global representation theorem: six generative quantities (closure invariant, flux quantization, Weyl phase, non-abelian Stokes, orientation visibility, terminal normalization) are evaluations of cohomology classes against closed chains — one syntax, six exact machine probes. The gatekeeper no-go theorems registered and discharged: Kochen–Specker (the framework posits no global valuation — outcomes are resolutions of constraint gaps, and basis-relative additivity blocks state-independent variants), Pusey–Barrett–Rudolph (the statistical envelope is over resolutions yet to happen, not values already held; machine witness: identical configuration ensembles with opposite relative phase have opposite statistics), Wallstrom (registered as an explicit input, structurally discharged in the finite-abelian sector), Hegerfeldt (the Lieb–Robinson exponential tail identified as the only claimable front form for the continuum limit), Frauchiger–Renner (cross-agent certainty transfer presupposes kernel-free record transport, exactly what exchange-incompleteness excludes), and no-cloning (inherited in the covered sectors); Conway–Kochen and Colbeck–Renner noted as concordant. Relativity sector Time from finite capacity plus genuine randomness: the staging theorem (Θ(d) dependency depth, unbounded chains via the Moore bound; the quantitative failure law conditionalized on measure-level independence with exact minimal witnesses); the temporal quantum defined combinatorially, no Hamiltonian input. The speed of light grounded in an exact discrete dependency cone (combinatorial, no quantum input), the Lieb–Robinson bound consumed at cross-check grade only; a boost no-go theorem reducing Lorentz invariance to named statistical conditions, with the Collins fine-tuning objection recorded as a gate. The trace-descent theorem group: the growth-history measure forced onto the Mazurkiewicz trace quotient; chain reflection positivity at every depth with completely explicit spectrum. An equation sector of nine verification families (Schrödinger, Dirac, Maxwell/U(1), canonical commutator, Yang–Mills SU(2), Regge calculus, Klein–Gordon, Noether loop, Cartan curvature), where every machine-precision claim is the shadow of a proved backbone lemma — including the factor ½ of H = ½L + V, identified as the Wilson Fourier coefficient fixed by the action theorem. Four experimentally testable predictions, including a discriminating flat-versus-sigmoid visibility protocol. The Standard Model gauge group, ħ, and G remain open, with attack routes registered. How to read Three entry paths are marked in the document. Path A (derivational core): axioms → group structure → Noether sector → least action → Born rule → positivity. Path B (physical mechanisms): constraint propagation → time as determination → wave function as gap envelope → fields. Path C (scope and limits): the Claim Ledger, then Discussion and Open Problems — for the reader who wants to know what is claimed and what is not before reading anything else. The Claim Ledger is the single authoritative register of claim status; wherever body text sounds stronger, the Ledger governs. New in Version 24.0 (Version 23.0 is the immediately preceding public release; this version supersedes the internal V12–V15 and V17–V19 series, whose numbers are not used externally.) Version 24.0 resolves the structure of the multi-cycle non-abelian obstruction — the sector's longest-standing open front — following an external research exchange with three rounds of independent re-verification (fourteen core figures, three commutant instruments implemented independently, exact integer arithmetic on the obstruction side) and two rounds of counter-review that corrected one quantifier on each side: A dichotomy of the multi-cycle obstruction (new remark). Word blindness on multi-cycle networks has two identified sources, jointly covering every tested case, with opposite behaviour as cycle rank grows. Type I (global; graded): a class-preserving outer automorphism yields a commutant element by an argument free of cycle-rank reference; its non-triviality holds at theorem grade for b₁ ≥ d(G) via generating tuples, is monotone in b₁, and the threshold is measured at b_σ = 2 on the tested Wall group (at b₁ = 1 the permutation is the identity — no universal claim is made, and the existence of groups with threshold above 2 is registered open). Type II (local; measured): Out_c trivial but fusion fails on a proper subgroup, the blind pairs sitting in non-generating sectors — the same mechanism that defeats Wilson generation; this blindness heals: on every verified group (A₄, SL(2,3), A₄×ℤ₂) the full commutant collapses to trivial at b₁ = 3 by exact computation on two independently implemented instruments, although word readouts remain blind — the shift primitives separate, above a threshold, and the natural explanation (tuples becoming generating) is refuted by the persistence of blind blocks. Three boundaries at full strength: no exhaustiveness proof (a third source is not excluded); no general healing theorem (the verified groups all heal at 3, cause unidentified); the verified Type II sample is narrow. The quotient proposition restated in stratified form. The operational-equivalence quotient's necessity is now part of the proposition's own statement, in three strata: necessary where Out_c ≠ 1 at every b₁ ≥ b_σ(G), with b_σ ≤ d(G) at theorem grade; one-dimensional at b₁ = 2 on the fusion-type groups (verified cases); trivial at b₁ ≥ 3 on every verified fusion-type group and at every tested b₁ for word-separation-complete groups (a fourteen-group scan of orders ≤ 31 found no third source). Pre-registered validation preceded the restatement: a stratification-exhaustiveness scan (pass), a new-witness search (no new witness in range, downgrade clause applied as pre-registered), and a zero-conflict audit of the three strata against all archived data. The residual open problem is now precisely located. What remains of multi-cycle completeness: the rare Out_c ≠ 1 groups (54 in the exhaustive range through order 96) above their measured thresholds, the low-b₁ fusion stratum, a general healing theorem, and whether a threshold above 2 occurs. The multiplicity-freeness conjecture's mechanism sentence is corrected with the two regimes stated separately (at b₁ = 2 the shifts do not separate — the blind-pair transposition commutes with every primitive, integer-exact — and reduce the commutant to
A spin-orbit Hamiltonian with an effective gauge structure carries two distinct loop objects that are routinely conflated: an energy-independent Wilson holonomy, which organizes interference and internal spin transport, and an energy-dependent monodromy, which quantizes the spectrum. We show that cleanly separating these objects supplies a precise, computable bridge between the loop/holonomy representation of gauge theories and condensed-matter spin-orbit transport. The construction maps a spin-orbit Hamiltonian to an effective U(1) plus internal non-Abelian connection, reduces it to a first-order transport problem, and reads physical predictions from holonomy, monodromy, curvature, and eigenphase data. Two rings make the separation explicit. For a Dirac (graphene) ring with Rashba coupling and Aharonov-Bohm flux, the total holonomy factorizes exactly into a commuting U(1) flux phase times an internal spin/pseudospin holonomy, and the spectrum follows from a holonomy-eigenvalue condition. For a Rashba-Dresselhaus ring, the internal SU(2) transport is genuinely non-Abelian away from the α=±β pure-gauge locus, where curvature controls path ordering; spectral quantization then requires an explicit first-order reduction obtained by phase-space doubling of the second-order Schrödinger problem. A non-Abelian Stokes formulation and Magnus expansion serve as ordering diagnostics rather than spectral tools. Spin-network ideas enter only as historical geometric motivation, not as a dynamical import into spintronics.
La mécanique quantique est à la base de la technologie moderne. Elle a donné lieu à de nombreuses applications en physique, en chimie et même en biologie.Introduction à la mécanique quantique 2 étudie l’équation de Schrödinger, les puits, les marches et les barrières de potentiel. Les propriétés des bases orthonormées dans l’espace des fonctions d’onde de carré sommable et les notations de Dirac relatives aux kets et aux bras dans l’espace des états sont également examinées.Cet ouvrage analyse les notions d’opérateur linéaire, d’opérateur hermitien, d’observable, de conjugaison hermitienne et de commutateur. La représentation des kets, des bras et des opérateurs dans l’espace des états est exposée, tout comme l’équation aux valeurs propres et l’équation caractéristique. Enfin, ce second volume présente la définition de la valeur moyenne d’une observable et son équation d’évolution, l’étude des systèmes conservatifs et l’établissement du théorème d’Ehrenfest.
Short description A conceptual and minimally formal note within the ZM framework proposing that spin may be interpreted as a topological holonomy associated with finite oriented subgraphs of a discrete relational network, with collective coherence yielding effective classical fields and mesoscopic extensions of spin. Abstract We present a minimal discrete framework in which spin is not postulated as a fundamental quantum attribute but arises as a topological invariant associated with SU(2) holonomies on finite oriented subgraphs representing stable excitations of an underlying relational network. Each excitation is characterized by non contractible internal cycles whose holonomy encodes global phase memory. A fermionic sector is defined by minimal loops with holonomy −I, giving rise to the SU(2) to SO(3) double covering and the characteristic 4π behavior without invoking continuum geometry or intrinsic spin postulates. We further show that when gauge invariant phase coherence is established between neighboring subgraphs, internal holonomies become phase locked and extend over mesoscopic and macroscopic domains. In this regime, spin ceases to be a purely local property and acquires the status of a collective orientation field. Classical electromagnetic fields are interpreted as continuous limits of extended coherence domains of discrete holonomies. Minimal U(1) coupling reproduces the Pauli interaction with g = 2 at leading order, while deviations arise from holonomy fluctuations. The framework is compact, mathematically defined, and falsifiable, offering a unified interpretation of spin, its electromagnetic coupling, and the quantum to classical transition as consequences of discrete geometry and collective coherence. Description (EN) This record presents a discrete and relational interpretation of spin within the ZM research program. It examines whether spin can be understood as a topological holonomy associated with finite oriented subgraphs representing stable excitations of an underlying network, rather than as a primitive intrinsic quantum attribute. The note proposes that a fermionic sector is characterized by minimal cycles with holonomy −I in SU(2), thereby yielding the double covering SU(2) to SO(3) and the characteristic 4π structure without invoking background continuum geometry. It further argues that gauge invariant phase locking across neighboring subgraphs gives rise to collective coherence, effective orientation fields, and classical electromagnetic responses as coarse-grained limits of extended holonomy domains. The purpose of the present deposit is not to provide a complete dynamical derivation. It is to formulate a compact and falsifiable module within the broader ZM framework, clarify its conceptual scope, and define a structured basis for future formal and empirical development. Description complémentaire (FR) Ce dépôt présente une interprétation discrète et relationnelle du spin au sein du programme de recherche ZM. Il examine si le spin peut être compris comme une holonomie topologique associée à des sous-graphes orientés finis représentant des excitations stables d’un réseau sous-jacent, plutôt que comme un attribut quantique intrinsèque et primitif. La note propose qu’un secteur fermionique soit caractérisé par des cycles minimaux d’holonomie −I dans SU(2), ce qui fait émerger le double recouvrement SU(2) vers SO(3) ainsi que la structure caractéristique en 4π sans invoquer de géométrie continue de fond. Elle soutient également qu’un verrouillage de phase invariant de jauge entre sous-graphes voisins conduit à une cohérence collective, à des champs d’orientation effectifs, et à des réponses électromagnétiques classiques comme limites de coarse-graining de domaines étendus d’holonomies. L’objectif du présent dépôt n’est pas de fournir une dérivation dynamique complète. Il est de formuler un module compact et falsifiable dans le cadre plus large de ZM, d’en clarifier la portée conceptuelle, et d’établir une base structurée pour de futurs développements formels et empiriques. Scope and disclosure This record is a conceptual and minimally formal theoretical note within the ZM research program. It does not claim a complete derivation of the full quantum formalism, nor a finished microscopic dynamics. It introduces no implementation protocol, no engineering architecture, and no operational system design. All statements are formulated at the level of discrete structure, topological interpretation, admissible effective consequences, and theoretical scope.