Academic literature on the topic 'Rule commutation'

We show that the series product, which serves as an algebraic rule for connecting state-based input–output systems, is intimately related to the Heisenberg group and the canonical commutation relations. The series product for quantum stochastic models then corresponds to a non-abelian generalization of the Weyl commutation relation. We show that the series product gives the general rule for combining the generators of quantum stochastic evolutions using a Lie–Trotter product formula.

In this paper an ansatz that the anti-commutation rules hold only as integrated average over time intervals and not at every instant giving rise to a time-discrete form of Klein-Gordon equation is examined. This coarse-grained validation of the anti-commutation rules enables us to show that the relativistic energy-momentum relation holds only over discrete time intervals, fitting well with the time-energy uncertainty relation. When this time-discrete scheme is applied to four vector notations in relativity, the line-element can be quantized and thereby how the physical attributes associated with time, space and matter can be quantized is sketched. This potentially enables us to discuss the Zeno’s arrow paradox within the classical limit. As the solutions of the Dirac equation can be used to construct solutions to the Klein-Gordon equation, this temporal quantization rule is applied to the Dirac equation and the solutions associated with the Dirac equation under such conditions are interpreted. Finally, the general relativistic effects are introduced to a line-element associated with a particle in relativistic motion and a time quantized line-element associated with gravity is obtained.

In this paper an ansatz that the anti-commutation rules hold only as integrated average over time intervals and not at every instant giving rise to a time-discrete form of Klein-Gordon equation is examined. This coarse-grained validation of the anti-commutation rules enables us to show that the relativistic energy-momentum relation holds only over discrete time intervals, fitting well with the time-energy uncertainty relation. When this time-discrete scheme is applied to four vector notations in relativity, the line-element can be quantized and thereby how the physical attributes associated with time, space and matter can be quantized is sketched. This potentially enables us to discuss the Zeno’s arrow paradox within the classical limit. As the solutions of the Dirac equation can be used to construct solutions to the Klein-Gordon equation, this temporal quantization rule is applied to the Dirac equation and the solutions associated with the Dirac equation under such conditions are interpreted. Finally, the general relativistic effects are introduced to a line-element associated with a particle in relativistic motion and a time quantized line-element associated with gravity is obtained.

AbstractThe Thomas–Reiche–Kuhn (TRK) sum rule is a fundamental consequence of the position–momentum commutation relation for an atomic electron, and it provides an important constraint on the transition matrix elements for an atom. Here, we propose a TRK sum rule for electromagnetic fields which is valid even in the presence of very strong light–matter interactions and/or optical nonlinearities. While the standard TRK sum rule involves dipole matrix moments calculated between atomic energy levels (in the absence of interaction with the field), the sum rule here proposed involves expectation values of field operators calculated between general eigenstates of the interacting light–matter system. This sum rule provides constraints and guidance for the analysis of strongly interacting light–matter systems and can be used to test the validity of approximate effective Hamiltonians often used in quantum optics.

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, probability density current, commutation and uncertainty relations, momentum operator, rules for including scalar and vector potentials and antiparticles can be derived from the definition of the mean values of powers of space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrödinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the theory. The limit case of localized probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many-particle systems are also discussed.

Using the representation of the quantum group SL q(2) by the Weyl operators of the canonical commutation relations in quantum mechanics, we construct and solve a new vertex model on a square lattice. Random variables on horizontal bonds are Ising variables, and those on the vertical bonds take half positive integer values. The vertex is subjected to a generalized form of the so-called “ice rule,” its property is studied in detail and its free energy calculated with the method of quantum inverse scattering. Remarkably, in analogy with the usual six-vertex model, there exists a “free-fermion” limit with a novel rich operator structure. The existing algebraic structure suggests a possible connection with a lattice neutral plasma of charges, via the fermion-boson correspondence.

Cette thèse s'intéresse à l'égalité des preuves et des formules en logique linéaire, avec des contributions en particulier dans le fragment multiplicatif-additif de cette logique. En logique linéaire, et dans de nombreuses autres logiques (telle que la logique intuitionniste), on dispose de deux transformations sur les preuves : l'élimination des coupures et l'expansion des axiomes. On souhaite très souvent identifier deux preuves reliées par ces transformations, étant donné qu'elles le sont sémantiquement (dans un modèle catégorique par exemple). Cette situation est similaire à celle du λ-calcul où les termes sont identifiés à β-réduction et η-expansion près, opérations qui, par le prisme de la correspondance de Curry-Howard, se rapportent respectivement à l'élimination des coupures et à l'expansion des axiomes. Nous montrons ici que cette identification des preuves correspond exactement à l'identification des preuves à commutation de règle près, qui est une troisième opération sur les preuves bien connue et plus facile à manipuler. Notre démonstration considère uniquement la logique linéaire multiplicative-additive, même si nous conjecturons que ce résultat est vrai pour la logique linéaire dans son entièreté.Non seulement des preuves peuvent être identifiées à élimination des coupures et expansion des axiomes près, mais aussi des formules. Deux formules sont isomorphes si elles sont reliées par des preuves dont les compositions donnent l'identité, toujours à élimination des coupures et expansion des axiomes près. Ces formules sont alors réellement considérées comme les mêmes, et toute utilisation de l'une peut être remplacée par une utilisation de l'autre. Nous donnons une théorie équationnelle caractérisant exactement les formules isomorphes dans la logique linéaire multiplicative-additive. Un problème généralisant les isomorphismes est celui des rétractions, qui intuitivement correspondent aux couples de formules où la première peut être remplacée par la seconde - mais pas nécessairement la seconde par la première, contrairement aux isomorphismes. Étudier les rétractions est bien plus complexe, et nous avons caractérisé les rétractions des atomes dans le fragment multiplicatif de la logique linéaire.Pour l'étude des deux problèmes précédents, la syntaxe usuelle des preuves du calcul des séquents semble mal adaptée, car on considère des preuves à commutation de règle prés. Une partie de la logique linéaire possède une meilleure syntaxe dans ce cas : les réseaux de preuve, qui sont des graphes représentant des preuves quotientées par les commutations de règle. Cette syntaxe fut un outil indispensable pour caractériser isomorphismes et rétractions. Malheureusement, les réseaux de preuve ne sont pas (ou mal) définis en présence des unités. Pour nos problèmes, cette restriction conduit à une étude du cas sans unité dans les réseaux avec le coeur de la démonstration, précédé d'un travail en calcul des séquents pour prendre en compte les unités.Cette thèse développe par ailleurs une partie de la théorie des réseaux de preuve en fournissant une simple preuve du théorème de séquentialisation, qui relie les deux syntaxes des réseaux de preuve et du calcul des séquents, justifiant qu'elles décrivent les mêmes objets sous-jacents. Cette nouvelle démonstration s'obtient comme corollaire d'une généralisation du théorème de Yeo. Ce dernier résultat s'exprime entièrement dans la théorie des graphes aux arêtes colorées, et permet de déduire des preuves de sequentialisation pour différentes définitions de réseaux de preuve. Enfin, nous avons aussi formalisé les réseaux de preuve du fragment multiplicatif de la logique linéaire dans l'assistant de preuve Coq, avec en particulier une implémentation de notre nouvelle preuve de séquentialisation
This study is concerned with the equality of proofs and formulas in linear logic, with in particular contributions for the multiplicative-additive fragment of this logic. In linear logic, and as in many other logics (such as intuitionistic logic), there are two transformations on proofs: cut-elimination and axiom-expansion. One often wishes to identify two proofs related by these transformations, as it is the case semantically (in a categorical model for instance). This situation is similar to the one in the λ-calculus where terms are identified up to β-reduction and η-expansion, operations that, through the prism of the Curry-Howard correspondence, are related respectively to cut-elimination and axiom-expansion. We show here that this identification corresponds exactly to identifying proofs up to rule commutation, a third well-known operation on proofs which is easier to manipulate. We prove so only in multiplicative-additive linear logic, even if we conjecture such a result holds in full linear logic.Not only proofs but also formulas can be identified up to cut-elimination and axiom-expansion. Two formulas are isomorphic if there are proofs between them whose compositions yield identities, still up to cut-elimination and axiom-expansion. These formulas are then really considered to be the same, and every use of one can be replaced with one use of the other. We give an equational theory characterizing exactly isomorphic formulas in multiplicative-additive linear logic. A generalization of an isomorphism is a retraction, which intuitively corresponds to a couple of formulas where the first can be replaced by the second -- but not necessarily the other way around, contrary to an isomorphism. Studying retractions is more complicated, and we characterize retractions to an atom in the multiplicative fragment of linear logic.When studying the two previous problems, the usual syntax of proofs from sequent calculus seems ill-suited because we consider proofs up to rule commutation. Part of linear logic can be expressed in a better adapted syntax in this case: proof-nets, which are graphs representing proofs quotiented by rule commutation. This syntax was an instrumental tool for the characterization of isomorphisms and retractions. Unfortunately, proof-nets are not (or badly) defined with units. Concerning our issues, this restriction leads to a study of the unit-free case by means of proof-nets with the crux of the demonstration, preceded by a work in sequent calculus to handle the units. Besides, this thesis also develops part of the theory of proof-nets by providing a simple proof of the sequentialization theorem, which relates the two syntaxes of proof-net and sequent calculus, substantiating that they describe the same underlying objects. This new demonstration is obtained as a corollary of a generalization of Yeo's theorem. This last result is fully expressed in the theory of edge-colored graphs, and allows to recover proofs of sequentialization for various definitions of proof-nets. Finally, we also formalized proof-nets for the multiplicative fragment of linear logic in the proof assistant Coq, with notably an implementation of our new sequentialization proof

Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.

AbstractAdding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system ($$\mathsf {acLL}_\varSigma $$ acLL Σ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of $$\mathsf {acLL}_\varSigma $$ acLL Σ .

Abstract This chapter covers the process of the consolidation of the ideas in Heisenberg’s Umdeutung paper into the Two-Man-Paper by Born and Jordan, the Three-Man-Paper (Dreimännerarbeit) by Born, Heisenberg, and Jordan, and a paper by Dirac. Both Born and Jordan recognized that Heisenberg’s peculiar non-commutative multiplication rule is recognized to be nothing but the standard multiplication rule for matrices. They also replaced Heisenberg’s quantization condition, the Thomas–Reiche–Kuhn sum rule, by the now-familiar commutation relations of position and momentum. The chapter then discusses how Dirac independently derived these commutation relations, relying on a profound analogy with the Poisson brackets of classical mechanics

Abstract H*(G, k) may be endowed with a multiplicative structure which turns it into a commutative graded ring. (Here, ‘commutative graded’ means that the product satisfies the graded commutation rule αβ = (‒1)Pq βα for α ∈ H P (G, k), β ∈ Hq(G, k).) Much of this monograph will be concerned with what is known about the structure of this ring and how it is related to the structure of G. At present, unfortunately, there is relatively little such knowledge.

Abstract The syntactic calculus, also known as ‘ bidirectional categorial grammar’, is a kind of logic without any structural rules, other than the obligatory reflexive law and cut-rule. It had been inspired by multilinear algebra and non-commutative ring theory and was developed with applications to linguistics in mind. Here we shall confine attention to the associative version, although a non-associative version has also been studied [L 1961, Kandulski 1988, Došen 1988, 1989]. It differs from a very rudimentary form of Girard’s linear logic [Girard 1987, 1989] by the absence of the interchange rule which licenses commutativity. Because of its roots in non-commutative algebra and syntax, all appearance of commutativity is forbidden. The notation, which goes back to a pre-historic collaboration with George Findlay, was carefully chosen to reflect this absence of commutativity: the order of two letters was never to be wantonly interchanged, yet all rules were to be preserved under left-right symmetry, the guiding slogan being ‘ symmetry without commutativity’. It must be admitted, however, that the notation, proposed for linguistics [L 1958] and ring theory [L 1966], caught on in neither field (with some recent exceptions in linguistics).

We apply the canonical and the path integral quantisation methods to scalar, spinor and vector fields. The scalar field is a generalisation to an infinite number of degrees of freedom of the single harmonic oscillator we studied in Chapter 9. For the spinor fields we show the need for anti-commutation relations and introduce the corresponding Grassmann algebra. The rules of Fermi statistics follow from these anti-commutation relations. The canonical quantisation method applied to the Maxwell field in a Lorentz covariant gauge requires the introduction of negative metric states in the Hilbert space. The power of the path integral quantisation is already manifest. In each case we expand the fields in creation and annihilation operators.

Theory of molecular Taylor-Aris dispersion (TAD) is an accepted framework describing tracer dispersion in suspension flows and determining effective diffusion coefficients. Our group reported a pseudo-Lagrangian method to study dispersion in suspension flows at FEDSM-2000. Tracer motions were studied in a steadily moving inertial reference frame (SMIRF) aligned with the flow direction; increments of change of axial position of individual tracers were collected to demonstrate how the tracer moved as they, individually, interacted with similar collections of other bodies brought to and from the region. First, individual tracers with no apparent axial velocity component (NAAVC) were identified; they exhibited fixed positions in video recordings of images collected in the SMIRF. Then, time increments were measured for tracers to pass at least 5, but usually 10 pre-selected, nested distances in the up- or downstream direction laid out with respect to the zero-site in the SMIRF. Such data were richer than measurements of tracer spread over time because stations along each path were serial first-passages (FP) with probabilistic meaning. Dispersion of various types of suspension and two transformation rules for combining velocity components are discussed herein. Traditional low-speed continuum theory and particle dynamics use Galilean transforms. Yet, to recognize the limited speed in laws for channel flows, Lorentzian transformations may be appropriate. In a four-space, deterministic paths would begin at NAAVC sites and continue in time-like conical regions of four-space. Distances in this space are measured using Minkowski’s metric; at the NAAVC site and on the boundary of the space-time cone, this metric has the format of the Fürth, Ornstein, and Taylor (FOT) equation when only terms to order t2 are used. Data shown at FEDSM-2000 can be reinterpreted as “prospective paths” in time-like regions that were consolidated in normalized cumulative probability distributions to provide retrospective descriptions. The ad hoc sign alteration of the FOT equation to fit the data of FEDSM-2000 is now taken as a part of measuring lengths using a Minkowski metric, which signifies a hyperbolic geometry, for which an inherent scaling constant is a negative curvature. The space also has an intrinsic distance of ℓ = Sτ, obtained from fitting parameters (S, τ) for the FOT equation. Integrals of the area under the FOT curve have units of volume, which are considered as describing an average volume of dispersion on S3, the 3-sphere. Path motion through this volume was kinematic dispersion, S2τ, which was the form for effective diffusivity in continuum theory used in FEDSM-2000. Weiner and Wilmer describe transformations in four-spaces in terms of commutating rotations on orthogonal planes, a concept readily linked to symmetries in the hyperbolic space typical of Lorentzian transformations; they also describe a second order ODE like the FOT equation.