Diagrammatic Version Research Articles

Purely virtual extension of quantum field theory for gauge invariant fields: quantum gravity

Quantum gravity is extended to include purely virtual “cloud sectors”, which allow us to define a complete set of point-dependent observables, including a gauge invariant metric and gauge invariant matter fields, and calculate their off-shell correlation functions perturbatively. The ordinary on-shell correlation functions and the S matrix elements are unaffected. Each extra sector is made of a cloud field, its anticommuting partner, a “cloud-fixing” function and a cloud Faddeev-Popov determinant. The additional fields are purely virtual, to ensure that no ghosts propagate. The extension is unitary. In particular, the off-shell, diagrammatic version of the optical theorem holds. The one-loop two-point functions of dressed scalars, vectors and gravitons are calculated. Their absorptive parts are positive, cloud independent and gauge independent, while they are unphysical if non purely virtual clouds are used. We illustrate the differences between our approach to the problem of finding a complete set of observables in quantum gravity and other approaches available in the literature.

Quantum field theory of physical and purely virtual particles in a finite interval of time on a compact space manifold: diagrams, amplitudes and unitarity

We provide a diagrammatic formulation of perturbative quantum field theory in a finite interval of time τ, on a compact space manifold Ω. We explain how to compute the evolution operator U(tf, ti) between the initial time ti and the final time tf = ti + τ, study unitarity and renormalizability, and show how to include purely virtual particles, by rendering some physical particles (and all the ghosts, if present) purely virtual. The details about the restriction to finite τ and compact Ω are moved away from the internal sectors of the diagrams (apart from the discretization of the three-momenta), and coded into external sources. Unitarity is studied by means of the spectral optical identities, and the diagrammatic version of the identity U†(tf, ti)U(tf, ti) = 1. The dimensional regularization is extended to finite τ and compact Ω, and used to prove, under general assumptions, that renormalizability holds whenever it holds at τ = ∞, Ω = ℝ3. Purely virtual particles are introduced by removing the on-shell contributions of some physical particles, and the ghosts, from the core diagrams, and trivializing their initial and final conditions. The resulting evolution operator Uph(tf, ti) is unitary, but does not satisfy the more general identity Uph(t3, t2)Uph(t2, t1) = Uph(t3, t1). As a consequence, Uph(tf, ti) cannot be derived from a Hamiltonian in a standard way, in the presence of purely virtual particles.

Purely virtual extension of quantum field theory for gauge invariant fields: Yang–Mills theory

We extend quantum field theory by including purely virtual “cloud” sectors, to define physical off-shell correlation functions of gauge invariant quark and gluon fields, without affecting the S matrix amplitudes. The extension is made of certain cloud bosons, plus their anticommuting partners. Both are quantized as purely virtual, to ensure that they do not propagate ghosts. The extended theory is renormalizable and unitary. In particular, the off-shell, diagrammatic version of the optical theorem holds. We calculate the one-loop two-point functions of dressed quarks and gluons, and show that their absorptive parts are gauge independent, cloud independent and positive (while they are generically unphysical if the cloud sectors are not purely virtual). A gauge/cloud duality simplifies the computations and shows that the gauge choice is just a particular cloud. It is possible to dress every field insertion with a different cloud. We compare the purely virtual extension to previous approaches to similar problems.

Another New Design for an Old Map

Another New Design for an Old Map

Developing and refining a programme theory for understanding how twenty mile per hour speed limits impact health

Background and aimTraffic speed is an important determinant of health, affecting both the frequency and severity of road traffic collisions and casualties. Reduced traffic speed may also improve conditions for walking and cycling. Twenty mile per hour (30km per hour) speed restriction schemes are one policy option for reducing traffic speeds. The aim of this study was to develop and refine a programme theory for understanding how such schemes may impact on health outcomes. MethodsThe programme theory was developed through an iterative process. Firstly, a logic model (diagrammatic version of the programme theory) was created through discussions with 20 mph programme delivery staff from the local authority and through reviewing literature. The second stage consisted of interviews with key stakeholders, and focus groups with the public to further develop and refine the logic model. Data were analysed using Framework Method. ResultsThis study first presents an early stage logic model (phase 1) of how the 20 mph speed limit is proposed to work. Qualitative data are then presented (phase 2), describing potential mechanisms as to how the scheme could influence health outcomes through changing i) driver behaviour; ii) casualties and casualty severity; iii) walking and cycling behaviour; iv) air quality; and v) street pleasantness. Participants identified a need for driver education and enforcement in order to reduce speeds, and the importance of improving perceptions of safety for achieving increased walking and cycling. A refined version of the logic model is presented. ImplicationsThe programme theory will be used as a framework to evaluate the implementation of 20 mph speed limit schemes in Edinburgh and Belfast, aiding in study design decisions such as outcome data to be collected and in the generation of specific testable hypotheses.

Algebraic cutting equations

The cutting equations are diagrammatic identities that are used to prove perturbative unitarity in quantum field theory. In this paper, we derive algebraic, upgraded versions of them. Differently from the diagrammatic versions, the algebraic identities also holds for propagators with arbitrary, nonvanishing widths. In particular, the cut propagators do not need to vanish off shell. The new approach provides a framework to address unsolved problems of perturbative quantum field theory and a tool to investigate perturbative unitarity in higher-derivative theories that are relevant to the problem of quantum gravity, such as the Lee–Wick models and the fakeon models.

Local unitary representation of braids and N-qubit entanglements

In this paper, by utilizing the idea of stabilizer codes, we give some relationships between one local unitary representation of braid group in N-qubit tensor space and the corresponding entanglement properties of the N-qubit pure state $|\Psi\rangle$, where the N-qubit state $|\Psi\rangle$ is obtained by applying the braiding operation on the natural basis. Specifically, we show that the separability of $|\Psi\rangle=\mathcal{B}|0\rangle^{\otimes N}$ is closely related to the diagrammatic version of the braid operator $\mathcal{B}$. This may provide us more insights about the topological entanglement and quantum entanglement.

Super q–Howe duality and web categories

We use super $q$-Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of $\mathfrak{gl}_N$-modules (and, more generally, $\mathfrak{gl}_{N|M}$-modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an application, we give a representation theoretic explanation and a diagrammatic version of a known symmetry of colored HOMFLY--PT polynomials.

Lambek vs. Lambek: Functorial vector space semantics and string diagrams for Lambek calculus

The Distributional Compositional Categorical (DisCoCat) model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, something which the other methods lack, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambekʼs pregroup grammar. Pregroup types form a posetal compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambekʼs previous type-logic, the Lambek calculus. The latter and its extensions have been extensively used to formalise and reason about various linguistic phenomena. Hence, the apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings, and also reflects the structure of the parse trees of the Lambek calculus.

The Diagrammatic Soergel Category and sl(N)‐Foams, for N ≥ 4

For each N ≥ 4, we define a monoidal functor from Elias and Khovanov′s diagrammatic version of Soergel′s category of bimodules to the category of sl(N) foams defined by Mackaay, Stošić, and Vaz. We show that through these functors Soergel′s category can be obtained from the sl(N) foams.

TANGLE AND BRAUER DIAGRAM ALGEBRAS OF TYPE Dn

A generalization of the Kauffman tangle algebra is given for Coxeter type D n. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the Birman–Murakami–Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which, in our set-up, occurs when the Coxeter type is A n - 1. The proof involves a diagrammatic version of the Brauer algebra of type D n of which the generalized Temperley–Lieb algebra of type D n is a subalgebra.

Translating Toulmin Diagrams: Theory Neutrality in Argument Representation

The Toulmin diagram layout is very familiar and widely used, particularly in the teaching of critical thinking skills. The conventional box-and-arrow diagram is equally familiar and widespread. Translation between the two throws up a number of interesting challenges. Some of these challenges (such as the relationship between Toulmin warrants and their counterparts in traditional diagrams) represent slightly different ways of looking at old and deep theoretical questions. Others (such as how to allow Toulmin diagrams to be recursive) are diagrammatic versions of questions that have already been addressed in artificial intelligence models of argument. But there are further questions (such as the relationships between refutations, rebuttals and undercutters, and the roles of multiple warrants) that are posed as a specific result of examining the diagram inter-translation problem. These three classes of problems are discussed. To the first class are addressed solutions based on engineering pragmatism; to the second class, are addressed solutions drawn from the appropriate literature; and to the third class, fuller exploration is offered justifying the approaches taken in developing solutions that offer both pragmatic utility and theoretical interest. Finally, these solutions are explored briefly in the context of the Araucaria system, showing the ways in which analysts can tackle arguments either using one diagrammatic style or another, or even a combination of the two.

Is There a Demand for Pain and Suffering Coverage?

There is a large literature on this question, almost all of which avers that economic theory rules out the possibility of demand for insurance coverage of nonpecuniary losses on a priori grounds. There is little hard evidence, and much controversy over the interpretation of what evidence there is (see, e.g., Croley and Hanson, Nonpecuniary Costs of Accidents: Pain-and-Suffering Damages in Tort Law, Harvard Law Review, 1995). Nevertheless, the American Law Institute's Reporter has accepted that view into the recommendations for the new Restatement of Torts. This paper argues that the existence or nonexistence of demand for pain and suffering coverage is a purely empirical issue and policy recommendations should view it as such. The underlying economic theory is too weak to support such a recommendation without, at least, some supporting evidence. I argue first that even taken on its own terms, orthodox theory omits various possible contingencies, such as pre- and post-mishap curves that cross, that some writers such as Ellen Smith Pryor, The Tort Law Debate, Efficiency, and the Kingdom of the Ill: A Critique of the Insurance Theory of Compensation, 79 Va. L.R. 91 (1993), point out are sometimes valid. Second, I argue that as a matter of economic theory, it is odd to rule out demand for anything without evidence, inasmuch as economics generally holds that there is no accounting for taste. It turns out, on close inspection, that the literature generally descends from Cook and Graham, The Demand for Insurance and Protection: The Case of Irreplaceable Commodities, 91 Q.J.Econ. 143 (1977), whose equalized marginal utility view in turn descends directly from the controversial independence axiom implicit in the original work of Von Neumann and Morgenstern and made explicit by other authors in the 1940s. A result with such a disputable base, and no other evidence, is a poor basis for a policy recommendation. Still further, I offer two economic models in which the demand for pain and suffering coverage cannot be ruled out. The first (see appendix, below) is based on the expected model, allowing for the lag between entering an insurance agreement and the realization of the random event. The second is a simpler diagrammatic version, which bypasses the expected model altogether, inspired by Mishan, Choices Involving Risk: Simple Steps toward an Ordinalist Analysis, 86 Econ. J. 759 (1976). In both models, it may be optimal for a person to over- or underinsure pain and suffering relative to the strict position taken by the orthodox view. The point is not that to claim that there is or is not demand for this insurance, but merely to demonstrate that economic theory, by itself, cannot answer the question.

The Development of the Theory of Exchange

The aim of this paper is to provide an outline of the development of the theory of exchange, concentrating on the less well-known development of the formal model which culminated in the contribution of Edgeworth. The importance of exchange, viewed as the central economic problem for the early neoclassical economists, is stressed. Instead of taking a chronological approach, non-utility approaches are first discussed. These include the extension by Walras of Cournot’s attempt to model trade between regions, and Whewell’s mathematical version of J. S. Mill’s international trade analysis, followed by Marshall’s diagrammatic version. Jevons’s and Walras’s utility approaches are then examined, showing the different paths they took from the same basic equations of exchange. After a very brief discussion of Edgeworth, the neglected but valuable contribution of Launhardt, along with the later work of Wicksell, are examined. Emphasis is placed on the similarity of the formal structure of the exchange model used by the various writers. This similarity has been obscured by the different forms of presentation used and the emphasis given to various aspects and results by each investigator.

Some remarks on quantum and braided group gauge theory

We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzeziński and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. The bosonisation of a

Quasilinearized form of the CPMET method for closed-shell molecules

A diagrammatic version of the quasilinearized (QL) form of the coupled-pair-many-electron-theory (CPMET) equations for closed-shell molecules is suggested. Using the Feynman–Goldstone diagrammatic technique, a special type of one- and two-particle effective interactions is introduced, and then the CPMET equations can be formally rewritten in a form very similar to the linear coupled-cluster approach (CCA) of Coester, Kümmel, and Čížek. The QL–CPMET facilitates the formulation of the coupled-electron-pair-approximation (CEPA) methods.

Quantum field theory in a time-dependent gravitational field

A quantum field theory of a charged scalar field with self-interactions in a Robertson-Walker metric is considered. The structure of the Hilbert space of the state vectors in this theory is investigated; and taking account of the change of the Fock space, a method of calculating physical quantities is constructed and its diagrammatic version is given. By practical calculation to second order in the coupling constant, it is shown that the theory is renormalizable and renormalization constants are identical to those in the theory in Minkowski spacetime.

Complete fourth-order calculation of correlation energy for closed-shell molecules by diagrammatic RSPT

Starting from the diagrammatic version of Wigner's (2n+1) rule the formulae for the correlation energy of closed-shell molecules are derived. This approach allows to formulate in the very compact way the correlation energy through the fourth order of perturbation theory, a repeating evaluation of the same intermediate terms is completely removed.

Response function theory of electron correlation

The full perturbation expansion for the response (or density—density correlation) function is examined in order to provide a useful general theory of excitation energies, oscillator strengths, dynamic polarizabilities, etc., that is more accurate than the random phase approximation. It is first shown how the formal partition of the diagrammatic version of the perturbation expansion into reducible and irreducible diagrams is generally useless as the latter category contains all the difficult terms which have heretofore resisted analysis in all but a haphazard form. It is then shown how the diagram for the response function can be partitioned into “correlated” and “uncorrelated” subsets. Restricting attention to the particle—hole blocks of the full response function, the “uncorrelated” diagrams desecribe the propagation of a particle—hole pair in an N-electron system where the particle and hole are each interacting with the remaining electrons but they are not interacting with each other. The “correlated” diagrams are those containing the hole—particle interactions, and, by defining a new class of reducible and irreducible diagrams, these are all summed to provide a perturbation expansion of the effective two-body hole—particle interaction that appears in the inverse of the response function. The “uncorrelated” diagrams are further partitioned into two sets, one of which is summed to all orders, while the other set is inverted in an order by order fashion. The final result presents a perturbation expansion for the inverse of the response function that is analogous to the Dyson equation for one-electron Green functions. Maintaining the perturbation expansion through first order for the inverse of the response function yields the eigenvalue equation of the familiar random phase approximation, while truncation at second order provides the most advanced theories that have been generated by the equations-of-motion method.

A non-perturbative open-shell theory for atomic and molecular systems: Application to transbutadiene

A non-perturbative theory is proposed in this article in which an energy independent effective Hamiltonian is obtained for open-shell systems. We have given a diagrammatic version of theory to facilitate the analysis of the problem. The theory has been applied to a model 4-π electron problem, for calculating the lowestπ-π* singlet and triplet energy levels of transbutadiene. Comparison with full Cl calculation indicates the excellent workability of the theory.