Relevant bibliographies by topics / Operatory Diraca

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  2. Dissertations / Theses
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  4. Book chapters
  5. Conference papers
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Academic literature on the topic 'Operatory Diraca'

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Published: 24 April 2022

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Journal articles on the topic "Operatory Diraca"

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KHACHIDZE, TAMARI T., and ANZOR A. KHELASHVILI. "AN "ACCIDENTAL" SYMMETRY OPERATOR FOR THE DIRAC EQUATION IN THE COULOMB POTENTIAL." Modern Physics Letters A 20, no. 30 (September 28, 2005): 2277–81. http://dx.doi.org/10.1142/s0217732305018505.

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On the basis of the generalization of the theorem about K-odd operators (K is the Dirac's operator), certain linear combination is constructed, which appears to commute with the Dirac Hamiltonian for Coulomb field. This operator coincides with the Johnson and Lippmann operator and is closely connected to the familiar Laplace–Runge–Lenz vector. Our approach guarantees not only derivation of Johnson–Lippmann operator, but simultaneously commutativity with the Dirac Hamiltonian.
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AVRAMIDI, IVAN G. "DIRAC OPERATOR IN MATRIX GEOMETRY." International Journal of Geometric Methods in Modern Physics 02, no. 02 (April 2005): 227–64. http://dx.doi.org/10.1142/s0219887805000636.

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We review the construction of the Dirac operator and its properties in Riemannian geometry, and show how the asymptotic expansion of the trace of the heat kernel determines the spectral invariants of the Dirac operator and its index. We also point out that the Einstein–Hilbert functional can be obtained as a linear combination of the first two spectral invariants of the Dirac operator. Next, we report on our previous attempts to generalize the notion of the Dirac operator to the case of Matrix Geometry, where, instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays a role of a "non-commutative" metric. We construct invariant first-order and second-order self-adjoint elliptic partial differential operators, which can be called "non-commutative" Dirac operators and non-commutative Laplace operators. We construct the corresponding heat kernel for the non-commutative Laplace type operator and compute its first two spectral invariants. A linear combination of these two spectral invariants gives a functional that can be considered as a non-commutative generalization of the Einstein–Hilbert action.
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Cojuhari, Petru, and Aurelian Gheondea. "Embeddings, Operator Ranges, and Dirac Operators." Complex Analysis and Operator Theory 5, no. 3 (April 13, 2010): 941–53. http://dx.doi.org/10.1007/s11785-010-0066-5.

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ARAI, ASAO. "HEISENBERG OPERATORS, INVARIANT DOMAINS AND HEISENBERG EQUATIONS OF MOTION." Reviews in Mathematical Physics 19, no. 10 (November 2007): 1045–69. http://dx.doi.org/10.1142/s0129055x07003206.

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An abstract operator theory is developed on operators of the form AH(t) := eitHAe-itH, t ∈ ℝ, with H a self-adjoint operator and A a linear operator on a Hilbert space (in the context of quantum mechanics, AH(t) is called the Heisenberg operator of A with respect to H). The following aspects are discussed: (i) integral equations for AH(t) for a general class of A; (ii) a sufficient condition for D(A), the domain of A, to be left invariant by e-itH for all t ∈ ℝ; (iii) a mathematically rigorous formulation of the Heisenberg equation of motion in quantum mechanics and the uniqueness of its solutions; (iv) invariant domains in the case where H is an abstract version of Schrödinger and Dirac operators; (v) applications to Schrödinger operators with matrix-valued potentials and Dirac operators.
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Aastrup, Johannes, and Jesper Møller Grimstrup. "The quantum holonomy-diffeomorphism algebra and quantum gravity." International Journal of Modern Physics A 31, no. 10 (April 6, 2016): 1650048. http://dx.doi.org/10.1142/s0217751x16500482.

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We introduce the quantum holonomy-diffeomorphism ∗-algebra, which is generated by holonomy-diffeomorphisms on a three-dimensional manifold and translations on a space of SU(2)-connections. We show that this algebra encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Furthermore, we show that semiclassical states exist on the holonomy-diffeomorphism part of the algebra but that these states cannot be extended to the full algebra. Via a Dirac-type operator we derive a certain class of unbounded operators that act in the GNS construction of the semiclassical states. These unbounded operators are the type of operators, which we have previously shown to entail the spatial three-dimensional Dirac operator and Dirac–Hamiltonian in a semiclassical limit. Finally, we show that the structure of the Hamilton constraint emerges from a Yang–Mills-type operator over the space of SU(2)-connections.
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MATSUTANI, SHIGEKI. "DIRAC OPERATOR ON A CONFORMAL SURFACE IMMERSED IN ℝ4: A WAY TO FURTHER GENERALIZED WEIERSTRASS EQUATION." Reviews in Mathematical Physics 12, no. 03 (March 2000): 431–44. http://dx.doi.org/10.1142/s0129055x00000149.

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In the previous report (J. Phys.A30 (1997) 4019–4029), I showed that the Dirac operator defined over a conformal surface immersed in ℝ3 by means of confinement procedure is identified with the differential operator of the generalized Weierstrass equation and the Lax operator of the modified Novikov–Veselov (MNV) equation. In this article, using the same procedure, I determine the Dirac operator defined over a conformal surface immersed in ℝ4, which is for a Dirac field confined in the surface. Then it is reduced to the Lax operators of the nonlinear Schrödinger and the MNV equations by taking appropriate limits. It means that the Dirac operator is related to the further generalized Weierstrass equation for a surface in ℝ4.
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Benameur, Moulay-Tahar, James L. Heitsch, and Charlotte Wahl. "An interesting example for spectral invariants." Journal of K-Theory 13, no. 2 (April 2014): 305–11. http://dx.doi.org/10.1017/is014002020jkt255.

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AbstractIn [HL99], the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then shown that the associated heat operator converges to the Chern character of the index bundle of the operator. In [BH08], this result was improved by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.
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DABROWSKI, LUDWIK, ANDRZEJ SITARZ, and ALESSANDRO ZUCCA. "DIRAC OPERATORS ON NONCOMMUTATIVE PRINCIPAL CIRCLE BUNDLES." International Journal of Geometric Methods in Modern Physics 11, no. 01 (December 16, 2013): 1450012. http://dx.doi.org/10.1142/s0219887814500121.

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We study spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low-dimensional noncommutative tori are analyzed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U(1)-bundle [Formula: see text].
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Wang, Yong. "A Kastler-Kalau-Walze Type Theorem and the Spectral Action for Perturbations of Dirac Operators on Manifolds with Boundary." Abstract and Applied Analysis 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/619120.

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We prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact manifolds.
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Akuamoah, Saviour Worlanyo, Aly R. Seadawy, and Dianchen Lu. "Energy and momentum operator substitution method derived from Schrödinger equation for light and matter waves." Modern Physics Letters B 33, no. 24 (August 30, 2019): 1950285. http://dx.doi.org/10.1142/s0217984919502853.

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In this paper, the energy and momentum operator substitution method derived from the Schrödinger equation is used to list all possible light and matter wave equations, among which the first light wave equation and relativistic approximation equation are proposed for the first time. We expect that we will have some practical application value. The negative sign pairing of energy and momentum operators are important characteristics of this paper. Then the Klein–Gordon equation and Dirac equation are introduced. The process of deriving relativistic energy–momentum relationship by undetermined coefficient method and establishing Dirac equation are mainly introduced. Dirac’s idea of treating negative energy in relativity into positrons is also discussed. Finally, the four-dimensional space-time representation of relativistic wave equation is introduced, which is usually the main representation of quantum electrodynamics and quantum field theory.
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Dissertations / Theses on the topic "Operatory Diraca"

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Kungsman, Jimmy. "Resonances of Dirac Operators." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-223841.

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This thesis consists of a summary of four papers dealing with resonances of Dirac operators on Euclidean 3-space. In Paper I we show that the Complex Absorbing Potential (CAP) method is valid in the semiclassical limit for resonances sufficiently close to the real line if the potential is smooth and compactly supported. In Paper II  we continue the investigations initiated in Paper I but here we study clouds of resonances close to the real line and show that in some sense the CAP method remains valid also for multiple resonances. In Paper III we study perturbations of Dirac operators with smooth decaying scalar potentials  and show that these possess many resonances near certain points related to the maximum and the minimum of the potential. In Paper IV we show a trace formula of Poisson type for Dirac operators having compactly supported potentials which is related to resonances. The techniques mainly stem from complex function theory and scattering theory.
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Ginoux, Nicolas. "Dirac operators on Lagrangian submanifolds." Universität Potsdam, 2004. http://opus.kobv.de/ubp/volltexte/2005/562/.

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We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge - de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.
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Fischmann, Matthias. "Conformally covariant differential operators acting on spinor bundles and related conformal covariants." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://dx.doi.org/10.18452/16703.

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Konforme Potenzen des Dirac Operators einer semi Riemannschen Spin-Mannigfaltigkeit werden untersucht. Wir präsentieren einen neuen Beweis, basierend auf dem Traktor Kalkül, für die Existenz von konformen ungeraden Potenzen des Dirac Operators auf semi Riemannschen Spin-Mannigfaltigkeiten. Desweiteren konstruieren wir eine neue Familie von konform kovarianten linearen Differentialoperatoren auf dem standard spin Traktor Bündel. Weiterhin verallgemeinern wir den Existenzbeweis für konforme ungerade Potenzen des Dirac Operators auf semi Riemannsche Spin-Mannigfaltigkeiten. Da die Existenzbeweise konstruktive sind, erhalten wir explizite Formeln für die konforme dritte und fünfte Potenz des Dirac Operators. Basierend auf den expliziten Formeln zeigen wir, dass die konforme dritte und fünfte Potenz des Dirac Operators formal selbstadjungiert (anti selbstadjungiert) bezüglich des L2-Skalarproduktes auf dem Spinorbündel ist. Abschliessend präsentieren wir neue Strukturen der konformen ersten, dritten und fünften Potenz des Dirac Operators: Es existieren lineare Differentialoperatoren auf dem Spinorbündel der Ordnung kleiner gleich eins, so dass die konforme erste, dritte und fünfte Potenz des Dirac Operators ein Polynom in jenen Operatoren ist.
Conformal powers of the Dirac operator on semi Riemannian spin manifolds are investigated. We give a new proof of the existence of conformal odd powers of the Dirac operator on semi Riemannian spin manifolds using the tractor machinery. We will also present a new family of conformally covariant linear differential operators on the standard spin tractor bundle. Furthermore, we generalize the known existence proof of conformal power of the Dirac operator on Riemannian spin manifolds to semi Riemannian spin manifolds. Both proofs concering the existence of conformal odd powers of the Dirac operator are constructive, hence we also derive an explicit formula for a conformal third- and fifth power of the Dirac operator. Due to explicit formulas, we show that the conformal third- and fifth power of the Dirac operator is formally self-adjoint (anti self-adjoint), with respect to the L2-scalar product on the spinor bundle. Finally, we present a new structure of the conformal first-, third- and fifth power of the Dirac operator: There exist linear differential operators on the spinor bundle of order less or equal one, such that the conformal first-, third- and fifth power of the Dirac operator is a polynomial in these operators.
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Stadtmüller, Christoph Martin. "Horizontal Dirac Operators in CR Geometry." Doctoral thesis, Humboldt-Universität zu Berlin, 2017. http://dx.doi.org/10.18452/18130.

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In dieser Dissertation beschäftigen wir uns mit angepassten Zusammenhängen und ihren (horizontalen) Dirac-Operatoren auf strikt pseudokonvexen CR-Mannigfaltigkeiten. Einen Zusammenhang nennen wir dann angepasst, wenn er die relevanten Daten parallelisiert. Wir beschreiben den Raum der angepassten Zusammenhänge, indem wir ihre Torsionstensoren studieren, von denen gewisse Teile durch die Geometrie der Mannigfaltigkeit festgelegt sind, während andere frei wählbar sind. Als Anwendung betrachten wir die Eigenschaften der Dirac-Operatoren, die zu diesen Zusammenhängen gehören. Weiter betrachten wir horizontale Dirac-Operatoren, die nur in Richtung des horizontalen Bündels H ableiten. Diese Operatoren sind besser an die Sub-Riemannsche Struktur einer CR-Mannigfaltigkeit angepasst als die vollen Dirac-Operatoren. Wir diskutieren, wann diese Operatoren formal selbstadjungiert sind und beweisen eine Weitzenböck-Typ-Formel. Wir konzentrieren uns dann auf den horizontalen Dirac-Operator zum Tanaka-Webster-Zusammenhang. Dieser ändert sich konform kovariant, wenn wir die Kontaktform konform ändern. Für diesen Operator betrachten wir weiterhin zwei Beispiele: Wir betrachten S^1-Bündel über Kähler-Mannigfaltigkeiten, insbesondere berechnen wir für Sphären einen Teil des Spektrums. Außerdem betrachten wir kompakte Quotienten der Heisenberggruppe und berechnen hier in den Dimensionen 3 und 5 das volle Spektrum. Die horizontalen Dirac-Operatoren sind nicht mehr elliptisch, sondern „elliptisch in Richtung von H“. Mithilfe des Heisenbergkalküls stellen wir fest, dass die horizontalen Dirac-Operatoren nicht hypoelliptisch sind. Im Fall des Tanaka-Webster-Zusammenhangs lässt sich aber zeigen, dass der zugehörige Operator auf gewissen Teilen des Spinorbündels hypoelliptisch ist. Dies genügt, um zu beweisen, dass er (nun auf dem gesamten Spinorbündel) ein reines Punktspektrum hat und die Eigenräume, bis auf den Kern, endlich-dimensional sind und aus glatten Eigenspinoren bestehen.
In the present thesis, we study adapted connections and their (horizontal) Dirac operators on strictly pseudoconvex CR manifolds. An adapted connection is one that parallelises the relevant data. We describe the space of adapted connections through their torsion tensors, certain parts of which are determined by the geometry of the manifold, while others may be freely chosen. As an application, we study the properties of the Dirac operators induced by these connections. We further consider horizontal Dirac operators, which only derive in the direction of the horizontal bundle H. These operators are more adapted to the essentially sub-Riemannian structure of a CR manifold than the full Dirac operators. We discuss the question of their self-adjointness and prove a Weitzenböck type formula for these operators. Focusing on the horizontal Dirac operator associated with the Tanaka-Webster connection, we show that this operator changes in a covariant way if we change the contact form conformally. Moreover, for this operator we discuss two examples: On S^1-bundles over Kähler manifolds, we can compute part of the spectrum and for compact quotients of the Heisenberg group, we determine the whole spectrum in dimensions three and five. The horizontal Dirac operators are not elliptic, but rather "elliptic in some directions". We review the Heisenberg Calculus for such operators and find that in general, the horizontal Dirac operators are not hypoelliptic. However, in the case of the Tanaka-Webster connection, the associated horizontal Dirac operator is hypoelliptic on certain parts of the spinor bundle and this is enough to prove that its spectrum consists only of eigenvalues and except for the kernel, the corresponding eigenspaces are finite-dimensional spaces of smooth sections.
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Yang, Fangyun Ph D. Massachusetts Institute of Technology. "Dirac operators and monopoles with singularities." Thesis, Massachusetts Institute of Technology, 2007. http://hdl.handle.net/1721.1/41723.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007.
Includes bibliographical references (p. 75-77).
This thesis consists of two parts. In the first part of the thesis, we prove an index theorem for Dirac operators of conic singularities with codimension 2. One immediate corollary is the generalized Rohklin congruence formula. The eta function for a twisted spin Dirac operator on a circle bundle over a even dimensional spin manifold is also derived along the way. In the second part, we study the moduli space of monopoles with singularities along an embedded surface. We prove that when the base manifold is Kahler, there is a holomorphic description of the singular monopoles. The compactness for this case is also proved.
by Fangyun Yang.
Ph.D.
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6

Savale, Nikhil Jr (Nikhil A. ). "Spectral asymptotics for coupled Dirac operators." Thesis, Massachusetts Institute of Technology, 2012. http://hdl.handle.net/1721.1/77804.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 137-139).
In this thesis, we study the problem of asymptotic spectral flow for a family of coupled Dirac operators. We prove that the leading order term in the spectral flow on an n dimensional manifold is of order r n+1/2 followed by a remainder of O(r n/2). We perform computations of spectral flow on the sphere which show that O(r n-1/2) is the best possible estimate on the remainder. To obtain the sharp remainder we study a semiclassical Dirac operator and show that its odd functional trace exhibits cancellations in its first n+3/2 terms. A normal form result for this Dirac operator and a bound on its counting function are also obtained.
by Nikhil Savale.
Ph.D.
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7

Kim, Yonne Mi. "Unique continuation theorems for the Dirac operator and the Laplace operator." Thesis, Massachusetts Institute of Technology, 1989. http://hdl.handle.net/1721.1/14469.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1989.
Title as it appeared in M.I.T. Graduate List, Feb. 1989: Carleman inequalities and strong unique continuation.
Includes bibliographical references (leaf 59).
by Yonne Mi Kim.
Ph.D.
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Le, Thu Hoai. "Hyperholomorphic structures and corresponding explicit orthogonal function systems in 3D and 4D." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola", 2014. http://nbn-resolving.de/urn:nbn:de:bsz:105-qucosa-150508.

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Die Reichhaltigkeit und breite Anwendbarkeit der Theorie der holomorphen Funktionen in der komplexen Ebene ist stark motivierend eine ähnliche Theorie für höhere Dimensionen zu entwickeln. Viele Forscher waren und sind in diese Aufgaben involviert, insbesondere in der Entwicklung der Quaternionenanalysis. In den letzten Jahren wurde die Quaternionenanalysis bereits erfolgreich auf eine Vielzahl von Problemen der mathematischen Physik angewandt. Das Ziel der Dissertation besteht darin, holomorphe Strukturen in höheren Dimensionen zu studieren. Zunächst wird ein neues Holomorphiekonzept vorgelegt, was auf der Theorie rechtsinvertierbarer Operatoren basiert und nicht auf Verallgemeinerungen des Cauchy-Riemann-Systems wie üblich. Dieser Begriff umfasst die meisten der gut bekannten holomorphen Strukturen in höheren Dimensionen. Unter anderem sind die üblichen Modelle für reelle und komplexe quaternionenwertige Funktionen sowie Clifford-algebra-wertige Funktionen enthalten. Außerdem werden holomorphe Funktionen mittels einer geeignete Formel vom Taylor-Typ durch spezielle Funktionen lokal approximiert. Um globale Approximationen für holomorphe Funktionen zu erhalten, werden im zweiten Teil der Arbeit verschiedene Systeme holomorpher Basisfunktionen in drei und vier Dimensionen mittels geeigneter Fourier-Entwicklungen explizit konstruiert. Das Konzept der Holomorphie ist verbunden mit der Lösung verallgemeinerter Cauchy-Riemann Systeme, deren Funktionswerte reellen Quaternionen bzw. reduzierte Quaternionen sind. In expliziter Form werden orthogonale holomorphe Funktionensysteme konstruiert, die Lösungen des Riesz-Systems bzw. des Moisil-Teodorescu Systems über zylindrischen Gebieten im R3, sowie Lösungen des Riesz-Systems in Kugeln des R4 sind. Um konkrete Anwendungen auf Randwertprobleme realisieren zu können wird eine orthogonale Zerlegung eines Rechts-Quasi-Hilbert-Moduls komplex-quaternionischer Funktionen unter gegebenen Bedingungen studiert. Die Ergebnisse werden auf die Behandlung von Maxwell-Gleichungen mit zeitvariabler elektrischer Dielektrizitätskonstante und magnetischer Permeabilität angewandt
The richness and widely applicability of the theory of holomorphic functions in complex analysis requires to perform a similar theory in higher dimensions. It has been developed by many researchers so far, especially in quaternionic analysis. Over the last years, it has been successfully applied to a vast array of problems in mathematical physics. The aim of this thesis is to study the structure of holomorphy in higher dimensions. First, a new concept of holomorphy is introduced based on the theory of right invertible operators, and not by means of an analogue of the Cauchy-Riemann operator as usual. This notion covers most of the well-known holomorphic structures in higher dimensions including real, complex, quaternionic, Clifford analysis, among others. In addition, from our operators a local approximation of a holomorphic function is attained by the Taylor type formula. In order to obtain the global approximation for holomorphic functions, the second part of the thesis deals with the construction of different systems of basis holomorphic functions in three and four dimensions by means of Fourier analysis. The concept of holomorphy is related to the null-solutions of generalized Cauchy-Riemann systems, which take either values in the reduced quaternions or real quaternions. We obtain several explicit orthogonal holomorphic function systems: solutions to the Riesz and Moisil-Teodorescu systems over cylindrical domains in R3, and solutions to the Riesz system over spherical domains in R4. Having in mind concrete applications to boundary value problems, we investigate an orthogonal decomposition of complex-quaternionic functions over a right quasi-Hilbert module under given conditions. It is then applied to the treatment of Maxwell’s equations with electric permittivity and magnetic permeability depending on the time variable
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Bär, Christian. "Das Spektrum von Dirac-Operatoren." Bonn : [s.n.], 1991. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=003506032&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.

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Afentoulidis-Almpanis, Spyridon. "Noncubic Dirac Operators for finite-dimensional modules." Electronic Thesis or Diss., Université de Lorraine, 2021. http://www.theses.fr/2021LORR0035.

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Cette thèse porte sur l’étude des opérateurs de Dirac non-cubiques dans le cadre de la théorie des représentations des groupes de Lie. Après avoir présenté des notions de la théorie de Lie et des algèbres de Clifford, nous rappelons les propriétés principales des opérateurs de Dirac cubiques D introduits par Kostant en 1999. Ces résultats ont rapidement suscité un vif intérêt. En particulier, à la fin des années 2000, Vogan introduit une cohomologie définie par l’opérateur de Kostant et suggère une classification cohomologique des représentations. La cohomologie de Dirac a été calculée pour diverses familles de représentations, telles que les séries discrètes, les modules Aq(>) ou les modules de dimension finie. Pour les modules de dimension finie, la cohomologie de Dirac coïncide avec le noyau de D. Il apparait que l’opérateur de Dirac de Kostant est une version algébrique d’un opérateur différentiel issu d’une famille continue d’opérateurs de Dirac géométriques introduits par Slebarski dans les années 1980 dans le cadre de fibrés au- dessus d’espaces homogènes G/H de groupes compacts. Ce qui distingue l’opérateur de Dirac de Kostant est qu’il est le seul membre de cette famille dont le carré, généralisant une formule de Parthasarathy, diffère de l’opérateur de Casimir à un scalaire près. Cette propriété a des applications importantes en théorie des représentations des groupes de Lie. Le carré des opérateurs de Dirac non-cubiques, i.e des autres membres de la famille d’opérateurs de Slebarski, a été calculé par Agricola qui a également établit des liens précis entre ces opérateurs non-cubiques et la théorie des cordes en physique. Par ailleurs, les opérateurs de Dirac non-cubiques sont des opérateurs différentiels invariants, et donc leur noyau est le siège de représentations (de dimension finie) de groupes compacts. Dans cette thèse nous étudions le noyau des opérateurs de Dirac non-cubiques, et nous montrons, sous certaines conditions sur les espaces homogènes G/H, que ce noyau contient le noyau de l’opérateur de Dirac cubique. Nous obtenons en fait une formule explicite pour le noyau que nous appliquons aux cas des algèbres de Lie classiques et des algèbres de Lie exceptionnelles. Nous constatons que certaines propriétés des opérateurs non-cubiques sont analogues à celles de l’opérateur de Dirac de Kostant, tel que l’indice. Nous déduisons également quelques observations sur les opérateurs de Dirac géométrique non-cubiques
This thesis focuses on the study of noncubic Dirac operators within the framework of representation theory of Lie groups. After recalling basic notions of Lie theory and Clifford algebras, we present the main properties of cubic Dirac operators D introduced by Kostant in 1999. These results quickly aroused great interest. In particular, in the late 1990’s, Vogan introduced a cohomology defined by Kostant operator D and suggested a cohomological classification of representations. Dirac cohomology was computed for various families of representations, such as the discrete series, Aq(>) modules or finite dimensional representations. It turns out that for finite dimensional modules, Dirac cohomology coincides with the kernel of D. It appears that Kostant’s Dirac operator is an algebraic version of a specific member of a continuous family of geometric Dirac operators introduced by Slebarski in the mid 1980’s in the context of bundles over homogeneous spaces G/H of compact groups. What distinguishes the cubic Dirac operator is that it is the only member of this family whose square, generalizing Parthasarathy’s formula, differs from the Casimir operator up to a scalar. This property has important applications in representation theory of Lie groups. The square of the noncubic Dirac operators, i.e. of the other members of Slebarski’s family, was calculated by Agricola who also established precise links between these noncubic operators and string theory in physics. Actually, noncubic Dirac operators are invariant differential operators, and therefore their kernels define (finite-dimensional) representations of compact groups. In this thesis we study the kernel of noncubic Dirac operators, and we show that, under certain conditions on the homogeneous spaces G/H, the kernel contains the kernel of the cubic Dirac operator. We obtain an explicit formula for the kernel which we apply to the case of classical Lie algebras and of exceptional Lie algebras. We remark that some properties of noncubic operators are analogous to those of Kostant cubic Dirac operator, such as the index. We also deduce some observations on noncubic geometric Dirac operators
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Books on the topic "Operatory Diraca"

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S, Sargsi͡a︡n I., ed. Operatory Shturma-Liuvilli͡a︡ i Diraka. Moskva: "Nauka," Glav. red. fiziko-matematicheskoĭ lit-ry, 1988.

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S, Sargsi͡a︡n I., ed. Sturm-Liouville and Dirac operators. Dordrecht: Kluwer Academic, 1991.

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3

Dirac operators and spectral geometry. Cambridge: Cambridge University Press, 1998.

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Thomas, Friedrich. Dirac operators in Riemannian geometry. Providence, R.I: American Mathematical Society, 2000.

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Berline, Nicole. Heat kernels and Dirac operators. 2nd ed. Berlin: Springer, 1996.

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Berline, Nicole. Heat kernels and Dirac operators. Berlin: Springer-Verlag, 1992.

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7

Levitan, B. M., and I. S. Sargsjan. Sturm—Liouville and Dirac Operators. Dordrecht: Springer Netherlands, 1991. http://dx.doi.org/10.1007/978-94-011-3748-5.

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Habermann, Katharina, and Lutz Habermann. Introduction to Symplectic Dirac Operators. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006. http://dx.doi.org/10.1007/b138212.

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Berline, Nicole, Ezra Getzler, and Michèle Vergne. Heat Kernels and Dirac Operators. Berlin, Heidelberg: Springer Berlin Heidelberg, 1992. http://dx.doi.org/10.1007/978-3-642-58088-8.

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Booß-Bavnbek, Bernhelm, and Krzysztof P. Wojciechowski. Elliptic Boundary Problems for Dirac Operators. Boston, MA: Birkhäuser Boston, 1993. http://dx.doi.org/10.1007/978-1-4612-0337-7.

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Book chapters on the topic "Operatory Diraca"

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Rabinovich, Vladimir. "Dirac Operators on $$ \mathbb {R}$$ with General Point Interactions." In Operator Algebras, Toeplitz Operators and Related Topics, 351–81. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-44651-2_21.

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Krähmer, Ulrich, and Elmar Wagner. "Twisted Dirac Operator on Quantum SU(2) in Disc Coordinates." In Operator Algebras, Toeplitz Operators and Related Topics, 233–53. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-44651-2_16.

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Friedrich, Thomas. "Dirac operators." In Graduate Studies in Mathematics, 57–90. Providence, Rhode Island: American Mathematical Society, 2000. http://dx.doi.org/10.1090/gsm/025/03.

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Cnops, Jan. "Dirac Operators." In An Introduction to Dirac Operators on Manifolds, 61–89. Boston, MA: Birkhäuser Boston, 2002. http://dx.doi.org/10.1007/978-1-4612-0065-9_3.

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Booß-Bavnbek, Bernhelm, and Krzysztof P. Wojciechowski. "Dirac Operators." In Elliptic Boundary Problems for Dirac Operators, 19–25. Boston, MA: Birkhäuser Boston, 1993. http://dx.doi.org/10.1007/978-1-4612-0337-7_3.

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Martin, Mircea. "Deconstructing Dirac operators. III: Dirac and semi-Dirac pairs." In Topics in Operator Theory, 347–62. Basel: Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-0346-0161-0_14.

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Friedrich, Thomas. "Dirac-Operatoren." In Dirac-Operatoren in der Riemannschen Geometrie, 63–99. Wiesbaden: Vieweg+Teubner Verlag, 1997. http://dx.doi.org/10.1007/978-3-322-80302-3_3.

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Gesztesy, Fritz, and Marcus Waurick. "Dirac-Type Operators." In The Callias Index Formula Revisited, 55–63. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-29977-8_6.

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Edmunds, David E., and W. Desmond Evans. "The Dirac Operator." In Springer Monographs in Mathematics, 281–301. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-02125-2_12.

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Nardone, Mary S. "Operator Training." In Direct Digital Control Systems, 181–89. Boston, MA: Springer US, 1999. http://dx.doi.org/10.1007/978-1-4615-4923-9_11.

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Conference papers on the topic "Operatory Diraca"

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Fishman, Louis. "Direct and Inverse Wave Propagation in the Frequency Domain via the Weyl Operator Symbol Calculus." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0660.

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Abstract Wave field splitting, invariant imbedding, and phase space methods reformulate the Helmholtz wave propagation problem in terms of an operator scattering matrix characteristic of the modeled environment. The equations for the reflection and transmission operators are first-order in range, nonlinear (Riccati-like), and, in general, nonlocal. The singularity structure of the corresponding Weyl operator symbols plays a crucial role in the development of both direct and inverse wave propagation algorithms.
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Arriola, E. Ruiz. "Anomalies for nonlocal dirac operators." In The international workshop on hadron physics of low energy QCD. AIP, 2000. http://dx.doi.org/10.1063/1.1303042.

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Korotyaev, Evgeny L., and Dmitrii S. Mokeev. "Dislocation problem for the Dirac operator." In 2019 Days on Diffraction (DD). IEEE, 2019. http://dx.doi.org/10.1109/dd46733.2019.9016424.

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Boitsev, A. A. "Boundary triplets approach for Dirac operator." In QMath12 – Mathematical Results in Quantum Mechanics. WORLD SCIENTIFIC, 2014. http://dx.doi.org/10.1142/9789814618144_0015.

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Tantalo, Nazario. "A mass preconditioning for lattice Dirac operators." In The XXVIII International Symposium on Lattice Field Theory. Trieste, Italy: Sissa Medialab, 2011. http://dx.doi.org/10.22323/1.105.0041.

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Li, Yao, and Thenkurussi Kesavadas. "Brain Computer Interface Robotic Co-Workers: Defective Part Picking System." In ASME 2018 13th International Manufacturing Science and Engineering Conference. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/msec2018-6655.

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Industrial robotic co-workers are robots that can work with human being in an unstructured environment. Such robots, must be able to assist human operators in a seamless way without receiving specific instructions. Robotic co-workers can open entirely new application fields in manufacturing as demonstrated in this paper. We designed such an industrial co-robot to pick up defective parts by simply monitoring a human operator directly through a brain computer interface (BCI). By constantly monitoring the operator using BCI sensors, the robotic co-worker can sense when an operator notices a defective part and then moves to remove the part from a moving conveyor with no direct instruction from the operator. The robot, equipped with an RGB camera, recognizes the part, tracks the position and generates accurate motion plan. We demonstrated the system using a human subject study.
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Kukushkin, Andrey A. "On homogenization of the periodic Dirac operator." In Days on Diffraction 2012 (DD). IEEE, 2012. http://dx.doi.org/10.1109/dd.2012.6402772.

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Di Renzo, Francesco, and M. Brambilla. "The Dirac operator spectrum: a perturbative approach." In The XXVII International Symposium on Lattice Field Theory. Trieste, Italy: Sissa Medialab, 2010. http://dx.doi.org/10.22323/1.091.0209.

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Baaske, Franka, and Swanhild Bernstein. "Scattering theory for a Dirac type operator." In 9TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2012. AIP, 2012. http://dx.doi.org/10.1063/1.4765467.

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Falomir, H. "Global boundary conditions for the Dirac operator." In Trends in theoretical physics CERN-Santiago de Compostela-La Plata meeting. AIP, 1998. http://dx.doi.org/10.1063/1.54693.

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Reports on the topic "Operatory Diraca"

1

Tolksdorf, Jurgen. Gauge Theories with Spontaneously Broken Gauge Symmetry, Connections and Dirac Operators. GIQ, 2012. http://dx.doi.org/10.7546/giq-3-2002-141-162.

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