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Journal articles on the topic 'Quantum principal bundles'

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Published: 14 March 2023

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1

Đurđević, Mićo. "Geometry of Quantum Principal Bundles II." Reviews in Mathematical Physics 09, no. 05 (July 1997): 531–607. http://dx.doi.org/10.1142/s0129055x9700021x.

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A general non-commutative-geometric theory of principal bundles is developed. Quantum groups play the role of structure groups and general quantum spaces play the role of base manifolds. A general conceptual framework for the study of differential structures on quantum principal bundles is presented. Algebras of horizontal, verticalized and "horizontally vertically" decomposed differential forms on the bundle are introduced and investigated. Constructive approaches to differential calculi on quantum principal bundles are discussed. The formalism of connections is developed further. The corresponding operators of horizontal projection, covariant derivative and curvature are constructed and analyzed. In particular the analogs of the basic classical algebraic identities are derived. A quantum generalization of classical Weil's theory of characteristic classes is sketched. Quantum analogs of infinitesimal gauge transformations are studied. Interesting examples are presented.

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2

Aschieri, Paolo. "Deformation quantization of principal bundles." International Journal of Geometric Methods in Modern Physics 13, no. 08 (September 2016): 1630010. http://dx.doi.org/10.1142/s0219887816300105.

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We outline how Drinfeld twist deformation techniques can be applied to the deformation quantization of principal bundles into noncommutative principal bundles and, more in general, to the deformation of Hopf–Galois extensions. First, we twist deform the structure group in a quantum group, and this leads to a deformation of the fibers of the principal bundle. Next, we twist deform a subgroup of the group of automorphisms of the principal bundle, and this leads to a noncommutative base space. Considering both deformations, we obtain noncommutative principal bundles with noncommutative fiber and base space as well.

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3

Brzeziński, Tomasz, and Simon A. Fairfax. "Bundles over Quantum RealWeighted Projective Spaces." Axioms 1, no. 2 (September 17, 2012): 201–25. http://dx.doi.org/10.3390/axioms1020201.

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The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free.

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4

Aschieri, Paolo, Rita Fioresi, and Emanuele Latini. "Quantum Principal Bundles on Projective Bases." Communications in Mathematical Physics 382, no. 3 (March 2021): 1691–724. http://dx.doi.org/10.1007/s00220-021-03985-4.

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AbstractThe purpose of this paper is to propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We study noncommutative principal bundles corresponding to $$G \rightarrow G/P$$ G → G / P , where G is a semisimple group and P a parabolic subgroup.

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5

Đurđević, Mićo. "Differential structures on quantum principal bundles." Reports on Mathematical Physics 41, no. 1 (February 1998): 91–115. http://dx.doi.org/10.1016/s0034-4877(98)80183-3.

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6

Brzeziński, Tomasz. "Translation map in quantum principal bundles." Journal of Geometry and Physics 20, no. 4 (November 1996): 349–70. http://dx.doi.org/10.1016/s0393-0440(96)00003-4.

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7

Hajac, Piotr M. "Strong connections on quantum principal bundles." Communications in Mathematical Physics 182, no. 3 (December 1996): 579–617. http://dx.doi.org/10.1007/bf02506418.

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8

Müller, Andreas. "Classifying spaces for quantum principal bundles." Communications in Mathematical Physics 149, no. 3 (October 1992): 495–512. http://dx.doi.org/10.1007/bf02096940.

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9

Durdević, Mićo. "Geometry of quantum principal bundles I." Communications in Mathematical Physics 175, no. 3 (February 1996): 457–520. http://dx.doi.org/10.1007/bf02099507.

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10

Budzyński, Robert J., and Witold Kondracki. "Quantum principal fibre bundles: Topological aspects." Reports on Mathematical Physics 37, no. 3 (June 1996): 365–85. http://dx.doi.org/10.1016/0034-4877(96)84074-2.

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11

Brzeziński, Tomasz, and Bartosz Zieliński. "Quantum principal bundles over quantum real projective spaces." Journal of Geometry and Physics 62, no. 5 (May 2012): 1097–107. http://dx.doi.org/10.1016/j.geomphys.2011.12.008.

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12

Đurđević, Mićo. "Quantum principal bundles and their characteristic classes." Banach Center Publications 40, no. 1 (1997): 303–13. http://dx.doi.org/10.4064/-40-1-303-313.

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13

ÐurÐevich, Micho. "General frame structures on quantum principal bundles." Reports on Mathematical Physics 44, no. 1-2 (August 1999): 53–70. http://dx.doi.org/10.1016/s0034-4877(99)80145-1.

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14

Durdevic, Mico. "Quantum principal bundles and corresponding gauge theories." Journal of Physics A: Mathematical and General 30, no. 6 (March 21, 1997): 2027–54. http://dx.doi.org/10.1088/0305-4470/30/6/025.

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15

kDurkdević, Mićo. "Quantum principal bundles and Tannaka-Krein duality theory." Reports on Mathematical Physics 38, no. 3 (December 1996): 313–24. http://dx.doi.org/10.1016/s0034-4877(97)84884-7.

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16

Calow, Dirk, and Rainer Matthes. "Connections on locally trivial quantum principal fibre bundles." Journal of Geometry and Physics 41, no. 1-2 (February 2002): 114–65. http://dx.doi.org/10.1016/s0393-0440(01)00050-x.

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17

Epstein, Marcelo. "Hilbert bundles as quantum-classical continua." Mathematics and Mechanics of Solids 25, no. 6 (June 2020): 1312–17. http://dx.doi.org/10.1177/1081286519888964.

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A hybrid quantum–classical model is proposed whereby a micro-structured (Cosserat-type) continuum is construed as a principal Hilbert bundle. A numerical example demonstrates the possible applicability of the theory.

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18

Mashford, John. "An Approach to Classical Quantum Field Theory Based on the Geometry of Locally Conformally Flat Space-Time." Advances in Mathematical Physics 2017 (2017): 1–15. http://dx.doi.org/10.1155/2017/8070462.

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This paper gives an introduction to certain classical physical theories described in the context of locally Minkowskian causal structures (LMCSs). For simplicity of exposition we consider LMCSs which have locally Euclidean topology (i.e., are manifolds) and hence are Möbius structures. We describe natural principal bundle structures associated with Möbius structures. Fermion fields are associated with sections of vector bundles associated with the principal bundles while interaction fields (bosons) are associated with endomorphisms of the space of fermion fields. Classical quantum field theory (the Dirac equation and Maxwell’s equations) is obtained by considering representations of the structure group K⊂SU(2,2) of a principal bundle associated with a given Möbius structure where K, while being a subset of SU(2,2), is also isomorphic to SL2,C×U(1). The analysis requires the use of an intertwining operator between the action of K on R4 and the adjoint action of K on su⁡(2,2) and it is shown that the Feynman slash operator, in the chiral representation for the Dirac gamma matrices, has this intertwining property.

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19

Kim, Minhyong. "Arithmetic gauge theory: A brief introduction." Modern Physics Letters A 33, no. 29 (September 20, 2018): 1830012. http://dx.doi.org/10.1142/s0217732318300124.

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Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles holds the key to an effective version of Faltings’ theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands program. In this paper, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory.

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20

Miković, A. "Quantum Field Theory of Open Spin Networks and New Spin Foam Models." International Journal of Modern Physics A 18, supp02 (October 2003): 83–96. http://dx.doi.org/10.1142/s0217751x0301797x.

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We describe how a spin-foam state sum model can be reformulated as a quantum field theory of spin networks, such that the Feynman diagrams of that field theory are the spin-foam amplitudes. In the case of open spin networks, we obtain a new type of state-sum models, which we call the matter spin foam models. In this type of state-sum models, one labels both the faces and the edges of the dual two-complex for a manifold triangulation with the simple objects from a tensor category. In the case of Lie groups, such a model corresponds to a quantization of a theory whose fields are the principal bundle connection and the sections of the associated vector bundles. We briefly discuss the relevance of the matter spin foam models for quantum gravity and for topological quantum field theories.

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21

ILIEV, BOZHIDAR Z. "FIBRE BUNDLE FORMULATION OF NONRELATIVISTIC QUANTUM MECHANICS V: INTERPRETATION, SUMMARY AND DISCUSSION." International Journal of Modern Physics A 17, no. 02 (January 20, 2002): 245–58. http://dx.doi.org/10.1142/s0217751x02005712.

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We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analog of the) Schrödinger equation. Now the dynamical variables and density operator are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this work. This is the fifth, closing, part of our investigation. We briefly discuss the observer's role in the theory and different realizations of the space–time model used as a base space in the bundle approach to quantum mechanics. The exact conditions for the equivalence of Hilbert bundle and Hilbert space formulations of the theory are pointed out. A table of comparison between both descriptions of nonrelativistic quantum mechanics is presented. We discuss some principal moments of the Hilbert bundle description and show that as a scheme it is more general than the Hilbert space one. Different directions for further research are pointed out.

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22

Sheu, Albert Jeu-Liang. "Projective Modules Over Quantum Projective Line." International Journal of Mathematics 28, no. 03 (March 2017): 1750022. http://dx.doi.org/10.1142/s0129167x17500227.

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Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces [Formula: see text] constructed from the multipullback quantum spheres introduced by Hajac and collaborators, we analyze the structure of the C*-algebra [Formula: see text] realized as a concrete groupoid C*-algebra, and find its [Formula: see text]-groups. Furthermore, after a complete classification of the unitary equivalence classes of projections or equivalently the isomorphism classes of finitely generated projective modules over the C*-algebra [Formula: see text], we identify those quantum principal [Formula: see text]-bundles introduced by Hajac and collaborators among the projections classified.

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23

IVANOV, O. M., and A. G. SAVINKOV. "QUANTUM ANYON-LIKE SYSTEMS WITH TOPOLOGICAL CHARGES ON U(1) BUNDLES." International Journal of Modern Physics A 08, no. 31 (December 20, 1993): 5505–36. http://dx.doi.org/10.1142/s0217751x93002174.

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We investigate some classes of locally Hamiltonian systems. To formulate a global Hamiltonian formalism a nontrivial principal U(1) bundle over a multi-particle dynamical space is constructed. Lift of dynamics to the total space is considered for the models. Absence of a transposition operator lifting to the bundle space forces us to review a statistics theme from a new standpoint. A topological invariant to characterize system properties is generalized by using geometrical methods and the anyon analogy.

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24

Canarutto, Daniel. "Special generalized densities and propagators: A geometric account." International Journal of Geometric Methods in Modern Physics 13, no. 01 (January 2016): 1530004. http://dx.doi.org/10.1142/s0219887815300044.

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Starting from a short review of spaces of generalized sections of vector bundles, we give a concise systematic description, in precise geometric terms, of Leray densities, principal value densities, propagators and elementary solutions of field equations in flat spacetime. We then sketch a partly original geometric presentation of free quantum fields and show how propagators arise from their graded commutators in the boson and fermion cases.

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25

Katabarwa, Amara, Sukin Sim, Dax Enshan Koh, and Pierre-Luc Dallaire-Demers. "Connecting geometry and performance of two-qubit parameterized quantum circuits." Quantum 6 (August 23, 2022): 782. http://dx.doi.org/10.22331/q-2022-08-23-782.

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Parameterized quantum circuits (PQCs) are a central component of many variational quantum algorithms, yet there is a lack of understanding of how their parameterization impacts algorithm performance. We initiate this discussion by using principal bundles to geometrically characterize two-qubit PQCs. On the base manifold, we use the Mannoury-Fubini-Study metric to find a simple equation relating the Ricci scalar (geometry) and concurrence (entanglement). By calculating the Ricci scalar during a variational quantum eigensolver (VQE) optimization process, this offers us a new perspective to how and why Quantum Natural Gradient outperforms the standard gradient descent. We argue that the key to the Quantum Natural Gradient's superior performance is its ability to find regions of high negative curvature early in the optimization process. These regions of high negative curvature appear to be important in accelerating the optimization process.

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Mashford, John. "Divergence-free quantum electrodynamics in locally conformally flat space–time." International Journal of Modern Physics A 36, no. 13 (May 6, 2021): 2150083. http://dx.doi.org/10.1142/s0217751x21500834.

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This paper describes an approach to quantum electrodynamics (QED) in curved space–time obtained by considering infinite-dimensional algebra bundles associated to a natural principal bundle [Formula: see text] associated with any locally conformally flat space–time, with typical fibers including the Fock space and a space of fermionic multiparticle states which forms a Grassmann algebra. Both these algebras are direct sums of generalized Hilbert spaces. The requirement of [Formula: see text] covariance associated with the geometry of space–time, where [Formula: see text] is the structure group of [Formula: see text], leads to the consideration of [Formula: see text] intertwining operators between various spaces. Scattering processes are associated with such operators and are encoded in an algebra of kernels. Intertwining kernels can be generated using [Formula: see text] covariant matrix-valued measures. Feynman propagators, fermion loops and the electron self-energy can be given well-defined interpretations as such measures. Divergence-free calculations in QED can be carried out by computing the spectra of these measures and kernels (a process called spectral regularization). As an example of the approach the precise Uehling potential function for the [Formula: see text] atom is calculated without requiring renormalization from which the Uehling contribution to the Lamb shift can be calculated exactly.

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27

Reyes, Armando, and Fabio Calderón. "Some interactions between Hopf Galois extensions and noncommutative rings." Universitas Scientiarum 27, no. 2 (August 10, 2022): 58–161. http://dx.doi.org/10.11144/javeriana.sc271.sibh.

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In this paper, our objects of interest are Hopf Galois extensions (e.g., Hopf algebras, Galois field extensions, strongly graded algebras, crossed products, principal bundles, etc.) and families of noncommutative rings (e.g., skew polynomial rings, PBW extensions and skew PBW extensions, etc.) We collect and systematize questions, problems, properties and recent advances in both theories by explicitly developing examples and doing calculations that are usually omitted in the literature. In particular, for Hopf Galois extensions we consider approaches from the point of view of quantum torsors (also known as quantum heaps) and Hopf Galois systems, while for some families of noncommutative rings we present advances in the characterization of ring-theoretic and homological properties. Every developed topic is exemplified with abundant references to classic and current works, so this paper serves as a survey for those interested in either of the two theories. Throughout, interactions between both are presented.

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Boeijink, Jord, Klaas Landsman, and Walter van Suijlekom. "Quantization commutes with singular reduction: Cotangent bundles of compact Lie groups." Reviews in Mathematical Physics 31, no. 06 (June 25, 2019): 1950016. http://dx.doi.org/10.1142/s0129055x19500168.

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We analyze the ‘quantization commutes with reduction’ problem (first studied in physics by Dirac, and known in the mathematical literature also as the Guillemin–Sternberg Conjecture) for the conjugate action of a compact connected Lie group [Formula: see text] on its own cotangent bundle [Formula: see text]. This example is interesting because the momentum map is not proper and the ensuing symplectic (or Marsden–Weinstein quotient) [Formula: see text] is typically singular.In the spirit of (modern) geometric quantization, our quantization of [Formula: see text] (with its standard Kähler structure) is defined as the kernel of the Dolbeault–Dirac operator (or, equivalently, the spin[Formula: see text]–Dirac operator) twisted by the pre-quantum line bundle. We show that this quantization of [Formula: see text] reproduces the Hilbert space found earlier by Hall (2002) using geometric quantization based on a holomorphic polarization. We then define the quantization of the singular quotient [Formula: see text] as the kernel of the twisted Dolbeault–Dirac operator on the principal stratum, and show that quantization commutes with reduction in the sense that either way one obtains the same Hilbert space [Formula: see text].

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MAGNOT, JEAN-PIERRE. "AMBROSE–SINGER THEOREM ON DIFFEOLOGICAL BUNDLES AND COMPLETE INTEGRABILITY OF THE KP EQUATION." International Journal of Geometric Methods in Modern Physics 10, no. 09 (August 30, 2013): 1350043. http://dx.doi.org/10.1142/s0219887813500436.

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In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose–Singer theorem that enlarges the one stated in [J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math.128 (2004) 513–529], and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frölicher (Lie) algebras such as formal q-series of the quantum algebra of pseudo-differential operators. These deformations can be defined for classical pseudo-differential operators but they are used here on formal pseudo-differential operators in order to get a differential geometric framework to deal with the KP hierarchy that is known to be completely integrable with formal power series. Here, we get an integration of the Zakharov–Shabat connection form by means of smooth sections of a (differential geometric) bundle with structure group, some groups of q-deformed operators. The integration obtained by Mulase [Complete integrability of the Kadomtsev–Petviashvili equation Adv. Math.54 (1984) 57–66], and the key tools he developed, are totally recovered on the germs of the smooth maps of our construction. The tool coming from (classical) differential geometry used in this construction is the holonomy group, on which we have an Ambrose–Singer-like theorem: the Lie algebra is spanned by the curvature elements. This result is proved for any connection a diffeological principal bundle with structure group a regular Frölicher Lie group. The case of a (classical) Lie group modeled on a complete locally convex topological vector space is also recovered and the work developed in [J.-P. Magnot, Difféologie du fibré d'Holonomie en dimension infinie, Math. Rep. Canadian Roy. Math. Soc.28(4) (2006); J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math. 128 (2004) 513–529] is completed.

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Emmrich, Claudio. "Local spin base invariance from a global differential–geometrical point of view." Journal of Mathematical Physics 63, no. 4 (April 1, 2022): 042302. http://dx.doi.org/10.1063/5.0081140.

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This article gives a geometric interpretation of the spin base formulation with local spin base invariance of spinors on a curved space-time and, in particular, of a central element, the global Dirac structure, in terms of principal and vector bundles and their endomorphisms. It is shown that this is intimately related to Spin and [Formula: see text] structures in the sense that the existence of one of those implies the existence of a Dirac structure and allows for an extension to local spin base invariance. Vice versa, as a central result, the existence of a Dirac structure implies the existence of a [Formula: see text] structure. Nevertheless, the spin base invariant setting may be considered more general, allowing for more physical degrees of freedom. Furthermore, arguments are given that the Dirac structure is a more natural choice as a variable for (quantum) gravity than tetrads/vielbeins.

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31

Smrz, PK. "Geometrical Models of Elementary Particles." Australian Journal of Physics 48, no. 6 (1995): 1045. http://dx.doi.org/10.1071/ph951045.

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A new concept of space and time, constructed from a de Sitter structured principal fibre bundle with a connection, is used to discuss a geometrical interpretation for the complex plane of the quantum theory and quantum behaviour of particles. In particular some features of a theory based on a torsion free metric linear connection in a five-dimensional base manifold are described.

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DJEGHLOUL, N., and M. TAHIRI. "AN ALTERNATIVE CONSTRUCTION OF THE QUANTUM ACTION FOR SUPERGRAVITY." Modern Physics Letters A 15, no. 20 (June 28, 2000): 1307–20. http://dx.doi.org/10.1142/s0217732300001729.

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We develop a method to derive the on-shell invariant quantum action of the simple supergravity in such a way that the quartic ghost interactions term is explicity determined. First, we reinvestigate the simple supergravity in terms of a principal superfiber bundle. This gives rise to the closed geometrical BRST algebra. Therefore we determine the open BRST algebra, which realizes the invariance of the classical action. Then, given a prescription to build the full quantum action, we obtain the quantum BRST algebra. Together with the constructed quantum action this allows us to recover the auxiliary fields and the invariant extension of the classical action.

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GEER, NATHAN, and BERTRAND PATUREAU-MIRAND. "G-LINKS INVARIANTS, MARKOV TRACES AND THE SEMI-CYCLIC Uq𝔰𝔩(2)-MODULES." Journal of Knot Theory and Its Ramifications 22, no. 11 (October 2013): 1350063. http://dx.doi.org/10.1142/s0218216513500636.

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Kashaev and Reshetikhin proposed a generalization of the Reshetikhin–Turaev link invariant construction to tangles with a flat connection in a principal G-bundle of the complement of the tangle. The purpose of this paper is to adapt and renormalize their construction to define invariants of G-links using the semi-cyclic representations of the non-restricted quantum group associated to 𝔰𝔩(2), defined by De Concini and Kac. Our construction uses a modified Markov trace. In our main example, the semi-cyclic invariants are a natural extension of the generalized Alexander polynomial invariants defined by Akutsu, Deguchi and Ohtsuki. Surprisingly, direct computations suggest that these invariants are actually equal.

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DZHUNUSHALIEV, V., H. J. SCHMIDT, and O. RURENKO. "SPHERICALLY SYMMETRIC SOLUTIONS IN MULTIDIMENSIONAL GRAVITY WITH THE SU(2) GAUGE GROUP AS THE EXTRA DIMENSIONS." International Journal of Modern Physics D 11, no. 05 (May 2002): 685–701. http://dx.doi.org/10.1142/s0218271802001925.

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The multidimensional gravity on the principal bundle with the SU(2) gauge group is considered. The numerical investigation of the spherically symmetric metrics with the center of symmetry is made. The solution of the gravitational equations depends on the boundary conditions of the "SU(2) gauge potential" (off-diagonal metric components) at the symmetry center and on the type of symmetry (symmetrical or antisymmetrical) of these potentials. In the chosen range of the boundary conditions it is shown that there are two types of solutions: wormhole-like and flux tube. The physical application of such kind of solutions as quantum handles in a spacetime foam is discussed.

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LANDSMAN, N. P. "INDUCED REPRESENTATIONS, GAUGE FIELDS, AND QUANTIZATION ON HOMOGENEOUS SPACES." Reviews in Mathematical Physics 04, no. 04 (December 1992): 503–27. http://dx.doi.org/10.1142/s0129055x92000212.

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We study representations of the enveloping algebra of a Lie group G which are induced by a representation of a Lie subgroup H, assuming that G/H is reductive. Such representations describe the superselection sectors of a quantum particle moving on G/H. It is found that the representatives of both the generators and the quadratic Casimir operators of G have a natural geometric realization in terms of the canonical connection on the principal H-bundle G. The explicit expression for the generators can be understood from the point of view of conservation laws and moment maps in classical field theory and classical particle mechanics on G/H. The emergence of classical geometric structures in the quantum-mechanical situation is explained by a detailed study of the domain and possible self-adjointness properties of the relevant operators. A new and practical criterion for essential self-adjointness in general unitary representations is given.

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36

Viennot, David. "Principal bundle structure of quantum adiabatic dynamics with a Berry phase which does not commute with the dynamical phase." Journal of Mathematical Physics 46, no. 7 (July 2005): 072102. http://dx.doi.org/10.1063/1.1940547.

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LUSANNA, LUCA. "CLASSICAL YANG-MILLS THEORY WITH FERMIONS II: DIRAC’S OBSERVABLES." International Journal of Modern Physics A 10, no. 26 (October 20, 1995): 3675–757. http://dx.doi.org/10.1142/s0217751x95001753.

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For pure Yang-Mills theory on Minkowski space-time, formulated in functional spaces where the covariant divergence is an elliptic operator without zero modes, and for a trivial principal bundle over the fixed time Euclidean space with a compact, semisimple, connected and simply connected structure Lie group, a Green function for the covariant divergence has been found. It allows one to solve the first class constraints associated with Gauss’ laws and to identify a connection-dependent coordinatization of the trivial principal bundle. In a neighborhood of the global identity section, by using canonical coordinates of the first kind on the fibers, one has a symplectic implementation of the Lie algebra of the small gauge transformations generated by Gauss’ laws and one can make a generalized Hodge decomposition of the gauge potential one-forms based on the BRST operator. This decomposition singles out a pure gauge background connection (the BRST ghost as Maurer-Cartan one-form on the group of gauge transformations) and a transverse gauge-covariant magnetic gauge potential. After an analogous decomposition of the electric field strength into the transverse and the longitudinal part, Dirac’s observables associated with the transverse electric and magnetic components are identified as their restriction to the global identity section of the trivial principal bundle. The longitudinal part of the electric field can be re-expressed in terms of these electric and magnetic transverse parts and of the constraints without Gribov ambiguity. The physical Lagrangian, Hamiltonian, non-Abelian and topological charges have been obtained in terms of transverse Dirac’s observables, also in the presence of fermion fields, after a symplectic decoupling of the gauge degrees of freedom; one has an explicit realization of the abstract “Riemannian metric” on the orbit space. Both the Lagrangian and the Hamiltonian are nonlocal and nonpolynomial; like in the Coulomb gauge they are not Lorentz-invariant, but the invariance can be enforced on them if one introduces Wigner covariance of the observables by analyzing the various kinds of Poincare orbits of the system and by reformulating the theory on suitable spacelike hypersurfaces, following Dirac. By extending to classical relativistic field theory the problems associated with the Lorentz noncovariance of the canonical (presymplectic) center of mass for extended relativistic systems, in the sector of the field theory with P2>0 and W2≠0 one identifies a classical invariant intrinsic unit of length, determined by the Poincare Casimirs, whose quantum counterpart is the ultraviolet cutoff looked for by Dirac and Yukawa: it is the Compton wavelength of the field configuration (in an irreducible Poincare representation) multiplied by the value of its spin.

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Man, Zhao-Hui. "Affine Connection Representation of Gauge Fields." Advances in High Energy Physics 2022 (May 17, 2022): 1–32. http://dx.doi.org/10.1155/2022/4573981.

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There are two ways to unify gravitational field and gauge field. One is to represent gravitational field as principal bundle connection, and the other is to represent gauge field as affine connection. Poincaré gauge theory and metric-affine gauge theory adopt the first approach. This paper adopts the second. In this approach, (i) gauge field and gravitational field can both be represented by affine connection; they can be described by a unified spatial frame. (ii) Time can be regarded as the total metric with respect to all dimensions of internal coordinate space and external coordinate space. On-shell can be regarded as gradient direction. Quantum theory can be regarded as a geometric theory of distribution of gradient directions. Hence, gauge theory, gravitational theory, and quantum theory all reflect intrinsic geometric properties of manifold. (iii) Coupling constants, chiral asymmetry, PMNS mixing, and CKM mixing arise spontaneously as geometric properties in affine connection representation, so they are not necessary to be regarded as direct postulates in the Lagrangian anymore. (iv) The unification theory of gauge fields that are represented by affine connection can avoid the problem that a proton decays into a lepton in theories such as S U 5 . (v) There exists a geometric interpretation to the color confinement of quarks. In the affine connection representation, we can get better interpretations to the above physical properties; therefore, to represent gauge fields by affine connection is probably a necessary step towards the ultimate theory of physics.

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39

Majid, Shahn, and Liam Williams. "Poisson Principal Bundles." Symmetry, Integrability and Geometry: Methods and Applications, January 13, 2021. http://dx.doi.org/10.3842/sigma.2021.006.

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We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q-Hopf fibration on the standard q-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

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Krutov, Andrey O., Réamonn Ó Buachalla, and Karen R. Strung. "Nichols Algebras and Quantum Principal Bundles." International Mathematics Research Notices, February 6, 2023. http://dx.doi.org/10.1093/imrn/rnac366.

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Abstract We introduce a general framework for associating to a homogeneous quantum principal bundle a Yetter–Drinfeld module structure on the cotangent space of the base calculus. The holomorphic and anti-holomorphic Heckenberger–Kolb calculi of the quantum Grassmannians are then presented in this framework. This allows us to express the calculi in terms of the corresponding Nichols algebras. The extension of this result to all irreducible quantum flag manifolds is then conjectured.

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Schwieger, Kay, and Stefan Wagner. "An Atiyah Sequence for Noncommutative Principal Bundles." Symmetry, Integrability and Geometry: Methods and Applications, March 7, 2022. http://dx.doi.org/10.3842/sigma.2022.015.

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We present a derivation-based Atiyah sequence for noncommutative principal bundles. Along the way we treat the problem of deciding when a given ∗-automorphism on the quantum base space lifts to a ∗-automorphism on the quantum total space that commutes with the underlying structure group.

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Fioresi, Rita, Maria A. Lledo, and Junaid Razzaq. "N = 2 quantum chiral superﬁelds and quantum super bundles." Journal of Physics A: Mathematical and Theoretical, August 16, 2022. http://dx.doi.org/10.1088/1751-8121/ac8a2a.

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Abstract We give the superalgebra of N = 2 chiral (and antichiral) quantum superﬁelds realized as a subalgebra of the quantum supergroup SLq(4|2). The multiplication law in the quantum supergroup induces a coaction on the set of chiral superﬁelds. We also realize the quantum deformation of the chiral Minkowski superspace as a quantum principal bundle.

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43

Carotenuto, Alessandro, Fredy Díaz García, and Reamonn Ó Buachalla. "A Borel–Weil Theorem for the Irreducible Quantum Flag Manifolds." International Mathematics Research Notices, July 20, 2022. http://dx.doi.org/10.1093/imrn/rnac193.

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Abstract We establish a noncommutative generalisation of the Borel–Weil theorem for the Heckenberger–Kolb calculi of the irreducible quantum flag manifolds ${\mathcal {O}}_q(G/L_S)$, generalising previous work for the quantum Grassmannians ${\mathcal {O}}_q(\textrm {Gr}_{n,m})$. As a direct consequence we get a novel noncommutative differential geometric presentation of the quantum coordinate rings $S_q[G/L_S]$ of the irreducible quantum flag manifolds. The proof is formulated in terms of quantum principal bundles, and the recently introduced notion of a principal pair, and uses the Heckenberger and Kolb first-order differential calculus for the quantum Possion homogeneous spaces ${\mathcal {O}}_q(G/L^{\,\textrm {s}}_S)$.

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"The Bundle of Simultaneously Diagonalizable N-tuples of Matrices." WSEAS TRANSACTIONS ON MATHEMATICS 19 (April 27, 2020). http://dx.doi.org/10.37394/23206.2020.19.21.

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In this paper, a review of the simultaneous diagonalization of n-tuples of matrices for its applications in sciences is presented. For example, in quantum mechanics, position and momentum operators do not have a shared base that can represent the states of the system because they not commute, which is why switching operators form a key element of quantum physics since they define quantities that are compatible, that is, defined simultaneously. We are going to study this kind of family of linear operators using geometric constructions such as the principal bundles and associating them with a cohomology class measuring the deviation of the local product structure from the global product structure.

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Pap, Eric J., Daniël Boer, and Holger Waalkens. "A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics." Symmetry, Integrability and Geometry: Methods and Applications, January 13, 2022. http://dx.doi.org/10.3842/sigma.2022.003.

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We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.

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Đurđevich, Micho. "Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle." Symmetry, Integrability and Geometry: Methods and Applications, May 30, 2013. http://dx.doi.org/10.3842/sigma.2013.040.

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47

Han, Xiao, and Giovanni Landi. "Gauge groups and bialgebroids." Letters in Mathematical Physics 111, no. 6 (November 9, 2021). http://dx.doi.org/10.1007/s11005-021-01482-2.

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AbstractWe study the Ehresmann–Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include: Galois objects of Taft algebras, a monopole bundle over a quantum sphere and a not faithfully flat Hopf–Galois extension of commutative algebras. For each of the latter two examples, there is in fact a suitable invertible antipode for the bialgebroid making it a Hopf algebroid.

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