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Journal articles on the topic 'Quantum theory – Mathematics; Group theory'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Murphy, G. J., and L. Tuset. "Aspects of compact quantum group theory." Proceedings of the American Mathematical Society 132, no. 10 (June 2, 2004): 3055–67. http://dx.doi.org/10.1090/s0002-9939-04-07400-3.

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2

Vitiello, Giuseppe. "Group Contraction in Quantum Field Theory." International Journal of Theoretical Physics 47, no. 2 (July 25, 2007): 393–414. http://dx.doi.org/10.1007/s10773-007-9461-8.

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3

Antoine, Jean-Pierre. "Group Theory: Mathematical Expression of Symmetry in Physics." Symmetry 13, no. 8 (July 26, 2021): 1354. http://dx.doi.org/10.3390/sym13081354.

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The present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity: Euclidean, Galilean, and Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the state space of a quantum system has an intrinsic structure of pre-Hilbert space that one completes into a genuine Hilbert space. In this framework, the description of the invariance under a group G is based on a unitary representation of G. Next, we survey the various domains of application: atomic and molecular physics, quantum optics, signal and image processing, wavelets, internal symmetries, and approximate symmetries. Next, we discuss the extension to gauge theories, in particular, to the Standard Model of fundamental interactions. We conclude with some remarks about recent developments, including the application to braid groups.

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4

Lev, Felix M. "Symmetries in Foundation of Quantum Theory and Mathematics." Symmetry 12, no. 3 (March 4, 2020): 409. http://dx.doi.org/10.3390/sym12030409.

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In standard quantum theory, symmetry is defined in the spirit of Klein’s Erlangen Program—the background space has a symmetry group, and the basic operators should commute according to the Lie algebra of that group. We argue that the definition should be the opposite—background space has a direct physical meaning only on classical level while on quantum level symmetry should be defined by a Lie algebra of basic operators. Then the fact that de Sitter symmetry is more general than Poincare symmetry can be proved mathematically. The problem of explaining cosmological acceleration is very difficult but, as follows from our results, there exists a scenario in which the phenomenon of cosmological acceleration can be explained by proceeding from basic principles of quantum theory. The explanation has nothing to do with existence or nonexistence of dark energy and therefore the cosmological constant problem and the dark energy problem do not arise. We consider finite quantum theory (FQT) where states are elements of a space over a finite ring or field with characteristic p and operators of physical quantities act in this space. We prove that, with the same approach to symmetry, FQT and finite mathematics are more general than standard quantum theory and classical mathematics, respectively: the latter theories are special degenerated cases of the former ones in the formal limit p → ∞ .

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5

Pollatsek, Harriet. "Quantum Error Correction: Classic Group Theory Meets a Quantum Challenge." American Mathematical Monthly 108, no. 10 (December 2001): 932. http://dx.doi.org/10.2307/2695416.

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6

Brzeziński, Thomasz, and Shahn Majid. "Quantum group gauge theory on quantum spaces." Communications in Mathematical Physics 157, no. 3 (November 1993): 591–638. http://dx.doi.org/10.1007/bf02096884.

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7

Brzeziński, T., and Shahn Majid. "Quantum group Gauge theory on quantum spaces." Communications in Mathematical Physics 167, no. 1 (January 1995): 235. http://dx.doi.org/10.1007/bf02099359.

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8

Strickland, Elisabetta. "Classical Invariant Theory for the Quantum Symplectic Group." Advances in Mathematics 123, no. 1 (October 1996): 78–90. http://dx.doi.org/10.1006/aima.1996.0067.

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9

Robinson, Derek W. "Commutator Theory on Hilbert Space." Canadian Journal of Mathematics 39, no. 5 (October 1, 1987): 1235–80. http://dx.doi.org/10.4153/cjm-1987-063-2.

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Commutator theory has its origins in constructive quantum field theory. It was initially developed by Glirnm and Jaffe [7] as a method to establish self-adjointness of quantum fields and model Hamiltonians. But it has subsequently proved useful for a variety of other problems in field theory [17] [15] [8] [3], quantum mechanics [5], and Lie group theory [6]. Despite all these detailed applications no attempt appears to have been made to systematically develop the theory although reviews have been given in [22] and [9]. The primary aim of the present paper is to partially correct this situation. The secondary aim is to apply the theory to the analysis of first and second order partial differential operators associated with a Lie group.

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10

BONAHON, FRANCIS. "QUANTUM TEICHMÜLLER THEORY AND REPRESENTATIONS OF THE PURE BRAID GROUP." Communications in Contemporary Mathematics 10, supp01 (November 2008): 913–25. http://dx.doi.org/10.1142/s0219199708003095.

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11

HAYASHI, Takahiro. "Face algebras I---A generalization of quantum group theory." Journal of the Mathematical Society of Japan 50, no. 2 (April 1998): 293–315. http://dx.doi.org/10.2969/jmsj/05020293.

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12

Planat, Michel, and Philippe Jorrand. "Group theory for quantum gates and quantum coherence." Journal of Physics A: Mathematical and Theoretical 41, no. 18 (April 18, 2008): 182001. http://dx.doi.org/10.1088/1751-8113/41/18/182001.

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13

Roy, Sutanu, and Stanisław Lech Woronowicz. "Landstad–Vaes theory for locally compact quantum groups." International Journal of Mathematics 29, no. 04 (April 2018): 1850028. http://dx.doi.org/10.1142/s0129167x18500283.

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Landstad–Vaes theory deals with the structure of the crossed product of a [Formula: see text]-algebra by an action of locally compact (quantum) group. In particular, it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of [Formula: see text]-dynamical system introducing the concept of weak action of quantum groups on [Formula: see text]-algebras. It is still possible to define crossed product (by weak action) and characterize the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications.

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14

Jiang, Lining, Maozheng Guo, and Min Qian. "The duality theory of a finite dimensional discrete quantum group." Proceedings of the American Mathematical Society 132, no. 12 (July 14, 2004): 3537–47. http://dx.doi.org/10.1090/s0002-9939-04-07397-6.

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15

Brody, Dorje C., and Lane P. Hughston. "Theory of quantum space-time." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2061 (July 25, 2005): 2679–99. http://dx.doi.org/10.1098/rspa.2005.1457.

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A generalized equivalence principle is put forward according to which space-time symmetries and internal quantum symmetries are indistinguishable before symmetry breaking . Based on this principle, a higher-dimensional extension of Minkowski space is proposed and its properties examined. In this scheme the structure of space-time is intrinsically quantum mechanical. It is shown that the causal geometry of such a quantum space-time (QST) possesses a rich hierarchical structure. The natural extension of the Poincaré group to QST is investigated. In particular, we prove that the symmetry group of this space is generated in general by a system of irreducible Killing tensors. After the symmetries are broken, the points of the QST can be interpreted as space-time valued operators . The generic point of a QST in the broken symmetry phase then becomes a Minkowski space-time valued operator. Classical space-time emerges as a map from QST to Minkowski space. It is shown that the general such map satisfying appropriate causality-preserving conditions ensuring linearity and Poincaré invariance is necessarily a density matrix.

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16

Hanaki, Akihide, Masahiko Miyamoto, and Daisuke Tambara. "Quantum Galois theory for finite groups." Duke Mathematical Journal 97, no. 3 (April 1999): 541–44. http://dx.doi.org/10.1215/s0012-7094-99-09720-x.

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17

Senashov, V. I. "Applications of group theory in crystallography." IOP Conference Series: Materials Science and Engineering 1230, no. 1 (March 1, 2022): 012018. http://dx.doi.org/10.1088/1757-899x/1230/1/012018.

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Abstract Group theory is a powerful tool for studying symmetric physical systems. Such systems include, in particular, molecules and crystals with symmetry. Group theory serves to explain the most important characteristics of atomic spectra. Group theory is also applied to the problems of atomic and nuclear physics. This paper gives examples of the use of the apparatus of group theory in research on crystallography, quantum mechanics, elementary particle physics. In particular, in these studies matrix groups and representations of unitary groups are actively used. For such groups we give an overview of the results on their recognition by the spectrum (by the orders of the elements of the group). This direction has been intensively developed in recent years both in our country and abroad. Recognition of finite simple non-Abelian groups by spectrum has been studied for last thirty years in Yekaterinburg at the Institute of Mathematics and Mechanics of the Ural Division of the Russian Academy of Sciences, in Chelyabinsk Federal University and in the Novosibirsk Institute of Mathematics of Siberian Division of the Russian Academy of Sciences. Some simple non-Abelian groups are not recognizable by their spectra. We have proposed an approach for recognizing groups by the bottom layer. The bottom layer of a group is the set of its elements of prime orders. A group is called recognizable by the bottom layer under additional conditions if it is uniquely restored by the bottom layer under these conditions. The paper considers some examples of simple non-Abelian finite groups that are not recognizable by spectra. For these examples, simultaneous recognition by spectrum and by the bottom layer is proved.

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18

Bach, Volker, Jürg Fröhlich, and Israel Michael Sigal. "Renormalization Group Analysis of Spectral Problems in Quantum Field Theory." Advances in Mathematics 137, no. 2 (August 1998): 205–98. http://dx.doi.org/10.1006/aima.1998.1733.

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19

Majid, Shahn. "Transmutation theory and rank for quantum braided groups." Mathematical Proceedings of the Cambridge Philosophical Society 113, no. 1 (January 1993): 45–70. http://dx.doi.org/10.1017/s0305004100075769.

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AbstractLet f: H1 → H2be any pair of quasitriangular Hopf algebras over k with a Hopf algebra map f between them. We construct in this situation a quasitriangular Hopf algebra B(H1, f, H2) in the braided monoidal category of H1-modules. It consists in the same algebra as H2 with a modified comultiplication and has a quasitriangular structure given by the ratio of those of H1 and H2. This transmutation procedure trades a non-cocommutative Hopf algebra in the category of k-modules for a more cocommutative object in a more non-commutative category. As an application, every Hopf algebra containing the group algebra of ℤ2 becomes transmuted to a super-Hopf algebra.

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20

Freslon, Amaury, and Rubén Martos. "Torsion and K-theory for Some Free Wreath Products." International Mathematics Research Notices 2020, no. 6 (April 13, 2018): 1639–70. http://dx.doi.org/10.1093/imrn/rny071.

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Abstract We classify torsion actions of free wreath products of arbitrary compact quantum groups by $S_{N}^{+}$ and use this to prove that if $\mathbb{G}$ is a torsion-free compact quantum group satisfying the strong Baum–Connes property then $\mathbb{G}\wr _{\ast }S_{N}^{+}$ also satisfies the strong Baum–Connes property. We then compute the K-theory of free wreath products of classical and quantum free groups by $SO_{q}(3)$.

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21

Parshall, Brian, and Jian-Pan Wang. "Cohomology of Quantum Groups: the Quantum Dimension." Canadian Journal of Mathematics 45, no. 6 (December 18, 1993): 1276–98. http://dx.doi.org/10.4153/cjm-1993-072-4.

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AbstractThis paper uses the notion of the quantum dimension to obtain new results on the cohomology and representation theory of quantum groups at a root of unity. In particular, we consider the elementary theory of support varieties for quantum groups.

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22

Banica, Teodor. "Higher transitive quantum groups: theory and models." Colloquium Mathematicum 156, no. 1 (2019): 1–14. http://dx.doi.org/10.4064/cm7473-3-2018.

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23

Banica, Teodor, and Issan Patri. "Maximal torus theory for compact quantum groups." Illinois Journal of Mathematics 61, no. 1-2 (2017): 151–70. http://dx.doi.org/10.1215/ijm/1520046213.

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24

Vergnioux, Roland, and Christian Voigt. "The $$K$$ -theory of free quantum groups." Mathematische Annalen 357, no. 1 (February 12, 2013): 355–400. http://dx.doi.org/10.1007/s00208-013-0902-9.

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25

CRANE, LOUIS. "STRING FIELD THEORY FROM QUANTUM GRAVITY." Reviews in Mathematical Physics 25, no. 10 (November 2013): 1343005. http://dx.doi.org/10.1142/s0129055x13430058.

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Recent work on neutrino oscillations suggests that the three generations of fermions in the standard model are related by representations of the finite group A(4), the group of symmetries of the tetrahedron. Motivated by this, we explore models which extend the EPRL model for quantum gravity by coupling it to a bosonic quantum field of representations of A(4). This coupling is possible because the representation category of A(4) is a module category over the representation categories used to construct the EPRL model. The vertex operators which interchange vacua in the resulting quantum field theory reproduce the bosons and fermions of the standard model, up to issues of symmetry breaking which we do not resolve. We are led to the hypothesis that physical particles in nature represent vacuum changing operators on a sea of invisible excitations which are only observable in the A(4) representation labels which govern the horizontal symmetry revealed in neutrino oscillations. The quantum field theory of the A(4) representations is just the dual model on the extended lattice of the Lie group E6, as explained by the quantum McKay correspondence of Frenkel, Jing and Wang. The coupled model can be thought of as string field theory, but propagating on a discretized quantum spacetime rather than a classical manifold.

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26

Binz, Ernst, Sonja Pods, and Walter Schempp. "Heisenberg groups—A unifying structure of signal theory, holography and quantum information theory." Journal of Applied Mathematics and Computing 11, no. 1-2 (May 2003): 1–57. http://dx.doi.org/10.1007/bf02935722.

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27

DE SIMONE, E., and A. KUPIAINEN. "The KAM theorem and renormalization group." Ergodic Theory and Dynamical Systems 29, no. 2 (April 2009): 419–31. http://dx.doi.org/10.1017/s0143385708080450.

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AbstractWe give an elementary proof of the analytic KAM theorem by reducing it to a Picard iteration of a certain PDE with quadratic nonlinearity, the so-called Polchinski renormalization group equation studied in quantum field theory.

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28

Olshanetsky, M. A. "Quantum-mechanical calculations in the algebraic group theory." Communications in Mathematical Physics 132, no. 2 (September 1990): 441–59. http://dx.doi.org/10.1007/bf02096657.

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29

IP, IVAN C. H. "THE CLASSICAL LIMIT OF REPRESENTATION THEORY OF THE QUANTUM PLANE." International Journal of Mathematics 24, no. 04 (April 2013): 1350031. http://dx.doi.org/10.1142/s0129167x13500316.

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We showed that there is a complete analogue of a representation of the quantum plane [Formula: see text] where |q| = 1, with the classical ax+b group. We showed that the Fourier transform of the representation of [Formula: see text] on [Formula: see text] has a limit (in the dual corepresentation) toward the Mellin transform of the unitary representation of the ax+b group, and furthermore the intertwiners of the tensor products representation has a limit toward the intertwiners of the Mellin transform of the classical ax+b representation. We also wrote explicitly the multiplicative unitary defining the quantum ax+b semigroup and showed that it defines the corepresentation that is dual to the representation of [Formula: see text] above, and also correspond precisely to the classical family of unitary representation of the ax+b group.

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30

WORONOWICZ, S. L. "QUANTUM 'az + b' GROUP ON COMPLEX PLANE." International Journal of Mathematics 12, no. 04 (June 2001): 461–503. http://dx.doi.org/10.1142/s0129167x01000836.

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'az + b' is the group of affine transformations of complex plane [Formula: see text]. The coefficients a, [Formula: see text]. In quantum version a, b are normal operators such that ab = q2ba, where q is the deformation parameter. We shall assume that q is a root of unity, more precisely [Formula: see text], where N is an even natural number. To construct the group we write an explicit formula for the Kac Takesaki operator W. It is shown that W is a manageable multiplicative unitary in the sense of [3, 18]. Then using the general theory we construct a C *-algebra A and a comultiplication Δ ∈ Mor (A, A ⊗ A). A should be interpreted as the algebra of all continuous functions vanishing at infinity on quantum 'az + b'-group. The group structure is encoded by Δ. The existence of coinverse also follows from the general theory [18]. In the appendix, we briefly discuss the case of real q.

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31

Scholz, Erhard. "Introducing groups into quantum theory (1926–1930)." Historia Mathematica 33, no. 4 (November 2006): 440–90. http://dx.doi.org/10.1016/j.hm.2005.11.007.

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32

Wu, A. C. T. "Hidden quantum group structure in Chern-Simons theory." Journal of Physics A: Mathematical and General 26, no. 18 (September 21, 1993): L941—L944. http://dx.doi.org/10.1088/0305-4470/26/18/010.

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33

WORONOWICZ, S. L. "FROM MULTIPLICATIVE UNITARIES TO QUANTUM GROUPS." International Journal of Mathematics 07, no. 01 (February 1996): 129–49. http://dx.doi.org/10.1142/s0129167x96000086.

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An alternative version of the theory of multiplicative unitaries is presented. Instead of the original regularity condition of Baaj and Skandalis we formulate another condition selecting manageable multiplicative unitaries. The manageability is the property of multiplicative unitaries coming from the quantum group theory. For manageable multiplicative unitaries we reproduce all the essential results of the original paper of Baaj and Skandalis and much more. In particular the existence of the antipode and its polar decomposition is shown.

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34

Fan, Zhaobing, Haitao Ma, and Husileng Xiao. "Equivariant K-Theory Approach to $\imath$-Quantum Groups." Publications of the Research Institute for Mathematical Sciences 58, no. 3 (July 26, 2022): 635–68. http://dx.doi.org/10.4171/prims/58-3-6.

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35

Demidov, E. E. "Some aspects of the theory of quantum groups." Russian Mathematical Surveys 48, no. 6 (December 31, 1993): 41–79. http://dx.doi.org/10.1070/rm1993v048n06abeh001091.

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36

Yettera, David N. "Quantum groups and representations of monoidal categories." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 2 (September 1990): 261–90. http://dx.doi.org/10.1017/s0305004100069139.

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This paper is intended to make explicit some aspects of the interactions which have recently come to light between the theory of classical knots and links, the theory of monoidal categories, Hopf-algebra theory, quantum integrable systems, the theory of exactly solvable models in statistical mechanics, and quantum field theories. The main results herein show an intimate relation between representations of certain monoidal categories arising from the study of new knot invariants or from physical considerations and quantum groups (that is, Hopf algebras). In particular categories of modules and comodules over Hopf algebras would seem to be much more fundamental examples of monoidal categories than might at first be apparent. This fundamental role of Hopf algebras in monoidal categories theory is also manifest in the Tannaka duality theory of Deligne and Mime [8a], although the relationship of that result and the present work is less clear than might be hoped.

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37

FRÖHLICH, J., and F. GABBIANI. "BRAID STATISTICS IN LOCAL QUANTUM THEORY." Reviews in Mathematical Physics 02, no. 03 (January 1990): 251–353. http://dx.doi.org/10.1142/s0129055x90000107.

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We present details of a mathematical theory of superselection sectors and their statistics in local quantum theory over (two- and) three-dimensional space-time. The framework for our analysis is algebraic quantum field theory. Statistics of superselection sectors in three-dimensional local quantum theory with charges not localizable in bounded space-time regions and in two-dimensional chiral theories is described in terms of unitary representations of the braid groups generated by certain Yang-Baxter matrices. We describe the beginnings of a systematic classification of those representations. Our analysis makes contact with the classification theory of subfactors initiated by Jones. We prove a general theorem on the connection between spin and statistics in theories with braid statistics. We also show that every theory with braid statistics gives rise to a “Verlinde algebra”. It determines a projective representation of SL(2, ℤ) and, presumably, of the mapping class group of any Riemann surface, even if the theory does not display conformal symmetry.

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38

Kulish, P. P. "Twists of quantum groups and noncommutative field theory." Journal of Mathematical Sciences 143, no. 1 (May 2007): 2806–15. http://dx.doi.org/10.1007/s10958-007-0166-6.

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39

Lining, Jiang. "Towards a quantum Galois theory for quantum double algebras of finite groups." Proceedings of the American Mathematical Society 138, no. 08 (August 1, 2010): 2793. http://dx.doi.org/10.1090/s0002-9939-10-10315-3.

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40

NICOLAIDIS, A. "CATEGORICAL FOUNDATION OF QUANTUM MECHANICS AND STRING THEORY." International Journal of Modern Physics A 24, no. 06 (March 10, 2009): 1175–83. http://dx.doi.org/10.1142/s0217751x09043079.

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The unification of quantum mechanics and general relativity remains the primary goal of theoretical physics, with string theory appearing as the only plausible unifying scheme. In the present work, in a search of the conceptual foundations of string theory, we analyze the relational logic developed by C. S. Peirce in the late 19th century. The Peircean logic has the mathematical structure of a category with the relation Rij among two individual terms Si and Sj, serving as an arrow (or morphism). We introduce a realization of the corresponding categorical algebra of compositions, which naturally gives rise to the fundamental quantum laws, thus indicating category theory as the foundation of quantum mechanics. The same relational algebra generates a number of group structures, among them W∞. The group W∞ is embodied and realized by the matrix models, themselves closely linked with string theory. It is suggested that relational logic and in general category theory may provide a new paradigm, within which to develop modern physical theories.

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41

Baaj, Saad, and Stefaan Vaes. "DOUBLE CROSSED PRODUCTS OF LOCALLY COMPACT QUANTUM GROUPS." Journal of the Institute of Mathematics of Jussieu 4, no. 1 (January 2005): 135–73. http://dx.doi.org/10.1017/s1474748005000034.

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For a matched pair of locally compact quantum groups, we construct the double crossed product as a locally compact quantum group. This construction generalizes Drinfeld’s quantum double construction. We study the modular theory and the $\mathrm{C}^*$-algebraic properties of these double crossed products, as well as several links between double crossed products and bicrossed products. In an appendix, we study the Radon–Nikodym derivative of a weight under a quantum group action (following Yamanouchi) and obtain, as a corollary, a new characterization of closed quantum subgroups. AMS 2000 Mathematics subject classification: Primary 46L89. Secondary 46L65

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42

Goldfain, Ervin. "Renormalization group and the emergence of random fractal topology in quantum field theory." Chaos, Solitons & Fractals 19, no. 5 (March 2004): 1023–30. http://dx.doi.org/10.1016/s0960-0779(03)00304-7.

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43

Goodearl, Kenneth R., Thomas H. Lenagan, and Laurent Rigal. "The first fundamental theorem of coinvariant theory for the quantum general linear group." Publications of the Research Institute for Mathematical Sciences 36, no. 2 (2000): 269–96. http://dx.doi.org/10.2977/prims/1195143104.

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44

Lambert-Mogiliansky, Ariane, and Adrian Calmettes. "Phishing for (Quantum-Like) Phools—Theory and Experimental Evidence." Symmetry 13, no. 2 (January 21, 2021): 162. http://dx.doi.org/10.3390/sym13020162.

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Quantum-like decision theory is by now a theoretically well-developed field (see e.g., Danilov, Lambert-Mogiliansky & Vergopoulos, 2018). We provide a first test of the predictions of an application of this approach to persuasion. One remarkable result entails that, in contrast to Bayesian persuasion, distraction rather than relevant information has a powerful potential to influence decision-making. We first develop a quantum decision model of choice between two uncertain alternatives. We derive the impact of persuasion by means of distractive questions and contrast them with the predictions of the Bayesian model. Next, we provide the results from a first test of the theory. We conducted an experiment where respondents choose between supporting either one of two projects to save endangered species. We tested the impact of persuasion in the form of questions related to different aspects of the uncertain value of the two projects. The experiment involved 1253 respondents divided into three groups: a control group, a first treatment group and the distraction treatment group. Our main result is that, in accordance with the predictions of quantum persuasion but in violation with the Bayesian model, distraction significantly affects decision-making. Population variables play no role. Some significant variations between subgroups are exhibited and discussed. The results of the experiment provide support for the hypothesis that the manipulability of people’s decision-making can to some extent be explained by the quantum indeterminacy of their subjective representation of reality.

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45

Kaushal Rana. "Homological Algebra and Its Application: A Descriptive Study." Integrated Journal for Research in Arts and Humanities 2, no. 1 (January 31, 2022): 29–35. http://dx.doi.org/10.55544/ijrah.2.1.47.

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Algebra has been used to define and answer issues in almost every field of mathematics, science, and engineering. Homological algebra depends largely on computable algebraic invariants to categorise diverse mathematical structures, such as topological, geometrical, arithmetical, and algebraic (up to certain equivalences). String theory and quantum theory, in particular, have shown it to be of crucial importance in addressing difficult physics questions. Geometric, topological and algebraic algebraic techniques to the study of homology are to be introduced in this research. Homology theory in abelian categories and a category theory are covered. the n-fold extension functors EXTn (-,-) , the torsion functors TORn (-,-), Algebraic geometry, derived functor theory, simplicial and singular homology theory, group co-homology theory, the sheaf theory, the sheaf co-homology, and the l-adic co-homology, as well as a demonstration of its applicability in representation theory.

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46

Gould, M. D. "Quantum double finite group algebras and their representations." Bulletin of the Australian Mathematical Society 48, no. 2 (October 1993): 275–301. http://dx.doi.org/10.1017/s0004972700015707.

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The quantum double construction is applied to the group algebra of a finite group. Such algebras are shown to be semi-simple and a complete theory of characters is developed. The irreducible matrix representations are classified and applied to the explicit construction of R-matrices: this affords solutions to the Yang-Baxter equation associated with certain induced representations of a finite group. These results are applied in the second paper of the series to construct unitary representations of the Braid group and corresponding link polynomials.

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47

D'Ariano, Giacomo Mauro, and Andrei Khrennikov. "Preface of the special issue quantum foundations: information approach." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374, no. 2068 (May 28, 2016): 20150244. http://dx.doi.org/10.1098/rsta.2015.0244.

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This special issue is based on the contributions of a group of top experts in quantum foundations and quantum information and probability. It enlightens a number of interpretational, mathematical and experimental problems of quantum theory.

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Garnerone, S., A. Marzuoli, and M. Rasetti. "Quantum automata, braid group and link polynomials." Quantum Information and Computation 7, no. 5&6 (July 2007): 479–503. http://dx.doi.org/10.26421/qic7.5-6-5.

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The spin--network quantum simulator model, which essentially encodes the (quantum deformed) $SU(2)$ Racah--Wigner tensor algebra, is particularly suitable to address problems arising in low dimensional topology and group theory. In this combinatorial framework we implement families of finite--states and discrete--time quantum automata capable of accepting the language generated by the braid group, and whose transition amplitudes are colored Jones polynomials. The automaton calculation of the polynomial of (the plat closure of) a link $L$ on $2N$ strands at any fixed root of unity is shown to be bounded from above by a linear function of the number of crossings of the link, on the one hand, and polynomially bounded in terms of the braid index $2N$, on the other. The growth rate of the time complexity function in terms of the integer $k$ appearing in the root of unity $q$ can be estimated to be (polynomially) bounded by resorting to the field theoretical background given by the Chern--Simons theory.

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Wocjan, P., and J. Yard. "The Jones polynomial: quantum algorithms and applications in quantum complexity theory." Quantum Information and Computation 8, no. 1&2 (January 2008): 147–80. http://dx.doi.org/10.26421/qic8.1-2-10.

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We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al.\ that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a \#P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

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Aniello, Paolo. "Quantum Stochastic Products and the Quantum Convolution." Geometry, Integrability and Quantization 22 (2021): 64–77. http://dx.doi.org/10.7546/giq-22-2021-64-77.

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A quantum stochastic product is a binary operation on the space of quantum states preserving the convex structure. We describe a class of associative stochastic products, the twirled products, that have interesting connections with quantum measurement theory. Constructing such a product involves a square integrable group representation, a probability measure and a fiducial state. By extending a twirled product to the full space of trace class operators, one obtains a Banach algebra. This algebra is commutative if the underlying group is abelian. In the case of the group of translations on phase space, one gets a quantum convolution algebra, a quantum counterpart of the classical phase-space convolution algebra. The peculiar role of the fiducial state characterizing each quantum convolution product is highlighted.

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