Journal articles: 'Algebraic quantum theory'

1

Ellis, Eugenia. "Algebraic quantum kk-theory." Communications in Algebra 46, no. 8 (February 8, 2018): 3642–62. http://dx.doi.org/10.1080/00927872.2018.1424877.

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KITAJIMA, Yuichiro. "Contextuality in Algebraic Quantum Theory." Journal of the Japan Association for Philosophy of Science 45, no. 1-2 (2018): 23–34. http://dx.doi.org/10.4288/kisoron.45.1-2_23.

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Petz, Dénes. "Algebraic complementarity in quantum theory." Journal of Mathematical Physics 51, no. 1 (January 2010): 015215. http://dx.doi.org/10.1063/1.3276681.

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Kravzova, G. A. "Algebraic quantum theory with maximal frequency." Theoretical and Mathematical Physics 208, no. 2 (August 2021): 1051–60. http://dx.doi.org/10.1134/s0040577921080055.

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Heunen, Chris, Nicolaas P. Landsman, and Bas Spitters. "A Topos for Algebraic Quantum Theory." Communications in Mathematical Physics 291, no. 1 (July 21, 2009): 63–110. http://dx.doi.org/10.1007/s00220-009-0865-6.

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6

Green, H. S. "Quantum Theory of Gravitation." Australian Journal of Physics 51, no. 3 (1998): 459. http://dx.doi.org/10.1071/p97084.

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It is possible to construct the non-euclidean geometry of space-time from the information carried by neutral particles. Points are identified with the quantal events in which photons or neutrinos are created and annihilated, and represented by the relativistic density matrices of particles immediately after creation or before annihilation. From these, matrices representing subspaces in any number of dimensions are constructed, and the metric and curvature tensors are derived by an elementary algebraic method; these are similar in all respects to those of Riemannian geometry. The algebraic method is extended to obtain solutions of Einstein’s gravitational field equations for empty space, with a cosmological term. General relativity and quantum theory are unified by the quantal embedding of non-euclidean space-time, and the derivation of a generalisation, consistent with Einstein"s equations, of the special relativistic wave equations of particles of any spin within representations of SO(3) ⊗ SO(4; 2). There are some novel results concerning the dependence of the scale of space-time on properties of the particles by means of which it is observed, and the gauge groups associated with gravitation.

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Timmermann, Thomas. "Integration on algebraic quantum groupoids." International Journal of Mathematics 27, no. 02 (February 2016): 1650014. http://dx.doi.org/10.1142/s0129167x16500142.

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In this paper, we develop a theory of integration on algebraic quantum groupoids in the form of regular multiplier Hopf algebroids, and establish the main properties of integrals obtained by Van Daele for algebraic quantum groups before — faithfulness, uniqueness up to scaling, existence of a modular element and existence of a modular automorphism — for algebraic quantum groupoids under reasonable assumptions. The approach to integration developed in this paper forms the basis for the extension of Pontrjagin duality to algebraic quantum groupoids, and for the passage from algebraic quantum groupoids to operator-algebraic completions, which both will be studied in separate papers.

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BENINI, MARCO, CLAUDIO DAPPIAGGI, and THOMAS-PAUL HACK. "QUANTUM FIELD THEORY ON CURVED BACKGROUNDS — A PRIMER." International Journal of Modern Physics A 28, no. 17 (July 10, 2013): 1330023. http://dx.doi.org/10.1142/s0217751x13300238.

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Goal of this paper is to introduce the algebraic approach to quantum field theory on curved backgrounds. Based on a set of axioms, first written down by Haag and Kastler, this method consists of a two-step procedure. In the first one, it is assigned to a physical system a suitable algebra of observables, which is meant to encode all algebraic relations among observables, such as commutation relations. In the second step, one must select an algebraic state in order to recover the standard Hilbert space interpretation of a quantum system. As quantum field theories possess infinitely many degrees of freedom, many unitarily inequivalent Hilbert space representations exist and the power of such approach is the ability to treat them all in a coherent manner. We will discuss in detail the algebraic approach for free fields in order to give the reader all necessary information to deal with the recent literature, which focuses on the applications to specific problems, mostly in cosmology.

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9

Plotnitsky, Arkady. "“The Heisenberg Method”: Geometry, Algebra, and Probability in Quantum Theory." Entropy 20, no. 9 (August 30, 2018): 656. http://dx.doi.org/10.3390/e20090656.

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The article reconsiders quantum theory in terms of the following principle, which can be symbolically represented as QUANTUMNESS → PROBABILITY → ALGEBRA and will be referred to as the QPA principle. The principle states that the quantumness of physical phenomena, that is, the specific character of physical phenomena known as quantum, implies that our predictions concerning them are irreducibly probabilistic, even in dealing with quantum phenomena resulting from the elementary individual quantum behavior (such as that of elementary particles), which in turn implies that our theories concerning these phenomena are fundamentally algebraic, in contrast to more geometrical classical or relativistic theories, although these theories, too, have an algebraic component to them. It follows that one needs to find an algebraic scheme able make these predictions in a given quantum regime. Heisenberg was first to accomplish this in the case of quantum mechanics, as matrix mechanics, whose matrix character testified to his algebraic method, as Einstein characterized it. The article explores the implications of the Heisenberg method and of the QPA principle for quantum theory, and for the relationships between mathematics and physics there, from a nonrealist or, in terms of this article, “reality-without-realism” or RWR perspective, defining the RWR principle, thus joined to the QPA principle.

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10

Schwarz, A. "Geometric and algebraic approaches to quantum theory." Nuclear Physics B 973 (December 2021): 115601. http://dx.doi.org/10.1016/j.nuclphysb.2021.115601.

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11

Earman, John, and Giovanni Valente. "Relativistic Causality in Algebraic Quantum Field Theory." International Studies in the Philosophy of Science 28, no. 1 (January 2, 2014): 1–48. http://dx.doi.org/10.1080/02698595.2014.915652.

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12

Kronz, Frederick M., and Tracy A. Lupher. "Unitarily Inequivalent Representations in Algebraic Quantum Theory." International Journal of Theoretical Physics 44, no. 8 (August 2005): 1239–58. http://dx.doi.org/10.1007/s10773-005-4683-0.

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13

Verch, Rainer. "Wavefront Sets in Algebraic Quantum Field Theory." Communications in Mathematical Physics 205, no. 2 (August 1, 1999): 337–67. http://dx.doi.org/10.1007/s002200050680.

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14

Benini, Marco, Alexander Schenkel, and Lukas Woike. "Homotopy theory of algebraic quantum field theories." Letters in Mathematical Physics 109, no. 7 (January 16, 2019): 1487–532. http://dx.doi.org/10.1007/s11005-018-01151-x.

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15

Kitajima, Yuichiro. "Imperfect Cloning Operations in Algebraic Quantum Theory." Foundations of Physics 45, no. 1 (October 16, 2014): 62–74. http://dx.doi.org/10.1007/s10701-014-9843-8.

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16

Witten, Edward. "Quantum field theory, Grassmannians, and algebraic curves." Communications in Mathematical Physics 113, no. 4 (December 1988): 529–600. http://dx.doi.org/10.1007/bf01223238.

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17

Benini, Marco, and Alexander Schenkel. "Higher Structures in Algebraic Quantum Field Theory." Fortschritte der Physik 67, no. 8-9 (May 6, 2019): 1910015. http://dx.doi.org/10.1002/prop.201910015.

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18

RECKNAGEL, ANDREAS. "FUSION RULES FROM ALGEBRAIC K-THEORY." International Journal of Modern Physics A 08, no. 07 (March 20, 1993): 1345–57. http://dx.doi.org/10.1142/s0217751x93000552.

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19

Niestegge, Gerd. "Quantum Probability’s Algebraic Origin." Entropy 22, no. 11 (October 23, 2020): 1196. http://dx.doi.org/10.3390/e22111196.

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Max Born’s statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. Although the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further physically meaningful and experimentally verifiable novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg’s and others’ uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.

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20

Varlamov, V. V. "Algebraic quantum mechanics: I. Basic definitions." Mathematical Structures and Modeling, no. 2 (54) (October 5, 2020): 4–23. http://dx.doi.org/10.24147/2222-8772.2020.2.4-23.

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21

Bédos, Erik, Gerard J. Murphy, and Lars Tuset. "Amenability and coamenability of algebraic quantum groups." International Journal of Mathematics and Mathematical Sciences 31, no. 10 (2002): 577–601. http://dx.doi.org/10.1155/s016117120210603x.

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We define concepts of amenability and coamenability for algebraic quantum groups in the sense of Van Daele (1998). We show that coamenability of an algebraic quantum group always implies amenability of its dual. Various necessary and/or sufficient conditions for amenability or coamenability are obtained. Coamenability is shown to have interesting consequences for the modular theory in the case that the algebraic quantum group is of compact type.

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22

Govindarajan, T. R. "Algebraic formulation of quantum theory, particle identity and entanglement." International Journal of Quantum Information 14, no. 06 (September 2016): 1640026. http://dx.doi.org/10.1142/s0219749916400268.

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Quantum theory as formulated in conventional framework using statevectors in Hilbert spaces misses the statistical nature of the underlying quantum physics. Formulation using operators [Formula: see text] algebra and density matrices appropriately captures this feature in addition leading to the correct formulation of particle identity. In this framework, Hilbert space is an emergent concept. Problems related to anomalies and quantum epistemology are discussed.

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23

Im, Bokhee, Alex W. Nowak, and Jonathan D. H. Smith. "Algebraic properties of quantum quasigroups." Journal of Pure and Applied Algebra 225, no. 3 (March 2021): 106539. http://dx.doi.org/10.1016/j.jpaa.2020.106539.

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24

SUN, C. P., X. F. LIU, and S. X. YU. "ALGEBRAIC CONSTRUCTION OF 't HOOFT'S QUANTUM EQUIVALENCE CLASSES." Modern Physics Letters A 16, no. 02 (January 20, 2001): 75–82. http://dx.doi.org/10.1142/s0217732301002869.

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Most recently 't Hooft has postulated (G. 't Hooft, Class. Quantum Grav.16, 3263 (1999)) that quantum states at the "atomic scale" can be understood as equivalence classes of primordial states governed by a dissipative deterministic theory underlying quantum theory at the "Planck scale". Defining invariant subspaces clearly for primordial states according to a given evolution, we mathematically reformulate 't Hooft's theory as a quotient space construction with the time-reversible evolution operator induced naturally. With this observation and some analysis, 't Hooft's theory is generalized beyond his case where the evolution at the "Planck scale" is a power of a one-time-step evolution or the time is discrete.

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25

Dütsch, M., and K. Fredenhagen. "Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion." Communications in Mathematical Physics 219, no. 1 (May 2001): 5–30. http://dx.doi.org/10.1007/pl00005563.

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26

KITAJIMA, Yuichiro. "Causation and Intervention in Algebraic Quantum Field Theory." Journal of the Japan Association for Philosophy of Science 38, no. 1 (2010): 9–15. http://dx.doi.org/10.4288/kisoron.38.1_9.

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27

Giesel, K., and T. Thiemann. "Algebraic quantum gravity (AQG): III. Semiclassical perturbation theory." Classical and Quantum Gravity 24, no. 10 (April 30, 2007): 2565–88. http://dx.doi.org/10.1088/0264-9381/24/10/005.

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28

Baez, J. C., I. E. Segal, Z. Zhou, and Mark A. Kon. "Introduction to Algebraic and Constructive Quantum Field Theory." Physics Today 46, no. 12 (December 1993): 43. http://dx.doi.org/10.1063/1.2809125.

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29

Arickx, F., J. Broeckhove, P. Van Leuven, V. Vasilevsky, and G. Filippov. "Algebraic method for the quantum theory of scattering." American Journal of Physics 62, no. 4 (April 1994): 362–70. http://dx.doi.org/10.1119/1.17579.

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30

Clifton, Rob. "The modal interpretation of algebraic quantum field theory." Physics Letters A 271, no. 3 (June 2000): 167–77. http://dx.doi.org/10.1016/s0375-9601(00)00364-9.

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31

Longo, Roberto, and Edward Witten. "An Algebraic Construction of Boundary Quantum Field Theory." Communications in Mathematical Physics 303, no. 1 (September 21, 2010): 213–32. http://dx.doi.org/10.1007/s00220-010-1133-5.

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32

Hawkins, Eli. "A Cohomological Perspective on Algebraic Quantum Field Theory." Communications in Mathematical Physics 360, no. 1 (February 15, 2018): 439–79. http://dx.doi.org/10.1007/s00220-018-3098-8.

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33

Oneda, Sadao, and Kunihiko Terasaki. "Algebraic Approach in Quantum Chromodynamics and Electroweak Theory." Progress of Theoretical Physics Supplement 82 (1985): 1–164. http://dx.doi.org/10.1143/ptps.82.1.

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34

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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35

Gerisch, Thomas, Reinhard Honegger, and Alfred Rieckers. "Algebraic Quantum Theory of the Josephson Microwave Radiator." Annales Henri Poincaré 4, no. 6 (December 2003): 1051–82. http://dx.doi.org/10.1007/s00023-003-0155-3.

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36

Schwarz, Albert. "Scattering in Geometric Approach to Quantum Theory." Universe 8, no. 12 (December 16, 2022): 663. http://dx.doi.org/10.3390/universe8120663.

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We define inclusive scattering matrix in the framework of a geometric approach to quantum field theory. We review the definitions of scattering theory in the algebraic approach and relate them to the definitions in the geometric approach.

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37

DELVAUX, L., A. VAN DAELE, and S. H. WANG. "BICROSSPRODUCTS OF ALGEBRAIC QUANTUM GROUPS." International Journal of Mathematics 24, no. 01 (January 2013): 1250131. http://dx.doi.org/10.1142/s0129167x12501315.

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Let A and B be two algebraic quantum groups. Assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. If the action and coaction are matched, it is possible to define a coproduct Δ# on the smash product A#B making the pair (A#B, Δ#) into an algebraic quantum group. In this paper we study the various data of the bicrossproduct A#B, such as the modular automorphisms, the modular elements, … and we obtain formulas in terms of the data of the components A and B. Secondly, we look at the dual of A#B (in the sense of algebraic quantum groups) and we show it is itself a bicrossproduct (of the second type) of the duals [Formula: see text] and [Formula: see text]. We give some examples that are typical for algebraic quantum groups. In particular, we focus on the extra structure, provided by the integrals and associated objects. It should be mentioned that with examples of bicrossproducts of algebraic quantum groups, we do get examples that are essentially different from those commonly known in Hopf algebra theory.

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38

Delvaux, L., A. Van Daele, and Shuanhong Wang. "A Note on the Antipode for Algebraic Quantum Groups." Canadian Mathematical Bulletin 55, no. 2 (June 1, 2012): 260–70. http://dx.doi.org/10.4153/cmb-2011-079-4.

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AbstractRecently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra.In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals.We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

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39

Podleś, P., and E. Müller. "Introduction to Quantum Groups." Reviews in Mathematical Physics 10, no. 04 (May 1998): 511–51. http://dx.doi.org/10.1142/s0129055x98000173.

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We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra theory, theory of compact (matrix) quantum groups and the theory of their actions on compact quantum spaces. We also provide the most important examples, including the classification of quantum SL(2)-groups, their real forms and quantum spheres. We also consider quantum SLq(N)-groups and quantum Lorentz groups.

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40

Derksen, Harm, Emmanuel Jeandel, and Pascal Koiran. "Quantum automata and algebraic groups." Journal of Symbolic Computation 39, no. 3-4 (March 2005): 357–71. http://dx.doi.org/10.1016/j.jsc.2004.11.008.

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41

HOLLANDS, STEFAN. "ALGEBRAIC APPROACH TO THE 1/N EXPANSION IN QUANTUM FIELD THEORY." Reviews in Mathematical Physics 16, no. 04 (May 2004): 509–58. http://dx.doi.org/10.1142/s0129055x04002072.

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The 1/N expansion in quantum field theory is formulated within an algebraic framework. For a scalar field taking values in the N by N hermitian matrices, we rigorously construct the gauge invariant interacting quantum field operators in the sense of power series in 1/N and the 't Hooft coupling parameter as members of an abstract *-algebra. The key advantages of our algebraic formulation over the usual formulation of the 1/N expansion in terms of Green's functions are (i) that it is completely local so that infrared divergencies in massless theories are avoided on the algebraic level and (ii) that it admits a generalization to quantum field theories on globally hypberbolic Lorentzian curved spacetimes. We expect that our constructions are also applicable in models possessing local gauge invariance such as Yang–Mills theories. The 1/N expansion of the renormalization group flow is constructed on the algebraic level via a family of *-isomorphisms between the algebras of interacting field observables corresponding to different scales. We also consider k-parameter deformations of the interacting field algebras that arise from reducing the symmetry group of the model to a diagonal subgroup with k factors. These parameters smoothly interpolate between situations of different symmetry.

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42

ADLER, C. "BRAID GROUP STATISTICS IN TWO-DIMENSIONAL QUANTUM FIELD THEORY." Reviews in Mathematical Physics 08, no. 07 (October 1996): 907–24. http://dx.doi.org/10.1142/s0129055x96000329.

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Within the framework of algebraic quantum field theory, we construct explicitly localized morphisms of a Haag-Kastler net in 1+1-dimensional Minkowski space showing abelian braid group statistics. Moreover, we investigate the scattering theory of the corresponding quantum fields.

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43

Ambainis, Andris, Martin Beaudry, Marats Golovkins, Arnolds Kikusts, Mark Mercer, and Denis Therien. "Algebraic Results on Quantum Automata." Theory of Computing Systems 39, no. 1 (November 29, 2005): 165–88. http://dx.doi.org/10.1007/s00224-005-1263-x.

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44

BUCHHOLZ, D., and C. D’ANTONI. "PHASE SPACE PROPERTIES OF CHARGED FIELDS IN THEORIES OF LOCAL OBSERVABLES." Reviews in Mathematical Physics 07, no. 04 (May 1995): 527–57. http://dx.doi.org/10.1142/s0129055x95000219.

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Within the setting of algebraic quantum field theory a relation between phase-space properties of observables and charged fields is established. These properties are expressed in terms of compactness and nuclearity conditions which are the basis for the characterization of theories with physically reasonable causal and thermal features. Relevant concepts and results of phase space analysis in algebraic quantum field theory are reviewed and the underlying ideas are outlined.

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45

De Commer, Kenny. "Galois objects for algebraic quantum groups." Journal of Algebra 321, no. 6 (March 2009): 1746–85. http://dx.doi.org/10.1016/j.jalgebra.2008.11.039.

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46

Jaffe, Arthur, Chunlan Jiang, Zhengwei Liu, Yunxiang Ren, and Jinsong Wu. "Quantum Fourier analysis." Proceedings of the National Academy of Sciences 117, no. 20 (April 30, 2020): 10715–20. http://dx.doi.org/10.1073/pnas.2002813117.

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Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform F, as a map between suitably defined Lp spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems.

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47

BELAVKIN, V. P., and X. DAI. "AN OPERATIONAL ALGEBRAIC APPROACH TO QUANTUM CHANNEL CAPACITY." International Journal of Quantum Information 06, no. 05 (October 2008): 981–96. http://dx.doi.org/10.1142/s0219749908004225.

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An elementary introduction into algebraic approach to unified quantum information theory and operational approach to quantum entanglement as generalized encoding is given. After introducing compound quantum state and two types of informational divergences, namely, Araki–Umegaki (a-type) and of Belavkin–Staszewski (b-type) quantum relative entropic information, this paper treats two types of quantum mutual information via entanglement and defines two types of corresponding quantum channel capacities as the supremum via the generalized encodings. It proves the additivity property of quantum channel capacities via entanglement, which extends the earlier results of Belavkin to products of arbitrary quantum channels for quantum relative entropy of any type.

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48

Conte, Elio, and Rich Norman. "Synopsis and Review: Algebraic Quantum Theory of Consciousness by Elio Conte." JOURNAL OF ADVANCES IN PHYSICS 16, no. 1 (March 8, 2019): 19–20. http://dx.doi.org/10.24297/jap.v16i1.8110.

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Algebraic Quantum Theory of Consciousness A Solution to the Problem of Quantum Collapse in Systems Having Three Anticommuting Elements ISBN 978-88-255-1707-1, formato 17 x 24 cm, 112 pagine, 14 euro

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49

Stumpf, H., B. Fauser, and W. Pfister. "Composite Particle Theory in Quantum Electrodynamics." Zeitschrift für Naturforschung A 48, no. 7 (July 1, 1993): 765–76. http://dx.doi.org/10.1515/zna-1993-0705.

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Abstract Without use of pathintegral formalism a composite particle effective dynamics is developed for spinor quantum electrodynamics. By algebraic evaluation of spinor quantum electrodynamics in Coulomb gauge a corresponding functional equation is derived. The commutation rules for the transversal electromagnetic field can be deduced as a consequence of this formalism. By application of weak mapping theorems the QED functional equation can be mapped onto a functional equation for composite particles with mutual interaction and interaction with the electromagnetic field. The formalism is demonstrated for positronium states. The incorporation of renormalization into this scheme is verified.

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50

HERDEGEN, ANDRZEJ. "QUANTUM BACKREACTION (CASIMIR) EFFECT WITHOUT INFINITIES: ALGEBRAIC ANALYSIS." International Journal of Modern Physics: Conference Series 14 (January 2012): 376–82. http://dx.doi.org/10.1142/s2010194512007490.

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Casimir effect, in most general terms, is the backreaction of a quantum system responding to an adiabatic change of external conditions. This backreaction is expected to be quantitatively measured by a change in the expectation value of a certain energy observable of the system. However, for this concept to be applicable, the system has to retain its identity in the process. Most prevailing tendencies in the analysis of the effect seem to ignore this question. In general, a quantum theory is defined by an algebra of observables, whose representations by operators in a Hilbert space define concrete physical systems described by the theory. A quantum system retains its identity if both the algebra as well as its representation do not change. We discuss the resulting restrictions for admissible models of changing external conditions. These ideas are applied to quantum field models. No infinities arise, if the algebraic demands are respected.

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Journal articles on the topic 'Algebraic quantum theory'

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