Schwinger’s picture of quantum mechanics. II: Algebras and observables. (English) Zbl 07804178
Summary: The kinematical foundations of Schwinger’s algebra of selective measurements were discussed in [the authors, Int. J. Geom. Methods Mod. Phys. 16, No. 8, Article ID 1950119, 31 p. (2019; Zbl 1421.81049)] and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analyzed. For that, the algebra generated by the observables, as well as the notion of state, are discussed, and the structure of the transition functions, that plays an instrumental role in Schwinger’s picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers a simple way to discuss the quantum-to-classical transition. Some basic examples, the qubit and the harmonic oscillator, are examined, and the relation with the standard Dirac-Schrödinger and Born-Jordan-Heisenberg pictures is discussed.
For Part III, see [the authors, Int. J. Geom. Methods Mod. Phys. 16, No. 11, Article ID 1950165, 37 p. (2019)].
For Part III, see [the authors, Int. J. Geom. Methods Mod. Phys. 16, No. 11, Article ID 1950165, 37 p. (2019)].
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 53C35 | Differential geometry of symmetric spaces |
| 57S20 | Noncompact Lie groups of transformations |
Keywords:
Schwinger algebra; groupoids; groupoid algebra; \(C^\ast\)-algebra; observables; states; Hamiltonian; dynamical flowCitations:
Zbl 1421.81049References:
| [1] | F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics I: Groupoids, to appear in IJGMMP · Zbl 1421.81049 |
| [2] | Schwinger, J.The algebra of microscopic measurementProc. Nat. Acad. Sci. USA4510195915421553 |
| [3] | J. Schwinger, Quantum Kinematics and DynamicsQuantum Kinematics and Dynamicsibid.Quantum Mechanics Symbolism of Atomic Measurements |
| [4] | J. Schwinger, The theory of quantized fields I, Phys. Rev.82Phys. Rev.91Phys. Rev.91Phys. Rev.92Phys. Rev.93Phys. Rev.94 |
| [5] | F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics III: The statistical interpretation, Preprint (2019) |
| [6] | Penrose, R.Angular momentum: An approach to combinatorial space-timeQuantum Theory and BeyondBastin, T.Cambridge University PressCambridge1971151180 |
| [7] | Rovelli, C.Smolin, L.L. Spin networks and quantum gravityPhys. Rev. D521019955743 |
| [8] | Weizsäcker, C. F. v.Die Quantentheorie der einfachen Alternative (Komplementarität und Logik II)Z. Naturforsch.131958245253 |
| [9] | Lyre, H.von Weizsäcker, C. F.Reconstruction of physics: Yesterday, today, tomorrowTime, Quantum and InformationSpringerBerlin, Heidelberg2003373383 |
| [10] | Sorkin, R. D.Forks in the road, on the way to quantum gravityInt. J. Theor. Phys.361219972759 |
| [11] | Frauca, A. M.Sorkin, R.How to measure the quantum measureInt. J. Theor. Phys.5612017232258 |
| [12] | Isham, C. J.A new approach to quantising space-time: I. Quantising on a general categoryAdv. Theor. Math. Phys.722003331367 |
| [13] | Connes, A.Noncommutative GeometryAcademic Press1994 · Zbl 0818.46076 |
| [14] | Balachandran, A. P.Ibort, A.Marmo, G.Martone, M.Quantum fields on noncommutative spacetimes: Theory and phenomenologySymmetry, Integr. Geom. Methods Appl.6201005222 |
| [15] | Doplicher, S.Fredenhagen, K.Roberts, J.The structure of spacetime at the Planck Scale and quantum fieldsCommun. Math. Phys.1721995187 |
| [16] | P. A. M. Dirac, The Lagrangian in quantum mechanics, Physikalische Zeitschrift der Sovietunion3A New Approach to Quantum Theory · JFM 59.1527.03 |
| [17] | Dirac, P. A. M.The Principles of Quantum Mechanics27Oxford University Press1981 |
| [18] | Ibort, A.Rodríguez, M. A.On the structure of finite groupoids and their representationsSymmetry112019414https://doi.org/10.3390/sym11030414 |
| [19] | Renault, J.A Groupoid Approach to \(C^\ast 793\) Springer-VerlagBerlin1980 |
| [20] | Landsman, N. P.Between classical and quantumHandbook of the Philosophy of ScienceNorth-HollandAmsterdam2007417553 |
| [21] | Sorkin, R. D.Quantum mechanics as quantum measure theoryMod. Phys. Lett. A933199431193127 |
| [22] | Di Cosmo, F.Ibort, A.Marmo, G.Groupoids and coherent statesOpen Systems and Information Dynamics2642019 |
| [23] | Connes, A.Noncommutative GeometryAcademic Press1994 · Zbl 0818.46076 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
