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Artículos de revistas sobre el tema "Phase Space Formulation"

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Publicado: 10 de marzo de 2023

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1

CIRELLI, RENZO, ALESSANDRO MANIÀ y LIVIO PIZZOCCHERO. "QUANTUM PHASE SPACE FORMULATION OF SCHRÖDINGER MECHANICS". International Journal of Modern Physics A 06, n.º 12 (20 de mayo de 1991): 2133–46. http://dx.doi.org/10.1142/s0217751x91001064.

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We develop a geometrical approach to Schrödinger quantum mechanics, alternative to the usual one, which is based on linear and algebraic structures such as Hilbert spaces, operator algebras, etc. The starting point of this approach is the Kähler structure possessed by the set of the pure states of a quantum system. The Kähler manifold of the pure states is regarded as a “quantum phase space”, conceptually analogous to the phase space of a classical hamiltonian system, and all the constituents of the conventional formulation, in particular the algebraic structure of the observables, are reproduced using a suitable “Kähler formalism”. We also show that the probabilistic character of the measurement process in quantum mechanics and the uncertainty principle are contained in the geometrical structure of the quantum phase space. Finally, we obtain a characterization for quantum phase spaces which can be interpreted as a statement of uniqueness for Schrödinger quantum mechanics.

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2

Chruściński, Dariusz. "Phase-Space Approach to Berry Phases". Open Systems & Information Dynamics 13, n.º 01 (marzo de 2006): 67–74. http://dx.doi.org/10.1007/s11080-006-7268-3.

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We propose a new formula for the adiabatic Berry phase which is based on phase-space formulation of quantum mechanics. This approach sheds a new light onto the correspondence between classical and quantum adiabatic phases — both phases are related with the averaging procedure: Hannay angle with averaging over the classical torus and Berry phase with averaging over the entire classical phase space with respect to the corresponding Wigner function.

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3

ZACHOS, COSMAS. "DEFORMATION QUANTIZATION: QUANTUM MECHANICS LIVES AND WORKS IN PHASE-SPACE". International Journal of Modern Physics A 17, n.º 03 (30 de enero de 2002): 297–316. http://dx.doi.org/10.1142/s0217751x02006079.

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Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (e.g. quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides — coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations.

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4

Wu, Xizeng y Hong Liu. "Phase-space formulation for phase-contrast x-ray imaging". Applied Optics 44, n.º 28 (1 de octubre de 2005): 5847. http://dx.doi.org/10.1364/ao.44.005847.

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5

Tosiek, J. y P. Brzykcy. "States in the Hilbert space formulation and in the phase space formulation of quantum mechanics". Annals of Physics 332 (mayo de 2012): 1–15. http://dx.doi.org/10.1016/j.aop.2013.01.010.

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6

Kalmykov, Yuri P. y William T. Coffey. "Transition state theory for spins: phase-space formulation". Journal of Physics A: Mathematical and Theoretical 41, n.º 18 (18 de abril de 2008): 185003. http://dx.doi.org/10.1088/1751-8113/41/18/185003.

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7

Batalin, I. A., K. Bering y P. H. Damgaard. "Superfield formulation of the phase space path integral". Physics Letters B 446, n.º 2 (enero de 1999): 175–78. http://dx.doi.org/10.1016/s0370-2693(98)01537-8.

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8

Rosato, J. "A quantum phase space formulation of radiative transfer". Physics Letters A 378, n.º 34 (julio de 2014): 2586–89. http://dx.doi.org/10.1016/j.physleta.2014.07.003.

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9

SOBOUTI, Y. y S. NASIRI. "A PHASE SPACE FORMULATION OF QUANTUM STATE FUNCTIONS". International Journal of Modern Physics B 07, n.º 18 (15 de agosto de 1993): 3255–72. http://dx.doi.org/10.1142/s0217979293003218.

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Allowing for virtual paths in phase space permits an extension of Hamilton’s principle of least action, of lagrangians and of hamiltonians to phase space. A subsequent canonical quantization, then, provides a framework for quantum statistical mechanics. The classical statistical mechanics and the conventional quantum mechanics emerge as special case of this formalism. Von Neumann’s density matrix may be inferred from it. Wigner’s functions and their evolution equation may also be obtained by a unitary transformation.

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10

Torre, C. G. "Covariant phase space formulation of parametrized field theories". Journal of Mathematical Physics 33, n.º 11 (noviembre de 1992): 3802–12. http://dx.doi.org/10.1063/1.529878.

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11

Pimpale, Ashok y M. Razavy. "Quantum-mechanical phase space: A generalization of Wigner phase-space formulation to arbitrary coordinate systems". Physical Review A 38, n.º 12 (1 de diciembre de 1988): 6046–54. http://dx.doi.org/10.1103/physreva.38.6046.

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12

Chung, E., J. Qian, G. Uhlmann y Hong-Kai Zhao. "A phase-space formulation for elastic-wave traveltime tomography". Journal of Physics: Conference Series 124 (1 de julio de 2008): 012018. http://dx.doi.org/10.1088/1742-6596/124/1/012018.

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13

Booth, Ivan y Stephen Fairhurst. "Canonical phase space formulation of quasi-local general relativity". Classical and Quantum Gravity 20, n.º 21 (29 de septiembre de 2003): 4507–31. http://dx.doi.org/10.1088/0264-9381/20/21/001.

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14

Dragoman, D. "The formulation of Fermi's golden rule in phase space". Physics Letters A 274, n.º 3-4 (septiembre de 2000): 93–97. http://dx.doi.org/10.1016/s0375-9601(00)00530-2.

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15

Freidel, Laurent, Marc Geiller y Jonathan Ziprick. "Continuous formulation of the loop quantum gravity phase space". Classical and Quantum Gravity 30, n.º 8 (2 de abril de 2013): 085013. http://dx.doi.org/10.1088/0264-9381/30/8/085013.

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16

De Gosson, Charlyne y Maurice A. De Gosson. "The Phase Space Formulation of Time-Symmetric Quantum Mechanics". Quanta 4, n.º 1 (23 de noviembre de 2015): 27. http://dx.doi.org/10.12743/quanta.v4i1.46.

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17

Barik, Debashis, Suman Kumar Banik y Deb Shankar Ray. "Quantum phase-space function formulation of reactive flux theory". Journal of Chemical Physics 119, n.º 2 (8 de julio de 2003): 680–95. http://dx.doi.org/10.1063/1.1579473.

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18

Freidel, Laurent, Robert G. Leigh y Djordje Minic. "Quantum gravity, dynamical phase-space and string theory". International Journal of Modern Physics D 23, n.º 12 (octubre de 2014): 1442006. http://dx.doi.org/10.1142/s0218271814420061.

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In a natural extension of the relativity principle, we speculate that a quantum theory of gravity involves two fundamental scales associated with both dynamical spacetime as well as dynamical momentum space. This view of quantum gravity is explicitly realized in a new formulation of string theory which involves dynamical phase-space and in which spacetime is a derived concept. This formulation naturally unifies symplectic geometry of Hamiltonian dynamics, complex geometry of quantum theory and real geometry of general relativity. The spacetime and momentum space dynamics, and thus dynamical phase-space, is governed by a new version of the renormalization group (RG).

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19

BARS, ITZHAK. "GAUGE SYMMETRY IN PHASE SPACE CONSEQUENCES FOR PHYSICS AND SPACE–TIME". International Journal of Modern Physics A 25, n.º 29 (20 de noviembre de 2010): 5235–52. http://dx.doi.org/10.1142/s0217751x10051128.

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Position and momentum enter at the same level of importance in the formulation of classical or quantum mechanics. This is reflected in the invariance of Poisson brackets or quantum commutators under canonical transformations, which I regard as a global symmetry. A gauge symmetry can be defined in phase space (XM, PM) that imposes equivalence of momentum and position for every motion at every instant of the worldline. One of the consequences of this gauge symmetry is a new formulation of physics in space–time. Instead of one time there must be two, while phenomena described by one-time physics in 3+1 dimensions appear as various "shadows" of the same phenomena that occur in 4+2 dimensions with one extra space and one extra time dimensions (more generally, d+2). The 2T-physics formulation leads to a unification of 1T-physics systems not suspected before and there are new correct predictions from 2T-physics that 1T-physics is unable to make on its own systematically. Additional data related to the predictions, that provides information about the properties of the extra 1-space and extra 1-time dimensions, can be gathered by observers stuck in 3+1 dimensions. This is the probe for investigating indirectly the extra 1+1 dimensions which are neither small nor hidden. This 2T formalism that originated in 1998 has been extended in recent years from the worldline to field theory in d+2 dimensions. This includes 2T field theories that yield 1T field theories for the Standard Model and General Relativity as shadows of their counterparts in 4+2 dimensions. Problems of ghosts and causality in a 2T space-time are resolved automatically by the gauge symmetry, while a higher unification of 1T field theories is obtained. In this paper the approach will be described at an elementary worldline level, and the current status of 2T-physics will be summarized.

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20

Voulis, Igor y Arnold Reusken. "A time dependent Stokes interface problem: well-posedness and space-time finite element discretization". ESAIM: Mathematical Modelling and Numerical Analysis 52, n.º 6 (noviembre de 2018): 2187–213. http://dx.doi.org/10.1051/m2an/2018053.

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In this paper a time dependent Stokes problem that is motivated by a standard sharp interface model for the fluid dynamics of two-phase flows is studied. This Stokes interface problem has discontinuous density and viscosity coefficients and a pressure solution that is discontinuous across an evolving interface. This strongly simplified two-phase Stokes equation is considered to be a good model problem for the development and analysis of finite element discretization methods for two-phase flow problems. In view of theunfitted finite element methods that are often used for two-phase flow simulations, we are particularly interested in a well-posed variational formulation of this Stokes interface problem in a Euclidean setting. Such well-posed weak formulations, which are not known in the literature, are the main results of this paper. Different variants are considered, namely one with suitable spaces of divergence free functions, a discrete-in-time version of it, and variants in which the divergence free constraint in the solution space is treated by a pressure Lagrange multiplier. The discrete-in-time variational formulation involving the pressure variable for the divergence free constraint is a natural starting point for a space-time finite element discretization. Such a method is introduced and results of numerical experiments with this method are presented.

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21

BIZDADEA, CONSTANTIN, MARIA-MAGDALENA BÂRCAN, MIHAELA TINCA MIAUTĂ y SOLANGE-ODILE SALIU. "SECOND-ORDER LAGRANGIAN DYNAMICS IN THE PHASE-SPACE: SOME EXAMPLES". Modern Physics Letters A 27, n.º 10 (28 de marzo de 2012): 1250062. http://dx.doi.org/10.1142/s0217732312500629.

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By means of a class of nondegenerate models with a finite number of degrees of freedom, it is proved that given a Hamiltonian formulation of dynamics, there exists an equivalent second-order Lagrangian formulation whose configuration space coincides with the Hamiltonian phase-space. The above result is extended to scalar field theories in a Lorentz-covariant manner.

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22

HARIKUMAR, E. y M. SIVAKUMAR. "ON THE EQUIVALENCE BETWEEN TOPOLOGICALLY AND NON-TOPOLOGICALLY MASSIVE ABELIAN GAUGE THEORIES". Modern Physics Letters A 15, n.º 02 (20 de enero de 2000): 121–31. http://dx.doi.org/10.1142/s0217732300000128.

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We analyze the equivalence between topologically massive gauge theory (TMGT) and different formulations of non-topologically massive gauge theories (NTMGTs) in the canonical approach. The different NTMGTs studied are Stückelberg formulation of (a) a first-order formulation involving one- and two-form fields, (b) Proca theory, and (c) massive Kalb–Ramond theory. We first quantize these reducible gauge systems by using the phase space extension procedure and using it, identify the phase space variables of NTMGTs which are equivalent to the canonical variables of TMGT and show that under this the Hamiltonian also get mapped. Interestingly it is found that the different NTMGTs are equivalent to different formulations of TMGTs which differ only by a total divergence term. We also provide covariant mappings between the fields in TMGT to NTMGTs at the level of correlation function.

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23

Pastorello, Davide. "Geometric Hamiltonian quantum mechanics and applications". International Journal of Geometric Methods in Modern Physics 13, Supp. 1 (octubre de 2016): 1630017. http://dx.doi.org/10.1142/s0219887816300178.

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Adopting a geometric point of view on Quantum Mechanics is an intriguing idea since, we know that geometric methods are very powerful in Classical Mechanics then, we can try to use them to study quantum systems. In this paper, we summarize the construction of a general prescription to set up a well-defined and self-consistent geometric Hamiltonian formulation of finite-dimensional quantum theories, where phase space is given by the Hilbert projective space (as Kähler manifold), in the spirit of celebrated works of Kibble, Ashtekar and others. Within geometric Hamiltonian formulation quantum observables are represented by phase space functions, quantum states are described by Liouville densities (phase space probability densities), and Schrödinger dynamics is induced by a Hamiltonian flow on the projective space. We construct the star-product of this phase space formulation and some applications of geometric picture are discussed.

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24

Shlivinski, A., E. Heyman, A. Boag y C. Letrou. "A Phase-Space Beam Summation Formulation for Ultrawide-Band Radiation". IEEE Transactions on Antennas and Propagation 52, n.º 8 (agosto de 2004): 2042–56. http://dx.doi.org/10.1109/tap.2004.832513.

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25

Ohba, I. "Stochastic Quantization of System under Constraints through Phase Space Formulation". Progress of Theoretical Physics 77, n.º 5 (1 de mayo de 1987): 1267–85. http://dx.doi.org/10.1143/ptp.77.1267.

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26

Milošević, D. B. "Phase space path-integral formulation of the above-threshold ionization". Journal of Mathematical Physics 54, n.º 4 (abril de 2013): 042101. http://dx.doi.org/10.1063/1.4797476.

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27

Gauntlett, J. P., J. Gomis y P. K. Townsend. "Supersymmetry and the physical-phase-space formulation of spinning particles". Physics Letters B 248, n.º 3-4 (octubre de 1990): 288–94. http://dx.doi.org/10.1016/0370-2693(90)90294-g.

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28

Calzetta, E. y B. L. Hu. "Wigner distribution function and phase-space formulation of quantum cosmology". Physical Review D 40, n.º 2 (15 de julio de 1989): 380–89. http://dx.doi.org/10.1103/physrevd.40.380.

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29

Rosato, J. "Phase space formulation of radiative transfer in optically thick plasmas". Annals of Physics 383 (agosto de 2017): 130–39. http://dx.doi.org/10.1016/j.aop.2017.05.010.

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30

CORTESE, IGNACIO y J. ANTONIO GARCÍA. "A VARIATIONAL FORMULATION OF SYMPLECTIC NONCOMMUTATIVE MECHANICS". International Journal of Geometric Methods in Modern Physics 04, n.º 05 (agosto de 2007): 789–805. http://dx.doi.org/10.1142/s0219887807002296.

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The standard lore in noncommutative physics is the use of first order variational description of a dynamical system to probe the space noncommutativity and its consequences in the dynamics in phase space. As the ultimate goal is to understand the inherent space noncommutativity, we propose a variational principle for noncommutative dynamical systems in configuration space, based on results of our previous work [18]. We hope that this variational formulation in configuration space can be of help to elucidate the definition of some global and dynamical properties of classical and quantum noncommutative space.

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31

ZUMINO, BRUNO. "DEFORMATION OF THE QUANTUM MECHANICAL PHASE SPACE WITH BOSONIC OR FERMIONIC COORDINATES". Modern Physics Letters A 06, n.º 13 (30 de abril de 1991): 1225–35. http://dx.doi.org/10.1142/s0217732391001305.

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We describe the one-parameter deformation of the phase space of a quantum mechanical system and show that this twisted phase space is covariant under the action of the symplectic quantum group. The analogous case of a system with fermionic coordinates is also considered and the phase space is shown to be covariant under the action of the orthogonal quantum group. Twisted commutation relations occur in the description of deformed spaces or superspaces as well as in the formulation of field theories with generalized statistics. The many-parameter case is briefly discussed.

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32

Liu, Jian, Xin He y Baihua Wu. "Unified Formulation of Phase Space Mapping Approaches for Nonadiabatic Quantum Dynamics". Accounts of Chemical Research 54, n.º 23 (10 de noviembre de 2021): 4215–28. http://dx.doi.org/10.1021/acs.accounts.1c00511.

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33

Ramírez, Rafael y Telesforo López-Ciudad. "Phase-Space Formulation of Thermodynamic and Dynamical Properties of Quantum Particles". Physical Review Letters 83, n.º 22 (29 de noviembre de 1999): 4456–59. http://dx.doi.org/10.1103/physrevlett.83.4456.

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34

Hu, Xu-Guang, Qian-Shu Li y Au-Chin Tang. "A new formulation of the potential scattering in quantum phase space". Physica Scripta 54, n.º 2 (1 de agosto de 1996): 129–36. http://dx.doi.org/10.1088/0031-8949/54/2/001.

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35

Dragoman, Daniela. "Phase-space formulation of filtering: insight into the wave-particle duality". Journal of the Optical Society of America B 22, n.º 3 (1 de marzo de 2005): 633. http://dx.doi.org/10.1364/josab.22.000633.

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36

Chen, B. X. y X. L. He. "A geometric-optical phase space formulation for paraxial light beam propagation". European Physical Journal D 64, n.º 2-3 (2 de agosto de 2011): 499–504. http://dx.doi.org/10.1140/epjd/e2011-10597-2.

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37

Dragoman, D. "Phase Space Formulation of Quantum Mechanics. Insight into the Measurement Problem". Physica Scripta 72, n.º 4 (1 de enero de 2005): 290–96. http://dx.doi.org/10.1238/physica.regular.072a00290.

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38

Ban, Masashi. "Phase-space representation of quantum systems in the relative-state formulation". International Journal of Theoretical Physics 35, n.º 9 (septiembre de 1996): 1947–92. http://dx.doi.org/10.1007/bf02302423.

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39

Shi-Dong, Liang y T. Harko. "Towards an Observable Test of Noncommutative Quantum Mechanics". Ukrainian Journal of Physics 64, n.º 11 (25 de noviembre de 2019): 983. http://dx.doi.org/10.15407/ujpe64.11.983.

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The conceptual incompatibility of spacetime in gravity and quantum physics implies the existence of noncommutative spacetime and geometry on the Planck scale. We present the formulation of a noncommutative quantum mechanics based on the Seiberg–Witten map, and we study the Aharonov–Bohm effect induced by the noncommutative phase space. We investigate the existence of the persistent current in a nanoscale ring with an external magnetic field along the ring axis, and we introduce two observables to probe the signal coming from the noncommutative phase space. Based on this formulation, we give a value-independent criterion to demonstrate the existence of the noncommutative phase space.

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40

DIAS, NUNO COSTA, MAURICE A. DE GOSSON y JOÃO NUNO PRATA. "METAPLECTIC FORMULATION OF THE WIGNER TRANSFORM AND APPLICATIONS". Reviews in Mathematical Physics 25, n.º 10 (noviembre de 2013): 1343010. http://dx.doi.org/10.1142/s0129055x13430101.

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We show that the cross Wigner function can be written in the form [Formula: see text] where [Formula: see text] is the Fourier transform of ϕ and Ŝ is a metaplectic operator that projects onto a linear symplectomorphism S consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl–Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform, we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schrödinger and the Bopp representations of quantum mechanics.

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41

Das, A. "Discrete phase space - I: Variational formalism for classical relativistic wave fields". Canadian Journal of Physics 88, n.º 2 (febrero de 2010): 73–91. http://dx.doi.org/10.1139/p09-089.

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This is the first in a series of papers dealing with a discrete phase space formulation for classical and quantum fields. This formulation leads to a representation of quantum field theory that is covariant and possesses singularity free S-matrix. In this paper, the classical relativistic wave equations are presented as partial difference equations in the arena of covariant discrete phase space. These equations are also expressed as difference-differential equations in discrete phase space and continuous time. The relativistic invariance and covariance of the equations in both versions are established. The partial difference and difference-differential equations are derived as the Euler–Lagrange equations from the variational principle. The difference and difference-differential conservation equations are derived. Finally, the total momentum, energy, and charge of the relativistic classical fields satisfying difference-differential equations are computed.

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42

Braasch, William F. y William K. Wootters. "A Classical Formulation of Quantum Theory?" Entropy 24, n.º 1 (17 de enero de 2022): 137. http://dx.doi.org/10.3390/e24010137.

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We explore a particular way of reformulating quantum theory in classical terms, starting with phase space rather than Hilbert space, and with actual probability distributions rather than quasiprobabilities. The classical picture we start with is epistemically restricted, in the spirit of a model introduced by Spekkens. We obtain quantum theory only by combining a collection of restricted classical pictures. Our main challenge in this paper is to find a simple way of characterizing the allowed sets of classical pictures. We present one promising approach to this problem and show how it works out for the case of a single qubit.

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43

Dossa, F. A., J. T. Koumagnon, J. V. Hounguevou y G. Y. H. Avossevou. "Некоммутативная задача Ландау о фазовом пространстве при наличии минимальной длины". Вестник КРАУНЦ. Физико-математические науки, n.º 4 (29 de diciembre de 2020): 188–98. http://dx.doi.org/10.26117/2079-6641-2020-33-4-188-198.

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The deformed Landau problem under a electromagnetic field is studied, where the Heisenberg algebra is constructed in detail in non-commutative phase space in the presence of a minimal length. We show that, in the presence of a minimal length, the momentum space is more practical to solve any problem of eigenvalues. From the Nikiforov-Uvarov method, the energy eigenvalues are obtained and the corresponding wave functions are expressed in terms of hypergeometric functions. The fortuitous degeneration observed in the spectrum shows that the formulation of the minimal length complements that of the non-commutative phase space. Изучается деформированная задача Ландау в электромагнитном поле, в которой алгебра Гейзенберга подробно строится в некоммутативном фазовом пространстве при наличии минимальной длины. Мы показываем, что при наличии минимальной длины импульсное пространство более практично для решения любой проблемы собственных значений. С помощью метода Никифорова-Уварова получаются собственные значения энергии, а соответствующие волновые функции выражаются через гипергеометрические функции. Случайное вырождение, наблюдаемое в спектре, показывает, что формулировка минимальной длины дополняет формулировку некоммутативного фазового пространства.

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44

Gonzalez, Diego, Daniel Gutiérrez-Ruiz y J. David Vergara. "Phase space formulation of the Abelian and non-Abelian quantum geometric tensor". Journal of Physics A: Mathematical and Theoretical 53, n.º 50 (18 de noviembre de 2020): 505305. http://dx.doi.org/10.1088/1751-8121/abc6c2.

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45

Fernando Barbero G., J. y Madhavan Varadarajan. "The phase space of (2 + 1)-dimensional gravity in the Ashtekar formulation". Nuclear Physics B 415, n.º 2 (marzo de 1994): 515–30. http://dx.doi.org/10.1016/0550-3213(94)90121-x.

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46

Davidović, Milena D., Ljubica D. Davidović y Milesa Srećković. "Time of Arrival in the Wigner Phase Space Formulation of Quantum Mechanics". Acta Physica Hungarica A) Heavy Ion Physics 26, n.º 3-4 (1 de noviembre de 2006): 253–60. http://dx.doi.org/10.1556/aph.26.2006.3-4.5.

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47

Wang, Lipo. "On the classical limit of phase‐space formulation of quantum mechanics: Entropy". Journal of Mathematical Physics 27, n.º 2 (febrero de 1986): 483–87. http://dx.doi.org/10.1063/1.527247.

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Narcowich, Francis J. "The problem of moments in the phase‐space formulation of quantum mechanics". Journal of Mathematical Physics 28, n.º 12 (diciembre de 1987): 2873–82. http://dx.doi.org/10.1063/1.527687.

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Wang, Lipo y R. F. O'Connell. "A precaution needed in using the phase-space formulation of quantum mechanics". Physica A: Statistical Mechanics and its Applications 144, n.º 1 (julio de 1987): 201–10. http://dx.doi.org/10.1016/0378-4371(87)90153-1.

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Dias, Nuno Costa y João Nuno Prata. "Stargenfunctions, generally parametrized systems and a causal formulation of phase space quantum mechanics". Journal of Mathematical Physics 46, n.º 7 (julio de 2005): 072107. http://dx.doi.org/10.1063/1.1948327.

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