Symplectic Field Theories: Scalar and Spinor Representations

R., Costa, Caroline S.; Tenser, Marcia; Amorim, R. G. G.; Fernandes, M. C. B.; Santana, A. E.; Vianna, J. D. M.

doi:10.1007/s00006-018-0840-4

Cited by 2 publications

(1 citation statement)

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“…Generalizing the 4D-Poincaré group to the nD case (with an arbitrary signature), we shall call them Poincaré groups. Various representations of the Poincaré groups have been proposed either in specific dimensions or signatures and are often expressed in terms of matrices [4,5,7,8,16,19,21,[24][25][26][27]. Yet, matrices are neither the only nor probably the best way to represent rotation groups.…”

Section: Introductionmentioning

confidence: 99%

Dual Hyperquaternion Poincaré Groups

Girard

Clarysse

Pujol

et al. 2021

Adv. Appl. Clifford Algebras

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A new representation of the Poincaré groups in n dimensions via dual hyperquaternions is developed, hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof). This formalism yields a uniquely defined product and simple expressions of the Poincaré generators, with immediate physical meaning, revealing the algebraic structure independently of matrices or operators. An extended multivector calculus is introduced (allowing an eventual sign change of the metric or of the exterior product). The Poincaré groups are formulated as a dual extension of hyperquaternion pseudo-orthogonal groups. The canonical decomposition and the invariants are discussed. As concrete example, the 4D Poincaré group is examined together with a numerical application. Finally, the hyperquaternion representation is compared to the quantum mechanical one. Potential applications include in particular, moving reference frames and computer graphics. Response to Reviewers:All reviewer comments have been taken into account.

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Section: Introductionmentioning

confidence: 99%

Dual Hyperquaternion Poincaré Groups

Girard

Clarysse

Pujol

et al. 2021

Adv. Appl. Clifford Algebras

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Quark-Antiquark Effective Potential in Symplectic Quantum Mechanics

Luz

Petronilo

et al. 2022

Advances in High Energy Physics

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In this paper, we study within the structure of Symplectic Quantum Mechanics a bidimensional nonrelativistic strong interaction system which represent the bound state of heavy quark-antiquark, where we consider a Cornell potential which consists of Coulomb-type plus linear potentials. First, we solve the Schrödinger equation in the phase space with the linear potential. The solution (ground state) is obtained and analyzed by means of the Wigner function related to Airy function for the c c ¯ meson. In the second case, to treat the Schrödinger-like equation in the phase space, a procedure based on the Bohlin transformation is presented and applied to the Cornell potential. In this case, the system is separated into two parts, one analogous to the oscillator and the other we treat using perturbation method. Then, we quantized the Hamiltonian with the aid of stars operators in the phase space representation so that we can determine through the algebraic method the eigenfunctions of the undisturbed Hamiltonian (oscillator solution), and the other part of the Hamiltonian was the perturbation method. The eigenfunctions found (undisturbed plus disturbed) are associated with the Wigner function via Weyl product using the representation theory of Galilei group in the phase space. The Wigner function is analyzed, and the nonclassicality of ground state and first excited state is studied by the nonclassicality indicator or negativity parameter of the Wigner function for this system. In some aspects, we observe that the Wigner function offers an easier way to visualize the nonclassic nature of meson system than the wavefunction does phase space.

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Symplectic Field Theories: Scalar and Spinor Representations

Cited by 2 publications

References 54 publications

Dual Hyperquaternion Poincaré Groups

Dual Hyperquaternion Poincaré Groups

Quark-Antiquark Effective Potential in Symplectic Quantum Mechanics

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