Symplectic Field Theory of the Galilean Covariant Scalar and Spinor Representations


  • G. X. A. Petronilo Centro Internacional de F´isica, Universidade de Bras´ilia
  • S. C. Ulhoa Centro Internacional de F´isica, Universidade de Bras´ilia
  • A. E. Santana Centro Internacional de F´isica, Universidade de Bras´ilia



Galilean covariance, star-product, phase space, symplectic structure


We explore the concept of the extended Galilei group, a representation for the symplectic quantum mechanics in the manifold G, written in the light-cone of a five-dimensional de Sitter space-time in the phase space. The Hilbert space is constructed endowed with a symplectic structure. We study the unitary operators describing rotations and translations, whose generators satisfy the Lie algebra of G. This representation gives rise to the Schr¨odinger (Klein–Gordon-like) equation for the wave function in the phase space such that the dependent variables have the position and linear momentum contents. The wave functions are associated to the Wigner function through the Moyal product such that the wave functions represent a quasiamplitude of probability. We construct the Pauli–Schr¨odinger (Dirac-like) equation in the phase space in its explicitly covariant form. Finally, we show the equivalence between the five-dimensional formalism of the phase space with the usual formalism, proposing a solution that recovers the non-covariant form of the Pauli–Schr¨odinger equation in the phase space.


Y. Takahashi. Towards the many-body theory with the Galilei invariance as a guide: Part I. Fortschr. Phys. 36, 63 (1988).

M. Omote, S. Kamefuchi, Y. Takahashi, Y. Ohnuki. Galilean covariance and the Schr?odinger equation. Fortschr. Phys. 37, 933 (1989).

A.E. Santana, F.C. Khanna, Y. Takahashi. Galilei covariance and (4,1) de Sitter space. Prog. Theor. Phys. 99, 327 (1998).

E. Wigner. ? Uber das ?uberschreiten von potentialschwellen bei chemischen reaktionen. Z. Phys. Chem. 19, 203 (1932).

H. Weyl. Quantenmechanik und gruppentheorie. Z. Phys. 46, 1 (1927).

J. Ville. Theorie et application de la notion de signal analytique. Cables et Transmissions 2, 61 (1948).

J.E. Moyal. Quantum mechanics as a statistical theory. Math. Proc. Cambr. Phil. Soc. 45, 99 (1949).

M.D. Oliveira, M.C.B. Fernandes, F.C. Khanna, A.E. Santana, J.D.M. Vianna. Symplectic quantum mechanics. Ann. Phys. 312, 492 (2004).

R.G.G. Amorim, F.C. Khanna, A.P.C. Malbouisson, J.M.C. Malbouisson, A.E. Santana. Quantum field theory in phase space. Int. J. Mod. Phys. A 1950037 (2019).

R.A.S. Paiva, R.G.G. Amorim. Quantum physics in phase space: An analysis of simple pendulum. Adv. Theor. Comp. Phys. 1, 1 (2018).

H. Dessano, R.A.S. Paiva, R.G.G. Amorim, S.C. Ulhoa, A.E. Santana. Wigner function and non-classicality for oscillator systems. Brazilian J. Phys. 1 (2019).




How to Cite

Petronilo, G. X. A., Ulhoa, S. C., & Santana, A. E. (2019). Symplectic Field Theory of the Galilean Covariant Scalar and Spinor Representations. Ukrainian Journal of Physics, 64(8), 719.



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