Symplectic Field Theory of the Galilean Covariant Scalar

https://doi.org/10.15407/ujpe64.8.719 G.X.A. PETRONILO, S.C. ULHOA, A.E. SANTANA Centro Internacional de Física, Universidade de Brasília (70910-900, Brasília, DF, Brazil; e-mail: [email protected]) SYMPLECTIC FIELD THEORY OF THE GALILEAN COVARIANT SCALAR AND SPINOR REPRESENTATIONS

We explore the concept of the extended Galilei group, a representation for the symplectic quantum mechanics in the manifold 풢, 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 풢. This representation gives rise to the Schrödinger(Klein– Gordon-like) equation for the wave function in the phase space such that the dependent vari- ables 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–Schrödinger(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–Schrödingerequation in the phase space. K e y w o r d s: Galilean covariance, star-product, phase space, symplectic structure.

1. Introduction tion 휓(푥). Thus, it maps the quantum density ma- In 1988, Takahashi et. al. [1] began a study of the trix onto the real phase space functions and opera- Galilean covariance, where it was possible to de- tors introduced by Hermann Weyl in 1927 [5] in a velop an explicitly covariant non-relativistic field the- context related to the theory of representations in ory. With this formalism, the Schrödinger equation mathematics (Weyl quantization in physics). Indeed, takes a similar form as the Klein–Gordon equation in this is the Wigner–Weyl transformation of the den- the light-cone of a (4,1) de Sitter space [2, 3]. With sity matrix; i.e., the realization of that operator in the advent of the Galilean covariance, it was possi- the phase space. It was later re-derived by Jean Ville ble to derive the non-relativistic version of the Dirac in 1948 [6] as a quadratic representation (in sign) theory, which is known in its usual form as the Pauli– of the local time frequency energy of a signal, effec- Schrödingerequation. The goal in the present work is tively a spectrogram. In 1949, JoséEnrique Moyal to derive a Wigner representation for such covariant [7], who independently derived the Wigner function, theory. as the functional generator of the quantum momentum, as a basis for an elegant codification of all ex- The Wigner quasiprobability distribution (also pected values and, therefore, of quantum mechanics called the Wigner function or the Wigner–Ville dis- in the phase-space formulation (phase-space repre- tribution in honor of Eugene Wigner and Jean–André sentation). This representation has been applied to a Ville) was introduced by Eugene Wigner in 1932 [4] in number of areas such as statistical mechanics, quan- order to study quantum corrections to classical sta- tum chemistry, quantum optics, classical optics, sig- tistical mechanics. The aim was to relate the wave nal analysis, electrical engineering, seismology, time- function that appears in the Schrödingerequation frequency analysis for music signals, spectrograms in to a probability distribution in the phase space. It biology and speech processing, and motor design. In is a generating function for all the spatial autocorre- order to derive a phase-space representation for the lation functions of a given quantum mechanical func- Galilean-covariant spin 1/2 particles, we use a symplectic representation for the Galilei group, which is ○c G.X.A. PETRONILO, S.C. ULHOA, associated with the Wigner approach [8–11]. A.E. SANTANA, 2019 ISSN 2071-0194. Ukr. J. Phys. 2019. Vol. 64, No. 8 719 G.X.A. Petronilo, S.C. Ulhoa, A.E. Santana

This article is organized as follows. In Section 2, That can be seen as a special case of scalar product the construction of the Galilean covariance is presen- in 퐺 denoted as 3 ted. The Schrödinger (Klein–Gordon-like) equation ∑︁ (푥|푦) = 푔휇휈 푥 푦 = 푥 푦 − 푥 푦 − 푥 푦 , (4) and the Pauli–Schrödinger(Dirac-like) equation are 휇 휈 푖 푖 4 5 5 4 푖=1 derived showing the equivalence between our formalism and the usual non-relativistic formalism. In Sec- 4 4 5 푥2 5 푦2 where 푥 = 푦 = 푡, 푥 = 2푡 e 푦 = 2푡 . Hence, the tion 4, a symplectic structure is constructed in the following metric can be introduced: Galilean manifold. Using the commutation relations, ⎛1 0 0 0 0 ⎞ the Schrödingerequation in five dimensions in the 0 1 0 0 0 phase space is constructed. With a proposed solu- ⎜0 0 1 0 0 ⎟ (푔휇휈 ) = ⎝ ⎠. (5) tion, the Schrödingerequation in the phase space 0 0 0 0 −1 0 0 0 −1 0 is restored to its non-covariant form in (3 + 1) dimensions. The explicitly covariant Pauli–Schrödinger This is the metric of a Galilean manifold 풢. In the equation is derived in Section 5. We study a Galilean sequence, this structure is explored in order to study spin 1/2 particle in a external potential, and the so- unitary representations. lutions are proposed and discussed. In Section 6, the final concluding remarks are presented. 3. Hilbert Space and Sympletic Structure Consider an analytical manifold 풢, where each point 2. Galilean Covariance is specified by the coordinates 푞휇, with 휇 = 1, 2, 3, 4, 5 The Galilei transformations are given by and the metric specified by (5). The coordinates of every point in the cotangent-bundle 푇 *풢 will be de- * x′ = 푅x + v푡 + a, (1) noted by (푞휇, 푝휇). The space 푇 풢 is equipped with a symplectic structure via the 2-form ′ 푡 = 푡 + 푏, (2) 휇 휔 = 푑푞 ∧ 푑푝휇 (6) where 푅 stands for the three-dimensional Euclid- called the symplectic form (sum over repeated indices ian rotations, 푣 is the relative velocity defining the is assumed). We consider the following bidifferential Galilean boosts, a and b stand for spatial and operator on 퐶∞(푇 *풢) functions, time translations, respectively. Consider a free par- ←− −→ ←− −→ ticle with mass 푚; the mass shell relation is given 휕 휕 휕 휕 2 Λ = 휇 − 휇 , (7) by 푃̂︀ − 2푚퐸 = 0. Then we can define a 5-vector, 휕푞 휕푝휇 휕푝 휕푞휇 휇 푖 푝 = (푝푥, 푝푦, 푝푧, 푚, 퐸) = (푝 , 푚, 퐸), with 푖 = 1, 2, 3. such that, for 퐶∞ functions, 푓(푞, 푝) and 푔(푞, 푝), we Thus, we can define a scalar product of the type have

휇휈 2 푝휇푝휈 푔 = 푝푖푝푖 − 푝5푝4 − 푝4푝5 = 푃̂︀ − 2푚퐸 = 푘, (3) 휔(푓Λ, 푔Λ) = 푓Λ푔 = {푓, 푔} (8)

휇휈 where where 푔 is the metric of the space-time to be con- 휕푓 휕푔 휕푓 휕푔 휇휈 휇 structed, e 푝휈 푔 = 푝 . {푓, 푔} = 휇 − 휇 . (9) 휕푞 휕푝휇 휕푝 휕푞휇 Let us define a set of canonical coordinates 푞휇 associated with 푝휇, by writing a five-vector in 푀, It is the Poisson bracket, and 푓Λ and 푔Λ are two 휇 4 5 푞 = (q, 푞 , 푞 ), q is the canonical coordinate asso- vector fields given by ℎΛ = 푋ℎ = −{ℎ, }. * ciated with 푃̂︀; 푞4 is the canonical coordinate associ- The space 푇 풢 endowed with this symplectic struc- ated with 퐸, and thus can be considered as the time ture is called the phase space and will be denoted by coordinate; 푞5 is the canonical coordinate associated Γ. In order to associate the Hilbert space with the 4 휇 phase space Γ, we will consider the set of square- with 푚 explicitly given in terms of q and 푞 , 푞 푞휇 = 휇 휈 2 4 5 2 5 q2 integrable complex functions, 휑(푞, 푝) in Γ such that 푞 푞 휂휇휈 = q − 푞 푞 = 푠 = 0. From this 푞 = 2푡 , or 5 q ∫︁ infinitesimally, we obtain 훿푞 = v 훿 2 . Therefore, the † fifth component is basically defined by the velocity. 푑푝푑푞 휑 (푞, 푝)휑(푞, 푝) < ∞ (10)

720 ISSN 2071-0194. Ukr. J. Phys. 2019. Vol. 64, No. 8 Symplectic Field Theory of the Galilean Covariant Scalar

is a real bilinear form. In this case, 휑(푞, 푝) = ⟨푞, 푝|휑⟩ = −푖(푔휈휌푀̂︁휇휎 − 푔휇휌푀̂︁휈휎 + 푔휇휎푀̂︁휈휌 − 푔휇휎푀̂︁휈휌). is written with the aid of 휇 ∫︁ Consider a vector 푞 ∈ 퐺 that obeys the set of linear 푑푝푑푞|푞, 푝⟩⟨푞, 푝| = 1, (11) transformations of the type

푞¯휇 = 퐺휇 푞휈 + 푎휇. (13) where ⟨휑| is the dual vector of |휑⟩. This symplectic 휈 Hilbert space is denoted by 퐻(Γ). A particular case of interest in these transformations is given by 4. Symplectic Quantum Mechanics and the Galilei Group 푖 푖 푗 푖 4 푖 푞¯ = 푅푗푞 + 푣 푞 + 푎 (14) In this section, we will study the Galilei group consid- 푞¯4 = 푞4 + 푎4 (15) ering 퐻(Γ) as the space of representation. To do so, consider the unit transformations 푈:ℋ(Γ) → ℋ(Γ) 5 5 푖 푗 1 2 4 푞¯ = 푞 − (푅푗푞 )푣푖 + v 푞 . (16) such that ⟨휓1|휓2⟩ is invariant. Using the Λ operator, 2 푖 Λ we define a mapping 푒 2 = ⋆:Γ × Γ → Γ called a In the matrix form, the homogeneous transformations Moyal (or star) product and defined by are written as [︃ (︃ ←− −→ ←− −→ )︃]︃ ⎛ 1 1 1 푖 ⎞ 푖 휕 휕 휕 휕 푅 1 푅 2 푅 3 푣 0 푓 ⋆ 푔 = 푓(푞, 푝)exp − 푔(푞, 푝), 2 2 2 2 휇 휇 ⎜ 푅 1 푅 2 푅 3 푣 0⎟ 2 휕푞 휕푝휇 휕푝 휕푞휇 휇 ⎜ 푅3 푅3 푅3 푣3 0⎟ 퐺 휈 = ⎜ 1 2 3 ⎟. (17) ⎝ 0 0 0 1 0⎠ it should be noted that we used = 1. The generators 푖 푖 푖 v2 ~ 푣푖푅 푣푖푅 푣푖푅 1 of 푈 can be introduced by the following (Moyal–Weyl) 푗 2 3 2 star-operators: We can write the generators as (︂ 푖 휕 푖 휕 )︂ 퐹̂︀ = 푓(푞, 푝)⋆ = 푓 푞휇 + , 푝휇 − . 1 2 휕푝휇 2 휕푞휇 퐽̂︀푖 = 휖푖푗푘푀̂︁푗푘, 퐶̂︀푖 = 푀̂︁4푖, 2 (18) To construct a representation of the Galilei algebra 퐾̂︀푖 = 푀̂︁5푖, 퐷̂︀ = 푀̂︁54. in ℋ, we define the operators Hence, the non-vanishing commutation relations can 푖 휕 푃̂︀휇 = 푝휇⋆ = 푝휇 − , (12a) be rewritten as 2 휕푞휇 [︁ ]︁ [︁ ]︁ 휇 휇 푖 휕 퐽̂︀푖, 퐽̂︀푗 = 푖휖푖푗푘퐽̂︀푘, 퐽̂︀푖, 퐾̂︀푗 = 푖휖푖푗푘퐾̂︀푘, 푄̂︀ = 푞⋆ = 푞 + . (12b) 2 휕푝휇 [︁ ]︁ [︁ ]︁ and 퐽̂︀푖, 퐶̂︀푗 = 푖휖푖푗푘퐶̂︀푘, 퐾̂︀푖, 퐶̂︀푗 = 푖훿푖푗퐷̂︀ + 푖휖푖푗푘퐽푘, [︁ ]︁ [︁ ]︁ 퐷, 퐾 = 푖퐾 , 퐶 , 퐷 = 푖퐶 , 푀̂︁휈휎 = 푀휈휎⋆ = 푄̂︀휈 푃̂︀휎 − 푄̂︀휎푃̂︀휈 , (12c) ̂︀ ̂︀푖 ̂︀푖 ̂︀푖 ̂︀ ̂︀푖 [︁ ]︁ [︁ ]︁ 푃̂︀4, 퐷̂︀ = 푖푃̂︀4, 퐽̂︀푖, 푃̂︀푗 = 푖휖푖푗푘푃̂︀푘, (19) where 푀̂︁휈휎 and 푃̂︀휇 are the generators of homogeneous and inhomogeneous transformations, respec- [︁ ]︁ [︁ ]︁ 푃̂︀푖, 퐾̂︀푗 = 푖훿푖푗푃̂︀5, 푃̂︀푖, 퐶̂︀푗 = 푖훿푖푗푃̂︀4, tively. From this set of unitary operators, we obtain, [︁ ]︁ [︁ ]︁ after some simple calculations, the following set of 푃̂︀4, 퐾̂︀푖 = 푖푃̂︀푖, 푃̂︀5, 퐶̂︀푖 = 푖푃̂︀푖. commutations relations: [︁ ]︁ 퐷,̂︀ 푃̂︀5 = 푖푃̂︀5, [︁ ]︁ 휎 휌 푃̂︀휇, 푀̂︁휌휎 = −푖(푔휇휌푃̂︀ − 푔휇휎푃̂︀ ), These relations have the Lie algebra of the Galilei [︁ ]︁ 3 푃̂︀휇, 푃̂︀휎 = 0, group as a subalgebra in the case of ℛ × ℛ, considering 퐽푖 the generators of rotations 퐾푖 of the pure and Galilei transformations, 푃휇 the spatial and temporal [︁ ]︁ translations. In fact, we can observe that Eqs. (14) 푀̂︁휇휈 , 푀̂︁휌휎 = and (15) are the Galilei transformations given by

ISSN 2071-0194. Ukr. J. Phys. 2019. Vol. 64, No. 8 721 G.X.A. Petronilo, S.C. Ulhoa, A.E. Santana

Eq. (1) and (2) with 푥4 = 푡. Equation (16) is the com- Using the fact that patibility condition which represents the embedding (︂ 푖 )︂ −푖(2푝4+훼)푡 −푖(2푝4+훼)푡 푃̂︀4Ψ = 푝4 − 휕푡 푒 = −퐸 푒 (︂ A2 )︂ 2 ℐ : A → 퐴 = A, 퐴 , ; A ∈ ℰ , 퐴 ∈ 풢. 4 2퐴 3 4 and The commutation of 퐾 and 푃 is naturally non- (︂ )︂ 푖 푖 푖 5 5 −푖(2푝5+훽)푞 −푖(2푝5+훽)푞 푃̂︀5Ψ = 푝5 − 휕5 푒 = −푚 푒 , zero in this context, so 푃5 will be related to the 2 mass, which is the extension parameter of the Galilei group or an invariant of the extended Galilei–Lie al- we can conclude that gebra. So, the invariants of this algebra in the light cone of the de Sitter space-time are 훼 = 2퐸, 훽 = 2푚. (25)

휇 퐼1 = 푃̂︀휇푃̂︀ (20) So, we have 2 퐼2 = 푃̂︀5 = −푚퐼 (21) 1 (︂ 1 )︂ (︂ 푘 )︂ 푝2 − 푖푝 ∇ − ∇2 Φ = 퐸 + Φ, 휇 2푚 4 2푚 퐼3 = 푊̂︁5휇푊̂︁5 , (22) where 퐼 is the identity operator, 푚 is the mass, which is the usual form of the Schr¨odingerequation 1 훼 훽휌 in the phase space for a free particle with mass 푚 and 푊휇휈 = 2 휖휇훼훽휌휈 푃 푀 is the 5-dimensional Pauli– 푘2 Lubanski tensor, and 휖휇휈훼훽휌 is the totally antisym- with an additional kinetic energy of 2푚 , that we can metric tensor in five dimensions. In the scalar repre- always set as the zero of energy. santation, we can defined 퐼3 = 0. Using the Casimir This equation and its complex conjugate can also invariants 퐼1 and 퐼2 and applying them to Ψ, we have be obtained by using the Lagrangian density in the 휇 phase space (we use 푑 = 푑/푑푞휇) 휇 2 푃̂︀휇푃̂︀ Ψ = 푘 Ψ, 휇 * 푖 휇 휇 * 푃̂︀5Ψ = −푚Ψ. ℒ = 휕 Ψ(푞, 푝)휕Ψ (푞, 푝) + 푝 [Ψ(푞, 푝)휕 Ψ (푞, 푝) − 2 We obtain [︂푝휇푝 ]︂ − Ψ*(푞, 푝)휕휇Ψ(푞, 푝)] + 휇 − 푘2 Ψ. (︂ 1 )︂ 4 푝2 − 푖p ∇ − ∇2 − 푘2 Ψ = 4 The association of this representation with the (︂ 푖 )︂ (︂ 푖 )︂ = 2 푝 − 휕 푝 − 휕 Ψ, Wigner formalism is given by 4 2 푡 5 2 5 푓 (푞, 푝) = Ψ(푞, 푝) ⋆ Ψ†(푞, 푝), and a solution of this equation is 푤

5 where 푓푤(푞, 푝) is the Wigner function. To prove this, −푖2푝5푞 5 −2푖푝4푡 Ψ = 푒 휌(푞 )푒 휒(푡)Φ(q, p). (23) we recall that Eq. (23) can be written as

Thus, 휇 2 푃̂︀휇푃̂︀ Ψ = 푝 ⋆ Ψ(푞, 푝). (︂ 1 )︂ 1 푝2Φ − 푖p ∇Φ − ∇2Φ − 푘2 = Multiplying the right-hand side of the above equation 4 Φ by Ψ†, we obtain 1 1 = (푖휕푡휒)(푖휕5휌) , 2 휒휌 (푝2 ⋆ Ψ) ⋆ Ψ† = 푘2Ψ ⋆ Ψ†. (26) which yields But Ψ† ⋆ 푝2 = 푘2Ψ†. Thus,

푖휕푡휒 = 훼휒, and 푖휕5휌 = 훽휌. Ψ ⋆ (Ψ† ⋆ 푝2) = 푘2Ψ ⋆ Ψ†. (27)

Thus, our solution for 휒 and 휌 is Subtracting (27) from (26), we have

−푖훼푡 −푖훽푞5 2 2 휒 = 푒 , 휌 = 푒 . (24) 푝 ⋆ 푓푤(푞, 푝) − 푝 ⋆ 푓푤(푞, 푝) = 0, (28)

722 ISSN 2071-0194. Ukr. J. Phys. 2019. Vol. 64, No. 8 Symplectic Field Theory of the Galilean Covariant Scalar

2 which is the Moyal brackets {푝 , 푓푤}푀 . In view of This describes an electron in an external field with Eq. (12a), Eq. (28) becomes the Pauli–Schrödingerequation given by [︂ (︂ )︂ ]︂ 푝휇휕푞휇 푓푤(푞, 푝) = 0, (29) 휇 푖 훾 푝휇 − 휕휇 − 푒퐴휇 − 푘 Ψ = 0. (35) where the Wigner function in the Galilean manifold 2 is a solution of this equation. In order to illustrate such result, let us consider a electron in an external field given by 퐴휇(A, 퐴4, 퐴5), with 5. Spin 1/2 Symplectic Representaion 퐴4 = −휑 and 퐴5 = 0. Considering the representation In order to study the representations of spin-1/2 par- of the 훾휇 matrices 휇 푖 ticles, we introduce 훾 푃̂︀휇, where 푃̂︀휇 = 푝휇 − 2 휕휇 in (︂ )︂ (︂ )︂ (︂ √ )︂ 휎푖 0 0 0 such a way that, acting on the 5-spinor in the phase 훾푖 = , 훾4 = √ , 훾5 = 0 − 2 . 0 −휎푖 2 0 0 0 space Ψ(푞, 푝), we have √ (︂ )︂ where 휎푖 are the Pauli matrices, and 2 is the iden- 휇 푖 √ 훾 푝휇 − 휕휇 Ψ(푝, 푞) = 푘Ψ(푝, 푞), (30) tity 2×2 matrix multiplied by 2. We can rewrite the 2 (︁휙)︁ object Ψ, as Ψ = , where 휙 and 휒 are 2-spinors which is the Galilean-covariant Pauli–Schrödinger 휒 dependent on 푥휇; 휇 = 1, ..., 5. Thus, in the represen- equation. Consequently, the mass shell condition is tation where 푘 = 0, the Eq. (35) becomes obtained by the usual steps: (︂ )︂ √ (︂ )︂ 휇 휈 2 푖 푖 (훾 푃̂︀휇)(훾 푃̂︀휈 )Ψ(푞, 푝) = 푘 Ψ(푞, 푝). (31) 휎 p − 휕푞 − 푒A 휙 − 2 푝5 − 휕5 휒 = 0, 2 2 Therefore, (36) √ (︂ 푖 )︂ (︂ 푖 )︂ 휇 휈 2 휇 2 푝4 − 휕푡 − 푒휑 휙 − 휎 p − 휕푞 − 푒A 휒 = 0. 훾 훾 (푃̂︀휇푃̂︀휈 ) = 푘 = 푃̂︀ 푃̂︀휈 . (32) 2 2 Solving the coupled equations, we get an equation for Since 푃̂︀휇푃̂︀휈 = 푃̂︀휈 푃̂︀휇, we have 휙 and 휒. Replacing the eigenvalues of 푃̂︀4 and 푃̂︀5, we 1 휇 휈 휈 휇 휇 have (훾 훾 + 훾 훾 )푃̂︀휇푃̂︀휈 = 푃̂︀ 푃̂︀휈 , (33) 2 [︃ 2 ]︃ so 1 (︂ (︂ 푖 )︂)︂ 휎 p − 휕 − 푒A + 푒휑 휙 = 퐸휙, 2푚 2 푞 {훾휇, 훾휈 } = 2푔휇휈 . (34) [︃ ]︃ 1 (︂ (︂ 푖 )︂)︂2 Equation (30) can be derived from the Lagrangian 휎 p − 휕 − 푒A + 푒휑 휒 = 퐸휒. density for spin-1/2 particles in the phase space, 2푚 2 푞 which is given by These are the non-covariant form of the Pauli– 푖 (︁ 휇 휇 )︁ 휇 Schrödingerequation in the phase space independent ℒ = − (휕휇Ψ)¯ 훾 Ψ − Ψ(¯ 훾 휕휇Ψ) − (푘 − 훾 푝휇)ΨΨ¯ , 4 of the time with 푖 (︁0 −푖)︁ where Ψ¯ = 휁Ψ†, with휁 = − √ {훾4 + 훾5} = . † † 2 푖 0 푓푤 = Ψ ⋆ Ψ¯ = 푖휙 ⋆ 휒 − 푖휒 ⋆ 휙 . In the Galilean-covariant Pauli–Schrödingerequation case, the association to the Wigner function is given This leads to by 푓 = Ψ ⋆ Ψ¯ , with each component satisfying (︃ )︃ 푤 푒퐵 1 푠 푘2 Eq. (29). 퐸푛 = 푛 + − − , Let us now examine the gauge symmetries in the 푚 2 2 2푚 phase space demanding the invariance of the La- where 푠 = ±1. It should be noted that the above grangian under a local gauge transformation given by expression represents the Landau levels which show Λ(푞,푝) 푒 Ψ. This leads to the minimum coupling, the spin-splitting feature. (︂ 푖 )︂ The above Figures 1 and 2 show the Wigner func- 푃̂︀휇Ψ → (푃̂︀휇 − 푒퐴휇)Ψ = 푝휇 − 휕휇 − 푒퐴휇 Ψ. 2 tions for the ground and first excited states, respec- ISSN 2071-0194. Ukr. J. Phys. 2019. Vol. 64, No. 8 723 G.X.A. Petronilo, S.C. Ulhoa, A.E. Santana

1. Y. Takahashi. Towards the many-body theory with the Galilei invariance as a guide: Part I. Fortschr. Phys. 36, 63 (1988). 2. M. Omote, S. Kamefuchi, Y. Takahashi, Y. Ohnuki. Ga- lilean covariance and the Schrödingerequation. Fortschr. Phys. 37, 933 (1989). 3. A.E. Santana, F.C. Khanna, Y. Takahashi. Galilei covariance and (4,1) de Sitter space. Prog. Theor. Phys. 99, 327 (1998). 4. E. Wigner. Uber¨ das überschreiten von potentialschwellen bei chemischen reaktionen. Z. Phys. Chem. 19, 203 (1932). 5. H. Weyl. Quantenmechanik und gruppentheorie. Z. Phys. Fig. 1. Wigner Function (cut in 푞1,푝1),Ground State 46, 1 (1927). 6. J. Ville. Theorie et application de la notion de signal ana- lytique. Cables et Transmissions 2, 61 (1948). 7. J.E. Moyal. Quantum mechanics as a statistical theory. Math. Proc. Cambr. Phil. Soc. 45, 99 (1949). 8. M.D. Oliveira, M.C.B. Fernandes, F.C. Khanna, A.E. San- tana, J.D.M. Vianna. Symplectic quantum mechanics. Ann. Phys. 312, 492 (2004). 9. 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). 10. R.A.S. Paiva, R.G.G. Amorim. Quantum physics in phase Fig. 2. Wigner Function (cut in 푞1,푝1), First Excited State space: An analysis of simple pendulum. Adv. Theor. Comp. Phys. 1, 1 (2018). tively, in the cut (푞1, 푝1). These are the same solu- 11. H. Dessano, R.A.S. Paiva, R.G.G. Amorim, S.C. Ulhoa, tions known in the literature using the usual Wigner A.E. Santana. Wigner function and non-classicality for os- method. cillator systems. Brazilian J. Phys. 1 (2019). Received 08.07.19 6. Concluding Remarks We study the spin-1/2 particle equation, the Pauli– Г. Петронiло, С. Уйоа, А. Сантана Schrödingerequation, in the context of the Galilean СИМПЛЕКТИЧНА ТЕОРIЯ ПОЛЯ covariance, considering a symplectic Hilbert space. ГАЛIЛЕЄВО-КОВАРIАНТНИХ СКАЛЯРНОГО We begin with a presentation on the Galilean mani- I СПIНОРНОГО ПРЕДСТАВЛЕНЬ fold which is used to review the construction of the Р е з ю м е Galilean covariance and the representations of quan- Ми дослiджуємо концепцiю розширеної групи Галiлея, tum mechanics in this formalism, namely, the spin- деякого представлення для симплектичної квантової ме- 1/2 and scalar representations and the Schrödinger ханiки на многовидi 풢, заданого на свiтловому конусi (Klein–Gordon-like) and Pauli–Schrödinger (Dirac- п’ятивимiрного простору-часу де Сiттера у фазовому про- сторi. Побудувано Гiльбертiв простiр, надiлений симпле- like) equations, respectively. ктичною структурою. Ми вивчаємо унiтарнi оператори, що The quantum mechanics formalism in the phase описують повороти i трансляцiї, генератори яких утворю- space is derived in this context of the Galilean cova- ють алгебру Лi в 풢. Це представлення породжує рiвняння riance giving rise to the representations of spin-0 Шредiнгера (типу Кляйна–Гордона) для хвильової функцiї and spin-1/2 equations. For the spin-1/2 equation у фазовому просторi, так що змiннi мають змiст положення (the Dirac-like equation), we study the electron in i лiнiйного iмпульсу. Хвильовi функцiї пов’язанi з функцiєю an external field. Solving it, we were able to re- Вiгнера через добуток Мойала, так що хвильовi функцiї ре- презентують квазiамплiтуду ймовiрностi. Ми будуємо рiв- cover the non-covariant Pauli–Schrödingerequation няння Паулi–Шредiнгера (типу рiвняння Дiрака) у фазо- in phase space and to analyze, in this context, the вому просторi в явно коварiантнiй формi. На завершення Landau levels. ми показуємо еквiвалентнiсть мiж п’ятивимiрним формалi- змом фазового простору i звичайним формалiзмом, пропо- This work was supported by CAPES and CNPq of нуючи розв’язок, що вiдновлює нековарiантну форму рiв- Brazil. няння Паулi–Шредiнгера у фазовому просторi.

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