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Symplectic Field Theory of the Galilean Covariant Scalar 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 F´isica, Universidade de Bras´ilia (70910-900, Bras´ilia, 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 quan- tum 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 gener- ators 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 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–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¨odingerequation in the phase space. K e y w o r d s: Galilean covariance, star-product, phase space, symplectic structure. 1. Introduction tion (x). 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 Schr¨odinger 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- Schr¨odingerequation. The goal in the present work is tively a spectrogram. In 1949, Jos´eEnrique Moyal to derive a Wigner representation for such covariant [7], who independently derived the Wigner function, theory. as the functional generator of the quantum momen- tum, 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–Andr´e 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 Schr¨odingerequation 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 sym- plectic 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 G denoted as 3 ted. The Schr¨odinger (Klein–Gordon-like) equation X (xjy) = g휇휈 x y = x y − x y − x y ; (4) and the Pauli–Schr¨odinger(Dirac-like) equation are 휇 휈 i i 4 5 5 4 i=1 derived showing the equivalence between our formal- ism and the usual non-relativistic formalism. In Sec- 4 4 5 x2 5 y2 where x = y = t, x = 2t e y = 2t . Hence, the tion 4, a symplectic structure is constructed in the following metric can be introduced: Galilean manifold. Using the commutation relations, 01 0 0 0 0 1 the Schr¨odingerequation in five dimensions in the 0 1 0 0 0 phase space is constructed. With a proposed solu- B0 0 1 0 0 C (g휇휈 ) = @ A: (5) tion, the Schr¨odingerequation in the phase space 0 0 0 0 −1 0 0 0 −1 0 is restored to its non-covariant form in (3 + 1) di- mensions. The explicitly covariant Pauli–Schr¨odinger This is the metric of a Galilean manifold G. 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 G; where each point 2. Galilean Covariance is specified by the coordinates q휇, 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 T *G will be de- * x0 = Rx + vt + a; (1) noted by (q휇; p휇). The space T G is equipped with a symplectic structure via the 2-form 0 t = t + b; (2) 휇 ! = dq ^ dp휇 (6) where R stands for the three-dimensional Euclid- called the symplectic form (sum over repeated indices ian rotations, v is the relative velocity defining the is assumed). We consider the following bidifferential Galilean boosts, a and b stand for spatial and operator on C1(T *G) functions, time translations, respectively. Consider a free par- − −! − −! ticle with mass m; the mass shell relation is given @ @ @ @ 2 Λ = 휇 − 휇 ; (7) by Pb − 2mE = 0. Then we can define a 5-vector, @q @p휇 @p @q휇 휇 i p = (px; py; pz; m; E) = (p ; m; E), with i = 1; 2; 3. such that, for C1 functions, f(q; p) and g(q; p), we Thus, we can define a scalar product of the type have 휇휈 2 p휇p휈 g = pipi − p5p4 − p4p5 = Pb − 2mE = k; (3) !(fΛ; gΛ) = fΛg = ff; gg (8) 휇휈 where where g is the metric of the space-time to be con- @f @g @f @g 휇휈 휇 structed, e p휈 g = p . ff; gg = 휇 − 휇 : (9) @q @p휇 @p @q휇 Let us define a set of canonical coordinates q휇 associated with p휇, by writing a five-vector in M, It is the Poisson bracket, and fΛ and gΛ are two 휇 4 5 q = (q; q ; q ), q is the canonical coordinate asso- vector fields given by hΛ = Xh = −{h; g. * ciated with Pb; q4 is the canonical coordinate associ- The space T G endowed with this symplectic struc- ated with E, and thus can be considered as the time ture is called the phase space and will be denoted by coordinate; q5 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 m explicitly given in terms of q and q , q q휇 = 휇 휈 2 4 5 2 5 q2 integrable complex functions, 휑(q; p) in Γ such that q q 휂휇휈 = q − q q = s = 0. From this q = 2t , or 5 q Z infinitesimally, we obtain 훿q = v 훿 2 . Therefore, the y fifth component is basically defined by the velocity. dpdq 휑 (q; p)휑(q; p) < 1 (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(q; p) = hq; pj휑i = −i(g휈휌Mc휇휎 − g휇휌Mc휈휎 + g휇휎Mc휈휌 − g휇휎Mc휈휌): is written with the aid of 휇 Z Consider a vector q 2 G that obeys the set of linear dpdqjq; pihq; pj = 1; (11) transformations of the type q¯휇 = G휇 q휈 + a휇: (13) where h휑j is the dual vector of j휑i. This symplectic 휈 Hilbert space is denoted by H(Γ). A particular case of interest in these transformations is given by 4. Symplectic Quantum Mechanics and the Galilei Group i i j i 4 i q¯ = Rjq + v q + a (14) In this section, we will study the Galilei group consid- q¯4 = q4 + a4 (15) ering H(Γ) as the space of representation. To do so, consider the unit transformations U:H(Γ) !H(Γ) 5 5 i j 1 2 4 q¯ = q − (Rjq )vi + v q : (16) such that h 1j 2i is invariant. Using the Λ operator, 2 i Λ we define a mapping e 2 = ?:Γ × Γ ! Γ called a In the matrix form, the homogeneous transformations Moyal (or star) product and defined by are written as " − −! − −! !# 0 1 1 1 i 1 i @ @ @ @ R 1 R 2 R 3 v 0 f ? g = f(q; p)exp − g(q; p); 2 2 2 2 휇 휇 B R 1 R 2 R 3 v 0C 2 @q @p휇 @p @q휇 휇 B R3 R3 R3 v3 0C G 휈 = B 1 2 3 C: (17) @ 0 0 0 1 0A it should be noted that we used = 1.
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