Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane

We study all the symmetries of the free Schrödinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetrie...

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Datum:	2014
Hauptverfasser:	Batlle, C., Gomis, J., Kamimura, K.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2014
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1468442019-02-12T01:23:55Z Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane Batlle, C. Gomis, J. Kamimura, K. We study all the symmetries of the free Schrödinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schrödinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches. 2014 Article Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane / C. Batlle, J. Gomis, K.Kamimura // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R60; 81S05; 83C65 DOI:10.3842/SIGMA.2014.011 http://dspace.nbuv.gov.ua/handle/123456789/146844 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We study all the symmetries of the free Schrödinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schrödinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.
format	Article
author	Batlle, C. Gomis, J. Kamimura, K.
spellingShingle	Batlle, C. Gomis, J. Kamimura, K. Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Batlle, C. Gomis, J. Kamimura, K.
author_sort	Batlle, C.
title	Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane
title_short	Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane
title_full	Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane
title_fullStr	Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane
title_full_unstemmed	Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane
title_sort	symmetries of the free schrödinger equation in the non-commutative plane
publisher	Інститут математики НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/146844
citation_txt	Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane / C. Batlle, J. Gomis, K.Kamimura // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 25 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT batllec symmetriesofthefreeschrodingerequationinthenoncommutativeplane AT gomisj symmetriesofthefreeschrodingerequationinthenoncommutativeplane AT kamimurak symmetriesofthefreeschrodingerequationinthenoncommutativeplane
first_indexed	2025-07-11T00:45:35Z
last_indexed	2025-07-11T00:45:35Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 011, 15 pages Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane? Carles BATLLE †, Joaquim GOMIS ‡ and Kiyoshi KAMIMURA § † Departament de Matemàtica Aplicada 4 and Institut d’Organització i Control, Universitat Politècnica de Catalunya - BarcelonaTech, EPSEVG, Av. V. Balaguer 1, 08800 Vilanova i la Geltrú, Spain E-mail: carles.batlle@upc.edu ‡ Departament d’Estructura i Constituents de la Matèria and Institut de Ciències del Cosmos, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain E-mail: gomis@ecm.ub.es § Department of Physics, Toho University, Funabashi, Chiba 274-8510, Japan E-mail: kamimura@ph.sci.toho-u.ac.jp Received August 29, 2013, in final form January 29, 2014; Published online February 08, 2014 http://dx.doi.org/10.3842/SIGMA.2014.011 Abstract. We study all the symmetries of the free Schrödinger equation in the non-commu- tative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non- relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schrödinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches. Key words: non-commutative plane; Schrödinger equation; Schrödinger symmetries; higher spin symmetries 2010 Mathematics Subject Classification: 81R60; 81S05; 83C65 1 Introduction and results The symmetries of a free massive non-relativistic particle and the associated Schrödinger equa- tion have been investigated. The projective symmetries of the Schrödinger equation induced by the transformation on the coordinates (t, ~x) are well known. They form the Schrödinger group [12, 19, 20, 23] that, apart from the Galilei symmetries, contains the dilatation and the expansion. Recently Valenzuela [24] (see also [4]) discussed higher-order symmetries of the free Schrödinger equation. These symmetry transformations form an infinite-dimensional Weyl alge- bra constructed from the generators of space-translation and the ordinary commuting Galilean boost. The extra symmetries that do not belong to the Schrödinger group correspond to higher spin symmetries. These transformations are not induced by the transformations on the coordi- nates but they map solutions into solutions of the Schrödinger equation. In the case of 2+1 dimensions, the Galilei group admits two central extensions [2, 5, 14, 15, 21], one associated to the exotic non-commuting boost and other appearing in the commutator of the ordinary boost and spatial translations. The non-relativistic particle in the non-commutative plane was introduced in [22] by considering a higher order Galilean invariant Lagrangian for the coordinates (t, ~x) of the particle. A first order Lagrangian depending on the coordinates (t, ~x) ?This paper is a contribution to the Special Issue on Deformations of Space-Time and its Symmetries. The full collection is available at http://www.emis.de/journals/SIGMA/space-time.html mailto:carles.batlle@upc.edu mailto:gomis@ecm.ub.es mailto:kamimura@ph.sci.toho-u.ac.jp http://dx.doi.org/10.3842/SIGMA.2014.011 http://www.emis.de/journals/SIGMA/space-time.html 2 C. Batlle, J. Gomis and K. Kamimura and extra coordinates ~v was introduced in [9]. For these Lagrangians there are two possible realizations, one with non-commuting (exotic) boosts, and the other with ordinary commuting boosts [5, 16] (see [15] for a review). In this paper we study all the infinitesimal Noether symmetries of a massive free particle in the (2 + 1)-dimensional non-commutative plane. The Noether symmetries are constructed from the Heisenberg algebra of commuting boosts Xi and the generators of translations Pi, {Xi, Pj} = δij , i, j = 1, 2, all of which are constants of motion and are written explicitly in terms of the initial conditions. The algebra of these symmetries is the infinite-dimensional Weyl algebra associated with the Heisenberg algebra. A general element of the Weyl algebra is given by G(Xi, Pj). The generators given by higher degree polynomials do not form a closed algebra for any finite degree. These infinite symmetries are the non-relativistic counterpart of all the symmetries of the free massless Klein–Gordon equation [10]. There is no known realization of this Weyl algebra for an Schrödinger equation with interaction. These symmetries could be useful to construct a non-relativistic analogue of Vasiliev’s higher spin theories [25]. The subset of generators constructed up to quadratic polynomials of (Xi, Pj) form a finite- dimensional sub-algebra, which in turn contains the 9-dimensional Schrödinger algebra. We study the realization of this algebra in the classical unreduced phase-space, as well as in the reduced one, the later appearing due to the presence of second class constraints. We also study all the symmetries of the free Schrödinger equation in the non-commutative plane. The symmetries are in one to one correspondence with the Noether symmetries of the free particle in the non- commutative plane. This analysis is done in the quantum reduced phase space, as well as in the extended one. In the extended space we impose non-hermitian combinations of the second class constraints. In this case we consider two representations for the physical states, namely a Fock representation [16] and a coordinate representation. We study the Schrödinger subalgebra in detail, and we show the equivalence between the reduced and extended space formulations. We show that, in general, the quadratic (and higher) generators in the extended space contain second order derivatives and hence do not generate point transformations for the coordinates. The organization of the paper is as follows. In Section 2 we construct all Noether symmetries of the massive particle in the non-commutative plane. Section 3 is devoted to the study of the quantum symmetries of the Schrödinger equation. 2 Classical symmetries of the non-relativistic particle Lagrangian in the non-commutative plane In this section we introduce a first order Lagrangian describing a particle in the non-commutative plane [9], and present the corresponding Hamiltonian formalism. The main result of the sec- tion is the construction of all the Noether symmetries of the non-relativistic particle in the non-commutative plane (equations (2.11)–(2.14) and the ensuing discussion). For the sake of completeness, we review the construction of the standard and exotic Galilei algebras and of the Schödinger generators [3, 5, 15, 16, 17]. We also perform the reduction of the second class constraints of the system for later use in the quantization in the reduced phase space. The first order Lagrangian of a non-relativistic particle in the non-commutative plane, see for example [9], is given by Lnc = m ( viẋi − v2 i 2 ) + κ 2 εijviv̇j , i, j = 1, 2, (2.1) where the overdot means derivative with respect to “time” t. This Lagrangian can be obtained using the NLR method [7, 6] applied to the exotic Galilei group in 2 + 1 dimensions1; see [1] 1Note that this Lagrangian is not dynamically equivalent to the higher order Lagrangian for a non-relativistic Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane 3 for the case of exotic Newton–Hooke whose flat limit gives (2.1). The coordinates xi’s are the Goldstone bosons of the transverse translations and vi’s are the Goldstone bosons of the broken boost. The vi’s and κ are dimensionless. The Lagrangian (2.1) gives two primary second class constraints Πi = πi + κ 2 εijvj ≈ 0, Vi = pi −mvi ≈ 0, (2.2) where pi and πi are the momenta canonically conjugate to xi and vi. The constraints (2.2) satisfy relations {Πi,Πj} = κεij , {Vi, Vj} = 0, {Πi, Vj} = mδij , and the Dirac Hamiltonian is H = p2 i 2m , (2.3) up to quadratic terms in the constraints. From the canonical pairs (x, v, p, π) we can get a new set of canonical pairs (x̃, ṽ, p̃, π̃) given by  x̃ p̃ ṽ π̃  =  1 − κ 2m2 ε κ 2m ε − 1 m 1 − 1 m 1 κ 2m ε 1   x p v π  . (2.4) In terms of the new variables the constraints (2.2) become a canonical pair, ṽi = − 1 m Vi ≈ 0, π̃i = Πi + κ 2m εijVj ≈ 0. (2.5) The position and momentum of the particle are expressed as xi = x̃i − κ 2m2 εij p̃j − κ 2m εij ṽj + 1 m π̃i, pi = p̃i, (2.6) and the Dirac Hamiltonian (2.3) is written as H = 1 2m p̃2 i . (2.7) The phase space is a direct product of two spaces. One is spanned by (ṽ, π̃) with the con- straints (2.5) ṽi ≈ 0, π̃i ≈ 0 (2.8) and thus classically trivial. The other one is spanned by (x̃, p̃) with the Hamiltonian (2.7). It is a system of a free non-relativistic particle in 2 + 1 dimensions but with the coordinates x̃i. In the classical reduced phase space defined by the second class constraints (2.8) the coordinates xi become non-commutative (see also Subsection 2.1), {xi, xj}∗ = { x̃i − κ 2m2 εikp̃k, x̃j − κ 2m2 εj`p̃` } = κ m2 εij . (2.9) particle proposed in [22]. It can be obtained from (2.1) using the inverse Higgs mechanism [18]. 4 C. Batlle, J. Gomis and K. Kamimura If we consider a point transformation (x, v)→ (y, u) yi = xi + κ 2m εijvj , ui = vi, (2.10) in the Lagrangian (2.1) it becomes L = m ( uiẏi − u2 i 2 ) , which is the Lagrangian of a free non-relativistic particle with the commutative coordinates yi. Although it has a form of free particle we keep xi as the “position coordinates” of this system. Local interactions would be introduced at the position xi rather than yi. The coordinates yi in (2.10) are identified with the commuting coordinates x̃i in (2.6), while xi are non-commutative as in (2.9). All the Noether symmetries are generated by constants of motion which are arbitrary func- tions G(Xi, Pj) of Xi = x̃i(0) = x̃i(t)− t m p̃i(t) and Pi = p̃i(0) = p̃i(t), (2.11) verifying {Pi, Pj} = 0, {Xi, Pj} = δij , {Xi, Xj} = 0. The Lagrangian (2.1) is quasi-invariant under the transformation generated by G(Xi, Pj). The canonical variations of (x, v) are δxi = ∂G ∂pi = ∂G ∂Pi − t m ∂G ∂Xi + κ 2m2 εij ∂G ∂Xj , δvi = ∂G ∂πi = − 1 m ∂G ∂Xi . (2.12) When computing the variation of the Lagrangian (2.1) under (2.12), the (pi, πi) are replaced, using the definition of momenta (2.2), by pi → mvi, πi → − κ 2 εijvj , Xi → xi − tvi + κ 2m εijvj . (2.13) It follows that the variation of the Lagrangian becomes a total derivative, δLnc = d dτ F(x, v, t), F(x, v, t) = [piδxi + πiδvi −G]pi=mvi, πi=−κ2 εijvj = [ mvi ( ∂G ∂Pi − t m ∂G ∂Xi ) −G ] pi=mvi, πi=−κ2 εijvj . (2.14) All these Noether symmetries generate an infinite-dimensional Weyl algebra. The Weyl alge- bra, denoted by [h∗2], can be defined [24] as the one generated by (the Weyl ordered) polynomials of the Heisenberg algebra generators, (Xi, Pi), that we indicate by G(Xi, Pj). [h∗2] is the infinite- dimensional algebra of a particle in the non-commutative plane. These infinite symmetries are the non-relativistic counterpart of the complete set of symmetries of the free massless Klein– Gordon equation [10]. The existence of a realization of this Weyl algebra for an interacting Schrödinger equation is an interesting open question. There are finite-dimensional subalgebras of the higher spin algebra whose generators are constructed from the product of generators Xi, Pj up to second order: h2 ⊂ Galilei ⊂ Sch(2) ⊂ h2 ⊕ sp(4) ⊂ [h∗2]. Sch(2) is the Schrödinger algebra2 in 2D, whose generators are those of the Galilean algebra Xi, Pi, H, J , together with the dilatation, D, and the expansion, C. 2A field theory realization of this algebra was given in [14]. Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane 5 Let us restrict now to Galilean and Schrödinger symmetries. We start by considering the Galilean symmetries of (2.1). The action is invariant under translations, x′i = xi + αi, v′i = vi, boosts, x′i = xi − βit, v′i = vi − βi, rotations, x′i = xi cosϕ+ εijxj sinϕ, v′i = vi cosϕ+ εijvj sinϕ, and time translations t′ = t− γ. The corresponding Noether charges of translations and boosts are given by Pi = pi, Ki = mxi − pit− πi + κ 2 εijvj = mXi + κ 2m εijPj , while the angular momentum is J = εij(xipj + viπj) = εij(XiPj + ṽiπ̃j). (2.15) Together with the total Hamiltonian (2.3), they generate the exotic Galilei algebra [2, 5, 14, 15, 21] {H,J} = 0, {H,Ki} = −Pi, {H,Pi} = 0, {J, Pi} = εijPj , {J,Ki} = εijKj , {Ki, Pj} = mδij , {Ki,Kj} = −κεij , {Pi, Pj} = 0. From this, it may seem that the Lagrangian (2.1) gives a phase space realization of the (2 + 1)- dimensional Galilei group with two central charges m, κ. However, one of the central charges is trivial since, if we modify the generator of the boost as in [5, 13], K̃i = Ki − κ 2m εijPj = mXi = mxi − πi + 1 2 κεijvj − κ 2m εijpj − pit, one gets that (H,P, K̃, J) verifies the standard Galilean algebra without κ.3 Physically, the result of changing the boost generators is a shift in the parameter of the translations αi → αi + κ 2m εijβj . Note that the modified boost generators K̃i are proportional to the coordinates at t = 0, Xi = x̃i(0), that verify {Xi, Xj} = 0, and we have a realization with only one non-trivial central charge associated to the mass of the particle4. The Schrödinger generators are those of the Galilean algebra Xi, Pi, H, J , and the dilata- tion, D, and the expansion, C, given by D = XiPi = xipi − t m p2 i − 1 m πipi + κ 2m εijpivj , 3If we introduce an interaction with a background field this statement is no longer true, since it depends on which coordinates (commutative or non-commutative) are used to define the interaction; see [8, 9, 15, 17]. Notice however that the background field will break, in general, part of the symmetries of the Galilei group. 4Note however that δKiL = δK̃i L = d dt (−mxi − κ 2 εijvj)βi, where βi is boost parameter. 6 C. Batlle, J. Gomis and K. Kamimura C = mXiXi = mx2 i + 1 m t2p2 i + 1 m π2 i + κ2 4m v2 i + κ2 4m3 p2 i − 2txipi − 2xiπi + κεijxivj − κ m εijxipj + 2 m tpiπi − κ m tεijpivj − κ m εijπivj + κ m2 εijπipj − κ2 2m2 vipi. In the same spirit, we also redefine the generator of rotations as J = εijXiPj = εijxipj − κ 2m2 p2 i + κ 2m vipi + 1 m εijpiπj , which, up to square of constraints, coincides with (2.15). The new, non-zero Poisson brackets are {D,C} = −2C, {D,H} = 2H, {H,C} = −2D, {D,Pi} = Pi, {D,Xi} = −Xi, {C,Pi} = 2mXi. The transformations of the coordinates xi, vi under dilatation and expansion are obtained from (2.12) as δDxi = α m (mxi − 2mtvi + κεijvj), δDvi = −αvi, δCxi = λ m ( 2mt2vi − 2mtxi + κεijxj − 2κtεijvj − κ2 2m vi ) , δCvi = λ m (−2mxi + 2mtvi − κεijvj), where α and λ are the corresponding infinitesimal parameters. 2.1 Reduction of second class constraints The classical symmetry algebra is also realized in the reduced phase space defined by the second class constraints Πi = Vi = 0. The Dirac bracket is {A,B}∗ = {A,B}+ {A,Πi} 1 m {Vi, B} − {A, Vi} 1 m {Πi, B} − {A, Vi} κεij m2 {Vj , B} and yields {xi, xj}∗ = κ m2 εij , {xi, pj}∗ = δij , {pi, pj}∗ = 0. (2.16) In this space, the symmetry transformations are generated using the Dirac bracket and the reduced generators, which can be obtained by substituting vi = pi/m, πi = −κ/(2m)εijpj into the standard ones. The infinite Weyl symmetries are generated by G(R)(xi, pj) = G(Xi, Pj)\|vi=pi/m, πi=−κ/(2m)εijpj . In particular the Schrödinger generators are given by [3] P (R) i = pi, (2.17) K (R) i = mxi − tpi + κ m εijpj (exotic Galilei), (2.18) K̃ (R) i = K (R) i − κ 2m εijP (R) j = mxi − tpi + κ 2m εijpj (standard Galilei), (2.19) H(R) = 1 2m p2 i , (2.20) Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane 7 J (R) = εijxipj + κ 2m2 p2 i , (2.21) D(R) = pixi − 1 m tp2 i , (2.22) C(R) = mx2 i + 1 m t2p2 i + κ2 4m3 p2 i − 2txipi + κ m εijxipj . (2.23) They generate the Schrödinger algebra with the Dirac bracket, since K̃ (R) i , P (R) i generate a Heisenberg algebra:{ K̃ (R) i , P (R) j }∗ = mδij , { P (R) i , P (R) j }∗ = 0, and { K̃ (R) i , K̃ (R) j }∗ = 0. Symmetry transformations are generated either using the Poisson brackets in the original phase space or using the Dirac brackets with the reduced generators, (2.17)–(2.23). For example the “exotic Galilei” generators Ki satisfy {Ki,Kj} = { K (R) i ,K (R) j }∗ = −κεij , and generate “standard(covariant) Galilei” transformation of (xi, pi) as δxi = {xi, β ·K} = { xi, β ·K(R) }∗ = −tβi, δpi = {pi, β ·K} = { pi, β ·K(R) }∗ = −mβi. The “standard Galilei” generators K̃i satisfy{ K̃i, K̃j } = { K̃ (R) i , K̃ (R) j }∗ = 0. and generate “exotic Galilei” (non-covariant) transformations of xi, pi, δxi = {xi, β · K̃} = { xi, β · K̃(R) }∗ = −tβi + κ 2m εijβj , δpi = {pi, β · K̃} = { pi, β · K̃(R) }∗ = −mβi. 3 Quantum symmetries of free Schrödinger equation in the non-commutative plane In this section we will study the quantization of the model at the level of the Schrödinger equation and their symmetries. We will quantize it in two approaches, one in the reduced phase space and the other in the extended phase space. 3.1 Quantization in the reduced phase space In the classical theory, xi has a nonzero Dirac bracket {xi, xj}∗ as in (2.16) in the reduced phase space. Since Dirac brackets are replaced by commutators in the canonical quantization, one cannot have a xi-coordinate representation of quantum states5. To discuss symmetries of Schrödinger equations we introduce new coordinates yi ≡ xi + κ 2m2 εijpj , qi = pi, (3.1) 5Since pi’s are commuting the momentum representation is possible [9]. 8 C. Batlle, J. Gomis and K. Kamimura such that {yi, yj}∗ = 0, {yi, qj}∗ = δij , {qi, qj}∗ = 0. The coordinate yi is the one introduced in (2.10) and qi is its conjugate. In these coordinates, the Schrödinger equation (i∂t−H)\|Ψ(t)〉 = 0 takes the form corresponding to a free particle for the wave function Ψ(y, t) = 〈y\|Ψ(t)〉, ŷi\|y〉 = yi\|y〉, 〈y\|y′〉 = δ2(y − y′), i.e. ( i∂t − 1 2m (−i∂y) 2 ) Ψ (y, t) = 0, and the inner product is 〈Ψ\|Ψ〉 = ∫ dyΨ (y, t)Ψ (y, t) . Note that yi are not covariant under exotic Galilei transformation generated by Ki δyi = {yi, β ·K} = { yi, β ·K(R) }∗ = −βit− κ 2m εijβj , but covariant under the Galilei transformation generated by K̃i δyi = {yi, β · K̃} = { yi, β · K̃(R) }∗ = −βit. The position operators, covariant under Ki, are x̂i = yi − κ 2m2 εij(−i∂yj ). They are hermitian since ŷi = yi, q̂i = −i∂yi , with appropriate boundary conditions on Ψ (y, t), are hermitian. Although in the free theory we are able to work with both the commutative ŷi = yi and the non-commutative x̂i = yi− κ 2m2 εij(−i∂yj ) position operators, this may not be the case in an interacting theory. For example, if we consider an interaction with a background electromagnetic field, which introduces couplings with a source at position xi, the non-commutative coordinates are naturally selected (see, for example, [8, 9, 15, 17]). If we denote generically by G(R)(t, x, p) = G(X,P )\|Π=V=0 the generators of the Weyl algebra in the reduced classical space, the generators in this quantization are given by Ĝ (1) i (t, y, q̂) = G (R) i ∣∣∣ xj=yj− κ 2m2 εjlq̂l, pj=q̂j = Gi ( y − t m q̂, q̂ ) , (3.2) with q̂i = −i∂/∂yi and with the appropriate dealing of operator ordering. The knowledge of all the symmetries of the Schrödinger equation in terms of the coordina- tes yi, ŷi is the non-commutative analog in 2 + 1 dimensions of the high spin symmetries of the relativistic massless Klein Gordon equation [10]. The Vasiliev [25] non-linear theory has these high spin symmetries. In this sense these high spin-nonrelativistic symmetries could be useful in order to construct a non-relativistic Vasiliev theory [24]. We consider next in detail the Schrödinger generators, given by P̂ (1) i = q̂i = −i ∂ ∂yi , (3.3) Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane 9 ˆ̃K (1) i = myi − tq̂i = myi + it ∂ ∂yi , (3.4) Ĥ(1) = 1 2m q̂2 i = − 1 2m ∂2 ∂yi2 , (3.5) Ĵ (1) = εijyiq̂j = −iεijyi ∂ ∂yj , (3.6) D̂(1) = yiq̂i − i− 1 m tq̂2 i = −iyi ∂ ∂yi + 1 m t ∂2 ∂yi2 − i, (3.7) Ĉ(1) = my2 i − 2tyiq̂i + 2it+ 1 m t2q̂2 i = my2 i + 2ityi ∂ ∂yi − 1 m t2 ∂2 ∂yi2 + 2it, (3.8) where a Weyl ordering has been used for D̂(1) and Ĉ(1). These generators are hermitian operators when acting on the wave functions Ψ(t, y). Furthermore, they obey the abstract quantum Schrödinger algebra off shell, with non-zero commutators given by[ ˆ̃Ki, P̂j ] = imδij , [ Ĵ , P̂i ] = iεijP̂j , [ Ĵ , ˆ̃Ki ] = iεij ˆ̃Kj , [ Ĥ, ˆ̃Ki ] = −iP̂i,[ D̂, Ĥ ] = 2iĤ, [ D̂, P̂i ] = iP̂i, [ D̂, ˆ̃Ki ] = −i ˆ̃Ki,[ D̂, Ĉ ] = −2iĈ, [ Ĥ, Ĉ ] = −2iD̂, [ Ĉ, P̂i ] = 2i ˆ̃Ki. (3.9) Using these, together with[ i∂t, ˆ̃K (1) i ] = −iP̂ (1) i , [ i∂t, D̂ (1) ] = −2iĤ(1), [ i∂t, Ĉ (1) ] = −2iD̂(1), (3.10) one can show that[ i∂t − Ĥ(1), Ĝ (1) i ] = 0 for all the generators Ĝ (1) i , which proves the invariance of the Schrödinger equation under the Schrödinger transformations in this reduced space quantization. Under a general Weyl transformation, the wave functions transform as Ψ′ (y, t) = eiαiĜ (1) i (t,y,(−i∂y))Ψ (y, t) , where the αi are the parameters of the transformations. In particular, for the on-shell Schrö- dinger transformations one has Ψ′ (y, t) = eA+iBΨ ( y′, t′ ) , where the coordinate transformations of (y, t) are those of the N = 1 conformal Galilean trans- formation, and the multiplicative factor is eA+iB, with A and B real functions of the coordinates and of the parameters of the transformation given by (see, for instance, [11, 23]) 1) H (time translation), t′ = t+ a, y′ = y, A = B = 0, 2) D (dilatation), t′ = eλt, y′ = e λ 2 y, A = λ 2 , B = 0, 10 C. Batlle, J. Gomis and K. Kamimura 3) C (expansion), t′ = t 1− κt , y′i = yi 1− κt , eA = 1 (1− κt) , B = − κmy2 2(1− κt) , 4) (spatial translations and boost) t′ = t, y′i = yi + ( β0 + t β1 m ) i , A = 0, B = −m ( yi + 1 2 ( β0 i + t β1 i m )) β1 i m , with [β0 i ] = L, [β1 i ] = L−1. The difference with respect to the transformation of the ordinary Schrödinger equation is that in the non-commutative case the coordinates that are transformed by conformal Galilean transformations are the canonical ones yi, and not the physical position of the particle, xi. The invariance of the solutions of the Schrödinger equation under a general element of the Weyl algebra can be proved using the invariance under the generators of the Heisenberg algebra and the commutators (3.10). 3.2 Quantization in the extended phase space 3.2.1 Fock representation In order to quantize the model in the extended phase space the second class constraints (2.2) are imposed as physical state conditions by taking their non-hermitian combinations as in [1]. We first consider the canonical transformation (2.4) that separates the second class constraints as new coordinates. It is realized at quantum level as a unitary transformation q̃ = U †qU, U = e i m pi(πi−κ2 εijvj). (3.11) For example, x̃i = U †xiU = xi − 1 m ( πi − κ 2 εijvj ) + 1 2 κ m εij ( −pj m ) . It is useful to introduce the complex combinations of the phase space variables π̃± = π̃1± iπ̃2 and ṽ± = ṽ1 ± iṽ2, which allow us to introduce two pairs of annihilation and creation operators ã± = i√ 2κ ( π̃± − i κ 2 ṽ± ) , ã†± = −i√ 2κ ( π̃∓ + i κ 2 ṽ∓ ) , with nonzero commutators [ã±, ã † ±] = 1. Using the Fock representation for (ṽ, π̃) and coordinate representation for (x̃, p̃), any state of this system is described by \|Ψ(t)〉 = ∑ n+≥0,n−≥0 ∫ dy \|n+, n−〉 ⊗ \|y〉Φn+n−(y, t), where \|n+, n−〉 is the eigenstate of Ñ± = ã†±ã± with eigenvalues n± ∈ N ∪ {0} and \|y〉 is the eigenstate of commuting operators x̃i with eigenvalue yi. They are normalized as 〈n+, n−\|n′+, n′−〉 = δn+n′+ δn−n′− , 〈y\|y′〉 = δ2(y − y′). The scalar product is given by 〈Ψ\|Ψ′〉 = ∑ n± ∫ dyΦn+n−(y, t)Φ′n+n−(y, t). Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane 11 In the quantization in the extended phase space the second class constraints (2.2) are imposed as physical state conditions by taking their non-hermitian combination, ã±\|Ψphys(t)〉 = 0. (3.12) This means that physical states are minimum uncertainty states in (ṽ, π̃). Condition (3.12) selects out only the n+ = n− = 0 state, so that Φn+n−(y, t) = 0 except for Φ0,0(y, t) ≡ Φ0(y, t), \|Ψphys(t)〉 = ∫ dy \|0, 0〉 ⊗ \|y〉Φ0(y, t). The Schrödinger equation is (i∂t −H)\|Ψphys(t)〉 = 0, H = ˆ̃p2 2m , and thus (i∂t −H)Φ0(y, t) = 0, H = 1 2m (−i∂yi) 2. The generators of the Weyl algebra are given in the extended space as polynomials G(X,P ) of the operator equivalent of (2.11), and, since they commute with ã± and ã†±, physical states remain physical6. They act on the physical states as Ψphys(t)〉 → \|Ψ′phys(t)〉 = eiG(X,P )\|Ψphys(t)〉 and it turns out that the transformation of the wave function Φ0(y, t) is Φ′0(y, t) = eiG(X,P )Φ0(y, t) = eiG(y−t(−i∂y),(−i∂y))Φ0(y, t). This transformation has the same form as the one in the reduced phase space generated by (3.2)–(3.8). Then the wave function in the reduced space Ψ(y, t) = 〈y\|Ψ(t)〉 and Φ0(y, t) = 〈y\| ⊗ 〈00\|Ψ(t)〉 that appear in the extended space quantization are identified. Note that in the former 〈y\| is eigenstate of ŷi = xi + κ 2m2 εijpj in (3.1) but 〈y\| in the latter is eigenstate of ˆ̃xi that are commuting in the extended space. We can see now how the non-commutativity of the position operators appears. x̂± = x1± ix2 are commuting in the extended phase space. Using (2.4) we write x+ = x̃+ + i κ 2m2 p̃+ + i √ 2κ m2 ã†−, x− = x̃− − i κ 2m2 p̃− − i √ 2κ m2 ã− = x†+. In the reduced space quantization procedure, the ã± are effectively put to zero and x± becomes a non-commutative operator on \|Ψ(t)〉. On the other hand in the quantization in the extended space, expectation values of the position operators between two physical states are given by 〈Ψ\|x̂±\|Ψ′〉 = ∫ dydy′Φ0(y, t)〈y\|〈0\| ( x̃± ± i κ 2m2 p̃± ± i √ 2κ m2 ( ã†− ã− )) \|0〉\|y′〉Φ′0(y′, t) = ∫ dyΦ0(y, t) ( y± ± i κ 2m2 (−2i∂y±) ) Φ′0(y, t). Commutative position operators x̂± on states \|Ψ〉 act as non-commutative operators (y± ± i κ 2m2 (−2i∂y±)) on the wave function Φ0(y, t). 6The angular momentum J in (2.15) contains a term depending on (v, π), but it commutes with ã±, ã†±. 12 C. Batlle, J. Gomis and K. Kamimura It is useful to consider the unitary transformation U in (3.11) on the creation and annihilation operators ã±, ã†±, ã+ = U †a+U = a+, ã− = U †a−U = a− − √ κ 2m2 p−. (3.13) The quantization in the extended phase space can be also done by considering the constraint equations (3.12) in terms of the operators a±, a†±. The physical state conditions (3.12) are a+\|Ψphys(t)〉 = 0, ( p− − √ 2m2 κ a− ) \|Ψphys(t)〉 = 0, and \|Ψphys〉 is a coherent state of a− with eigenvalue √ κ 2m2 p− [16]. In this representation, the Schrödinger generators are X (2) ± = ( x± ∓ i κ 2m2 p± ) − t m p± ± i κ m2 ( p± − √ 2m2 κ ( a†− a− )) , P (2) ± = p± = −2i∂x∓ , [x±, p∓] = 2i, D(2) = 1 2 (( x+p− + p+x− − 2t m p+p− ) + i κ m2 ( p+ − √ 2m2 κ a†− ) p− − i κ m2 p+ ( p− − √ 2m2 κ a− )) , C(2) = 1 2 (( x+ − i κ 2m2 p+ )( x− + i κ 2m2 p− ) − t m (( x+ − i κ 2m2 p+ ) p− + p+ ( x− + i κ 2m2 p− )) + t2 2m2 p+p− + 1 2 (( x+ − i κ 2m2 p+ ) − t m p+ )( −i κ m2 )( p− − √ 2m2 κ a− ) + 1 2 i κ m2 ( p+ − √ 2m2 κ a†− )(( x− + i κ 2m2 p− ) − t m p− ) + 1 2 ( i κ m2 ( p+ − √ 2m2 κ a†− ))( −i κ m2 ( p− − √ 2m2 κ a− ))) . J (2) = i 2 (( x+p− − p+x− − i κ m2 p+p− ) + i κ m2 ( p+ − √ 2m2 κ a†− ) p− + i κ m2 p+ ( p− − √ 2m2 κ a− )) . These generators commute with the constraint equations and with the Schrödinger operator i∂t−H. Notice that the set of generators do not depend on a+, a†+, and therefore the transition to the Fock space used in [16] is recovered. The Fock expression of a generic element of the Weyl algebra G(X,P ) can be obtained using the expression of the operators X and P given by (2.11). Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane 13 3.2.2 Coordinate representation In the representation of coordinates the time Schrödinger equation and the constraint equa- tions (3.13) in the non-commutative plane becomes [1] Ŝ1Ψ ≡ ( ∂ ∂v− + κ 4 v+ ) Ψ (x, v, t) = 0, Ŝ2Ψ ≡ ( ∂ ∂x+ − i m 4 v− − i m κ ∂ ∂v+ ) Ψ (x, v, t) = 0, Ŝ3Ψ ≡ ( i ∂ ∂t + 2 m ∂2 ∂x+∂x− ) Ψ (x, v, t) = 0. In this representation, the operators associated to the generators of the Heisenberg algebra are P̂1 = −i ∂ ∂x+ − i ∂ ∂x− , P̂2 = ∂ ∂x+ − ∂ ∂x− , ˆ̃K1 = m 2 (x+ + x−) + ( it− κ 2m ) ∂ ∂x+ + ( it+ κ 2m ) ∂ ∂x− + κ 4i (v+ − v−) + i ∂ ∂v+ + i ∂ ∂v− , ˆ̃K2 = m 2i (x+ − x−)− ( t+ i κ 2m ) ∂ ∂x+ + ( t− i κ 2m ) ∂ ∂x− − κ 4 (v+ + v−)− ∂ ∂v+ + ∂ ∂v− , or, in covariant form, P̂i = −i ∂ ∂xi , ˆ̃Ki = mxi + it ∂ ∂xi + i κ 2m εij ∂ ∂xj + κ 2 εijvj + i ∂ ∂vi , which, indeed, satisfy [P̂i, ˆ̃Kj ] = −imδij , with all the other commutators equal to zero. It is immediate to check that the operators P̂i, ˆ̃Ki commute with all of Ŝ1, Ŝ2 and Ŝ3, and hence that they generate Schrödinger symmetries for the free particle in the non-commutative plane. The rest of generators of the Schrödinger algebra are given by Ĥ = − 2 m ∂2 ∂x+ ∂x− = − 1 2m ∂2 ∂xi2 , Ĵ = −iεijxi ∂ ∂xj + κ 2m2 ∂2 ∂xi2 − i κ 2m vi ∂ ∂xi − 1 m εij ∂2 ∂xi ∂vj , D̂ = −ixi ∂ ∂xi + 1 m t ∂2 ∂xi2 + 1 m ∂2 ∂xi ∂vi + i κ 2m εijvi ∂ ∂xj − i, Ĉ = 2itxi ∂ ∂xi + i κ2 2m2 vi ∂ ∂xi + i κ m εijxi ∂ ∂xj − i κ m tεijvi ∂ ∂xj − i κ m εijvi ∂ ∂vj + 2ixi ∂ ∂vi − 1 m t2 ∂2 ∂xi2 − κ2 4m3 ∂2 ∂xi2 − 1 m ∂2 ∂vi2 − 2 m t ∂2 ∂xi ∂vi + κ m2 εij ∂2 ∂xi ∂vj +mx2 i + κεijxivj + κ2 4m v2 i + 2it. Using these expressions, one can check explicitly the commutators (3.9), and also that these quadratic generators commute with Ŝ1, Ŝ2 and Ŝ3 (this also follows from the derivation properties of the commutators and the corresponding commutation of the linear generators P̂i, ˆ̃Ki, and this proves that the Schrödinger equation for the free particle in the noncommutative plane has the Schrödinger algebra as a symmetry. Notice, however, that in this coordinate representation of the non-reduced quantum space the quadratic operators contain second order derivatives, and 14 C. Batlle, J. Gomis and K. Kamimura hence do not generate point transformations for the coordinates x, v. This is in agreement with the results obtained in the reduced space quantization and the Fock space representation. In any case, the fact that the linear generators commute with Ŝ1, Ŝ2 and Ŝ3 allows to prove that the quadratic ones also commute, and thus generate symmetries of the Schrödinger equation of the free particle in the non-commutative plane. Acknowledgments We thank Jorge Zanelli for collaboration in some parts of this work and Mikhail Plyushchay for reading the manuscript. We also thank Adolfo Azcárraga and Jurek Lukierski for discussions, and Rabin Banerjee for letting us know about the results in [3]. CB was partially supported by Spanish Ministry of Economy and Competitiveness project DPI2011-25649. 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Phys. 5 (2004), 1101–1109, hep-th/0409260. http://dx.doi.org/10.1016/0550-3213(68)90190-9 http://dx.doi.org/10.1006/aphy.1997.5729 http://arxiv.org/abs/hep-th/9612017 http://arxiv.org/abs/0912.0789 http://dx.doi.org/10.1016/j.crhy.2004.10.005 http://arxiv.org/abs/hep-th/0409260 1 Introduction and results 2 Classical symmetries of the non-relativistic particle Lagrangian in the non-commutative plane 2.1 Reduction of second class constraints 3 Quantum symmetries of free Schrödinger equation in the non-commutative plane 3.1 Quantization in the reduced phase space 3.2 Quantization in the extended phase space 3.2.1 Fock representation 3.2.2 Coordinate representation References

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