From Noncommutative Sphere to Nonrelativistic Spin

Reparametrization invariant dynamics on a sphere, being parameterized by angular momentum coordinates, represents an example of noncommutative theory. It can be quantized according to Berezin-Marinov prescription, replacing the coordinates by Pauli matrices. Following the scheme, we present two semi...

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Опубліковано в: :	Symmetry, Integrability and Geometry: Methods and Applications
Дата:	2010
Автор:	Deriglazov, A.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2010
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/146154
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Цитувати:	From Noncommutative Sphere to Nonrelativistic Spin / A.A. Deriglazov // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 23 назв. — англ.

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spelling	Deriglazov, A.A. 2019-02-07T19:15:26Z 2019-02-07T19:15:26Z 2010 From Noncommutative Sphere to Nonrelativistic Spin / A.A. Deriglazov // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R05; 81R60; 81T75 https://nasplib.isofts.kiev.ua/handle/123456789/146154 Reparametrization invariant dynamics on a sphere, being parameterized by angular momentum coordinates, represents an example of noncommutative theory. It can be quantized according to Berezin-Marinov prescription, replacing the coordinates by Pauli matrices. Following the scheme, we present two semiclassical models for description of spin without use of Grassman variables. The first model implies Pauli equation upon the canonical quantization. The second model produces nonrelativistic limit of the Dirac equation implying correct value for the electron spin magnetic moment. This paper is a contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries (June 18–20, 2009, Prague, Czech Republic). The full collection is available at http://www.emis.de/journals/SIGMA/ISQS2009.html. The work has been supported by the Brazilian foundations CNPq (Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico - Brasil) and FAPEMIG. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications From Noncommutative Sphere to Nonrelativistic Spin Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	From Noncommutative Sphere to Nonrelativistic Spin
spellingShingle	From Noncommutative Sphere to Nonrelativistic Spin Deriglazov, A.A.
title_short	From Noncommutative Sphere to Nonrelativistic Spin
title_full	From Noncommutative Sphere to Nonrelativistic Spin
title_fullStr	From Noncommutative Sphere to Nonrelativistic Spin
title_full_unstemmed	From Noncommutative Sphere to Nonrelativistic Spin
title_sort	from noncommutative sphere to nonrelativistic spin
author	Deriglazov, A.A.
author_facet	Deriglazov, A.A.
publishDate	2010
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	Reparametrization invariant dynamics on a sphere, being parameterized by angular momentum coordinates, represents an example of noncommutative theory. It can be quantized according to Berezin-Marinov prescription, replacing the coordinates by Pauli matrices. Following the scheme, we present two semiclassical models for description of spin without use of Grassman variables. The first model implies Pauli equation upon the canonical quantization. The second model produces nonrelativistic limit of the Dirac equation implying correct value for the electron spin magnetic moment.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/146154
citation_txt	From Noncommutative Sphere to Nonrelativistic Spin / A.A. Deriglazov // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 23 назв. — англ.
work_keys_str_mv	AT deriglazovaa fromnoncommutativespheretononrelativisticspin
first_indexed	2025-11-25T01:47:31Z
last_indexed	2025-11-25T01:47:31Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 016, 8 pages From Noncommutative Sphere to Nonrelativistic Spin? Alexei A. DERIGLAZOV Dept. de Matematica, ICE, Universidade Federal de Juiz de Fora, MG, Brazil E-mail: alexei.deriglazov@ufjf.edu.br Received November 12, 2009, in final form January 26, 2010; Published online February 04, 2010 doi:10.3842/SIGMA.2010.016 Abstract. Reparametrization invariant dynamics on a sphere, being parameterized by angular momentum coordinates, represents an example of noncommutative theory. It can be quantized according to Berezin–Marinov prescription, replacing the coordinates by Pauli matrices. Following the scheme, we present two semiclassical models for description of spin without use of Grassman variables. The first model implies Pauli equation upon the cano- nical quantization. The second model produces nonrelativistic limit of the Dirac equation implying correct value for the electron spin magnetic moment. Key words: noncommutative geometry; nonrelativistic spin 2010 Mathematics Subject Classification: 81R05; 81R60; 81T75 1 Introduction In their pioneer work [1], Berezin and Marinov have suggested semiclassical description of a spin based on anticommuting variables. Their prescription can be shortly resumed as follows. For nonrelativistic spin, an appropriate Lagrangian reads 1 2(ẋi)2 + i 2ξiξ̇i, where the spin inner space is constructed from vector like Grassmann variables ξi, ξiξj = −ξjξi. Since the Lagrangian is linear on ξ̇i, their conjugate momenta coincide with ξ, πi = ∂L ∂ξ̇i = iξi. The relations represent second class constraints of a Hamiltonian formulation and are taken into account by transition from Grassmann Poisson bracket to the Dirac one, the latter reads {ξi, ξj} = iδij . (1) Dealing with the Dirac bracket, one can resolve the constraints, excluding the momenta from consideration. It gives very economic scheme for description of a spin: there are only three spin variables ξi. In accordance with equation (1), canonical quantization is performed replacing the variables by Pauli σ-matrices, [σi, σj ]+ = 2δij , acting on two dimensional spinor space Ψα. By this way, formal application of the Dirac method to Grassmann mechanics with constraints allows one to describe both nonrelativistic and relativistic spinning particle on an external electromagnetic field, see [2] for a review. It is naturally to ask whether a similar scheme can be realized with use of commuting variables only (in this relation, let us point out that there is no generalization of Grassmann mechanics on higher spins [3]). While description of a spin without Grassmann variables is a problem with a long history [4], search for spinning particle that would give reasonable classic and quantum theory remains under investigation up to date [5]. To apply the Berezin–Marinov prescription to commuting variables, one needs to construct a dynamical system with an inner space endowed with an algebra that can be realized by σ- matrices (we discuss the nonrelativistic spin). Due to symmetry properties of a Dirac bracket, it ?This paper is a contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Sys- tems and Quantum Symmetries (June 18–20, 2009, Prague, Czech Republic). The full collection is available at http://www.emis.de/journals/SIGMA/ISQS2009.html mailto:alexei.deriglazov@ufjf.edu.br http://dx.doi.org/10.3842/SIGMA.2010.016 http://www.emis.de/journals/SIGMA/ISQS2009.html 2 A.A. Deriglazov is not possible to arrive at the anticommutator bracket (1) working with commuting variables, say Ji. Instead, one can try to produce a bracket of the form {Ji, Jj} = εijkJk, the latter can also be realized quantum mechanically by σ-matrices. We discuss such a kind possibility in the present work. Our aim will be to construct a dynamical system that, at the end, admit three degrees of freedom Ji as the spin space basic variables, the latter obey SO(3) algebra with fixed value of Casimir operator. One notices that the spin inner space equipped with SO(3)-algebra represents an example of noncommutative (NC) system. Idea of noncommutativity became quite popular after the observation made in [6] that in certain limits string theory can be formulated as an effective field theory in NC space-time. Some well-known physical systems can be also treated from NC point of view, an example is a charged particle confined to the plane perpendicular to an external magnetic field. The space becomes noncommutative in the lowest Landau level [7]. Incorporation of NC geometry into the field theory framework can be naturally achieved using the Dirac method for analysis the constrained theories: NC geometry arises due to the Dirac bracket, after taking into account the constraints presented in the model. It seems to be reasonable approach, since at least the pioneer NC models [7, 8, 6] all can be properly treated by this way [9, 8]. Moreover, following this way, with any classical mechanical system one associates its NC version [10, 11, 12, 13, 14]. Since NC geometry turns out to be useful tool for reformulation and investigation of some problems in classical and quantum mechanics [10, 11, 12, 12, 14, 15, 16] as well as in QFT [17], there are attempts to formulate it on a more fundamental level (see [18] for the recent review). One of the barriers here is relativistic (Galilean) invariance. Except a couple of specific models, compatibility of NC geometry with the symmetries remains an open problem. In 2+1 dimensions, the Landau problem [7] and the Lukierski–Stichel–Zakrzewski higher derivative NC particle [8] are compatible with the Galilean invariance. In four dimensions situation is less promising. Similarly to the Snyder NC space [19], d = 4 NC particle turns out to be compatible with relativistic invariance realized only as a dynamical symmetry [10, 20, 21]. In the present work we avoid the problem since noncommutativity is associated with the inner space, being not only compatible but responsible for appearance the spin representation of SO(3) group. In this work we concentrate mainly on algebraic construction of the inner spin space, the latter presented in Section 2. Its dynamical realizations are only sketched, technical details will be presented elsewhere. As the dynamical realizations we discuss two different D = 3 + 1 spinning particles in Section 3. The first model implies Pauli equation upon the canonical quantization. The second model produces nonrelativistic limit of the Dirac equation thus leading to correct value for the electron spin magnetic moment. 2 Noncommutative sphere algebra Consider canonical pairs vi, πj , i, j = 1, 2, 3 with the Poisson bracket algebra being {vi, πj} = δij . To arrive at the desired SO(3) algebra, we restrict the initial system to lie on some d = 2 surface of the six dimensional phase space. It will be made in two steps. First we constrain the coordinates to lie on d = 4 surface specified by v2 = a2, vπ = 0, a = const. (2) From Noncommutative Sphere to Nonrelativistic Spin 3 The constraints form a second class system, {v2−a2, vπ} = 2v2. So one takes them into account by transition from the Poisson to Dirac bracket, { , }D1, the latter reads {A,B}D1 = {A,B} − {A, vπ} 1 2v2 { v2, B } + { A, v2 } 1 2v2 {vπ, B}. (3) Here A, B are phase space functions. For the phase space coordinates it implies the algebra1 {vi, vj}D1 = 0, {vi, πj}D1 = δij − 1 v2 vivj , {πi, πj}D1 = − 1 v2 (viπj − vjπi). (4) Constraints are consistent with the Dirac algebra, that is {A, v2− a2}D1 = 0, {A, vπ}D1 = 0 for any phase space function A(v, π). So one can resolve the constraints, keeping four independent coordinates and the corresponding algebra. We introduce coordinates that turn out to be convenient for canonical quantization of the system. Consider the quantities Ji ≡ εijkvjπk, π̃1 = π1, π̃2 = π2, s = viπi, (5) where εijk is three dimensional Levi-Civita tensor, ε123 = 1, εijk = ε[ijk]. The quantities Ji obey SO(3) angular momentum algebra with respect to both Poisson and Dirac brackets. Equa- tions (5) can be inverted π1 = π̃1, π2 = π̃2, π3 = −J1π̃1 + J2π̃2 J3 , (6) vi = 1 π2 (εijkπjJk + sπi), where πi in the last equality are given by (6). Hence Ji, π̃1, π̃2, s can be used as coordinates of the six dimensional space instead of vi, πi. Equations of the surface (2) in these coordinates acquire the form J2 = a2π̃2, s = 0. One can compute the Dirac bracket (3) for the new coordinates2. To keep the manifest SO(3) covariance of the formalism, the equation (6) prompts to introduce π̃3 ≡ −J1π̃1+J2π̃2 J3 (or, equiva- lently, Jπ̃ = 0). Then the Dirac brackets of Ji, π̃i are {Ji, Jj}D1 = εijkJk, (7) {π̃i, π̃j}D1 = − 1 a2 εijkJk, {Ji, π̃j}D1 = εijkπ̃k, (8) while the Dirac bracket of s with any other coordinate vanishes. Since s = 0 on the constraint surface, it can be omitted from consideration. In resume, the surface (2) is now described by coordinates Ji, π̃i that are constrained by Jπ̃ = 0, J2 = a2π̃2, (9) and obey the algebra (7), (8). When a2 = 1, it is just the Lorentz group algebra written in terms of the rotation J and the Lorentz boost π̃ generators. Notice also that Dirac bracket of the constraints (9) with any phase space quantity vanishes. 1Equation (4) is analogous to the Snyder noncommutative algebra. I am grateful to the referee for this observation. 2Equivalently, one can start with Poisson bracket of the new variables and construct the Dirac bracket corre- sponding the constraints J2 = a2π̃2, s = 0. 4 A.A. Deriglazov One has already the desired algebra (7), but in the system with four independent variables. To improve this, we impose two more second class constraints and construct the corresponding Dirac bracket. To guarantee that the bracket does not modify its form for Ji, the equation (7), one of constraints must give vanishing D1-bracket with Ji. The only possibility is to take (a function of) Casimir operator of SO(3) algebra. As another constraint, one takes any phase space function that forms second class system with the Casimir operator. The ambiguity in choosing the second constraint suggests that corresponding dynamical realization will be locally invariant theory, see below. Let us take, for example, the constraints J2 − 3~2 4 = 0, ε3jkπ̃jJk = 0 (that is v3 = 0), Using their bracket, {J2 − 3~2 4 , εijkπ̃jJk}D1 = −2π̃3J 2, one obtains D2-Dirac bracket {A,B}D2 = {A,B}D1 − { A, J2 } D1 1 2π̃3J2 {ε3jkπ̃jJk, B}D1 + {A, ε3jkπ̃jJk}D1 1 2π̃3J2 { J2, B } D1 . In the result we have d = 2 surface determined by the constraints Jπ̃ = 0, π̃2 = 3~2 4a2 , π̃1J2 − π̃2J1 = 0, (10) J2 = 3~2 4 . (11) Equations (10) can be used to exclude all π̃i. Then one deal with the remaining variables Ji obeying SO(3) algebra (7) and subject the constraint (11). Finite dimensional irreducible representations of the angular momentum algebra are num- bered by the spin s. The condition (11) fixes the spin s = 1 2 . So, J are quantized by Ji −→ Ĵi = ~ 2 σi, where σi states for the Pauli matrices. They act on space of two dimensional spinors Ψα, α = 1, 2. As a consequence of the σ-matrix commutator algebra, [σi, σj ] = 2iεijkσk, the operators Ĵi obey the quantum counterpart of the classical algebra (7) [Ĵi, Ĵj ] = i~εijkĴk, as well as the constraint (11). 3 Dynamical realizations Here we suggest two different spinning particle models that realize the algebraic construction described above. Nonrelativistic spinning particle implying the Pauli equation. Let us start with discussion of spinning part of a Lagrangian. The constraints (2) as well as the second constraint from equation (10) prompt to write Lspin = 1 2g v̇2 + g b2 2a2 + 1 φ ( v2 − a2 ) , b2 = 3~2 4 . (12) From Noncommutative Sphere to Nonrelativistic Spin 5 Here g(t), φ(t) are auxiliary degrees of freedom. The corresponding Hamiltonian is given by H = g 2 ( π2 − b2 a2 ) − 1 φ ( v2 − a2 ) + λgπg + λφπφ. where πg, πφ are conjugate momenta for g, φ and λ represent lagrangian multipliers for the primary constraints πg = 0, πφ = 0. The variables g, φ are subject to second class constraints πφ = 0, gb2 a2 + 2a2 φ = 0 and to the primary first class constraint πg = 0. The latter is associated with local symmetry of the action (12) (with a parameter being an arbitrary function α(t)) δvi = αv̇i, δg = (αg)., δφ = αφ̇− α̇φ. (13) Imposing the gauge g = 1, the variables g, φ, πg, πφ can be omitted from consideration. Besides one obtains the desirable constraints v2 = a2, vπ = 0, (14) π2 − 3~2 4a2 = 0. (15) The pair (14) is of second class, while (15) represents the first class constraint3. Consider now the spinning particle action S = ∫ dt ( m 2 ẋ2 + e c Aiẋ i − eA0 + 1 2g (v̇i − e mc εijkvjBk)2 + g b2 2a2 + 1 φ ( v2 − a2 )) . Here xi, i = 1, 2, 3, states for spatial coordinates of the particle, and B = ∇ × A. Second and third terms represent minimal interaction with the vector potential A0, Ai of an external electromagnetic field, while the fourth term contains interaction of spin with a magnetic field. At the end, it produces the Pauli term in quantum mechanical Hamiltonian. As it has been discussed above, constraints presented in the model allows one to describe it in terms of the variables xi, its conjugate momenta pi, and the spin vector Ji = εijkvjπk (let us point out that in contrast to vi, πi, the spin vector Ji turns out to be gauge invariant quantity with respect to the local symmetry (13)). One notices that the action leads to reasonable classical theory. In the variables x, J , classical dynamics is governed by the Lagrangian equations mẍi = eEi + e c εijkẋjBk − e mc Jk∂iBk, (16) J̇i = e mc εijkJjBk. (17) It has been denoted E = −1 c ∂A ∂t −∇A0. Since J2 ≈ ~2, the J-term disappears from equation (16) in the classical limit ~ → 0. Then equation (16) reproduces the classical motion in an external electromagnetic field. Notice also that in absence of interaction, the spinning particle does not experience an undesirable self acceleration. Equation (17) describes the classical spin precession in an external magnetic field. In the Hamiltonian formulation, taking into account the presented constraints, one obtains the only non vanishing Dirac brackets {xi, pj} = δij , {Ji, Jj} = εijkJk, and the Hamiltonian H = 1 2m ( pi − e c Ai )2 − e mc JiBi + eA0. 3More exactly, first class constraint is given by the combination π2 − 3~2 4a2 + 3~2 4a4 ( v2 − a2 ) = 0. 6 A.A. Deriglazov Hence canonical quantization of the model implies the Pauli equation i~ ∂Ψ ∂t = ( 1 2m ( p̂− e c A )2 + eA0 − e~ 2mc σB ) Ψ. (18) Nonrelativistic spinning particle that produces quantum mechanical Hamiltonian corresponding nonrelativistic limit of the Dirac equation. One remarkable property of the Dirac equation minimally coupled to the vector potential is that it yields the correct gyromagnetic ratio g = 2 for the electron spin magnetic moment [22]. Technically it happens due to the fact that in nonrelativistic limit arises the Hamiltonian 1 2m(pσ)2 instead of 1 2m(p)2. Then the minimal substitution p→ p− e cA leads automatically to the Pauli term with correct value of the spin magnetic moment4 (see the last term in equation (18)). It is interesting to look for a spinning particle action that would lead to the nonrelativistic limit of the Dirac equation. We propose the action S = ∫ dt [ 3 2 ( ~ √ m 2v2 εijkẋiv̇jvk ) 2 3 + φ ( v2 − a2 )] . The conventional degree, 2 3 , leads at the end to the desired Hamiltonian. The factor ~ √ m implies correct dimension of the action. In the Hamiltonian formulation one obtains the constraints vπ = 0, v2 − a2 = 0, (19) vp = 0. (20) The pair (19) is of second class, while (20) represents the first class constraint associated with the local symmetry δxi = αvi. Let us take the gauge5 J2 = 3~2 4 for the constraint (20). On the constraint surface the classical Hamiltonian is given by H = − 2 m~2 (pJ)2. Canonical quantization p → p̂ = −i~∇, J → Ĵ = ~ 2σ leads to nonrelativistic limit of the Dirac equation H = 1 2m (p̂σ)2. It implies correct value of the spin magnetic moment after the minimal substitution p̂→ p̂− e cA. Acknowledgments The work has been supported by the Brazilian foundations CNPq (Conselho Nacional de De- senvolvimento Cient́ıfico e Tecnológico - Brasil) and FAPEMIG. 4Anomalous magnetic moment is discussed in the quantum mechanics framework in [23]. 5It is admissible gauge for the motions with pπ 6= 0. From Noncommutative Sphere to Nonrelativistic Spin 7 References [1] Berezin F.A., Marinov M.S., Particle spin dynamics as the Grassmann variant of classical mechanics, Ann. Physics 104 (1977), 336–362. [2] Gitman D.M., Path integrals and pseudoclassical description for spinning particles in arbitrary dimensions, Nuclear Phys. B 488 (1997), 490–512, hep-th/9608180. 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From Noncommutative Sphere to Nonrelativistic Spin

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