Searching for effects of spatial noncommutativity via a Chern-Simons process

Jian-Zu ZhangInstitute for Theoretical Physics, East China University of Science and Technology, Box 316, Shanghai 200237, P. R. China

(Received 6 September 2005; revised manuscript received 24 September 2006; published 4 December 2006)

The possibility of testing spatial noncommutativity in the case of both position-position andmomentum-momentum noncommuting via a Chern-Simons’ process is explored. A Chern-Simonsprocess can be realized by an interaction of a charged particle in special crossed electric and magneticfields, in which the Chern-Simons term leads to nontrivial dynamics in the limit of vanishing kineticenergy. Spatial noncommutativity leads to the spectrum of the orbital angular momentum possessingfractional values. Furthermore, in both limits of vanishing kinetic energy and subsequent vanishingmagnetic field, the Chern-Simons term leads to this system having nontrivial dynamics again, and thedominant value of the lowest orbital angular momentum being @=4, which is a clear signal of spatialnoncommutativity. An experimental verification of this prediction by a Stern-Gerlach-type experiment issuggested.

DOI: 10.1103/PhysRevD.74.124005 PACS numbers: 04.20.Gz

I. INTRODUCTION

Studies of low energy effective theory of superstringsshow that space is noncommutative [1–8]. Spatial non-commutativity is apparent near the Planck scale. Its mod-ifications to ordinary quantum theory are extremely small.We ask whether one can find some low energy detectablerelics of physics at the Planck scale by current experi-ments. Such a possibility is inferred from the incompletedecoupling between effects at high and low energy scales.For the purpose of clarifying phenomenological low en-ergy effects, quantum mechanics in noncommutative space(NCQM) is available. If NCQM is a realistic physics, allthe low energy quantum phenomena should be reformu-lated in it. In literature, NCQM have been studied in detail[9–15]; many interesting topics, from the Aharonov-Bohmeffect to the quantum Hall effect have been considered[16–25]. Recent investigations of the deformedHeisenberg-Weyl algebra (the NCQM algebra) in noncom-mutative space explore some new features of effects ofspatial noncommutativity [15]. The possibility of testingspatial noncommutativity via Rydberg atoms is explored.But there are two problems in the suggested experiment ofRydberg atoms: (1) The special arrangement of the electricfield required in the experiment is difficult to realized inlaboratories; (2) The measurement depends on a extremelyhigh characteristic frequency which may be difficult toreach by current experiments.

In this paper we show a possibility of testing spatialnoncommutativity via a Chern-Simons process. Chern-Simons’ processes [26–28] exhibit interesting propertiesin physics. In laboratories a Chern-Simons process can berealized by an interaction of a charged particle in specialcrossed electric and magnetic fields, in which the experi-mental situation is different from one in the experiment ofRydberg atoms. Properties of the Chern-Simons process atthe level of NCQM are investigated. Spatial noncommuta-tivity leads to the spectrum of the orbital angular momen-

tum possessing a fractional zero-point angular momentum.In the limit of vanishing kinetic energy the Chern-Simonsterm leads to this system having nontrivial dynamics. Forthe case of both position-position and momentum-momentum noncommuting in a further limit of the subse-quent diminishing magnetic field this system possessesnontrivial dynamics again, and the dominant value of thelowest orbital angular momentum in the process is @=4.This result is a clear signal of spatial noncommutativity,and can be verified by a Stern-Gerlach-type experiment, inwhich two difficulties in the experiment of Rydberg atomsare resolved.

In Ref. [18] other electromagnetic effects of spatialnoncommutativity was explored.

II. THE DEFORMED HEISENBERG-WEYLALGEBRA

In the following we review the background of the de-formed Heisenberg-Weyl algebra. In order to develop theNCQM formulation we need to specify the phase space andthe Hilbert space on which operators act. The Hilbert spacecan consistently be taken to be exactly the same as theHilbert space of the corresponding commutative system[9].

As for the phase space we consider both position-position noncommutativity (position-time noncommuta-tivity is not considered) and momentum-momentum non-commutativity. There are different types ofnoncommutative theories, for example, see a review paper[8].

In the case of both position-position and momentum-momentum noncommuting the consistent deformedHeisenberg-Weyl algebra [15] is:

�xI; xJ� � i�2�IJ; �xI; pJ� � i@�IJ;

�pI; pJ� � i�2�IJ; �I; J � 1; 2; 3�;(1)

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where �IJ and �IJ are the antisymmetric constant parame-ters, independent of the position and momentum. We de-fine �IJ � �IJK�K (Henceforth the summation conventionis used), where �IJK is a three-dimensional antisymmetricunit tensor. We put �3 � � and the rest of the�-components to zero (which can be done by a rotationof coordinates), then we have �ij � �ij� (i; j � 1; 2),where �ij3 is rewritten as a two-dimensional antisymmetricunit tensor �ij, �12 � ��21 � 1, �11 � �22 � 0. Similarly,we have �ij � �ij�. In Eqs. (1) the scaling factor � is

� ��1�

1

4@2 ����1=2

: (2)

It plays a role for guaranteeing consistent representationsof �xi; pj� in terms of the undeformed canonical variables�xi; pj� (See Eqs. (7)).

In noncommutative space questions about whether theconcept of identical particles being still meaningful andwhether Bose-Einstein statistics being still maintainedshould be answered. Bose-Einstein statistics can be inves-tigated at two levels: the level of quantum field theory andthe level of quantum mechanics. On the fundamental levelof quantum field theory the annihilation and creation op-erators appear in the expansion of the (free) field operator��x� �

Rd3kak�t��k�x� � H:c:. The consistent multipar-

ticle interpretation requires the usual (anti)commutationrelations among ak and ayk . Introduction of the Moyaltype deformation of coordinates may yield a deformationof the algebra between the creation and annihilation op-erators [29]. Whether the deformed Heisenberg-Weyl al-gebra is consistent with Bose-Einstein statistics is an openissue at the level of quantum field theory.

In this paper our study is restricted in the context ofnonrelativistic quantum mechanics. Following the standardprocedure in the ordinary quantum mechanics in commu-tative space we construct the deformed annihilation-creation operators �ai; a

yi � which are related to the de-

formed canonical variables �xi; pi�. In order to maintainthe physical meaning of ai and ayi the relations among�ai; a

yi � and �xi; pi� should keep the same formulation as

the ones in commutative space. For a system with mass �and frequency ! � !p=2 (Here the reason of introducing!p=2 is that in the Hamiltonian (13) the potential energytakes the same form as one of harmonic oscillator) the aireads

a i �������������!p

4@

r �xi � i

2

�!ppi

�: (3)

From Eq. (3) and the deformed Heisenberg-Weyl algebra(1) we obtain the commutation relation between the opera-tors ai and aj: �ai; aj� � i�2�!p�ij��� 4�=�2!2

p�=4@.When the state vector space of identical bosons is con-structed by generalizing one-particle quantum mechanics,in order to maintain Bose-Einstein statistics at the de-

formed level described by ai the basic assumption is thatoperators ai and aj should be commuting. This require-ment leads to a consistency condition

� � 14�

2!2p�: (4)

which puts constraint between the parameters � to �. Thecommutation relations of ai and ayj are

�ai; ayj � � �ij � i

1

2@�2�!p��ij; �ai; aj� � 0: (5)

Here, the three equations �a1; ay1 � � �a2; a

y2 � � 1,

�a1; a2� � 0 are the same boson algebra as the one incommutative space. The equation

�a1; ay2 � � i

1

2@�2�!p� (6)

is a new type. Different from the case in commutativespace, it correlates different degrees of freedom to eachother, so it is called the correlated boson commutationrelation. It encodes effects of spatial noncommutativity atthe deformed level described by �ai; a

yj �, and plays essen-

tial roles in dynamics [15].It is worth noting that Eq. (6) is consistent with all

principles of quantum mechanics and Bose-Einsteinstatistics.

If momentum-momentum were commuting, � � 0, wecould not obtain �ai; aj� � 0. It is clear that in order tomaintain Bose-Einstein statistics for identical bosons at thedeformed level we should consider both position-positionnoncommutativity and momentum-momentum noncom-mutativity. In this paper momentum-momentum noncom-mutativity means the intrinsic noncommutativity. It differsfrom the momentum-momentum noncommutativity in anexternal magnetic field; In that case the correspondingnoncommutative parameter is determined by the externalmagnetic field. Here both parameters � and � should beextremely small, which is guaranteed by the consistencycondition (4).

The deformed Heisenberg-Weyl algebra (1) has differ-ent realizations by undeformed variables �xi; pi� [14]. Weconsider the following consistent ansatz of a linear repre-sentation of the deformed variables �xi; pj� by the unde-formed variables �xi; pj�:

x i � ��xi �

1

2@��ijpj

�; pi � �

�pi �

1

2@��ijxj

�:

(7)

where xi and pi satisfy the undeformed Heisenberg-Weylalgebra �xi; xj� � �pi; pj� � 0, �xi; pj� � i@�ij. It is worthnoting that the scaling factor � is necessary for guarantee-ing that the Heisenberg commutation relation �xi; pj� �i@�ij is maintained by Eq. (7).

The last paper in Ref. [15] clarified that though thedeformed xi and pj are related to the undeformed xi and

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pj by the linear transformation (7), the deformedHeisenberg-Weyl algebra is related to the undeformedone by a similarity transformation with a nonorthogonalreal matrix and a unitary similarity transformation whichtransforms two algebras to each other does not exist, thustwo algebras are not unitarily equivalent.

III. CHERN-SIMONS’ INTERACTIONS

Physical systems confined to a space-time of less thanfour dimensions show a variety of interesting properties.There are well-known examples, such as the quantum Halleffect, high Tc superconductivity, cosmic string in planargravity, etc. In many of these cases the Chern-Simonsinteraction [26–28] which exists in 2� 1 dimensionsand is associated with the topologically massive gaugefields, plays a crucial role. In laboratories a Chern-Simons’ process can be realized by an interaction of acharged particle in special crossed electric and magneticfields, an example is a Penning trap [30–32], in which ananalog of the Chern-Simons term reads

�ijxipj:

This term leads to nontrivial dynamics in the limit ofvanishing kinetic energy, and in turn a testable effect ofspatial noncommutativity.

The objects trapped in a Penning trap are charged par-ticles or ions. The trapping mechanism combines an elec-trostatic quadrupole potential

� �V0

2d2

��

1

2x2i � x

23

�; �i � 1; 2�; (8)

and a uniform magnetic field B aligned along the z axis.The vector potential Ai corresponding to the uniform mag-netic field B reads

A i �12�ijBxj: (9)

The parameters V0�>0� and d are the characteristic voltageand length. The particle oscillates harmonically with anaxial frequency !z � �qV0=�d

2�1=2 (charge q > 0) alongthe axial direction (the z-axis), and in the (1, 2)–plane,executes a superposition of a fast circular cyclotron motionof a cyclotron frequency!c � qB=�cwith a small radius,and a slow circular magnetron drift motion of a magnetronfrequency !m � !2

z=2!c in a large orbit. Typically thequadrupole potential superimposed upon the magneticfield is a relatively weak addition in the sense that thehierarchy of frequencies is

!m !z !c: (10)

The Hamiltonian H of this system can be decomposed intoa two-dimensional Hamiltonian H2 and a one-dimensionalharmonic Hamiltonian Hz:

H �1

2�

�pi �

qcAi

�2� q� � H2 � Hz; (11)

H z �1

2�p2

3 �1

2�!2

z x23; (12)

and H2 is [30,31]

H 2 �1

2�p2i �

1

8�!2

px2i �

1

2!c�ijxipj; (13)

where � is the particle mass, !p � !c�1–4!m=!c�1=2. If

NCQM is a realistic physics, low energy quantum phe-nomena should be reformulated in this framework. In theabove the noncommutative Hamiltonian (13) is obtainedby reformulating the corresponding commutative oneH2 � p2

i =2���!2px2

i =8�!c�ijxipj=2 in commutativespace in terms of the deformed canonical variables xi andpi.

In Eq. (13) the term !c�ijxipj=2 plays an interestingrole of realizing analogs of the Chern-Simons theory [26–28].

In order to explore the new features of such a system, ourattention is focused on the investigation of H2 and the zcomponent of the orbital angular momentum. There aredifferent ways to define the deformed angular momentumin noncommutative space.

(i) As a generator of rotations at the deformed level thedeformed angular momentum J0z should transform xi andpj as two-dimensional vectors [14]:

�J0z; xi� � i�ijxj; �J0z; pi� � i�ijpj:

Comparing to the case in commutative space, the deformedangular momentum J0z acquires �- and �-dependent scalarterms xixi and pipi,

J 0z �@

2

@2 � �4��

��ijxipj �

�2�2@

xixi ��2�2@

pipi

�: (14)

(ii) The quantum mechanical system described by thedeformed Hamiltonian Eq. (13), or equivalently Eq. (18),possesses a full rotational symmetry in (1, 2)–plane. Thegenerator of those rotations is given as

Jz � �ijxipj; (15)

i.e., all quantities xi, pi, xi, pi transforms as two-dimensional vectors.

(iii) The third point of view is as follows: If NCQM is arealistic physics, all deformed observables (the deformedHamiltonian, the deformed angular momentum, etc.) innoncommutative space can be obtained by reformulatingthe corresponding undeformed ones in commutative spacein terms of deformed canonical variables. Thus the de-formed angular momentum Jz, like the deformedHamiltonian (13), keeps the same representation as theundeformed one Jz, but is reformulated in terms of xi

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and pi, i.e., the Chern-Simons term

J z � �ijxipj: (16)

Our starting point is at the deformed level. Because ofthe scalar terms in J0z have nothing to do with the angularmomentum, so in this paper we prefer to take Eq. (16) asthe definition of Jz. Equation (6) modifies the commutationrelations between Jz and xi, pi. From the NCQM algebra(1) we obtain

�Jz; xi� � i�ijxj � i�2�pi;

�Jz; pi� � i�ijpj � i�2�xi:

Comparing with the commutative case, the above commu-tation relations acquires �- and �-dependent terms whichrepresent effects in noncommutative space. From the abovecommutation relations we conclude that Jz plays approxi-mately the role of the generator of rotations at the de-formed level.

All quantities Jz, J0z, Jz and H2 commute each other, thus

have common eigenstates. For example, from Eqs. (1), (4),(13), and (16) it follows that

�Jz; H2� � 0: (17)

Here cancellations between �- and �- dependent terms,provided by the consistency condition (4), is crucial forobtaining Eq. (17).

The �- and �- dependent terms of the Chern-Simonsterm Jz have no direct relation to the angular momentum,see Eq. (20). The essential point is whether the interactionwith magnetic field (the Stern-Gerlach part of the appara-tus) is mediated by Jz or Jz. The definition of a generator ofrotations elucidates that the interaction with magnetic fieldis mediated by Jz. It is worth noting that in both limits ofvanishing kinetic energy and subsequent vanishing mag-netic field Jz is proportional to Jz, see Eqs. (39)–(41). Thismeans that in the particular limit the eigenvalue of theChern-Simons term Jz should appear in the spectrum ofangular momentum.H2 and Jz constitute a complete set of observables of the

two-dimensional subsystem. Using Eqs. (7) theHamiltonian H2 is represented by undeformed variablesxi and pi as

H 2 �1

2M

�pi �

1

2MG�ijpixj

�2�

1

2Kx2

i

�1

2Mp2i �

1

2MG�ijpixj �

1

8M�2

px2i ; (18)

where the effective parametersM,G, �p andK are definedas

1=M � �2�b21=�� qV0�

2=8d2@

2�;

G=M � �2�2b1b2=�� qV0�=2d2@�;

M�2p � �2�4b2

2=�� 2qV0=d2�;

K � �G2=M�M�2p�=4;

(19)

and b1 � 1� qB�=4c@, b2 � qB=2c� �=2@. The pa-rameter K consists of the difference of two terms. It isworth noting that the dominant value of K is qV0=2d2 ��!2

z=2, which is positive.Similarly, from Eq. (7) and (16) the Chern-Simons term

Jz is rewritten as

J z � �ijxipj �1

2@�2��pipi � �xixi�

� Jz �1

2@�2��pipi � �xixi�: (20)

The deformed Heisenberg-Weyl algebra and the unde-formed one are, respectively, the foundations of noncom-mutative and commutative quantum theories. Because ofthe unitary unequivalency between two algebras it is ex-pected that the spectrum of deformed observables (theHamiltonian H2, the angular momentum Jz, etc.) may bedifferent from the spectrum of the corresponding unde-formed ones (H2, Jz, etc.).

IV. DYNAMICS IN THE LIMITING CASE OFVANISHING KINETIC ENERGY

In the following we are interested in the system (18) forthe limiting case of vanishing kinetic energy. In this limitthe Hamiltonian (18) has nontrivial dynamics, and thereare constraints which should be carefully considered[15,33,34]. For this purpose it is more convenient towork in the Lagrangian formalism. The limit of vanishingkinetic energy in the Hamiltonian formalism identifies withthe limit of the mass M ! 0 in the Lagrangian formalism.In Eq. (18) in the limit of vanishing kinetic energy, 1

2M

�pi �12G�ijxj�

2 � 12M _xi _xi ! 0, the Hamiltonian H2 re-

duces to

H0 � �12Kxixi: (21)

The Lagrangian corresponding to the Hamiltonian (18) is

L � 12M _xi _xi �

12G�ij _xixj �

12Kxixi: (22)

In the limit of M ! 0 this Lagrangian reduces to

L0 � �12G�ij _xixj �

12Kxixi: (23)

From L0 the corresponding canonical momentum is p0i �

@L0=@ _xi � �12G�ijxj; and the corresponding Hamiltonian

is H00 � p0i _xi � L0 � �12Kxixi � H0: Thus we identify

the two limiting processes. Here the point is that when thepotential is velocity-dependent, the limit of vanishing ki-netic energy in the Hamiltonian does not correspond to the

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limit of vanishing velocity in the Lagrangian. If the veloc-ity approached zero in the Lagrangian, there would be noway to define the corresponding canonical momentum,thus there would be no dynamics.

The massless limit have been studied by Dunne, Jackiw,and Trugenberger [28].

The first equation of (18) shows that in the limit M ! 0there are constraints

Ci � pi �12G�ijxj � 0; (24)

which should be carefully treated. In this example theFaddeev-Jackiw’s symplectic method [35] leads to thesame results as the Dirac method for constrained quantiza-tion, and the representation of the symplectic method ismuch streamlined. In the following we adopt the Diracmethod [36]. The Poisson brackets of the constraints arefCi; CjgP � G�ij � 0, so that the corresponding Diracbrackets of the canonical variables xi, pj can be deter-mined,

fxi; pjgD �1

2�ij; ; fx1; x2gD � �

1

G;

fp1; p2gD � �1

4G:

(25)

The Dirac brackets of Ci with any variables xi and pj arezero so that the constraints are strong conditions, and canbe used to eliminate the dependent variables. If we selectx1 and p1 as the independent variables, from the con-straints we obtain x2 � �2p1=G, p2 � Gx1=2. We intro-duce new canonical variables q �

���2px1 and p �

���2pp1

which satisfy the Heisenberg quantization condition�q; p� � i@, and define the effective mass �� and theeffective frequency !� as

�� �G2

2K; !� �

KG

(26)

then the Hamiltonian H2 reduces to

H0 � �

�1

2��p2 �

1

2��!�2q2

�: (27)

We define an annihilation operator

A �

��������������!�

2@

sq� i

�����������������1

2@��!�

sp; (28)

The annihilation and creation operators A and Ay satisfies�A; Ay� � 1, and the eigenvalues of the number operatorN � AyA is n0 � 0; 1; 2; � � � .

The Hamiltonian H0 is rewritten as

H0 � �@!��AyA� 1

2�: (29)

Similarly, the angular momentum Jz and the Chern-Simonsterm Jz reduce, respectively, to the following J0z and J0z

J0z � @�AyA� 12�; J0z � @J ��AyA� 1

2� � J �J0z;

(30)

where

J � � 1� �2

�G�4@�

�G@

�: (31)

The eigenvalues of H0 and Jz are, respectively,

E�n � �@!��n0 � 1

2�; (32)

J �n � @J ��n0 � 1

2�; (33)

The eigenvalue of H0 is negative, thus unbound. Thismotion is unstable. It is worth noting that the dominantvalue of!� is the magnetron frequency!m, i.e. in the limitof vanishing kinetic energy the surviving motion is mag-netronlike, which is more than adequately metastable[30,31].

The �- and �-dependent terms of J � take fractionalvalues. Thus the Chern-Simons term Jz possesses frac-tional eigenvalues and fractional intervals.

Using the consistency condition (4) we rewrite the J � inEq. (31) as J � � 1�O���. From Eq. (33) it follows thatthe zero-point value J �0 reads

J �0 �

12@�O���: (34)

For the case of both position-position and momentum-momentum noncommuting we can consider a further limit-ing process. After the sign of V0 is changed, the definitionof �p shows that the limit of magnetic field B! 0 ismeaningful, and the survived system also has nontrivialdynamics. In this limit the frequency !p reduces to ~!p ����

2p!z, the consistency condition (4) becomes a reduced

consistency condition

� � 12�

2!2z�; (35)

and the scaling parameter � in Eq. (2) reduces to

~� ��1�

1

8@2 �2!2

z�2

��1=2

� 1�O��2�: (36)

The effective parameters M, G, �p and K reduce, respec-tively, to the following effective parameters ~M, ~G, ~�p and~K, which are defined by

~M ��

~�2�

1

��

1

8@2 �!2z�

2

���1� �;

~G~M� ~�2

���@�

1

2@�!2

z���

1

@�!2

z��O��3�;

~�2p � ~�2

��2

�2@

2 � 2!2z

�� 2!2

z �O��2�;

~K �1

4�

~G2

~M� ~M ~�2

p� � �1

2�!2

z �O��2�:

(37)

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In this limit H0 and J0z reduce, respectively, to the follow-ing ~H0 and ~Jz:

~H 0 � �12

~Kx2i � @ ~!� ~Ay ~A� 1

2�; (38)

~J z � @ ~J � ~Ay ~A� 12� �

~J J0z: (39)

In this limit the angular momentum J0z is not changed, butcan be rewritten as

J0z � @� ~Ay ~A� 12�: (40)

In Eq. (39)

~J � 1� ~�2� ~G�

4@�

�~G@

�: (41)

In the above the annihilation operator is defined as

~A �

���������~� ~!2@

sq� i

��������������1

2@ ~� ~!

sp; (42)

and the effective mass ~� and the effective frequency ~! are

~� � �~G2

2 ~K�>0�; ~! �

~K~G: (43)

The annihilation and creation operators ~A and ~Ay satisfies� ~A; ~Ay� � 1, and the eigenvalues of the number operator~N � ~Ay ~A is n � 0; 1; 2; � � � .

Equations (38)–(41) show that ~H0, ~Jz and J0z commuteeach other, thus have common eigenstates. The eigenvaluesof ~H0, ~Jz and J0z are, respectively,

~E n � @ ~!�n� 12�; (44)

~J n � @ ~J �n� 12�; J 0n � @�n� 1

2�: (45)

It is worth noting that in both limits of vanishing kineticenergy and subsequent vanishing magnetic field we have~J n � ~JJ 0n.

Now we estimate the dominant value of the constant ~J .A dominant value of an observable means its �- and �-independent term. Generally, the dominant value is just thevalue in commutative space. In some special case theconsistency condition (4) or the reduced consistency con-dition (35) may provides a cancellation between � and � insome term of an observable. This leads to that the dominantvalue is different from one in commutative space.

In the third term of ~J in Eq. (41), unlike the term�=G@ � O��� of J � in Eq. (31), the reduced consistencycondition (35) provides a fine cancellation between � and�. Using Eqs. (35)–(37), this term reads �= ~G@ � 1=2,which leads to

~J � 12�O��

2�: (46)

From Eqs. (45) and (46) it follows that the zero-point value~J 0 is

~J 0 �14@�O��

2�; (47)

and the interval � ~J n of the Chern-Simons term reads

� ~J n �12@�O��

2�: (48)

The dominant values of the zero-point value and the inter-val of the Chern-Simons term are, respectively, @=4 and@=2, which are different from the values in commutativespace. These unusual results explore the essential newfeature of spatial noncommutativity.

V. TESTING SPATIAL NONCOMMUTATIVITY VIAA PENNING TRAP

The dominant value @=4 of the lowest Chern-Simonsterm in a Penning trap can be measured by a Stern-Gerlach-type experiment. The experiment consists of twoparts: the trapping region and the Stern-Gerlach experi-mental region. The trapping region serves as a source of theparticles for the Stern-Gerlach experimental region. Afterestablishing the trap, the experiment includes three steps.

(i) Taking the limit of vanishing kinetic energy. In anappropriate laser trapping field the speed of atomscan be slowed to the extent that the kinetic energyterm may be removed [37]. In the limit of vanishingkinetic energy the situations of the cyclotron motion,the harmonic axial oscillation and the magnetronlikemotion in a Penning trap are different [30,31]. Theenergy in the cyclotron motion is almost exclusivelykinetic energy. The energy in the harmonic axialoscillation alternates between kinetic and potentialenergy. Reducing the kinetic energy in either of thesemotions reduces their amplitude. In contrast to thesetwo motions, the energy in the magnetronlike motionis almost exclusively potential energy. Thus in thelimit of vanishing kinetic energy the harmonic axialoscillation and the cyclotron motion disappear, onlythe magnetronlike motion survives. Any process thatremoves energy from the magnetronlike motion in-creases the magnetron radius until the particle strikesthe ring electrode and is lost from the trap. Themagnetronlike motion is unstable. Fortunately, itsdamping time is on the order of years [30], so thatit is more than adequately metastable. In this limit,the survived magnetronlike motion slowly drifts in alarge orbit in the (1, 2)–plane. At the quantum level,in the limit of vanishing kinetic energy the modewith the frequency !� survives. As we noted before,the dominant value of!� is the magnetron frequency!m, i.e. the surviving mode is magnetronlike.

(ii) Changing the sign of the voltage V0 and subse-quently diminishing the magnetic field B to zero.The voltage V0 is weak enough so that when themagnetic field B approaches zero the trapped parti-cles can escape along the tangent direction of thecircle from the trapping region and are injected intothe Stern-Gerlach experimental region.

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(iii) Measuring the z-component of the lowest Chern-Simons term in the Stern-Gerlach experimental re-gion. As noticing before, the commutation relationsbetween Jz and xi; pi show that Jz plays approxi-mately the role of the generator of rotations at thedeformed level. Equations (38), (41), and (45) elu-cidate that the lowest dominant value @=4 of theChern-Simons term ~Jz can be read out from spec-trum of the angular momentum which are measuredfrom the deflection of the beam in the Stern-Gerlachexperimental region.

VI. DISCUSSIONS

As is well-known, a direct measurement of the magne-tism and the gyromagnetic ratio for free electrons areimpossible. Thus in the above suggested experiment, thetrapped object are chosen as ions.

When ions are injected into the Stern-Gerlach experi-mental region, in order to avoid a disturbance of theLorentz motion in the inhomogeneous magnetic field,they should first go through a region of revival and arerestored to neutral atoms. We should choose ions of thefirst class atoms in periodic table of the elements. Anadvantage of choosing such ions is that in the ordinarycase of commutative space the revived atoms are in theS-state.

Now we estimate the possibility of changing the state ofrevived atoms by effects of spatial noncommutativity. Theeffective frequency ~! in Eq. (43) depends on noncommu-tative parameters. There are different bounds on the pa-rameter � set by experiments. The existing experiments onthe Lorentz symmetry violation placed strong bounds on �[38]: �=�@c�2 �10 TeV��2; Measurements of the Lambshift [9] give a weaker bound; Clock-comparison experi-ments [39] claim a stronger bound. The magnitudes of �and � are surely extremely small. From Eq. (43) it follows

that the dominate value of the frequency ~! reads ~! �j ~Kj= ~G � @=�2���. If we take �c2 � 2 GeV and�=�@c�2 �104 GeV��2 we obtain ~! � 1032 Hz.Equation (44) shows that the corresponding energy interval� ~En is extremely large. Thus revived atoms can not transitto higher exciting states, they are definitely preserved in theground state.

The result obtained in this paper is different from resultsobtained in literature. All effects of spatial noncommuta-tivity explored in literature depend on extremely smallnoncommutative parameters � and/or �, thus can not betested in the foreseeable future. Because of a direct pro-portionality between � and � provided by the reducedconsistency condition (35), in Eq. (41) there is a finecancellation between � and �. This leads to a �- and�-independent effect of spatial noncommutativity whichcan be tested by current experiments.

In both limits of vanishing kinetic energy and subse-quent diminishing magnetic field for the case of onlyposition-position noncommuting dynamics of thePenning trap is trivial; But for the case of both position-position and momentum-momentum noncommuting itsdynamics is nontrivial, and the dominant value @=4 of thelowest Chern-Simons term in the Penning trap is differentfrom the value in commutative space. The above suggestedexperiment can distinguish the case of both position-position and momentum-momentum noncommuting fromthe case of only position-position noncommuting.

ACKNOWLEDGMENTS

The author would like to thank Jean-Patrick Connerade,bai-Wen Li, Ming-Sheng Zhan, Ting-Yun Shi, Si-Hong Guand Qing-Yu Cai for helpful discussions. This work hasbeen partly supported by the National Natural ScienceFoundation of China under the Grant No. 10575037 andby the Shanghai Education Development Foundation.

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