Dirac Quantization Condition for Monopole in Noncommutative Space-Time
aa r X i v : . [ h e p - t h ] F e b HIP-2009-02/TH
Dirac Quantization Condition for Monopolein Noncommutative Space-Time
Masud Chaichian , , ∗ Subir Ghosh , † Miklos L˚angvik , ‡ and Anca Tureanu , § Department of Physics, University of Helsinki,P.O. Box 64, FIN-00014 Helsinki, Finland Helsinki Institute of Physics, P.O. Box 64, FIN-00014 Helsinki, Finland and Physics and Applied Mathematics Unit,Indian Statistical Institute, Kolkata-700108, India
Abstract
Since the structure of space-time at very short distances is believed to get modified possiblydue to noncommutativity effects and as the Dirac Quantization Condition (DQC), µe = N ~ c ,probes the magnetic field point singularity, a natural question arises whether the same conditionwill still survive. We show that the DQC on a noncommutative space in a model of dynamicalnoncommutative quantum mechanics remains the same as in the commutative case to first orderin the noncommutativity parameter θ , leading to the conjecture that the condition will not alterin higher orders. PACS numbers: 14.80.Hv, 02.40.Gh, 03.65.-w ∗ Electronic address: masud.chaichian@helsinki.fi † Electronic address: subir˙ghosh2@rediffmail.com ‡ Electronic address: miklos.langvik@helsinki.fi § Electronic address: anca.tureanu@helsinki.fi . INTRODUCTION The idea of magnetic monopoles - the so-far hypothetical particles carrying magneticcharge - is one of the most influential in modern theoretical physics. The first effectivetheoretical proposal that magnetic charge should exist was made by Dirac [1], who arguedthat in quantum mechanics the unobservability of phase permits singularities which manifestthemselves as sources of magnetic fields. The Dirac Quantization Condition (DQC), µe = N ~ c , is a topological property, independent of space-time points, that tells us that the mereexistence of a single magnetic monopole would imply that the electric charge is quantized.Before Dirac, the surprising asymmetry in Maxwell’s equations made Poincar´e and J. J.Thompson introduce the magnetic charge in the theory as an artefact for simplifying thecomputation, while P. Curie suggested even the actual existence of magnetic charge [2]. Theidea of magnetic monopoles was later extended by the discovery of monopole solutions ofclassical non-Abelian theories [3, 4] and the introduction of the concept of dyons - particlescarrying both electric and magnetic charge [5], and with its significant influence, eventuallyleading to the concept of duality [6] and on the string theory.In recent years magnetic monopole structures have created a lot of interest in condensedmatter physics. In studying the Anomalous Hall Effect, magnetic monopole structures in momentum space have been experimentally observed and theoretically explained in [7, 8].However, till date no magnetic monopole has been found (for a comprehensive review ofmagnetic monopole searches, see e.g. [9]), but the theoretical interest has stayed, as nothingin the theory has ever been found to contradict the DQC.In the recent decade there has been a growing interest in the research concerning non-commutative spaces mainly due to the results in [10] and [11]. In [11] it was shown thatstring theory in a constant background field leads to a noncommutative field theory as a lowenergy limit.Moreover, the result of [10] has encouraged many to believe that noncommutative fieldtheory is a step towards a more complete description of physics. In the ”gedanken exper-iment” of [10], it was argued that in the process of measurement of space points, as theenergy grows, eventually black holes are formed and consequently objects of smaller extentthan the diameter of the black holes cannot be observed and one can think of space-time”points” as operators obeying a Heisenberg-like uncertainty principle from which it follows2hat space-time is homogeneous and can be interpreted as being noncommutative.Although the main interest in this field lies in the formulation of a consistent field theoryon a noncommutative space-time, it is also interesting to apply the noncommutative space topure quantum mechanics to see whether it is possible to extend ordinary quantum mechanicsto the noncommutative case. Specifically, the result [10] is quantum mechanical in natureand some results such as the DQC [1], which we will be exploring in this Letter, are notobtained directly from field theory.In the noncommutative case, the space-time is particularly sensitive to the short-distanceeffects. Since the DQC in its essence probes the singularity structure of the magnetic field,one would think that this condition could no longer remain valid in the noncommutativecase. This is the main motivation for the present work.We shall start by briefly reviewing one known method for deriving the DQC in thecommutative case. Then, starting from a classical Lagrangian corresponding to a dynamicalmodel of noncommutative quantum mechanics, we shall derive the DQC to first order inthe noncommutativity parameter θ , and finally we shall discuss the result and its possiblegeneralizations. II. ONE WAY OF DERIVING THE DQC
In the commutative case, there is an ingenious way to derive the DQC, first introducedby Jackiw [12], which uses a gauge-invariant algebra, dependent only on the magnetic field.The derivation is an example of the three-cocycles which appear when a representationof a transformation group is nonassociative; in particular, when the translations group isrepresented by gauge-invariant operators in the presence of a magnetic monopole, the Jacobiidentity among the translation generators fails. The restoration of the associativity of finite translations leads to the DQC. A sketch of the derivation [12] will be quite illuminating: fora nonrelativistic particle with the electric charge e , moving in a magnetic field B ( x ), onestarts by finding the non-canonical quantum brackets[ x i , x j ] = 0 , [ x i , π j ] = i ~ δ ij , (1)[ π i , π j ] = i ~ ec ǫ ijk B k ( x ) , (2)3here one defines the operators π i in the x -representation as π i = − i ~ ∂ i − ec A i ( x ) , (3)with A i ( x ) being the vector potential [20]. These commutation relations, along with theHamiltonian H = π m , π = m ˙ x , (4)yield the well-known Lorentz-Heisenberg equations of motion ˙ x = i ~ [ H, x ] = π m , (5) ˙ π = i ~ [ H, π ] = e mc [ π × B − B × π ] , (6)where π is the gauge-invariant mechanical momentum. So far there is no restriction on B but the following Jacobi identity violation,12 ǫ ijk [[ π i , π j ] , π k ] = e ~ c ∇ · B , (7)indicates that the magnetic field has to be source-free. Otherwise, ∇ · B = 0 will lead to aloss of associativity of the translation operators T ( a ) ≡ exp (cid:16) − i ~ a · π (cid:17) , (cid:16) T ( a ) T ( a ) (cid:17) T ( a ) = exp (cid:16) − ie ~ c ω ( x ; a , a , a ) (cid:17) × T ( a ) (cid:16) T ( a ) T ( a ) (cid:17) . (8)Here a i are constant vectors and the non-trivial phase factor turns out to be the magneticflux coming out of the tetrahedron formed by a i :exp (cid:16) − ie ~ c ω ( x ; a , a . a ) (cid:17) , (9)which is nonzero if a magnetic monopole is enclosed by the tetrahedron. The phase factor(9) becomes 1 and thus the associativity of finite translations in the presence of the magneticmonopoles can be re-established for Z d x ∇ · B = 2 π ~ ce N, (10)where N is an integer. This condition, together with the Gauss equation for a monopole ofmagnetic charge µ , ∇ · B = 4 πµδ ( x ), yields the celebrated DQC µe = 12 N ~ c. (11)Note that the Jacobi identity is still violated at the location of each monopole and thesepoints are conventionally excluded from the manifold.4 II. THE NONCOMMUTATIVE DQC
The extension of the approach in [12] to the noncommutative case can be achieved onceone finds the algebra of coordinate and gauge-invariant momentum operators for a chargedquantum mechanical particle in motion in a magnetic field in the noncommutative space-time, i.e. the analogue of the non-canonical algebra (2). It is expected that the noncommu-tativity of space-time coordinates would change the dynamics of the charged particle in themagnetic field (i.e. the Lorentz force), and this in turn will require a change in the commuta-tion relations (2). However, we can find the new noncommutative algebra by starting from aclassical Lagrangian, for example the one for the model [13], and deriving the correspondingDirac brackets and then quantizing them. We therefore consider a Lagrangian of the form L = (cid:16) P i + ec A i (cid:17) ˙ X i − ǫ ijk P i ˙ P j θ k − m P + eA , (12)where P i is the momentum, θ k - the noncommutativity parameter, of dimension(length) /action, and A i , A - the magnetic and electric potential, respectively. The La-grangian (12) is a straightforward generalization to three-dimensions of the one consideredin [14], which is a Lagrangian for the model [13].The Lagrangian (12) is a Lagrangian of dynamical noncommutativity of the space co-ordinates. This claim is better understood once we derive the Dirac brackets from thisLagrangian. For this, we need the canonical momenta which are given by π i = ∂L∂ ˙ X i = P i + ec A i , π Pi = ∂L∂ ˙ P i = 12 ǫ ijk P j θ k . These lead to the constraints η ≡ π i − P i − ec A i , ψ i ≡ π Pi − ǫ ijk P j θ k . In the classical framework, with { X i , π j } = δ ij , { P i , π Pj } = δ ij , we calculate the constraintalgebra, { η i , η j } = ec ( ∂ i A j − ∂ j A i ) = ec F ij = ec ǫ ijk B k , { ψ i , ψ j } = − ǫ ijk θ k , { η i , ψ j } = − δ ij . (13)From this algebra we find that the constraints are second class and, performing the Dirac5onstraint analysis [15, 16], we obtain the classical Dirac brackets as { X i , X j } = ǫ ijk θ k − ec θ · B { X i , P j } = δ ij − ec B i θ j − ec θ · B , { P i , P j } = ǫ ijk ec B k − ec θ · B . (14)This is exactly how interactions have been introduced in the model of [14]. In this modelthe noncommutativity of coordinate operators is dynamical in the sense that it is generatedwithin the system. Thus the gauge field cannot be affected by the noncommutativity whichemerges upon quantization. Therefore the field A i is the Abelian U (1) gauge field in thismodel of noncommutativity.Our next step is to quantize the brackets. We do this by promoting the classical variables X i and P i in the Dirac brackets (14) to the status of operators ˆ X i , ˆ P i and multiplying theright-hand side of the Dirac brackets by i ~ . This is the standard procedure [15, 16]. Weconsider the Dirac brackets (14) expanded to first order in θ and hereafter we perform allour calculations to this order only. We resort to this approximation because we will need tofind representations for our operators in order to have a well-defined quantum theory [16],and this is a task that is difficult to do exactly for the algebra (14). The quantization of(14) gives [ ˆ X i , ˆ X j ] = i ~ ǫ ijk θ k + O ( θ ) , (15)[ ˆ X i , ˆ P j ] = i ~ (cid:2) δ ij − ec B i ( ˆ X ) θ j + ec δ ij θ · B ( ˆ X ) (cid:3) + O ( θ ) , [ ˆ P i , ˆ P j ] = i ~ ec ǫ ijk B k ( ˆ X ) (cid:2) ec θ · B ( ˆ X ) (cid:3) + O ( θ ) . The algebra (15) poses a twofold problem. Firstly, the operator ˆ P j in (15) does not representthe translation generator, since there are extra terms on the right-hand side of [ ˆ X i , ˆ P j ], otherthan i ~ δ ij . Secondly, we face the problem of how to represent the operators ˆ X i , since theydo not commute to first order in θ . This problem becomes much simpler if we are able todefine some new operators x i , in terms of the old ones ˆ X i and ˆ P i , such that they commuteto first order in θ . An appropriate definition for our purpose is x i = ˆ X i + 12 ǫ ijk ˆ P j θ k . (16)6hen the functions of the operator ˆ X i can be expanded in terms of the new coordinateoperator x i as, e.g., B i ( ˆ X ) = B i ( x ) − ǫ njk θ k ˆ P j ∂ n B i ( x ) + O ( θ ) . (17)We use the operator x i (16) and the expansion (17) to obtain an intermediate algebra of x i and ˆ P j , and further define the generator of translations corresponding to x i : p j = ˆ P j − ec (cid:16) ˆ P j ( B · θ ) − ˆ P · B θ j (cid:17) . (18)The newly-defined operators p i and x i obey the algebra[ x i , x j ] = 0 + O ( θ )[ x i , p j ] = i ~ δ ij + O ( θ ) , (19)[ p i , p j ] = i ~ ec ǫ ijk B k − e c h i ~ (cid:16) p [ j ∂ i ] ( B · θ ) + p [ i θ j ] ∇ · B + p · θ [ i ∂ j ] B (cid:17) + p [ j [ p i ] , B ] · θ + p · [ B , p [ i ] θ j ] i + O ( θ ) , where the indices in brackets are anti-symmetrized.To have properly quantized the algebra (19), we need a representation of its operators.From the similarity of the algebras (19) and (2), we infer that in the x -representation, wecan realize the translation generators as (3) plus an extra term involving the first ordernoncommutativity contribution. Explicitly, p i = − i ~ ∂ i − ec A i ( x ) + T i ( θ, x ) + O ( θ ) . (20)Inserting (20) into the commutator [ p i , p j ] of the algebra (19), it simplifies to[ p i , p j ] = i ~ ec ǫ ijk B k + 12 ec h ( i ~ ∂ [ i + ec A [ i ) θ j ] ∇ · B i + O ( θ ) . (21)By directly computing the commutator of the operators p i in the representation (20), wehave to reproduce the result (21), which holds true if we set T i ( θ, x ) = − ec θ i ∇ · B + G i , (22)where ∂ j G i = 12 ~ (cid:16) ec (cid:17) A j θ i ∇ · B . (23)Thus, the quantized algebra (19) is given by[ x i , x j ] = 0 + O ( θ ) , [ x i , p j ] = i ~ δ ij + O ( θ ) , (24)[ p i , p j ] = i ~ ec ǫ ijk B k + e c h ( i ~ ∂ [ i + ec A [ i ) θ j ] ∇ · B i + O ( θ )7n the x -representation.We can now calculate the Jacobi identities of the algebra (24), and find that the onlynon-vanishing one is:12 ǫ ijk [[ p i , p j ] , p k ] = − ~ ec ∇ · B + i ~ (cid:16) ec (cid:17) ǫ ijk ∂ k ( A i θ j ∇ · B ) + O ( θ ) . (25)Since the nonvanishing terms in the right-hand side of (25) are proportional to ∇ · B , fora divergenceless magnetic field there are no Jacobi indentity violations. However, if themagnetic field is produced by monopoles, ∇ · B = 4 πµδ ( x ), the Jacobi identity (25) isviolated, meaning nonassociativity of the translation generators p i .We would like to remark at this point that although the Lagrangian (12) contains nomagnetic sources, the algebra (24) is valid whether the magnetic field is source-free or not.The reason is simply that the Lorentz-force describes the movement of electrically chargedparticles in a magnetic field, but does not set any requirement on how the magnetic field isproduced.Thus in the noncommutative space we end up with a Jacobi identity violation consisting ofthe original commutative space term plus a θ -dependent total-derivative term. Let us recallthat DQC appears in the commutative case [12] through a volume integration (see (9)) overthe tetrahedron formed by the three translation vectors a , a , a . Now, the θ -term in (25),being a total derivative, should contribute at the boundary of the tetrahedron. However, thiscontribution will be necessarily zero, because the integrand contains the δ -function comingfrom ∇ · B = 4 πµδ ( x ), which has support only at the origin, i.e. on the monopole. Hencethese two features conspire to cancel the effect of the θ -term. DQC remains unchanged inthe presence of spatial noncommutativity, since the argument for restoring the associativityof the noncommutative translation operators goes through in the same manner as in thecommutative case [12], but now with the translation operators T NC ( a ) = exp (cid:16) − i ~ a · p (cid:17) , (26)generated by p as the element of the algebra (24), valid to first order in θ and with the x -representation (20). 8 V. SUMMARY AND DISCUSSION
We have explicitly shown that the DQC (11) remains unaltered in noncommutative spaceto first order in θ . Based on the structure of the classical algebra (14) and the representationof the quantum algebra (20), and also considering the fact that the form of any topologicalcorrection is strongly constrained, we conjecture that the DQC will hold true in all ordersin the noncommutative space-time. We intend to elucidate this issue in the future.It would be interesting to obtain the same kind of indication of a DQC to first order in θ using a noncommutative non-Abelian vector potential [11, 17], especially since a gauge-covariant noncommutative Aharonov-Bohm effect has been formulated in [18]. However, thisformulation gives the required phase factor with the help of non-Abelian noncommutativeWilson lines which are notoriously tedious to work with even to first order in θ , due to thepath ordering appearing in the Wilson line. Therefore, obtaining a possible DQC in thisapproach stands as a challenge for the future.Our conclusion is that the DQC remains unchanged in the noncommutative case, to thefirst order in θ and expectably to all orders. This is of significance, since a vast amountof work has been devoted to studying various effects of noncommutative space only tothe lowest order in θ . Finally, we would like to mention that our work reinforces similartopological results in the noncommutative case for other nonperturbative monopole-, soliton-and dyon-solutions [19]. Acknowledgments
We are indebted to Claus Montonen and Shin Sasaki for illuminating discussions.A. T. acknowledges Projects Nos. 121720 and 127626 of the Academy of Finland. [1] P. A. M. Dirac, Proc. Roy. Soc. Lond. A , 60 (1931).[2] P. Curie, S´eances Soc. Phys. (Paris), 76-77 (1894).[3] T. T. Wu and C. N. Yang, Phys. Rev. D , 3845 (1975).[4] G. ’t Hooft, Nucl. Phys. B79 , 276 (1974);A. M. Polyakov, JETP Lett. , 194 (1974); . Nambu, Phys. Rev. D , 4262 (1974).[5] J. Schwinger, Science , 757 (1969);B. Julia and A. Zee, Phys. Rev. D , 2227 (1975);M. K. Prasad and C. M. Sommerfeld, Phys. Rev. Lett. , 760 (1975);E. B. Bogomol’nyi, Sov. J. Nucl. Phys. , 449 (1976).[6] C. Montonen and D. Olive, Phys. Lett. B , 117 (1977).[7] M. Onoda and N. Nagaosa, J. Phys. Soc. Jpn. , 19 (2002).[8] Z. Fang et. al., Science 302, 92 (2003).[9] G. Giacomelli and L. Patrizii, Magnetic Monopole Searches , Bologna preprint DFUB-2003-1(2003). In
Trieste 2002, Astroparticle physics and cosmology (ICTP, Trieste 2003) p. 121;K. A. Milton, Rept. Prog. Phys. , 1637 (2006).[10] S. Doplicher, K. Fredenhagen, and J. E. Roberts, Phys. Lett. B , 39 (1994);S. Doplicher, K. Fredenhagen, and J. E. Roberts, Commun. Math. Phys. , 187 (1995),hep-th/0303037.[11] N. Seiberg and E. Witten, J. High Energy Phys. 09 (1999) 032, hep-th/9908142.[12] R. Jackiw, Phys. Rev. Lett. , 159 (1985);R. Jackiw, Int. J. Mod. Phys. A , 137 (2004), hep-th/0212058.[13] C. Duval and P. A. Horvathy, Phys. Lett. B , 284 (2000), hep-th/0002233.[14] J. Lukierski, P. C. Stichel, and W. J. Zakrzewski, Annals Phys. , 78 (2003),hep-th/0207149.[15] P. A. M. Dirac, Lectures on Quantum Mechanics (Yeshiva University Press, New York, 1964).[16] D. M. Gitman and I. V. Tyutin,
Quantization of fields with constraints (Springer, 1990);M. Henneaux and C. Teitelboim,
Quantization of gauge systems (Princeton University Press,1991).[17] M. Hayakawa, Phys. Lett. B , 394 (2000), hep-th/9912094.[18] M. Chaichian, M. L˚angvik, S. Sasaki, and A. Tureanu, Phys. Lett. B , 199 (2008),arXiv:0804.3565 [hep-th].[19] A. Hashimoto and K. Hashimoto, J. High Energy Phys. 11 (1999) 005, hep-th/9909202;D. J. Gross and N. A. Nekrasov, J. High Energy Phys. 03 (2001) 044, hep-th/0010090;L. Cieri and F. A. Schaposnik, Res. Lett. Phys. (2008) 890916, arXiv:0706.0449 [hep-th].[20] When deriving the DQC using gauge-variant vector potentials, one still defines the magnetic eld as B = ∇ × A even though this cannot now remain true in the whole space without thepotential having singularities [3]. Instead one may consider the vector potential as being de-fined in many overlapping regions of space connected by singularity-free gauge transformations[3].even though this cannot now remain true in the whole space without thepotential having singularities [3]. Instead one may consider the vector potential as being de-fined in many overlapping regions of space connected by singularity-free gauge transformations[3].