Particle-like solutions to classical noncommutative gauge theory
aa r X i v : . [ h e p - t h ] A ug Particle-like solutions to classical noncommutative gauge theory
A. Stern
Department of Physics, University of Alabama,Tuscaloosa, Alabama 35487, USA
ABSTRACT
We construct perturbative static solutions to the classical field equations of noncommuta-tive U (1) gauge theory for the three cases: a) space-time noncommutativity, b) space-spacenoncommutativity and c) both a) and b). The solutions tend to the Coulomb solution atspatial infinity and are valid for intermediate values of the radial coordinate r . They yield aself-charge inside a sphere of radius r centered about the origin which increases with decreasing r for case a), and decreases with decreasing r for case b). For case a) this may mean that theexact solution screens an infinite charge at the origin, while for case b) it is plausible that thecharge density is well behaved at the origin, as happens in Born-Infeld electrodynamics. Forboth cases a) and b) the self-energy in the intermediate region grows faster as r tends to theorigin than that of the Coulomb solution. It then appears that the divergence of the classicalself-energy is more severe in the noncommutative theory than it is in the corresponding com-mutative theory. We compute the lowest order effects of these solutions on the hydrogen atomspectrum and use them to put experimental bounds on the space-time and space-space non-commutative scales. For the former, we get a significant improvement over previous bounds.We find that cases a) and b) have different experimental signatures.1 Introduction
Recent quantum field theory investigations of noncommutative gauge theories have not beenstraightforward due to UV/IR mixing and related problems of renormalization. (For reviews,see [1],[2].) On the other hand, important progress has been made in understanding the renor-malization of other noncommutative field theories, in particular the noncommutative φ -model[3]. There has also been a very recent attempt to apply similar methods to noncommutativegauge theories [4]. Concerning UV-IR mixing, it is absent from some theories upon using anapproach based on the twisted action of the Poincare group [5],[6], and it can be understoodfor noncommutative gauge theories in terms of an induced gravity action [7],[8]. Nevertheless,the renormalization of noncommutative gauge theories is yet to be fully understood.In light of the difficulties with the quantum field theory it may be useful to have a closer lookat the classical field theory, and more specifically, for some behavior which may provide clues tothese difficulties. Classical aspects of noncommutative gauge theories, and in particular theirsolutions, have been of recent interest. There has been much work done on the noncommutativeanalogues of vortex, monopole and instanton solutions ∗ , as well as the solutions of generalrelativity. † Here we shall be concerned with classical aspects of noncommutative U (1) gauge theory.One possible signal of regularization difficulties in the quantum theory could come from theclassical self-energy of a charged particle, which may exhibit more singular behavior in thenoncommutative gauge theory than appears in the corresponding commutative theory. Weare, of course, referring to the infinite self-energy of the Coulomb solution. Thus there ismotivation for studying properties of the noncommutative analogues of the Coulomb solution.The behavior of the fields and current density of the noncommutative solutions at the origin isof particular interest. As position eigenstates do not occur in theories with space-space non-commutativity, there can be no intrinsic notion of points for such theories. It has then beenargued that the point charges of commutative gauge theories become smeared in noncommu-tative gauge theories [24], but this has not been demonstrated explicitly. It is noteworthy thatsuch a desired behavior is seen in Born-Infeld electrodynamics [25]. The latter deformation ofMaxwell theory has particle-like electrostatic solutions (or bions) which are characterized by afinite size. Bions are characterized by a finite self-energy and a smooth charge density at theorigin (although a singularity in the electric field does remain at the origin). The Born-InfeldLagrangian can be given explicitly and a simple expression results for the static solution. Un-fortunately, this is not true for the case of noncommutative electrodynamics, or more precisely,for its equivalent description in terms of commutative gauge fields. Here we are referring to ∗ For a small sample, see [9],[10],[11],[12],[13],[14]. † Many different noncommutative deformations of general relativity have been found [15],[16],[17],[18]. Theyhave been used to compute corrections to black hole solutions. Approaches have ranged from smearing outthe point singularity[19], to perturbatively solving noncommutative analogues of general relativity[20],[21],[22].Corrections to conical singularities have also been computed [23]. θ . This then leads to an order-by-order expan-sion for the effective Lagrangian expressed in terms of commutative fields. The zeroth orderterm (in the case of Abelian gauge fields) is the Maxwell action. Consequently, at best, onecan obtain an order-by-order expansion for the field equations and any particle-like solutions.The expansion in θ corresponds to an expansion in one over the radial coordinate r for thesesolutions, and a Coulomb-like behavior is expected as r → ∞ . By going to higher orders in θ one can probe the intermediate region of the solutions. Although the nature of the solutionsat the origin remains uncertain in this approach, the behavior of the fields along with theassociated self-energy and charge density in intermediate region may provide clues to whetheror not a singularity is present at r = 0.In this article we shall construct static solutions to the classical field equations for noncom-mutative U (1) gauge theory for three cases:a) space-time noncommutativity,b) space-space noncommutativity andc) both a) and b).The solutions will tend to the Coulomb solution for r → ∞ and are valid for intermediate val-ues of r . Constant noncommutativity θ is assumed for all three cases. The operator algebra isrealized on Minkowski space-time using the standard Gronewald-Moyal star product [27],[28].After applying the inverse Seiberg-Witten map [26], we obtain the commutative gauge fieldsassociated with the solutions. We analyze the self-charge and energy distributions of thesesolutions for intermediate values of r , and compare the results with Born-Infeld, as well as,Maxwell electrodynamics. In order to see deviations from the self-charge and energy distribu-tions of the Coulomb solution it is necessary to carry out the expansion of the Seiberg-Wittenmap up to second order in θ . ‡ The results appear to confirm the above speculation that theclassical self-energy singularity in noncommutative electrodynamics is more severe than thatof the commutative theory. For case b), results from the intermediate region also appear toindicate that the charge density is better behaved at the origin than it is in the commutativetheory. On the other hand, the opposite appears to be true for case a).We shall also examine the effect on the hydrogen atom spectrum of replacing the Coulombpotential by a space-time noncommutative solution. Here effects are seen at first order in θ . (The first order effects on the spectrum of the space-space noncommutative solution wereexamined previously in [30], and were was used to put experimental bounds on the noncommu-tative scale.) We shall treat the electron in the standard fashion, i.e., applying the standard(commutative) Schr¨odinger equation. (It was shown in [31], [32] that quantum mechanical ‡ It was shown in [29] using a different approach that the Coulomb nature of the potential is preserved innoncommutative electrodynamics if one only examines first order effects. θ . We apply it in section 3 to obtain the secondorder corrections to commutative U (1) gauge theory Lagrangian and Hamiltonian densities,while the static solutions are given in section 4. In section 5 we remark on a possible exactexpression for the electrostatic Lagrangian for the case a) of space-time noncommutativity. Ifthe Lagrangian holds to all orders in the noncommutativity parameter, then we can show thatthe electrostatic fields of the particle-like solution must be singular at the origin. In section6 we apply the lowest order results to the hydrogen atom spectrum. Concluding remarks aremade in section 7. We review the Born-Infeld solution in appendix A, while we give expressionsfor the nonlinear field equations and the energy density of noncommutative electrodynamicsin appendix B. As it will be essential for the analysis that follows, we here review the Seiberg-Witten mapto second order in the noncommutativity parameter. We specialize to U (1) gauge theory, anddenote the commutative potentials by a µ , with gauge variations δa µ = ∂ µ λ , and field strengths f µν = ∂ µ a ν − ∂ ν a µ . Seiberg and Witten [26] showed it can be mapped to noncommutative U (1) gauge theory expressed in terms of noncommutative potentials A µ ( a, λ ) → (cid:16) A = A ( a ) , Λ = Λ( λ, a ) (cid:17) , (2.1)with gauge variations δA µ = ∂ µ Λ − ie [ A µ , Λ] ⋆ , (2.2)and field strengths F µν = ∂ µ A ν − ∂ ν A µ − ie [ A µ , A ν ] ⋆ , (2.3)where e is the coupling constant and [ , ] ⋆ is here defined as the star commutator associatedwith the Groenewald-Moyal star product[27],[28] ⋆ = exp (cid:26) i θ µν ←− ∂ µ −→ ∂ ν (cid:27) (2.4) θ µν = − θ νµ are constant matrix elements and ←− ∂ µ and −→ ∂ µ are left and right derivatives, respec-tively, with respect to coordinates x µ on some manifold M . Thus for two function f and g on M , [ f, g ] ⋆ ≡ f ⋆ g − g ⋆ f . The coordinates x µ are associated with constant noncommutativitysince [ x µ , x ν ] ⋆ = iθ µν (2.5)4he map (2.1) is required to satisfy A µ ( a + ∂λ ) − A µ ( a ) = ∂ µ Λ( λ, a ) − ie [ A µ ( a ) , Λ( λ, a )] ⋆ , (2.6)for infinitesimal transformation parameters Λ and λ . Solutions for the potentials, fields andtransformation parameters can be obtained in terms of expansions in θ µν (or equivalently e ) A µ ( a ) = A (0) µ ( a ) + eA (1) µ ( a ) + e A (2) µ ( a ) + · · · F µν ( a ) = F (0) µν ( a ) + eF (1) µν ( a ) + e F (2) µν ( a ) + · · · Λ( λ, a ) = Λ (0) ( λ, a ) + e Λ (1) ( λ, a ) + e Λ (2) ( λ, a ) + · · · , (2.7)where the zeroth order correspond to the commutative theory A (0) µ ( a ) = a µ F (0) µν ( a ) = f µν Λ (0) ( λ, a ) = λ (2.8)Explicit expressions up to the second order in θ µν have been found by various authors [33],[34],[35],[36],[37],[38].Up to homogeneous terms, the first and second order solutions are given by A (1) µ ( a ) = 12 θ ρσ a ρ ( f µσ − ∂ σ a µ ) F (1) µν ( a ) = − θ ρσ ( a ρ ∂ σ f µν + f µρ f σν )Λ (1) ( λ, a ) = 12 θ µν ∂ µ λa ν (2.9)and A (2) µ ( a ) = 12 θ ρσ θ ηξ a ρ ( a η ∂ ξ f σµ + ∂ ξ a µ ∂ σ a η + f ση f ξµ ) F (2) µν ( a ) = 12 θ ρσ θ ηξ (cid:26) a ρ (cid:16) ∂ σ ( a η ∂ ξ f µν ) + ∂ ξ f µν f ση + 2 f µξ ∂ σ f νη − f νξ ∂ σ f µη (cid:17) + 2 f µσ f νξ f ρη (cid:27) Λ (2) ( λ, a ) = 12 θ ρσ θ ηξ ∂ ξ λa ρ ∂ σ a η , (2.10)respectively. Here we give the effective Lagrangian and current density for noncommutative U (1) gaugetheory up to second order in θ . Although the first order results have been reported previously,the same is not true for the second order results which will be needed for discussions in sec.4. The details of the second order analysis appear in Appendix B. Here we also report on analternative Lagrangian approach using auxiliary fields.5he considerations in the previous section do not involve dynamics. In general, (2.1) will notmap solutions of a commutative gauge theory on a manifold M to solutions of the correspondingnoncommutative gauge theory. Here we choose M to be four-dimensional Minkowski space-time. Solutions of the free commutative Maxwell equations ∂ µ f µν = 0 on M are not in generalmapped to solutions of the free noncommutative field equations ∂ µ F µν − ie [ A µ , F µν ] ⋆ = 0 , (3.1)on M .The converse is also true. Solutions to the free noncommutative field equations (3.1) on M are not in general mapped to solutions to the free commutative Maxwell equations on M bythe inverse Seiberg-Witten map( A, Λ) → (cid:16) a = a ( A ) , λ = λ (Λ , A ) (cid:17) (3.2)The field equations (3.1) are recovered from the Lagrangian density L [ A ] = − F µν ⋆ F µν , (3.3)which up to boundary terms § is − F µν F µν . From (2.7) it can be expanded in θ µν (or equiva-lently e ) and expressed in terms of the commutative fields, giving the effective Lagrangian L [ A ( a )] = L [ a ] = L (0) [ a ] + e L (1) [ a ] + e L (2) [ a ] + · · · , (3.4)We find that, up to second order, the effective U (1) Lagrangian can be written as a functionof only f µν (and θ µν ), and not higher derivatives of the fields. At zeroth order we of courserecover the free Maxwell Lagrangian L (0) [ a ] = 14 Tr f , (3.5)while up to total derivatives, the first and second orders are given by L (1) [ a ] = −
12 Tr f θ + 18 Tr f Tr f θ , (3.6)and L (2) [ a ] = 14 Tr( f θ ) + 12 Tr f ( f θ ) −
14 Tr f θ Tr f θ + 132 Tr f (Tr f θ ) −
116 Tr f Tr( f θ ) (3.7)The first order correction L (1) is well known [39],[33],[40]. The field equations resulting fromvariations of a in (3.4) state that there is a divergenceless field B µν , ∂ µ B µν = 0 , (3.8) § In the next section we will examine solutions with a singularity at the origin. The boundary terms willaffect the singularity. However, we shall only be concerned with the behavior of the fields away from the origin. θ µν (or equivalently e ) and the field strengths f µν : B µν = B (0) µν + eB (1) µν + e B (2) µν + · · · , (3.9)where the zeroth order is just B (0) µν = f µν . The first and second orders are computed inappendix B. Thus solutions of the noncommutative field equations (3.1) are mapped under(3.2) to solutions of (3.8).Alternatively, the field equations can be re-expressed in terms of Maxwell equations for f with an effective conserved current j µ associated with the noncommutative self-interactions ∂ µ f µν = j ν (3.10)Some properties of these currents have been examined previously in [41],[42]. After some workwe find rather simple expressions for the current in an expansion in θ µν (or equivalently e ) upto order the second order: j ν = ej (1) ν + e j (2) ν + · · · j (1) ν = ( f θ ) ρσ ∂ ( ρ f σ ) ν j (2) ν = − (cid:16) ( f θ ) + 12 θf θ (cid:17) ρσ ∂ ( ρ f σ ) ν , (3.11)where parenthesis indicate a symmetrization of indices.The field equations (3.8) and (3.10) can also be obtained from the Lagrangian L ′ ( B, a ) = 12 Tr Bf − L ′′ ( B ) , (3.12)where here B as well as a are treated as independent variables and L ′′ ( B ) only depends on B .Up to first order in θ µν (or equivalently e ), the latter is given by L ′′ ( B ) = 14 Tr B + e B θ − e B Tr Bθ + O ( θ ) (3.13)The Hamiltonian density is obtained from (3.4) in the standard way H = ∂ L ∂ ( ∂ a i ) ∂ a i − L = H (0) + e H (1) + e H (2) + · · · , (3.14)where as usual only the spatial components of the potential a i are dynamical. The zerothorder term H (0) is the Maxwell Hamiltonian, while the first two corrections are computed inappendix B. 7 Static solutions
Here we look for static perturbative solutions to the field equations (3.8) or (3.10) on four-dimensional Minkowski space M . (More precisely, M is defined with the spatial origin re-moved.) The perturbative expansion in θ µν also corresponds to an expansion in one over theradial coordinate r for the solutions. The result can therefore be considered valid for interme-diate values of r . Rather than solve the commutative field equations (3.8) or (3.10) directly,it is simplest to first solve (3.1) for the noncommutative potentials A µ and then apply theinverse Seiberg-Witten map (3.2) to get the commutative potentials a µ . This is the approachwe follow below. Here we consider θ i = 0 and θ ij = 0, where i is a space index and 0 the time index. We shallbe concerned with electrostatic fields. The star commutator vanishes when acting between anystatic fields in this case, and so the noncommutative gauge field equations (3.1) reduce to thecommutative Maxwell equations. The Coulomb solution A = − er A i = 0 , (4.1)is then exact in this case. Upon performing the inverse Seiberg-Witten map (3.2) of thissolution one gets the following results for the commutative potentials and field strengths a = − er − e θ i ˆ x i r + e r (cid:16) ( θ i ) − θ i ˆ x i ) (cid:17) + · · · (4.2) f i = e ˆ x i r (cid:18) e r θ j ˆ x i − e r (cid:16) ( θ i ) − θ j ˆ x j ) (cid:17) + · · · (cid:19) − e θ i r (cid:18) e r θ j ˆ x i + · · · (cid:19) , along with a i = f ij = 0. (The hat denotes a unit vector ˆ x i = x i /r .) For this solution we canidentify the current density in the Maxwell equation (3.10) with j = − e r θ j ˆ x j + 5 e r (cid:16) ( θ i ) − θ i ˆ x i ) (cid:17) + O ( θ ) , j i = 0 , r > , (4.3)which breaks rotational invariance. From Gauss’ law the resulting effective charge inside asphere of radius r centered about the origin is14 π Z d Ω r ˆ x i f i = e (cid:16) e r ( θ i ) + O ( θ ) (cid:17) , (4.4)which increases with decreasing r . Ω is the solid angle. This is in contrast to what happensfor the electrostatic solution (bion) of Born-Infeld theory [see (A.10)]. The bion charge insidea sphere of radius r goes smoothly to zero as r →
0. In contrast, a singular source may be ageneral feature of gauge theories with space-time noncommutativity. Moreover, the singularity8ppears to be more severe than that which occurs in the commutative theory. The result (4.4)may indicate that the exact solution screens an infinite charge at the origin.For the case of θ ij = 0 and electrostatic fields, the Lagrangian density (3.4) and energydensity (3.14), simplify to L θ ij =0 es = 12 ( f i ) (cid:16) − e ( θ j f j ) + e ( θ j f j ) + · · · (cid:17) (4.5) H θ ij =0 es = 12 ( f i ) (cid:16) − e ( θ j f j ) + 3 e ( θ j f j ) + · · · (cid:17) , (4.6)respectively. Substituting the solution (4.2) in the energy density (4.6) and averaging over asphere of radius r gives 14 π Z d Ω H = e r (cid:18) e r ( θ i ) + O ( θ ) (cid:19) (4.7)This is also in contrast to what happens for the bion [see (A.11)], which has a finite self-energy.For the noncommutative solution (4.2), the self-energy in the intermediate region grows fasterthan that for a Coulomb point charge for decreasing r .An alternative approach to obtaining solution (4.2) would be to solve the field equations(3.8) directly. Since we are interested in electrostatic field configurations on R minus theorigin we may try setting the divergenceless field B µν equal to the Coulomb solution. Moregenerally, one can (actually, must) add homogeneous terms, in analogy to a multi-momentexpansion: B i = e ˆ x i r + c e r (cid:16) θ i − θ j ˆ x j ˆ x i (cid:17) + c e r (cid:16) ( θ j ) ˆ x i + 4 θ j ˆ x j θ i − θ j ˆ x j ) ˆ x i (cid:17) + · · · , (4.8)where c a are arbitrary coefficients and B ij = 0. Starting from the Lagrangian density (4.5) forelectrostatic fields one obtains the following expansion up to second order for the divergencelessfields B i in terms of f i B i = f i (cid:16) − e ( θ j f j ) + e ( θ j f j ) + · · · (cid:17) − eθ i ( f k ) (cid:16) − e ( θ j f j ) + · · · (cid:17) (4.9)We can invert this expression to solve for the field strengths f i for the general solution (4.8).The result is f i = e ˆ x i r + (cid:16) c + 12 (cid:17) e r θ i + (1 − c ) e r θ j ˆ x j ˆ x i + (cid:16) c + c + 12 (cid:17) e r ( θ j ) ˆ x i +(4 c + 1) e r θ j ˆ x j θ i + (1 − c − c ) e r ( θ j ˆ x j ) ˆ x i + · · · , (4.10)along with f ij = 0. Lastly, if we impose the Bianchi identities, which here means ∂ i f j = ∂ j f i ,we obtain the unique result up to second order that c = c = − , and thus recover thesolution (4.2). 9 .2 Space-space noncommutativity Here we consider θ i = 0 and θ ij = 0. A static solution of the free noncommutative fieldequations (3.1) on R (minus a point) × time was obtained perturbatively in [30]. It is givenby A = − er + e r (cid:26) − ( θ ki ˆ x i ) + 15 ( θ ij ) (cid:27) + O ( θ ) A i = e θ ij ˆ x j r + O ( θ ) (4.11)Upon performing the inverse Seiberg-Witten map (3.2) of this solution to the commutativepotentials and field strengths one gets a ( A ) = − er + e ( θ ij ) r + O ( θ ) a i ( A ) = e θ ij ˆ x j r + O ( θ ) , (4.12) f i = e ˆ x i r − e ( θ ij ) ˆ x i r + O ( θ ) f ij = e r (cid:16) − θ ij + θ ik ˆ x k ˆ x j − θ jk ˆ x k ˆ x i (cid:17) + O ( θ )Unlike in the previous case, this is not an electrostatic solution of the field equations (3.10)as magnetic fields are present (both in the commutative and noncommutative theory). Up toorder θ , the charge density is rotationally invariant, while the current density is not j = e r ( θ ij ) + O ( θ ) j i = e r θ ij ˆ x j + O ( θ ) , r > r centered about the origin is14 π Z d Ω r ˆ x i f i = e (cid:16) − e ( θ ij ) r + O ( θ ) (cid:17) , (4.14)which here decreases with decreasing r . This behavior is similar to that of the bion (A.10). Itis then a possibility that the charge distribution for the theory with space-space noncommu-tativity is smooth at the origin, and that one can associate the size of the distribution with e p | θ ij | , as has been done previously [24].Substituting into the energy density computed in the appendix (B.5) and averaging over asphere of radius r now gives14 π Z d Ω H = e r (cid:18) e r ( θ ij ) + O ( θ ) (cid:19) (4.15)Once again, in contrast to the energy density of the Born-Infeld solution (A.11), it grows fasterthan the self-energy of a Coulomb point charge as r tends to the origin. The solution (4.11) to the free noncommutative field equations (3.1) applies in the general casewhere both θ i and θ ij are nonzero. Upon performing the inverse Seiberg-Witten map (3.2) of(4.11) in this case, one gets a ( A ) = − er − e θ i ˆ x i r + e r (cid:16)
110 ( θ ij ) + ( θ i ) − θ i ˆ x i ) (cid:17) + O ( θ )10 i ( A ) = e θ ij ˆ x j r + e r θ j (cid:16) θ ji − θ jk ˆ x i ˆ x k + 8 θ ik ˆ x j ˆ x k (cid:17) + O ( θ ) (4.16)The charge density for the general case is simply the sum of (4.3) and (4.13) up to secondorder, while the current density contains mixed terms at second order; i.e. terms proportionalto the product of both θ ij and θ k . To obtain the effective charge inside a sphere of radius r up to second order in the general case, one adds the contributions (4.4) and (4.14). Thesesecond order contributions cancel when ( θ i ) = ( θ ij ) . Further analysis of the general caseis quite involved and does not appear to lead to novel results. θ i = 0 , θ ij = 0 ? This section is speculative in nature and offers a possible nonperturbative treatment for casea), and if correct, gives the solution for all r . Having an exact Lagrangian could also lead toan investigation of solutions which do not possess a commutative limit.From the second order results for the electrostatic Lagrangian (4.5) and energy density(4.6) for the case θ i = 0, θ ij = 0, it is tempting to surmise that the exact expressions for theelectrostatic Lagrangian and energy density in this case are L θ ij =0 es = ( f i ) eθ j f j (5.17)and H θ ij =0 es = ( f i ) (1 + eθ j f j ) , (5.18)respectively. The field equation following from (5.17) states that B i = f i eθ j f j − e θ i ( f k ) (1 + eθ j f j ) (5.19)has zero divergence (3.8). This can be inverted to obtain an expression for the field strength f i = B i + θ i e ( θ n ) (cid:16) − eθ m B m (cid:17)r(cid:16) − eθ j B j (cid:17) − e ( θ ℓ ) ( B k ) − θ i e ( θ ℓ ) (5.20)The field equations can then also be obtained from the Lagrangian L ′ θ ij =0 es ( B, a ) = 1 e ( θ n ) r(cid:16) − eθ j B j (cid:17) − e ( θ ℓ ) ( B k ) + θ i B i e ( θ n ) + B i f i , (5.21)where B and a are treated as independent variables as in (3.12). From (5.20) a well definedsolution should everywhere satisfy e | θ ℓ || B k | + eθ j B j ≤ B i proportional to the Coulombterm ˆ x i /r . The solution must include homogeneous terms as in (4.8). This is also needed for f i to satisfy the Bianchi identity.Although we have not found a nontrivial exact solution to the electrostatic field equation ∂ i B i = 0 which goes to the Coulomb solution as r → ∞ , we can show that that there are noregular solutions near the origin. The proof is by contradiction. Assume a power law behaviorfor the electrostatic potential near r = 0 a ∼ γ r ( θ i ˆ x i ) e ( θ k ) + βr n ( θ i ˆ x i ) m , as r → , (5.23)where γ and β are constants and n > m ≥
0. Then to leading order in r , the fieldstrength tends to a constant f i ∼ γθ i e ( θ k ) + β r n − h ( n − m )( θ j ˆ x j ) m ˆ x i + m ( θ j ˆ x j ) m − θ i i , as r → r , we get ∂ i B i = βr n − ( θ i ˆ x i ) m − γ ( ( n − m )( n + m + 1)( θ j ˆ x j ) + m ( m − θ j ) − (2 + γ ) γ (1 + γ ) ( θ k ) (cid:16) ( n − m )( n − m − θ j ˆ x j ) +( n − m )(1+2 m )( θ j ˆ x j ) ( θ ℓ ) + m ( m − θ ℓ ) (cid:17)) (5.25)A regular asymptotic solution requires that all the coefficients of ( θ j ˆ x j ) N in the braces vanish.From the vanishing of the N = 4 coefficient one gets a) n = m or b) n = m + 2. From thevanishing of the N = 0 coefficient one gets m = 0 or 1, which for a) is inconsistent with n > γ not real (after demanding that the N = 2 coefficient vanishes). Thereare then no nontrivial asymptotic solutions of the form (5.23) near the origin. The effect of a space-space noncommutative source on the hydrogen atom at lowest order in θ ij was considered in [30], and it was used to put experimental bounds on the noncommutativescale. We first briefly review the argument below and then make analogous arguments for thecase of space-time noncommutativity. We obtain a bound for the space-time noncommutativityparameter which is a substantial improvement over previous attempts.We start with the standard nonrelativistic Hamiltonian for a charged spinning particle H = 12 m e ( p i − ea i ) + ea − µ B ǫ ijk f ij S k , (6.26)12here m e is the electron mass, µ B = e m e denotes the Bohr magneton and we use natural units ~ = c = 1. For the case of space-space noncommutativity at leading order, a is the Coulombpotential and a i and f ij are the first order expressions given in (4.12). Then H = H + H ( ss )1 + H ( ss )2 , (6.27)where H is the nonrelativistic hydrogen atom Hamiltonian H = 12 m e p i − αr , (6.28)and H ( ss )1 and H ( ss )2 are the leading perturbations H ( ss )1 = α m e ~θ · ~Lr , H ( ss )2 = α m e ~S · ˆ x )( ~θ · ˆ x ) − ~S · ~θr (6.29)where θ ij = ǫ ijk θ k and α is the fine structure constant. H ( ss )1 leads to corrections to the Lambshifts of the ℓ = 0 states, while H ( ss )2 induces splittings in the 1 s states. After choosing ~θ inthe third direction θ i = θ ss δ i , one gets the following diagonal matrix elements < H ( ss )1 > P ± / / = α θ ss m e (cid:28) L z r (cid:29) P ± / / = ± α m e θ ss
144 (6.30) < H ( ss )2 > S ± / / = − α θ ss m e (cid:28) S z r (cid:29) S ± / / = ∓ α m e θ ss Λ QCD , (6.31)using spectroscopic notation nℓ m j j . To get a finite answer for (6.31) we inserted the Λ QCD cutoff, taking into account the finite size of the nucleus. (This appears to be the best onecan do without having a consistent treatment of noncommutative QCD.) According to [43]the current theoretical accuracy on the 2 P Lamb shift is about 0 .
08 kHz. From the splitting(6.30), this then gives the bound θ ss < ∼ (30 MeV) − (6.32)The current theoretical accuracy on the 1 S shift is about 14 kHz [43]. From the splitting (6.31)and Λ QCD ∼
200 MeV, this then gives the improved bound of θ ss < ∼ (4 GeV) − (6.33)In the above treatment of the hydrogen atom we considered the gauge fields generated bythe proton to be noncommutative, but we applied it to the standard Schr¨odinger equation,rather than its noncommutative counterpart. The relative coordinates of the Schr¨odingerequation were treated as commuting. In this regard, it has been noted that although thenoncommutativity parameter is fixed in quantum field theory, this is not necessarily the case innoncommutative quantum mechanics where the different particle coordinates may be associatedwith different θ µν [44],[45]. For a multi-particle quantum system, the commutation relationsfor the different particles should, in principle, be derived starting from the noncommutative13eld theory. In [44] starting from a noncommutative version of QED the authors found thattwo Dirac particles of opposite charge have opposite noncommutativity, while the relativecoordinates commute. As pointed out in [45], the correct approach for the hydrogen atomwould have to include noncommutative QCD, which unfortunately is not well understood. Apragmatic approach would be to instead set separate bounds on the noncommutativity of theelectron and nucleus.In contrast to the above approach, the spatial coordinates do not commute in the space-space noncommutative version of the Schr¨odinger equation. The latter was considered previ-ously for the hydrogen atom in [45], and it led to the additional perturbation H ( e ) = − α ~θ · ~Lr (6.34)to H and a further correction to the 2 P Lamb shift. This gives yet another bound on thenoncommutativity parameter θ i = θ e δ i , here associated with the electron. The bound wascorrectly computed in [30] to be θ e < ∼ (6 GeV) − (6.35)It remains to find the effect of a space-time noncommutative source on the hydrogen atom.Here we replace the standard Coulomb potential for the hydrogen atom with ea , with a given by (4.2). We shall again only be interested in leading order effects. The hydrogen atomHamiltonian is then H = H + H ( st ) , H ( st ) = − α θ i ˆ x i r (6.36)Unlike with the case of space-space noncommutativity, there are no diagonal matrix elementsamongst the orbital angular momentum eigenstates for the perturbative term H ( st ) . Choosing θ i = θ st δ i , the latter gives rise to nonvanishing matrix elements between states with ∆ ℓ = 1and ∆ m = 0. The matrix element between the degenerate 2s and 2p m = 0 states is < n = 2 , ℓ = 0 , m L = 0 | H ( st ) | n = 2 , ℓ = 1 , m L = 0 > = − α m e θ st
27 (6.37)Upon including the electron spin (and the fine structure), this implies a mixing of the fourdegenerate n = 2 , j = states (i.e., 2 s / and 2 p / ) < s ± / / | H ( st ) | p ± / / > = ± α m e θ st √ , (6.38)Consequently, there is an energy splitting of the levels equal to∆ E = 2 α m e θ st √ , (6.39)and two sets of doubly-degenerate levels result. Then from the above mentioned currenttheoretical accuracy on the 2 P Lamb shift and (6.39), one gets the bound θ st < ∼ ( . − (6.40)14revious bounds on the space-time noncommutativity parameter have been found usinggravitational quantum well experiments [46],[47],[48],[49]. (6.40) is a significant improvement.Unlike with the case of space-space noncommutativity at lowest order, all of the n = 2 , j = hydrogen atom states are shifted. Thus the experimental signature for space-time noncommu-tativity differs from that for space-space noncommutativity.As before we considered the gauge fields generated by the proton to be noncommutative,but we applied the standard (commutative) Schr¨odinger equation for the electron. x i doesnot commute with t in the space-time noncommutative version of the Schr¨odinger equation.However, it was shown in [31],[32] that the quantum mechanical spectrum is unaffected by thereplacement of the commutative Schr¨odinger equation with the one associated with noncom-muting time and space coordinates (provided that the spatial coordinates commute). We have constructed perturbative static solutions to the classical field equations of noncom-mutative U (1) gauge theory up to second order in θ for the three cases: a) space-time noncom-mutativity, b) space-space noncommutativity c) both a) and b). They tend to the Coulombsolution as r → ∞ . For case a) the solution is electrostatic and the associated self-charge insidea sphere of radius r centered about the origin increases with decreasing r . This may signalthat the exact solution screens an infinite charge at the origin. We proposed an exact expres-sion for the Lagrangian and Hamiltonian in this case and, if it is correct, have shown that nononsingular solutions exist at the origin. Magnetic as well as electric fields are present for thecase b) solution, and here the self-charge inside a sphere of radius r centered about the origindecreases with decreasing r . It then becomes plausible that the charge density is well behavedat the origin, as happens in Born-Infeld electrodynamics. If so, the guess that charges becomesmeared in gauge theories with space-space noncommutativity would be valid. For both casesa) and b) the self-energy of the solutions in the intermediate region grows faster than that fora Coulomb point charge as r tends to the origin. It thus appears that the noncommutativesolutions have infinite self-energy, contrary to the case of Born-Infeld solution, and that thedivergence of the classical self-energy in the noncommutative theory is more severe than itscounterpart in the commutative theory.We have also looked for the lowest order effects of these solutions on the hydrogen atomspectrum and used them to put experimental bounds on the space-time and space-space non-commutative scales. We found that the two different cases have different experimental signa-tures.It is known that the star product realization of any given operator algebra on a noncom-mutative space is not unique. For example, for the case of constant noncommutativity onecan also apply the Voros star product which is based on coherent states [50],[51]. More gen-erally, the star product belongs to a very large equivalence class of star products [52]. The15ifferent star products in the equivalence class are related by gauge transformations, and toolshave been developed for writing down gauge theories based on these equivalence classes [53].Since the analysis in this article relies on one particular choice of the star product; i.e., theGroenewald-Moyal star product, it is important to know how sensitive the above results areto this choice. It would be especially puzzling if the results obtained for the hydrogen atomspectrum depended on the choice of star product. Acknowledgment
I am grateful to Aleksandr Pinzul for useful discussions.
Appendix A Comparison with the bion
It is useful to compare the asymptotic behavior of the charge density and self-energy ofthe solutions found in section 4 with that of a known deformation of Maxwell theory, i.e.,Born-Infeld theory. The Born-Infeld Lagrangian [25] on four-dimensional Minkowski space isconstructed from the determinant of the matrix h = η + κf , (A.1)where f = [ f µν ] is the field tensor, η = [ η µν ] = diag( − , , ,
1) is the flat metric tensor and κ is a dimensionful constant. (In string theory the latter is identified with 2 π times the stringconstant.) The Lagrangian density is L BI = κ − (cid:18) − √− det h (cid:19) , det h = − − κ f µν f µν + κ
64 ( ǫ µνρσ f µν f ρσ ) (A.2)One recovers the Maxwell action at lowest order in the expansion in κ . The field equationsresulting from (A.2) state that there is a divergenceless field B BIµν , analogous to B µν in (3.8), ∂ µ B BIµν = 0 , B
BIµν = 1 √− det h (cid:16) f µν − κ
16 ( ǫ αβγδ f αβ f γδ )( ǫ µνρσ f ρσ ) (cid:17) (A.3)Alternatively, the field equations can be re-cast as Maxwell equations for the field strength f µν with an effective conserved current j BIν as in (3.10) ∂ µ f µν = j BIν , (A.4)where here j BIν = ∂ µ ( f µν − B BIµν ). The energy density is given by H BI = 1 √− det h (cid:16)
12 ( f ij ) + 1 κ (cid:17) − κ , (A.5)and it is easy to check that the Maxwell energy density is recovered at zeroth order in a κ expansion. 16or the case of electrostatics, (A.2) and (A.5) reduce to L BIes = κ − (cid:18) − p − κ ( f i ) (cid:19) H BIes = κ − (cid:18) p − κ ( f i ) − (cid:19) , (A.6)respectively. Substituting the Coulomb solution for B BIµν on R (minus a point ) × time B BIi = e ˆ x i r B BIij = 0 , (A.7)gives the bion solution for f µν f i = e ˆ x i √ r + κ e f ij = 0 , (A.8)which satisfies the Bianchi identities ∂ i f j = ∂ j f i . The associated electrostatic potential canbe expressed in terms of a hypergeometric function.[54] Using (A.4) one then gets the followingcontinuous charge distribution for the solution j BI = 2 κ e r ( r + κ e ) / (A.9)and the following effective charge inside a sphere of radius r centered about the origin14 π Z d Ω r ˆ x i f i = e q κer ) → e (cid:18) − κ e r + · · · (cid:19) as r → ∞ r κ (cid:18) − r κ e + · · · (cid:19) as r → , (A.10)which decreases monotonically to zero as r goes to zero. Substituting into (A.5) gives the bionenergy density H BIes = 1 κ (cid:18)r (cid:16) κer (cid:17) − (cid:19) → e r (cid:18) − κ e r + · · · (cid:19) as r → ∞ , (A.11)whose integral is finite. Appendix B Sourceless field B µν and Hamiltonian density Here we give explicit expressions for i) the sourceless field B µν in the noncommutative fieldtheory, along with ii) the Hamiltonian density in section 3, up to second order in θ µν .i) Varying the commutative field strengths f µν in the Lagrangian density (3.4) gives δ L = 12 Tr Bδf , (B.1)17nd the resulting field equations state that B µν satisfies (3.8). B µν is expanded in terms of θ µν ( or equivalently e ) in (3.9). The zeroth order is just B (0) = f , while B (1) = − ( f θ + f θf + θf ) + 12 f Tr f θ + 14 θ Tr f B (2) = θf θ + f θf θ + θf θf + f ( f θ ) + ( θf ) f + ( f θ ) f −
12 ( f θ + f θf + θf ) Tr f θ − θf θ Tr f − θ Tr f θ + 18 θ Tr f θ Tr f + 18 f (Tr f θ ) − f Tr( f θ ) (B.2)ii) Up to the first class constraints, the Hamiltonian density in (3.14) can be re-expressedas H = 2 ∂ L ∂f i f i − L (B.3)The coefficients H ( n ) in the expansion given in (3.14) are then given by H ( n ) = − ( f B ( n ) ) − L ( n ) , (B.4)and so H (0) = 12 ( f i ) + 14 ( f ij ) H (1) = ( f θ + f θf + f θf ) + 12 Tr f θ + 12 H (0) Tr f θ −
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