Mechanics and Quantum Supermechanics of a Monopole Probe 8 Including a Coulomb Potential 0 0 2 Steven G. Avery and Jeremy Michelson ∗ † n a J Department of Physics The Ohio State University 4 1040 Physics Research Building ] 191 West Woodruff Avenue h t Columbus, Ohio 43210-1117 - p e U.S.A. h [ 3 v 1 4 3 0 . 2 1 7 0 Abstract : v i X A supersymmetric Lagrangian used to study D-particle probes in a D6-brane background is r exactly soluble. We present an analysis of the classical and quantum mechanics of this theory, a including classical trajectories in the bosonic theory, and the exact quantum spectrum and wave- functions, including both bound and unbound states. ∗Electronic address: avery@mps,ohio-state,edu †Electronic address: jeremy@mps,ohio-state,edu 1 Contents I. Introduction 3 II. The Model 4 III. Conserved Quantities 5 A. The Classical Symmetry Algebra 6 B. The Quantum Symmetry Algebra 7 IV. Classical Trajectories 8 V. The Quantum Spectrum 12 A. Supersymmetric Ground States 12 B. Nonsupersymmetric Bound States 13 1. Angular Momentum Basis of Nonsupersymmetric Bound States 15 C. Marginally Bound States 16 D. Unbound States 16 VI. Wavefunctions 17 A. Angular Momentum Eigenfunctions 17 B. Ground State Wavefunctions 19 C. Bound State Wavefunctions 20 1. Bosonic Bound States 20 2. Fermionic Bound States 21 D. Unbound States 22 1. Bosonic Unbound States 22 2. Fermionic Unbound States 23 E. Marginally Bound States 23 1. Bosonic States 23 2. Fermionic States 24 Acknowledgments 24 A. Conventions 24 B. Fermionic Wavefunctions 25 C. Unitary Representations of SO(3,1) and its Contraction 28 D. Normalization of the Unbound Bosonic States 30 References 32 2 I. INTRODUCTION The best understanding we have of black holes comes from string theory. Within string theory, those black holes which are constructed by wrapping D-branes on cycles of a com- pact special holonomy manifold have provided our deepest insights. Nevertheless, there is still much to be understood. For example, while there is a good understanding of four- dimensional black holes obtained by wrapping D2-branes on 2-cycles of a Calabi-Yau 3-fold, an analysis which includes the addition of D6-branes wrapping the entire 3-fold has been elusive. AnadditionalcomplicationhasbeenthestabilitypropertiesofD-branes. Theappropriate basis of D-brane charges is a function of the Calabi-Yau moduli. Upon crossing lines of marginal stability in moduli space, D-branes may break up into pieces corresponding to the new stable basis. Moreover, the attractor mechanism—or just the fact that the moduli will vary between infinity and the black hole horizon—can mean that lines of marginal stability are crossed in a black hole spacetime, thus leading the microscopic description of the black hole to be more complicated than naively thought. This paper will not discuss these issues. These issues have been raised and discussed in [1, 2]. One of the consequences of the analysis in [2] is that it becomes interesting to un- derstand the motion of D-particle probes in a D6-branebackground. This is equivalently the supersymmetric quantum mechanics of a charged particle in the background of a magnetic monopole. Because D0’s and D6’s are naively not mutually supersymmetric—though they can form a supersymmetric bound state [3, 4]—the supersymmetric quantum mechanics has a nontrivial potential, which turns out to include a Coulumb potential (see Eq. (II.1)). Remarkably, this supersymmetric quantum mechanics is amenable to exact analysis. In particular, in addition to the usual conserved quantities, energy and angular momentum, there is an additional conserved vector quantity. The additional conserved charges are not unlike the Laplace-Runge-Lenz vector found in the hydrogen atom, and so we give it the same name. It is also interesting that attempting to understand black holes following [2] leads one to study a supersymmetric quantum mechanics. The near horizon limit of a black hole space- time has an AdS factor, whose conformal field theory dual should be a supersymmetric 2 quantum mechanics.[5, 6, 7] Refs. [8, 9] were able to find a supersymmetric quantum me- chanics by studying the D-particles induced by D2-branes in the presence of a background Ramond-Ramond field. Interestingly, however, although the supersymmetric quantum me- chanics of [6, 7] was, following [10], “Type B” and had fermions which are worldline spinors and target space vectors, the supersymmetric quantum mechanics of [2, 8, 9] is similar to that of [11] in that the fermions are target space spinors. The purpose of this paper, then, is, with the help of the Laplace-Runge-Lenz vector, to present the exact spectrum and wavefunctions for the theory (II.1). Some of this was already done over 20 years ago, for this theory or for related theories. References include [2, 3 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. However, we have not seen all of the wavefunctions previously, and many of the other results are scattered throughout the literature. So while not all of the results in this paper are new, we have merged our new results with the old results in a self-contained and complete way. As interesting as it is to have a nontrivial supersymmetric quantum mechanics which is amenable to exact analysis, our ultimate motivation is, of course, an understanding of black holes. We intend for the results presented here to be useful to that end. It would also be interesting to understand—perhaps similar to the relationship between the Neveu-Schwarz-Ramond superstring and the Green-Schwarz superstring—if the super- symmetric quantum mechanics studied here can be related to those Type B ones studied in e.g. [6, 7]. This paper is organized as follows. The Lagrangian of the model studied in this paper is presented in II. The conserved quantities and their classical and quantum algebra are § given in III. The classical mechanics is studied in section IV. The quantum spectrum § § for both bound and unbound states is derived in V. The corresponding wavefunctions are § presented in VI. Our conventions areoutlined inAppendix A. Aderivation ofthe fermionic § superpartners to a bosonic wavefunction is given in Appendix B. An outline of the unitary representations of SO(3,1)andits contraction, used to study unbound andmarginallybound states, is given in Appendix C. Finally, a derivation of the δ-function normalization of the unbound bosonic states is given in Appendix D. II. THE MODEL In this paper we study the mechanics defined by the Lagrangian m κ κ L = ~x˙2 +D2 +2iλ¯λ˙ +θ D κA~ ~x˙ ~x λ¯~σλ. (II.1) 2 − 2r − · − 2r3 · (cid:16) (cid:17) (cid:16) (cid:17) This and related theories have previously been studied in [2, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. The Lagrangian (II.1) describes a particle of mass m and integer charge κ in the background of a magnetic monopole with unit charge.1 There is also a Coulomb potential whose strength is parameterized by the dimensionful (with units of inverse length, in c = ~ = 1 units) parameter θ. As the model is a three Euclidean dimensional one, the position of the test particle, ~x = (x,y,z), is a three-vector whose norm is r = ~x . The | | fermion λ is a two-component spinor whose conjugate is λ¯; see Appendix A for conventions. D is an auxiliary field. The vector potential A~ determines the magnetic field B~ = ~ A~, ∇× 1 Actually, only the product of the electric charge of the test particle and the magnetic charge of the monopole appears in the action. This is κ. 4 and is the vector potential for a unit magnetic monople, A~ = 12 1− zr xxy2ˆ−+yyx2ˆ, zr > −sinǫ, B~ = ~ A~ = 1 ~x, (II.2) 1 1+ z xyˆ yxˆ, z < sinǫ, ∇× 2r3 (−(cid:0)2 (cid:1)r x2−+y2 r in terms of the unit(cid:0)vector(cid:1)s xˆ, yˆ, zˆ. The magnetic monopole vector potential is defined in patches which overlap in the region sinǫ < z < sinǫ, 0 < ǫ < π [26]. The difference − r 2 between the two vector potentials in the overlap region is pure gauge, 1 y z A~(z > rsinǫ) A~(z < rsinǫ) = ~ tan 1 , sinǫ < < sinǫ (II.3) − − − ∇2 x − r Also, the choice of gauge (II.2) gives ~ A~ = 0 in both patches. ∇· III. CONSERVED QUANTITIES In terms of the spin angular momentum which is not conserved, m ~s = λ¯~σλ, (III.1) 2 the conserved quantities are 1 m κ H = (p~+κA~)2 + D2 + ~x ~s, (III.2) 2m 2 mr3 · κ J~ = ~x (p~+κA~)+ ~x+~s, (III.3) × 2r κθ κ κ K~ = (p~+κA~) J~ i(p~+κA~)+ ~x+ (~x ~s)~x ( +θ)~s+(p~+κA~) ~s, (III.4) × − 2r r3 · − r × Q = mDλ i(p~+κA~) ~σλ, Q¯ = mDλ¯ +iλ¯~σ (p~+κA~), (III.5) − − · − · These quantities are respectively the Hamiltonian, angular momentum, Laplace-Runge-Lenz vector, andthe supercharges. The second term onthe right-handside of (III.4) isa quantum correction, also needed for hermiticity of the operator, and should be omitted classically. Otherwise, these expressions are valid both classically and quantum mechanically. p~ is the canonical momentum p~ = ∂L, and the on-shell value of D is ∂~x˙ 1 κ D = +θ . (III.6) m 2r (cid:16) (cid:17) The first two terms of the angular momenta (III.3) can be considered to be the orbital angular momentum, and the last term is the spin angular momentum. Note, however, that spin angular momentum and orbital angular momentum are not separately conserved. The Laplace-Runge-Lenz vector (III.4) is the Noether charge associated with the trans- formation κ δ~x = m~x ξ~ ~x˙ +mξ~ ~x ~x˙ + ξ~ ~x+2ξ~ ~s, (III.7a) × × × × 2r × × (cid:16) (cid:17) κ (cid:16) (cid:17) i κ δλ = imξ~ ~x˙ ~σ λ+i (ξ~ ~x)(~x ~σ)λ +θ ξ~ ~σλ, (III.7b) · × 2r3 · · − 2 r · δD = κ ξ~(cid:16)~x˙ + (cid:17)κ (ξ~ ~x)(~x ~x˙)+ κ ξ~ (~x (cid:16)~s), (cid:17) (III.7c) −2r · 2r3 · · mr3 · × 5 ~ parameterized by the vector ξ. All but the first three terms of the Laplace-Runge-Lenz vector (III.4) are bilinear in the fermions and so vanish in the bosonic theory. The fermion bilinears ensure that K~ is conserved in the supersymmetric theory. The supercharges (III.5) are the Noether charges associated with the transformation, δ~x = iλ¯~σξ iξ¯~σλ, δD = λ¯˙ξ ξ¯λ˙, δλ = ~x˙ ~σξ +iDξ, (III.8) − − − · where the spinor ξ and its conjugate ξ¯parameterize the supersymmetry. The spinorial supercharges can be combined to form a third conserved vector, 1 S~ = Q¯~σQ. (III.9) 4H Aconvenient normalizationhasbeenchosenwhich, however, onlymakesthedefinition(III.9) well-defined away from configurations or quantum states of zero energy. In the nonzero energy sector of the theory, it will be convenient to use modified angular momentum and Laplace-Runge-Lenz vectors J~˜= J~ S~, K~˜ = K~ +θS~. (III.10) − A. The Classical Symmetry Algebra As is standard, the fermions of the theory are first order and therefore they and their mo- mentaareconstrained. ThisisalsotrueoftheauxiliaryfieldD anditscanonicalmomentum. It is straightforward to find the nonzero Dirac brackets κ i xi,p = δi, [p~,D] = ~x, λ ,λ¯β = δβ. (III.11) j D.B. j D.B. 2mr3 α D.B. −m α (cid:2) (cid:3) (cid:8) (cid:9) As a result one finds, for the unmodified quantities, the classical symmetry algebra Q,Q = 0 = Q¯,Q¯ , Q,Q¯ = 2iH1l, (III.12a) { }D.B. D.B. D.B. − i i Q,J~ = ~σ(cid:8)Q, (cid:9) (cid:8)Q¯,J~(cid:9) = Q¯~σ, (III.12b) D.B. −2 D.B. 2 h i i h i i Q,K~ = θ~σQ, Q¯,K~ = θQ¯~σ, (III.12c) D.B. 2 D.B. −2 hJi,Jji = ǫijkJk, hJi,Kji = ǫijkKk, (III.12d) D.B. D.B. m (cid:2) (cid:3) Ki,Kj = θ2 2mH ǫij(cid:2)kJk + (cid:3) ǫijkQ¯σkQ. (III.12e) D.B. − 2 (cid:2) (cid:3) (cid:0) (cid:1) Away from zero energy, we can consider the conserved vector (III.9) and the modified an- gularmomentum andLaplace-Runge-Lenzvectors(III.10). These satisfyanSU(2) Spin(4), × 6 algebra2 Si,Sj = ǫijkSk, Si,J˜j = 0, Si,K˜j = 0, (III.13a) D.B. D.B. D.B. h i h i (cid:2)J˜i,J˜j(cid:3) = ǫijkJ˜k, J˜i,K˜j = ǫijkK˜k, K˜i,K˜j = (θ2 2mH)ǫijkJ˜k. D.B. D.B. D.B. − h i h i h i (III.13b) The Spin(4) Casimirs are classically given by ~ ~ κ2θ ~ ~ κ2 J˜ K˜ = , (2mH θ2)J˜2 K˜2 = (θ2 mH). (III.14) · 4 − − − 2 − B. The Quantum Symmetry Algebra Quantum mechanically, the Dirac brackets (III.11) become the nonzero commutators κ 1 xi,p = iδi, [~p,D] = i ~x, λ ,λ¯β = δβ. (III.15) j j 2mr3 α m α (cid:2) (cid:3) (cid:8) (cid:9) Thus the quantum symmetry algebra is essentially identical to the classical one, Q,Q = 0 = Q¯,Q¯ , Q ,Q¯β = 2Hδβ, (III.16a) { } α α 1 1 J~,Q = ~σ(cid:8)Q, (cid:9) (cid:8) J~,Q¯(cid:9) = Q¯~σ, (III.16b) −2 2 h i θ h i θ K~,Q = ~σQ, K~,Q¯ = Q¯~σ, (III.16c) 2 −2 hJi,Jji = iǫijkJk, hJi,Kji = iǫijkKk, (III.16d) Ki,Kj = iǫijk (θ2 2mH)Jk + mQ¯σkQ . (III.16e) (cid:2) (cid:3) (cid:2) (cid:3) − 2 (cid:2) (cid:3) (cid:2) (cid:3) Similarly, away from zero energy,3 Si,Sj = iǫijkSk, J˜i,Sj = 0, K˜i,Sj = 0, (III.17a) (cid:2) S~,Q(cid:3) = 1~σQ, h J~˜,Qi = 0, h K~˜,Qi = 0, (III.17b) −2 hS~,Q¯i = 1Q¯~σ, hJ~˜,Q¯i = 0, hK~˜,Q¯i = 0, (III.17c) 2 h i h i h i J˜i,J˜j = iǫijkJ˜k, J˜i,K˜j = iǫijkK˜k, K˜i,K˜j = iǫijk(θ2 2mH)J˜k. (III.17d) − h i h i h i 2 Tobe precise,whetherthe real algebrageneratedbyJ~andK~ is Spin(4)orSpin(3,1)depends onwhether one is considering the sector of the theory with energies 0<E < θ2 or E > θ2 . 2m 2m 3 These expressions have been givenby D’Hoker and Vinet [12, 13, 14, 16, 17], who state that they require a tedius calculation. We agree that they could require a tedius calculation, for we attempted to verify them with Mathematicar, but found that 2GB of RAM was not enough. Nevertheless, we have at least verified these expressions on constant test spinors. An elegant derivation is given in [21]. 7 However, one of the Casimirs is quantum mechanically modified in an important way,3 ~ ~ κ2θ ~ ~ κ2 J˜ K˜ = , K˜2 = (2mH θ2)(J˜2 +1) (mH θ2), (III.18) · 4 − − 2 − Also, 3 S~2 =~s2 = [(mλ¯λ)2 2mλ¯λ]; (III.19) −4 − the second term is a quantum correction. IV. CLASSICAL TRAJECTORIES In this section we consider the classical bosonic theory by setting λ 0. ≡ Poincar´e [27] has demonstrated that trajectories of charged particles in the presence of a magnetic monopole are always confined to a cone whose tip lies at the monopole. The addition of a radial potential does not effect this. Dotting ~x into Eq. (III.3) yields κ J~ ~x = r. (IV.1) · 2 Spherical symmetry allows us to choose the z-axis to be parallel to the angular momentum, so that J~ = J~ zˆ. Then (IV.1) reads | | z κ ~ = = const, J J , (IV.2) r 2J ≡ | | which is the equation of a cone whose slope is κ . Thus the dynamics are constrained √4J2 κ2 to the positive (negative) z axis for positive (negati−ve) κ. Since z r, Eq. (IV.2) implies ≤ κ J | |. (IV.3) ≥ 2 The particular radial potential in this problem admits a conserved Laplace-Runge-Lenz vector which, in this case, restricts the trajectories to also lie in a plane. Without the quantum term in Eq. (III.4), and with the fermions set to zero, K~ ~n J~ = ~x+ 1J~ (p~+κA~), (IV.4) ≡ − θ θ × (cid:16) (cid:17) is conserved and orthogonal to the velocity vector. Therefore, the trajectories must be confined to the plane whose normal is ~n. Thus, the trajectories not only live on the surface of a cone, they are also conic sections; however, the plane of motion does not generally contain the origin. To be precise, use the Casimirs (III.14) (since the modified vectors coincide with the angular momentum and Laplace-Runge-Lenz vectors when the fermions are set to zero) to see that 2 K~ 2mH κ2 ~n 2 = J~ = J~2 . (IV.5) | | (cid:12) − θ (cid:12) θ2 − 4 (cid:18) (cid:19) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 8 Notice that this is nonzero except for orbits with minimal angular momentum. (Orbits with J = κ lie entirely on the z-axis, by (IV.2).) When ~n is nonzero, the orthogonal distance 2 from the plane of motion to the origin is ~n J~2 κ2 z ~x = − 4 , (IV.6) ′ ≡ · ~n s 2mE | | where E is the energy. We can complete the definition of the primed coordinates (x,y ,z ). First complete the ′ ′ ′ definitionoftheunprimedcoordinatesbychoosing they direction(so faronlythez direction was chosen) so that the normal vector ~n has no y component. Then choose x to be only a ′ rotation of the x and z directions, i.e., n n n n xˆ′ = ~nz xˆ− ~nxzˆ, xˆ = ~nz xˆ′ + ~nxzˆ′, | | | | | | | | yˆ′ = yˆ, yˆ= yˆ′, (IV.7) ~n n n zˆ′ = ~n , zˆ= − ~nxxˆ′ + ~nz zˆ′ | | | | | | Since J~ Jzˆ, ≡ J~ κ2 2mE κ2 κ2 n = ~n = J , n = ~n2 n2 = 1+ J2 . (IV.8) z J · − 4J x − z s θ2 − 4J2 − 4 (cid:18) (cid:19)(cid:18) (cid:19) p It is shown below (see remark 2 on p. 11) that the first factor under the square root is indeed positive, as required for consistency. Using r = x2 +y2 +z2 and z = (xxˆ +y yˆ +z zˆ) zˆ where z is the constant (IV.6), ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ · the equation for the cone (IV.2) reads, p (1 ǫ2)(x x )2 2ǫr (x x )+y2 = r2, (IV.9a) − ′ − ′0 − 0 ′ − ′0 ′ 0 where 2J2 κ2θ2 J2 θ2 J2 κ2 4J2 κ2 ǫ = 2mE θ2 + , x = − 2mE − 4 , r = − . sκ2mE − 4J2 ′0 − q θ √2(cid:0)mE (cid:1) 0 2 κ √2mE (cid:18) (cid:19) | |− | | (IV.9b) This is easily recognized as a conic section in the (x,y )-plane. ǫ is the eccentricity of the ′ ′ orbit, r is the semi-latus rectum of the conic section, and the foci are offset along the x 0 ′ J2 θ2 J2 κ2 axis, with one at x = x and the other at x = x + 2ǫ r = r −2mE“ − 4 ”. In the ′ ′0 ′ ′0 1 ǫ2 0 − θ+√2mE − | | unprimed coordinate system, the foci are located at 1 θ2 + κ2θ2 J2 κ2 1 κ2 θ ~x = − 2mE 8mE − 4 xˆ+ J2 4 | | zˆ. (IV.10) q √2mE qθ J√2mE ± − √2mE θ ! ±| | ±| | 9 (a) (b) (c) (d) FIG. 1: Two representative orbits, one bound and one unbound, with J = 3, κ = 1 and units in 2 which θ = 2. (a) and (b) show a bound orbit with mE = 1.92089 with and without explicitly − exhibiting the plane and the cone on which the orbit lies. In particular, the foci are visible in (b). (c) and (d) are similar with mE = 2.2. A bound and an unbound orbit are shown in Fig. 1. Tofindtheexplicit time-dependence oftheparticleonitsorbit,wereturntotheunprimed coordinates. Upon using cylindrical coordinates (ρ = x2 +y2,φ = tan 1 y,z), Eq. (IV.2) − x yields p √4J2 κ2 4J2 ρ(t) = − z(t) = 1 z(t) . (IV.11) κ κ2 − | | r Thus, knowledge of the trajectory reduces to finding the two coordinates z(t) and φ(t). They are found by using the conserved energy and angular momentum to yield first order differential equations. Recalling the choice of z-axis parallel to J~, the z-component of angular momentum is ˙ ~ (since m~x = ~p+κA) κz κ2 J = mρ2φ˙ + z2φ˙ = . (IV.12) 2r ⇔ 4mJ 10