DECOUPLING OF TRANSLATIONAL AND ROTATIONAL MODES FOR A QUANTUM SOLITON A.Dubikovsky and K.Sveshnikov1 Department of Physics, Moscow State University, Moscow 119899, Russia 4 9 9 1 A set of integral relations for rotational and translational zero modes in n the vicinity of the soliton solution are derived from the particle-like properties a of the latter and verified for a number of models (solitons in 1+1-dimensions, J skyrmeons in 2+1- and 3+1-dimensions, non-abelian monopoles). It is shown, 9 1 that by consistent quantization within the framework of collective coordinates these relations ensure the correct diagonal expressions for the kinetic and cen- 1 v trifugal terms in the Hamiltonian in the lowest orders of the perturbation 9 expansion. The connection between these properties and virial relations is 8 0 also determined. 1 0 4 Motivated by success of the skyrmeon baryon models [1], there was 9 / recently much interest in quantization of hedgehog-type configurations in- h t cluding translationaland rotationaldegrees of freedom by means of suitable - p collective coordinates [2]. Generally, in this approach the corresponding e h quantum Hamiltonian contains a full bilinear form in conjugated momenta : v with nontrivial couplings between different collective variables [3, 4, 5]. i X However, for a particle-like classical solution one should expect additive r a diagonal contributions of kinetic and centrifugal terms to the Hamiltonian, at least to the lowest orders in the a ppropriate weak coupling expansion. Actually, it is a part of the general problem of decoupling upon quantiza- tion of various soliton degrees of freedom, which takes place for any type of field models with classical solutions. In this paper we’ll present a consistent general analysis of this problem for translational and rotational variables, based on the particle-like properties of the classical solution combined with Lorentz covariance and virial relations. Let us consider a field theory in d+1 space-time dimensions described by the Lagrangean density (ϕ), which possesses a classical particle-like L solution ϕ (x). For brevity the Lorentz and internal symmetry group in- c dices are suppressed. It is generally accepted, that if in the rest frame ϕ c is static with finite and localized energy density, then in quantum version of the theory such configuration describes an extended particle. Now we’ll 1E-mail: [email protected] show, that there exists a set of nontrivial integral relations, fulfilled by ϕ (x), which provide the validity of these assumptions. c For a given static solution ϕ (~x) the moving one is constructed via c Lorentz boost , what results in the replacement xi Λ 1ixν (1) − ν → in the arguments of ϕ , where Λµ is the corresponding Lorentz matrix. The c ν momentum of the moving solution is Pµ = Tµ0(ϕ (~x,x0)) d~x, (2) c Z where Tµν(ϕ ) is the energy–momentum tensor. Transforming the r.h.s. in c (2) to the rest frame, one gets Pµ = Λµ Λ0 Tµ′ν′(ϕ (ξ~)) J dξ~, (3) µ′ ν′ c Z where J = Λ0−1 is the Jacobian of transition from d~x to the rest frame spa- 0 tial variable. On the other hand, the l.h.s. of (2) should be the momentum of a particle with the mass M, that is Pµ = ΛµM. (4) 0 Then it follows from eqs.(3) and (4) that Tµν(ϕ (ξ~)) dξ~ = δµδνM. (5) c 0 0 Z For ν = 0 we get P0 = M, Pi = 0, just that we should expect for a static solution. However, for µ = i, ν = j we obtain ∂ (ϕ ) c ~ ~ L ∂ ϕ (ξ) dξ = Mδ . (6) ∂∂jϕ (ξ~) i c ij Z c (Henceforth the derivative ∂ acts on the indicated argument of the func- i tion.) So we get the first set of conditions (6), which holds for a particle-like ~ classical configuration ϕ (ξ) in the rest frame. c Now let us consider the orbital part of the 4-rotation tensor (without the spin term) Lµν = xνTµ0(ϕ (~x,x0)) xµTν0(ϕ (~x,x0)) d~x. (7) c c − Z (cid:16) (cid:17) Returning to the rest frame, we can write similarly Lµν = Λµ Λν Λ0 ξν′Tµ′σ(ϕ (ξ~)) ξµ′Tν′σ(ϕ (ξ~)) J dξ~. (8) − Z (cid:18) (cid:19) On the other hand, for a static particle-like solution in the rest frame one obviously has Lij = 0, while L0i coincide with the center-of-mass coordi- nates and therefore can be made vanish by a spatial translation. But since L µν is a tensor, then it must vanish in any other Lorentz system. Then it follows from eq.(8) that 0 = ξνTµσ(ϕ (ξ~)) ξµTνσ(ϕ (ξ~)) dξ~. (9) c c − Z (cid:18) (cid:19) For σ = 0 these relations mean, that in the rest frame Lµν = 0, just that we expected to have. When µ = 0 or ν = 0 but σ = 0, then due to eq.(5) 6 one obtains from (9) that ξiT0j(ϕ )dξ = 0, what gives an identity pro- c vided by symmetry of Tµν. The latter statement is valid even for theories R with Chern–Simons terms, since these terms do not contribute to Tµν [6]. However, for µ = i, ν = j, σ = k we get from (9) the following relations (for definiteness, we take d = 3) ∂ (ϕ ) ~ c ~ ~ ε ξ ∂ ϕ (ξ) L dξ = ε ξ (ϕ ) dξ = 0, (10) lij i j c ∂∂kϕ (ξ~) lik i L c Z c Z since L0k vanish in the rest frame by assumption. This is the second set ~ of relations on ϕ (ξ), following from the Lorentz covariance and particle- c likeness of the classical solution. So each particle-like solution should be subject of conditions (6) and (10). It should be noted, that the relation (4) for µ = 0 reproduces noth- ing else but the relativistic mass-energy relation. For the moving ϕ4-kink solution this relation has been explicitly verified in [5], and for the moving skyrmeon — in (7) by direct calculations. However, the eqs.(6) are more general and, moreover, the eqs.(10) also take place. Note also, that these relations, being consistent with the field equations and conservation laws, do not be the direct consequences of the latters, and should be considered separately. As a direct result of these relations we get the orthogonality of the zero–frequency eigenfunctions in the neighborhood of the classical particle- like solution cite8. Let us discuss the theory of a nonlinear scalar field in 3 spatial dimensions, described by the Lagrangean density 1 2 = (∂ ϕ) U(ϕ), (11) µ L 2 − which possesses a classical static solution ϕ (x) = u(~x). According to the c virial theorem such solutions are unstable in more then one spatial dimen- sion, but for our purposes this is not so important compared to simplicity of presentation. In the general case the non-spherical configuration u(~x) yields 6 zero-frequency modes — three translational ones ψ (~x) = ∂ u(~x), (12) i i and three rotational f = ε x ∂ u(~x). (13) i ijk j k Then from eqs.(6) and (10) one immediately obtains ~ ~ ~ dξ ψ (ξ) ψ (ξ) = Mδ , (14) i j ij Z ~ ~ ~ dξ f (ξ) ψ (ξ) = 0. (15) i j Z Further, by spatial rotations one can always achieve that ~ ~ ~ dξ f (ξ) f (ξ) = Ω = Ω δ , (16) i j ij i ij Z where Ω are the moments of inertia of the classical configuration. Ob- i viously, the relations (14) and (15) remain unchanged. As a result, the normalized set of translational and rotational zero-modes can always be written as ψ (~x)/√M , f (~x)/√Ω . i i i { } So the particle-likeness of the classical solution results in the diagonal- ity of the zero-frequency scalar product matrix. This diagonality plays an essential role in the procedure of quantization in the vicinity of a classical soliton solution by means of collective coordinates [3, 4, 5]. Following the conventionalprocedure[4], let us consider the field ϕ(~x) in the Schroedinger picture in the vicinity of the solution u(~x). The substitution, introducing translational and rotational collective coordinates, reads [8] ϕ(~x) = u R 1(~c)(~x ~q) + Φ R 1(~c)(~x ~q) , (17) − − − − (cid:16) (cid:17) (cid:16) (cid:17) where Φ is the meson field, R(~c) is the rotation matrix, ~q and ~c are the translational and rotational collective coordinates correspondingly. For our purposes the parametrization of the rotation group by means of the vector–parameter [9] is the most convenient. In this parametrization the rotation matrix R(~c) is taken in the form c + c 2 1 c2 + 2c + 2c c × × × R(~c) = 1 + 2 = − · , (18) 1 + c2 1 + c2 where c = ε c , (c c) = c c , c2 = ~c ~c. The compositionlaw for vector– ×ab adb d · ab a b ~ parameters, corresponding to product of rotations R(~a) R(b) = R(~c), is given by ~ ~ ~a + b +~a b ~ ~c = ~a, b = × . (19) h i ~ The generators of infinitesimal rotations are i ∂ ~ S = 1 + c c + c , (20) × −2 · ∂~c (cid:16) (cid:17) while the finite rotations U(~a), defined so that U+(~a)~b U(~a) = ~a, ~b , take h i the form ~ U(~a) = exp 2i~aS . (21) − (cid:26) (cid:27) Returning to the decomposition (17), one finds that the total momen- tum of the field is now represented as ∂ ~ P = i , (22) − ∂~q and the total angular momentum is equal to ~ ~ ~ J = L + S, (23) ~ ~ ~ where L = ~q P is the orbital angular momentum, and the spin S is × defined by relation (20). In order to keep the number of degrees of freedom we impose on the field Φ(~y) 6 subsidiary conditions, which in the theory of a weak coupling are usually taken as linear combinations d~y N(α)(~y) Φ(~y) = 0, α = 1,...,6. (24) Z The set N(α)(~y) should ensure the condition of orthogonality of the me- { } son field Φ(~y) to zero-frequency modes and is chosen in the following way. Let us denote M(α)(~y) = ψ (~y), f (~y) . In the general case [4] N(α)(~y) are i i { } given by linear combinations of M(β)(~y) subject of relations d~y N(α)(~y) M(β)(~y) = δ . (25) αβ Z In our case the system of zero-frequency modes is orthogonal, so one im- mediately gets N(α)(~y) = ψ (~y)/M, f (~y)/Ω . (26) i i i { } It is the relation (26), that ensures the additive form of the collective coordinate part of the Hamiltonian within the weak coupling expansion in powers of the meson fields. Let us consider the condition (24) as relation, defining ~q and ~c as functionals of ϕ(~x). A straightforward calculation gives N(α) R 1(~c)(~x ~q) + − − (cid:16)∂R ∂c (cid:17) +R jk l d~y N(α)(~y) y ∂ (u(~y) + Φ(~y)) + (27) ji k i ∂c ∂ϕ(~x) l Z ∂q +R j d~y N(α)(~y) ∂ (u(~y) + Φ(~y)) = 0. ji i ∂ϕ(~x) Z By means of eqs.(18)–(21) one can easily verify that ∂R 2 jk R = ε (1 c ) , ji ikm × ml ∂c −1 + c2 − l (28) 1 + c c + c (1 c ) 1 = · ×. × − − 1 + c2 The simplicity of these relations demonstrates the convenience of vec- tor parametrization (18)–(21) for such type of problems [8]. Then from eqs.(27), (28) we immediately get the following lowest-order expressions for ∂~q/∂ϕ(~x) and ∂~c/∂ϕ(~x) ∂~q ψ~ R 1(~c)(~x ~q) = MR 1 , − − − − ∂ϕ(~x) (cid:16) (cid:17) (29) 1 c ∂~c f~ R 1(~c)(~x ~q) = 2 Ω − × . − − − 1 + c2 ∂ϕ(~x) (cid:16) (cid:17) Calculating the conjugate momentum π(~x) = iδ/δϕ(~x) as a composite − derivative δ ∂~c ∂ π(~x) = i = i + − δϕ(~x) ∂ϕ(~x) − ∂~c! ∂~q ∂ δΦ(~y) δ + i + d~y i , (30) ∂ϕ(~x) − ∂~q! δϕ(~x) − δΦ(~y) Z   and using the relations (29), we obtain to the leading order the following result π(~x) = Π R 1(~c)(~x ~y) − − 1(cid:16) (cid:17) ψ~ R 1(~c)(~x ~q) K~ + d~y ((∂~Φ) Π)(~y) (31) − − M − (cid:16) (cid:17)(cid:18) Z (cid:19) f~ R 1(~c)(~x ~q) Ω 1 I~+ d~y (([~y ∂~] Φ) Π)(~y) . − − − − × (cid:16) (cid:17) (cid:18) Z (cid:19) In eq.(31) K~ = R 1(~c)P~ and I~ = R 1(~c)S~ are the momentum and the − − spin of the field, corresponding to the rotating frame, and the meson field momentum Π(~y) is defined as δ Π(~y) = d~z A(~z,~y) i , (32) − δΦ(~z) Z   where A(~x,~y) is the projection matrix on the subspace, orthogonal to zero- frequency modes A(~x,~y) = δ(~x ~y) M(α)(~x) N(α)(~y). (33) − − X Inserting eqs.(17) and (31) in the Hamiltonian 1 1 2 H = d~x π2(~x) + (∂~ ϕ) + U(ϕ) , (34) (2 2 ) Z we obtain finally the following lowest-order expression 1 1 2 1 H = M + d~y Π2 + (∂~Φ) + U (u(~y))Φ2 (~y) ′′ (2 2 2 ) Z 1 2 ~ ~ + K + d~y (∂Φ) Π(~y) (35) 2M (cid:18) Z (cid:19) 2 ~ I + d~y ([~y ∂] Φ) Π(~y) 1 i i × + (cid:18) (cid:19) . 2 R Ω i i X It is indeed such form of the Hamiltonian, that provides to interpret the resulting ground state as non-relativistic particle with the mass M and moments of inertia Ω . So the correct form of the Hamiltonian with addi- i tive kinetic and centrifugal terms, that means the absence of correlations between translational and rotational degrees of freedom, is ensured by the diagonality of zero–frequency scalar product matrix (14)–(16). In turn, this is a direct consequence of relations (6) and (10). Note also, that this result will be actually valid for any field model in the neighborhood of the suitable soliton solution. These general considerationscan be easily illustratedby concrete mod- els. Firstly, we consider the theory of a scalar field in 1+1-dimensions, described by the Lagrangean density (11). In this case we have only one relation (6) 2 dx (ϕ (x)) = M, (36) ′ Z where the mass M is given by 1 2 M = dx (ϕ (x)) + dx U(ϕ(x)). (37) ′ 2 Z Z Performing the dilatation ϕ(x) ϕ(λx) and demanding for the solution → dM(λ) at λ = 1, i.e. = 0, we find the well-known Hobart–Derrick virial dλ (cid:18) (cid:19)λ=1 relation [10] 1 2 dx (ϕ (x)) = dx U(ϕ(x)), (38) ′ 2 Z Z owing to which the ”particle-likeness condition” (36) is fulfilled automati- cally. In more spatial dimensions the situation with the model (11) is more complicated. Namely, it is a trivial task to verify by the same arguments, that for each i (∂ ϕ)2d~x = M. (39) Z However, the orthogonalityconditions between different spatial derivatives, predicted by eqs.(14) and (15), cannot be derived by such simple consid- erations. So here the additional arguments, used by derivation of relations (6) and (10), are crucial. In 2+1-dimensions, the solitons in CP -models are interesting exam- N ples with such particle-like properties. As it is well-known, for N = 1 the CP -model is reduced to O(3)-model [11], described by N 1 = ∂ ϕa∂µϕa (40) µ L 2 with subsidiary condition ϕaϕa = 1. (41) This theory is a planar analog of the Skyrme model [12]. The standard one-particle solution of the model is given by [13] ϕ1 = φ(r)cosnϑ, ϕ2 = φ(r)sinnϑ, ϕ3 = (1 φ2)1/2, (42) − where r,ϑ are polar coordinates and 4rn φ(r) = , (43) r2n + 4 anddescribes the ”baby-skyrmeon”configurationwiththetopologicalcharge Q = n and the mass M = 4πQ. The direct insertion of expression (42) into conditions (6) and (10) yields ∂ ϕa∂ ϕad2x = 4πnδ = Mδ , (44) i j ij ij Z and ε x ∂ ϕa∂ ϕad2x = 0, (45) ij i j k Z that means the particle-likeness of the solution (42) in the way described above. As a more nontrivial example, we consider the SU(2)-Skyrme model in 3+1-dimensions [1, 14], including the break–symmetry pion mass term f2 1 m2 = π tr L2 + tr [L L ]2 + π tr (U + U+ 2) L − 4 µ 32g2 µ ν 4 − = (2) + (4) + , (46) B L L L where, as usually, L = U 1∂ U is the left chiral current and U = σ+iτaπa µ − µ is the quaternion field. In the quaternion representation one has f2 f2 (2) = π(∂ σ)2 + π(∂ πa)2, µ µ L 1 (4) = ((∂ πa)4 (∂ πa∂ πa)2 (47) µ µ ν L −4g2 − +2((∂ σ)2(∂ πa)2 ∂ σ∂ σ∂ πa∂ πa)), µ ν µ ν µ ν − = m2(σ 1). B π L − Supposing the conventional ”hedgehog” Ansatz ra σ = cosφ(r), πa = sinφ(r) (48) r we find for the mass of the skyrmeon M = M(2) + M(4) + M , B 1 2 M(2) = 4πf2 r2dr φ 2 + sin2φ , (49) π 2 ′ r2 ! Z 4π sin2φ sin2φ M(4) = r2dr + 2φ 2 , ′ g2 2r2  r2  Z φ  M = 4πm2 r2dr 2sin2 . B π 2 Z A straightforward calculation gives ∂ 2 L ∂ uAd3x = δ (M(2) + 2M(4)), (50) i ij ∂∂juA 3 Z here uA = σ, πa . Inserting eqs.(49) and (50) into (6), we obtain the first { } ”particle-likeness condition” for the skyrmeon M(2) M(4) + 3M = 0. (51) B − On the other hand, the scaling uA(~x) uA(λ~x) yields → 1 1 M(λ) = M(2) + λM(4) + M . (52) B λ λ3 dM(λ) It is easy to verify, that the requirement of = 0 coincides with dλ (cid:18) (cid:19)λ=1 the eq.(51) for the skyrmeon. So the particle-likeness condition (51) for the skyrmeon is fulfilled due to virial relation. Further, inserting the sub- stitution (48) into eqs.(10) we find in the same way, that the second set of relations for the skyrmeon is provided by the symmetry properties. So we’ll get upon quantization, that the full bilinear form considered in [2], auto- matically simplifies up to a diagonal construction similar to eq.(35), and therefore the quantized skyrmeon describes an extended non-relativistic particle. Finally, we consider the ’t Hooft–Polyakov monopole for the SU(2)- Yang–Mills–Higgs theory [5, 15], described by the Lagrangean 1 1 = (Fa )2 + (D φ)2 V (φ), (53) µν µ L −4 2 − where λ V (φ) = (φaφa η2)2, 4 − Fa = ∂ Aa ∂ Aa + gεabcAbAc, (54) µν µ ν ν µ µ ν − D φa = ∂ φa + gεabcAbφc. µ µ µ The monopole solution is given by 1ra 1 r φa = H(r), Aa = ε j(1 K(r)), Aa = 0. (55) gr2 i g aijr2 − 0 The mass of the monopole is equal to η dξ 1 1 M = 4π ξ2K 2 + (1 K2)2 + (H ξH )2 ′ ′ g ξ2 2 − 2 − Z (cid:20) λ +K2H2 + (H2 ξ2)2 , (56) 4g2 − (cid:21) where the integration variable ξ is connected with the radial coordinate r via ξ = gηr. The l.h.s of condition (6) reads ∂ η 2 dξ L ∂ uAd3x = 4π δ ξ2K 2 + i ij ′ ∂∂juA g · 3 ξ2 Z Z (cid:20) 1 +2(1 K)2 + (1 K)3 + (H ξH )2 + H2K , (57) ′ − − 2 − (cid:21) here uA = Aa, φa . Now we take into account of the virial considerations. µ { } Performing the scaling uA(~x) uA(λ~x), we find → 1 1 1 1 M(λ) = M + M + M + M , 2 1 0 V λ λ2 λ3 λ3 1 M = d3x (∂ Aa)2 ∂ Aa∂ Aa (∂ φa)2 = 2 µ ν µ ν ν µ µ 2 − − Z h i η dξ 1 = 4π ξ2K 2 + 2(1 K)2 + (H ξH )2 + H2 , ′ ′ g ξ2 " − 2 − # Z M = d3x g εabc∂ AaAbAc εabc∂ φaAbφc = (58) 1 µ ν µ ν µ µ − Z h i η dξ = 4π 2(1 K)3 2H2(1 K) , g ξ2 − − − − Z h i