Collective dynamics of solitons and inequivalent quantizations

The collective dynamics of solitons with a coset space G / H as moduli space is studied. It is shown that the collective band for a vibrational state is given by the inequivalent coset space quantization corresponding to the representation of H carried by the vibration. To leading order the collective dynamics is free motion in G / H coupled to background gauge ﬁelds determined by the vibrational state.


I. INTRODUCTION
Solitons arise as static finite energy solutions to the equations of motion of nonlinear field theories. In general, a given soliton depends upon a set of parameters or moduli, and is a point in the manifold of solutions of equal energy, or moduli space. In many cases this manifold is simply an homogeneous or coset space G/H, where G is the group of symmetries of the action and HʚG is the symmetry of the solitonic solution.
Around a soliton there are two kinds of quantum excitations. The first corresponds to collective motion in the moduli space. The second are vibrational ͑intrinsic͒ excitations out of it. If the energy for the collective excitations is much lower than that for the vibrational ones the low energy spectrum can be approximately described by collective bands associated with each vibrational state. These bands can be described by an effective quantum mechanical problem given by the motion of a particle in the moduli space. However, as is well known from molecular and nuclear rotational bands, different vibrational states may have different collective bands. It is the purpose of this paper to show that in the case when the moduli space of the soliton is a coset space G/H a simple description of the collective bands of vibrational states can be given in terms of inequivalent coset space quantizations introduced by Mackey, 1 and more recently studied by Landsman and Linden 2 and McMullan and Tsutsui, 3 among others.
Since the soliton is invariant under the subgroup H, vibrational excitations fit into irreducible representations ͑irreps͒ of H. Below we show that the collective band corresponding to a vibrational state in a representation of H realizes a representation of G induced by . This representation of G is reducible, and when it is broken into irreducible representations the whole collective band is obtained. This is equivalent to saying that the collective band for a vibrational state is given by the inequivalent quantization of G/H corresponding to the irrep of H carried by the vibration. In this way we find that collective motion is a physical example of the inequivalent coset space quantizations.
The lowest energy collective band of the soliton is that of the ground state of the vibrations. If the ground state is in the trivial representation of H, our results provide nothing new for determining this band, which is obtained from the usual ͑trivial͒ quantization of free motion in G/H. However, although the soliton is invariant under H, the quantum ground state need not be in the trivial representation because of topological factors which may arise when adiabatically performing the operations of H. For example, in the SU ͑3͒ Skyrmion the Wess-Zumino-Witten term 4 contributes with an extra phase. The same happens with the SU ͑2͒ Skyrmion 5 when it is quantized as a fermion. For the Bϭ1 Skyrmion the result is the well-known fact that the ground state has spin an isospin 1/2. The outline of this paper is as follows: In Sec. II we introduce a model 6 which generically describes a soliton model. The action is invariant under a group G of transformation of the fields but the soliton only under a subgroup H, so that the moduli space is a coset space G/H. In Sec. III we summarize Mackey's 1 approach to the quantization on a coset space, and the more recent results by Landsman and Linden 2 and McMullan and Tsutsui. 3 Section IV is the main section of this paper. There we demonstrate the relation between the wave functions and energies of the collective bands of vibrational states and the corresponding ones of the inequivalent coset space quantizations. Finally, in Sec. V we give our conclusions. We also discuss an example of a possible application of our results.

II. THE MODEL
Let us consider a general bosonic field theory in dϩ1 dimensions. The fields s (x,t) are real and take their values in an internal space T. The variables x and t are space and time coordinates, and sϭ1...dim(T ) is an internal index. At any given time the fields s (x,t) determine a configuration of the system, given by a map from R d ͑or a compactification of it͒ into T. The set of these maps corresponds to the configuration space C of the theory. The action is taken to be quadratic in the time derivatives of the fields, namely,

͑1͒
where sums over repeated indices are implicit and, are functions of the fields and a finite number of its spatial derivatives. For example if the action corresponds to a nonlinear -model,

͑2͒
the functionals G st and V are given by, that is, the spatial derivatives are included in V. In other cases, for example the Skyrme model, 5 the function G st depend also on the spatial derivatives of the field. A useful way of thinking about the system is to consider it as a particle moving in the infinite dimensional space of configurations C ͑i.e., of maps from R d into T ͒ with a potential V and a metric given by g st (x,y)ϭG st ␦ d (x Ϫy). The equations of motion following from the action ͑1͒ have static solutions which satisfy For our present purpose we assume that Eqs. ͑4͒ have soliton solutions. We consider the case in which the action S is invariant under an unbroken finite-dimensional compact group G of transformations of the fields, This group usually includes spatial rotations and internal transformations of the fields. The invariance of the action under these transformations is expressed as,

͑7͒
The first condition can be rephrased saying that the transformations are isometries of the space C. Let us consider a soliton solution (x) of ͑4͒, i.e., a minimum of the potential ͐V͓͔d d x. In general, (x) is only invariant under a subgroup H of the symmetry group G of the action. Due to the invariance of the potential under G, any configuration R g ͓ (x)͔ obtained by acting with G on (x) is also a solution. The whole family of solutions, known as the moduli space of the soliton, is thus given by the orbit of (x) under G, and is therefore an homogeneous or coset space, It is possible that there exist other zero modes not related to symmetries of the action, but we shall ignore that possibility here. Consider now fluctuations around the soliton, (x)→ (x) ϩ(x,t). The linearized equations for the fluctuations read, where G st is evaluated on the solution, G st ϭG st ͓ (x)͔ and, The fluctuations can be understood as infinitesimal vectors in the tangent space to C at the point (x). The ͑infinite͒ set of those normal to the moduli space M are massive excitations, since M is the ''valley'' of minima of the static energy. They satisfy,

͑11͒
where n labels the modes, n 0, and n s (x) are the corresponding normalized eigenfunctions, The fluctuations tangent to M are massless, and are usually known as zero modes. They satisfy, where ␣ϭ1,...,dim(G/H) label the zero modes, which are given by the nonvanishing infinitesimal transformations ͑5͒ of the soliton, ␣ s (x)ϭ␦ ␣ s (x). The norm of the zero modes corresponds to the inertia tensor of the soliton,

͑13͒
The massive and zero modes satisfy the completeness relation, where I ␣␤ is the inverse of the inertia tensor. Quantum-mechanically, the massive modes correspond to vibrations of the soliton, while the zero modes to collective motion in M. We are interested in the case in which the collective energy is much smaller than the vibrational one. Schematically, this is given by where I is the inertia for the collective motion and the frequency for the intrinsic excitations. In this regime it is natural to treat the collective motion exactly and the vibrations in perturbation theory. The relation ͑15͒ implies that the small parameter is proportional to ប, and therefore the perturbative expansion is an expansion in loops ͑in powers of ប͒. In the following បϭ1, so that the expansion parameter is 1/I. In order to proceed with the quantization around the soliton (x) it is necessary to treat differently the massive vibrations from the zero modes. One possible approach, which will prove useful to our purposes, is to introduce collective coordinates by performing a transformation of the fields s (x,t) with time dependent parameters, 7-10,6 where ␣ a (t) (aϭ1,...,dim(G)) parameterize G. The action ͑1͒ is not symmetric under this time dependent transformations, and it changes to,

͑17͒
The covariant time derivatives are given by, The transformed action ͑17͒ can be understood as describing the problem from an arbitrary moving frame of reference, its motion given by the collective coordinates.
The classical Hamiltonian corresponding to the action ͑17͒ is given by where s (x,t) are the canonical momenta conjugate to s (x,t), i.e., ͕ s (x,t), t (y,t)͖ ϭϪ␦ s t ␦ d (xϪy), and a (t) are Lagrange multipliers which impose the first class constraints ⌽ a ϭJ a ϪI a , which generate the gauge transformations ͑19͒. 11 We display also the explicit dependence of G st and V on the fields. The operators J a are the generators for the infinitesimal transformations of the fields, and I a are the right generators for the collective coordinates, I a ϭ⌰ a b P b , where P a are the conjugate momenta to the collective coordinates, ͕␣ a ,P b ͖ϭ␦ b a . While the gauge symmetry ͑19͒ acts from the right on the collective coordinates, the original symmetry G acts from the left. The operators that generate the left infinitesimal transformations of the collective coordinates are correspondingly given by L a ϭ⌰ a b P b , where a b (␣) are components of the right-invariant Cartan- and ⌰ ϭ Ϫ1 . The gauge algebra g, generated by the constraints ⌽ a ϭJ a ϪI a , can be decomposed into g ϭhϩp, where the subalgebra h is generated by ⌽ i ϭJ i ϪI i (iϭ1,...,dim(H)), while the complement p is generated by ⌽ ␣ ϭJ ␣ ϪI ␣ (␣ϭdim(H)ϩ1,...,dim(G)). The operators J i generate the subgroup of transformations of the fields that leave the soliton (x) invariant, while J ␣ generate the transformations which change it. Since H is a subgroup, the operators J i close under commutation, ͓J i ,J j ͔ϭiC i j k J k , where C ab c are the structure constants of G. Furthermore, as G is compact the generators J ␣ can be chosen in such a way that which shows that J ␣ transform in a representation ͑possibly reducible͒ of H.
As is usual in this kind of problems, the idea is to replace the zero mode excitations of the fields (x,t) by the collective coordinates ␣(t), by means of a suitable gauge fixing of action ͑17͒. It is important to note that the number of zero modes is equal to the dimension of G/H, while the number of collective variables is dim(G). By gauge fixing the constraints ⌽ ␣ we will eliminate the zero modes in favor of the collective coordinates ␣ ␣ on G/H. We will make use of the remaining H-gauge symmetry to prove that the effective collective dynamics of the soliton can be understood in terms of inequivalent coset space quantizations.

III. INEQUIVALENT COSET SPACE QUANTIZATIONS
In this section we review briefly the approach developed by Mackey to the quantization of a system whose configuration space is a coset space G/H. 1 This must be distinguished from the quantization of G/H as a phase space, which is discussed for example in Ref.
where gG, hH, and (h) are the matrices of the representation of H. The left regular representation of G is defined on functions of G acting as, Equation ͑23͒ is invariant under this action. Therefore, the set of functions f (g) satisfying condition ͑23͒ transforms under the left action of G in a representation called the representation of G induced by . See Refs. 1 and 13 for the definition and properties of induced representations. Given a local section :G/H→G every element gG can be uniquely written as, where G/H and hH. This allows to define a one-to-one correspondence between the set of functions f (g) and functions F :G/H→V such that, The functions F () transform under G in the induced representation as, where g Ϫ1 ϭ(g Ϫ1 ()) defines the action of G on G/H. Landsman and Linden 2 studied the dynamical consequences of the inequivalent quantizations for the motion of a particle in G/H. They discovered that in the nontrivial quantum sectors the particle couples to a background gauge field A ␣ , known as the H-connection, which takes values in the representation of the sub-algebra (h). The Hamiltonian is given by, where ٌ ␣ is the covariant derivative constructed out of the metric g ␣␤ on G/H. Due to the H-connection the Hamiltonian is matrix valued ͑in the trivial representation of H it reduces to minus one-half the Laplacian Ϫ 1 2 ⌬ G/H ϭϪ 1 2 g ␣␤ ٌ ␣ ‫ץ‬ ␤ ͒. McMullan and Tsutsui 3 developed a different approach to the inequivalent quantizations. Using the fact that in Eq. ͑25͒ hH can be further decomposed as hϭr(h)s(h), where s belongs to the Cartan subgroup of H, they showed that instead of Mackey's functions f (g) one can use its highest weight component f evaluated at gϭr ͑i.e., sϭ1͒. The condition ͑23͒ implies that these functions are annihilated by the raising operators E Ͼ0 in the Chevalley basis ͕H ␣ ,E ͖ of h. In this case the wave functions are scalars, which allows for a simpler definition of the corresponding path integral.

IV. COLLECTIVE DYNAMICS
In this section we will prove that the collective band associated with an intrinsic vibrational state can be obtained from the inequivalent quantization corresponding to the representation of the subgroup H carried by the vibration. To see this we consider the canonical quantization of the action ͑17͒ introduced in Sec. II. This action has a G-gauge invariance due to the introduction of the collective coordinates as additional variables. Gauge fixing the G/H part of the gauge symmetry allows to eliminate the zero modes in favor of the collective coordinates. The remaining H-gauge invariance can be treated with the Dirac method of imposing the constraints on the wave functions. 11 In our case this becomes Mackey's condition ͑23͒ for the collective wave functions associated to a given vibrational state. Alternatively, the collective coordinates which parameterize H can be eliminated, fixing the H-gauge symmetry. In this case the wave functions become F (x) of Eq. ͑27͒ and the Hamiltonian that of Landsman and Linden.

A. Elimination of the zero modes
We start by eliminating the zero modes in favor of collective coordinates on G/H. Let us expand the fields in terms of fluctuations around the soliton. The fluctuations can be written as linear combinations of the normal modes ͑11͒ and ͑12͒, where the time dependent coefficients q A (t) ͑A stands both for n and ␣͒ are now the dynamical degrees of freedom. The conjugate momenta to the fields read, where p A are conjugate to q A . Inserting these expressions into the Hamiltonian ͑20͒, and making use of the equations for the normal modes ͑11͒, ͑12͒ and their orthogonality relations, we obtain up to quadratic order in the fluctuations, HϭE clas ϩ 1 2 ͑ p n 2 ϩ n 2 q n 2 ͒ϩ 1 2 I ␣␤ p ␣ p ␤ Ϫ a ⌽ a ϩO͑q 3 ,pq 2 ͒. ͑31͒ The first term is the classical soliton energy E clas ϭ͐d d xV͓ (x)͔. The second are the ͑infinite͒ harmonic oscillators corresponding to the vibrations of the soliton. The third term are the zero modes, and is purely kinetic. Anharmonic terms have been omitted.
Expanding similarly the constraints we get, where we have kept up to quadratic terms in the fluctuations but the collective operators are treated exactly. The matrices (D a ) AB are defined as, where ␦ a A t (x) are the infinitesimal transformations ͑5͒ of the eigenfunctions, and correspond to the representation of G under which the fluctuations transform.
We want to eliminate the degrees of freedom associated with the zero energy fluctuations (q ␣ ,p ␣ ). We achieve this by choosing to fix the gauge invariance generated by ⌽ ␣ the following gauge fixing conditions, ͕A,B͖ D ϭ͕A,B͖Ϫ͕A,q ␣ ͖͑␦ ␣␤ ϪD ␤␣n q n ͕͒⌽ ␤ ,B͖,

͑36͒
in order to treat the gauge conditions and constraints as operator identities. We are then able to solve p ␣ from the equations ⌽ ␣ ϭ0, p ␣ ϭI ␣ Ϫ͑D ␣ ͒ nm P n q m ϩO͑ pq 2 ,Iq͒. ͑37͒ Replacing (q ␣ , p ␣ ) by Eqs. ͑35͒, ͑37͒ in the Hamiltonian ͑31͒ we obtain, HϭE clas ϩ 1 2 ͑ p n 2 ϩ n 2 q n 2 ͒ϩ 1 2 I ␣␤ I ␣ I ␤ ϪI ␣␤ I ␣ ͑ D ␤ ͒ nm p n q m Ϫ i ͑ t ͒⌽ i ͑ t ͒ϩO͑ q 3 ,pq 2 ,Ipq 2 ͒. ͑38͒ The zero modes have been completely eliminated. Instead we have a kinetic term for the collective coordinates and a coupling between the collective coordinates and the vibrations. Anharmonic terms in the vibrations and higher order vibration-collective couplings have been omitted.

B. Collective energy eigenfunctions
The Hamiltonian ͑38͒ describes the dynamics of the system in the 1-loop approximation for the vibrations, while the collective motion is treated exactly. To this order, the spectrum corresponds to vibrational states and on top of each state a collective band. There is still the H-gauge symmetry.
Since the soliton is invariant under the subgroup H, the vibrations split into irreducible representations ͑irreps͒ of H. Consider now a given vibrational state classified by the irrep of H, and lets derive the effective collective dynamics for this state from the Hamiltonian ͑38͒. In other words, we restrict to energies which can only excite collective modes, but not any other vibrational ones. Taking into account this restriction, the wave functions for a state in the collective band has the general form, The functions (q) (ϭ1,...,dim()) form a basis for the irrep of the vibration, while the collective functions f (␣) are arbitrary and have to be determined. We still have to take into account the H-gauge invariance. As mentioned above, to deal with it we apply Dirac's method of imposing the constraints on the wave functions. 11 This restricts the wave functions ͑39͒ to satisfy ⌽ i (q,␣)ϭ0᭙ i , which imposes on the collective functions the conditions, Here (T i ) are the infinitesimal generators of H in the representation. This is the infinitesimal version of Eq. ͑23͒. This condition also holds for the discrete transformations hH, which together with ͑40͒ imply for all hH, where R h stands for the right action of h on the group element parameterized by ␣, g(␣) →g(␣)h. Therefore, the collective functions f (␣) satisfy Mackey's condition ͑23͒, i.e., they transform under the left action of G in the representation of G induced by the representation of H. 13 In other words, the states in the collective band are those of the inequivalent quantization of G/H given by the representation of H carried by the intrinsic state. In order to diagonalize the Hamiltonian ͑38͒ we can make use of the original symmetry G. The induced representation of G under which the collective functions in ͑39͒ transform is reducible. It can be broken into irreducible components. In fact, by the Peter-Weyl theorem 13 the collective functions f (␣) can be rewritten as, where the sum is over all irreps I of G defined by the matrices D MN I (␣). Under the left action of G each term of the sum transforms in the corresponding representation I of G. Under the right action of H each term in ͑42͒ transforms in the representation I of G considered as a representation of H. This representation of H is in general reducible and can be broken in irreducible pieces, I͉ H ϭ 1 ϩ¯ϩ n , where i are irreps of H. In this decomposition, the representation carried by the vibrations appears a number of times we denote by d I , that is, I͉ H ϭd I ϩother irreps. Condition ͑41͒ implies that the linear combination ͑42͒ must be restricted to where k runs over the d I representations k ϭ, and k indicates the th component of irrep k . This means that each representation I of G appears in the collective band a number of times equal to d I . The collective band of the vibration is given by the restriction of the Hamiltonian ͑38͒ to the subspace of functions ͑39͒, which satisfy the condition ͑41͒. The first two terms in ͑38͒ are constant throughout the band, and the last one vanishes on physical states. The collective Hamiltonian reads, where M ␣ stand for the restriction of the operators (D ␣ ) nm p n q m to the subspace span by the vibrational functions (q), that is, M ␣ ϭ͗ ͉(D ␣ ) nm p n q m ͉ ͘. The collective Hamiltonian is invariant under the action of G by the left, and so does not mix different representations I. Therefore, it can be diagonalized in subspaces of dimension d I . It is easy to check that it also commutes with the simultaneous action of H on the intrinsic states (J i ) and on the collective coordinates from the right (I i ). This ensures that it preserves the physical condition ⌽ i ϭ0 which the states of Eq. ͑39͒ satisfy.
The diagonalization of H coll in each subspace of dimension d I Ͼ1 must be performed case by case. However, some more information can be obtained using the H-invariance. The restricted matrices M ␣ no longer satisfy the G algebra, but they still satisfy, which means that they transform under H in the same representation as J ␣ . By the Wigner-Eckart theorem in the group H, they are determined by Clebsch-Gordan coefficients up to as many independent constants as irreps of H are contained in this representation. The number of irreps also gives the number of independent inertia moments in I ␣␤ . This is as far as we can go using symmetry arguments in the general case.
If G/H is a symmetric space a considerable simplification arises. In this case, there is an involutive automorphism :g→g ( 2 ϭid g ) under which the generators satisfy, 13 We assume that acts on the fields s (x,t) as :

C. Landsman and Linden Hamiltonian
In the previous section we found the collective wavefunctions by treating the H-gauge invariance by means of the Dirac method of imposing the constraints on the states. However, the dynamical consequences of the Mackey condition ͑41͒ on the collective functions are best understood by fixing the H-gauge symmetry.
Let us make the natural gauge choice of setting the collective coordinates related to motion on the subgroup ␣ i ϭ0. This fixes the H-gauge symmetry generated by ⌽ i , i.e., ͕␣ i ,⌽ j ͖Ϸ␦ j i . Proceeding as before, we eliminate the pairs (␣ i ,P i ) by solving P i from the equations ⌽ i ϭ0. This gives where we have used that ⌰ i j (0)ϭ␦ j i . Since the vibrations fall into irreps of H, if we restrict to a given vibrational state classified by , the above equation reduces to, where (T i ) are the generators of H in the irrep . Replacing in the collective operators I ␣ we get, The second term corresponds to Landsman-Linden Hamiltonian ͑28͒, with the metric on G/H 0 given by g ␣␤ ()ϭI ␥␦ ␣ ␥ () ␤ ␦ (), and the H 0 -connection by, The third term in ͑53͒ gives the coupling to an extra background field, which comes from the ''Coriolis'' terms in ͑44͒. In general, for real representations, this is an SO͑dim) connection, and is similar to the induced connections studied in Ref. 14.
The H 0 -connection ensures that Hamiltonian ͑53͒ applied to functions independent of ␣ i gives the same result as Hamiltonian ͑44͒ acting on functions over G which satisfy Mackey condition ͑40͒. When H 0 H the functions F () are still restricted by condition ͑41͒ for the discrete elements of H. This restriction can be lifted including a pure gauge connection which associates a discrete element of H with each nontrivial path in ⌸ 1 (G/H) ͑holonomy factors͒. The relation between inequivalent quantizations and holonomy factors in the path integral is discussed in Ref. 15.

V. CONCLUSIONS
We have discussed the collective bands of intrinsic states found when quantizing around a soliton solution with moduli space isomorphic to G/H. The result is that the collective band of an intrinsic vibrational state realizes an inequivalent coset space quantization given by the representation of H under which the intrinsic state transforms. The collective Hamiltonian is that of Landsman and Linden, 2 which describes free motion on G/H coupled to a background H-gauge field. Besides, there may be other background gauge fields coming from the Coriolis terms. The extra degrees of freedom associated with the nontrivial quantizations are given by the intrinsic coordinates. In this way, we have given a physical example of the inequivalent quantizations studied in Refs. 1-3.
Some of the consequences for the collective quantization of a soliton with moduli space G/H are the following. The states in the collective band of a vibration carrying the representation of H are classified by the irreducible representations I of the symmetry group G. Each irrep I appears in the band as many times as is contained in its decomposition into irreps of H. This also gives to lowest order the dimension of the collective Hamiltonian matrix within this subspace, from which the energies of the states are obtained. Determining the collective bands is analogous to calculating the rotational spectra of polyatomic molecules, 16 and our results can be understood as a generalization of this problem to a general symmetry group.
This work may be of interest for obtaining the spins and isospins bands of the ground state of multiskyrmions 17,18 and their excited states, which have been found for topological charges B ϭ2 and Bϭ4 by Barnes et al. 19,20 The symmetry group of the Skyrme model is the direct product of the spatial and isospatial rotations and the combined parity ͑without considering spatial trans-lations͒ GϭSO͑3͒ S ϫSO͑3͒ I ϫ P. As described in Ref. 21, the symmetry group of a Skyrmion HʚG is given by pairs (h,D(h)), where h is an element of the spatial group O͑3͒ S , and D(h) an element of the isospin group O͑3͒ I . The mapping D:h‫ۋ‬D(h) is a three-dimensional real representation of H, and det hϭdet D(h). For example, for the Bϭ2 case HϭD ϱ ͑not considering parity͒. The ground state is in the non trivial one-dimensional representation ⌺ Ϫ if the Skyrmion is quantized as a fermion. Therefore, the lowest allowed state of the band are ͑Iϭ1, Sϭ0͒ or ͑I ϭ0, Sϭ1͒, since they are the lowest irreps of G containing ⌺ Ϫ in its decomposition. Similarly, the other states of the band can be obtained. The Bϭ2 case has already been considered in Ref. 22, but we expect the more systematic treatment presented here will be useful in more complicated situations, as for BϾ2, where the subgroup H is discrete. 23