Scalar soliton quantization with generic moduli

We canonically quantize multi-component scalar field theories in the presence of solitons. This extends results of Tomboulis [1] to general soliton moduli spaces. We derive the quantum Hamiltonian, discuss reparameterization invariance and explicitly show how, in the semiclassical approximation, the dynamics of the full theory reduce to quantum mechanics on the soliton moduli space. We emphasize the difference between the semiclassical approximation and a truncation of the dynamical variables to moduli. Both procedures produce quantum mechanics on moduli space, but the two Hamiltonians are generically different.


Introduction and summary
The topic of quantization in soliton sectors is a rich one with a long list of applications. Foundational work on this subject was carried out in the mid 70's and includes [1][2][3][4][5][6][7][8][9]. For thoroughly pedagogical reviews we refer the reader to [10,11].
In [1] Tomboulis quantized a simple two-dimensional scalar theory in the one-soliton sector by introducing a canonical transformation from the original fields to a dynamical modulus -i.e. a collective coordinate -plus fluctuations around the classical soliton solution, while imposing a set of constraints which preserves the total number of degrees of freedom. A key assumption in that work was that the soliton solution has a single modulus associated with translations in the spatial direction, as is the case e.g. for a kink in φ 4 theory. This greatly simplifies some conceptual and calculational aspects of the analysis. Similarly, other contemporary approaches to soliton quantization around static classical solutions, including canonical transformations with unconstrained variables as well as path integral techniques, primarily dealt with systems in which all collective coordinate degrees of freedom correspond to translational modes. 1 Several years later, a more geometrical framework for understanding collective coordinates emerged from studies of the Bogomolny equation, describing 't Hooft-Polyakov monopoles [12,13] in four-dimensional Yang-Mills-Higgs theory in the BPS limit of vanishing potential [14,15]. In this theory, the minimal-energy solution set of the static field equations with fixed boundary conditions, corresponding to a particular topological charge sector, is a finite-dimensional Riemannian manifold, (M, G). The metric is the natural one induced from the (flat) metric on field configuration space. The framework suggested by Manton [16] is that, for slowly varying field configurations, the dynamics of the full system is well approximated by promoting the moduli to time-dependent variables -the collective coordinates -in which case the field theory equations of motion reduce to the geodesic equation on M. The usefulness of this framework was beautifully demonstrated by Atiyah and Hitchin's analysis of two-to-two monopole scattering [17].
Generically, M has curvature and not all moduli correspond to broken symmetries such as translations. Nevertheless in asymptotic regions of M, corresponding to field configurations with well-separated and localized lumps of energy, one can associate the parameters with locations and internal phases of constituent solitons. This suggests that Manton's paradigm of motion on moduli space is applicable in any theory admitting static multi-soliton solutions.
It should be emphasized that Manton's prescription is for constructing approximate time-dependent solutions to classical field equations. However, as demonstrated earlier by Gervais, Jevicki, and Sakita [8], it is also natural to assume small velocities for the collective coordinates in the semiclassical analysis of a soliton sector of a quantum theory. The approximate classical solution provides an approximate saddle point for the semiclassical expansion of the path integral. One would like the corrections coming from performing the saddle-point approximation to be comparable to those due to expanding around an approximate solution; the latter are controlled by the collective coordinate velocities. 2 Hence the geometry (M, G) provides a natural starting point for the quantum analysis of a soliton sector, and the Manton approximation is incorporated as part of the semiclassical expansion.
This point of view was first considered in [19] and has since been used to great effect, e.g. in the context of N = 2 supersymmetric four-dimensional gauge theory [20][21][22] where semiclassical results can be compared against the quantum-exact ones of Seiberg and Witten [23,24]. In these analyses one typically truncates the classical degrees of freedom to the collective coordinates and then quantizes the resulting finite-dimensional system, yielding a (supersymmetric) quantum mechanical sigma model with target M. This is sufficient for answering basic questions about the original quantum field theory, such as the existence of soliton states and what charges these carry. The first corrections to masses and charges, obtained from one-loop determinants, have also been considered [25,26]. However, to our knowledge, the exact quantum Hamiltonian describing the full dynamics of a quantum field theory around a (multi-) soliton sector has not been studied within the general geometrical framework.
In this work we extend the canonical transformation of [1] to multi-component scalar field theories with general multi-soliton moduli spaces. This naturally requires using geo-JHEP06(2014)003 metric quantities on the moduli space of classical solutions. Our primary goal is to extract the quantum Hamiltonian for this system, which may be useful in various contexts, such as the study of scattering processes involving both solitons and perturbative particles [27].
Our secondary goal is to establish a formalism that facilitates extending this inquiry to (supersymmetric) theories with gauge fields and fermions, e.g. involving monopoles in four dimensions or instanton-solitons in five dimensions. While we intend to return to this in the near future, the restriction to scalar fields helps highlight the main qualitative results against the added technical details required for those applications.
In that vein, we emphasize several conceptual points as they arise in the explicit analysis. We demonstrate how one recovers a reparameterization-invariant theory for the dynamical moduli when the fluctuations are switched off. This sector has knowledge of both the intrinsic and extrinsic geometry of the system. Furthermore, we show how the full quantum Hamiltonian of the field theory can be expanded in the perturbative coupling, when one additionally requires the solitons to be slowly moving. We organize and present this semiclassical expansion to the first few orders and briefly discuss how Lorentz invariance can be recovered in perturbation theory. Finally, we exhibit how, when restricting to incoming and outgoing states which do not involve perturbative excitations, the leadingorder dynamics reduce to quantum mechanics on the soliton moduli space.
We would like to point out that a detailed reduction to the quantum mechanics on the soliton moduli space has also been performed from the point of view of embedded submanifolds [28,29]. This approach is based on the formalism developed for studying quantum mechanical systems constrained to move on a sumbanifold of their full configuration space [30,31]. It can be extended to quantum field theory in the presence of a soliton, by viewing the moduli space as embedded inside the infinite-dimensional space of field configurations. The quantum mechanics on moduli space then emerges by solving the functional Schrödinger equation in the Born-Oppenheimer (or 'thin-layer') approximation. This effectively integrates out the massive modes and produces geometric corrections due to both the intrinsic and extrinsic curvature, which are missed by a naive truncation. Our quantum mechanics, obtained via conventional second quantization, exactly agrees with that of [28,29].
The rest of this paper is organized as follows: In section 2 we set up the background, introduce the change of variables and canonically quantize the theory. In section 3 we obtain the quantum Hamiltonian. Section 4 deals with the reparameterization invariance of the collective coordinate sector, while in section 5 we present the semiclassical expansion. Finally, section 6 discusses the reduction to quantum mechanics on the moduli space.

The change of variables
We begin with a general class of real scalar field theories with classical Lagrangian We assume that Φ, Π at fixed time t are Darboux coordinates on phase space where δ(x − y) is a (D − 1)-dimensional Dirac delta function and the Poisson bracket is given by .
In the quantum theory, Φ, Π are promoted to operators 4Φ ,Π and the Poisson bracket to a commutator We consider a fixed topological sector and assume there exists a finite-dimensional smooth family of classical static soliton solutions, parameterized by moduli U M , where M runs over the dimension of the moduli space dim R M, such that

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The inequality ∆(U ) ≥ 0 is meant to signify that ∆(U ) is a positive operator, such that all of its eigenvalues are non-negative, and the notation is to emphasize that this operator depends on where we are in moduli space. In order to study the behavior of the theory around the soliton configuration (2.7), one makes a change of variables from the original field Φ(x) to collective coordinates U M = U M (t) and fluctuations χ(x; U M (t)) about the solution: To preserve the number of degrees of freedom, there should be as many constraints on these new variables as there are coordinates U M . We note that ∂ M φ will be a zero-mode of the linear differential operator ∆. One would like to exclude such zero-frequency modes from the mode expansion of χ. This can be done by imposing the constraints We introduce momentum variables (p M , π(x; U M )) conjugate to (U M , χ) and extend this transformation to phase space. We treat these as Darboux coordinates in an extended phase space We also extend the coordinate transformation (2.9) to a phase space transformation with the ansatz where the functionals Π M 0 will be determined below. In analogy with (2.10) we impose 14) The constraints are second-class as the Poisson brackets are non-vanishing: Here G M N (U ) is the metric on the moduli space of soliton solutions. Restriction of the dynamics to the constraint surface is achieved through the introduction of Dirac brackets,

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Geometrically, the Dirac bracket is the pullback of the Poisson bracket to the constraint surface and satisfies all the properties of the ordinary Poisson bracket. The appearance of the moduli space metric G M N in the Dirac bracket is quite natural and can be viewed as a motivation for choosing the momentum constraint as in (2.14).
One can straightforwardly work out the Dirac brackets of our Darboux coordinates. The nonzero brackets with the constraints are with the rest vanishing. Here we suppressed all non-essential arguments of the fields. These brackets appear complicated at first, but we have not yet specified the functional dependence of χ, π on U M . One can freely do this, since the degrees of freedom contained in χ should comprise a basis for where ≈ denotes 'upon restriction to the constraint surface'; see appendix A for details. Having done so, the non-vanishing Dirac brackets become We remind that for systems with second-class constraints it is the Dirac bracket that is promoted to the commutator in the quantum theory In order for the quantum theory in the old and new variables to be equivalent, we must require that the transformation (Φ; Π) → (U M , χ; p M , π) defined by (2.9) and (2. be used to fix the functionals Π M 0 in (2.13). In order to implement this requirement we inserting the change of variables (2.9), (2.13) and using the brackets (2.20). In the process, we find Π M 0 such that the results are consistent with (2.3). The full computations are tedious but straightforward; some intermediate results are recorded in appendix B for the reader interested in the derivation.
We summarize these results as follows. The [Φ,Φ] commutator is trivial sinceÛ ,χ are commuting operators. Thus The calculation of [Φ,Π] ′ fixes the form of Π M 0 . At the classical level one finds At the quantum level one must be careful about operator ordering in eq. (2.13). The symmetrized ansatẑ provides a natural generalization of the ansatz in [1]. Here we have begun using the shorthand dzπ(z; U )·∂ N χ(z; U ) = π·∂ N χ. It is also useful to introduce the combination where All terms beyond the first two result from the evaluation of two commutators and should be thought of as O( 2 ). We have also introduced to define Christoffel symbols on the moduli space. At this point we note that, in the special case where the moduli space consists of a single modulus associated with translations, M = R, all terms in the second and third lines vanish. The terms in the first line then reproduce the analogous result in [1], including the 'quantum correction' term ∫ (∂ 2 φ) 2 .
The full Hamiltonian follows trivially from (3.1) by adding the potential, which can be expanded around the solution Φ = φ: and V I (χ) denotes cubic and higher-order interaction terms in the fluctuations χ coming from the original potential. If φ(x, U ) parameterizes a family of exact static solutions then: a) v(U ) will be a constant, by definition the classical soliton mass and b) the source term s(x; U ) will vanish. However, we will see shortly that it is natural to also allow for a small deviation from an exact solution. In that case M is not really a true moduli space, as evidenced by the appearance of the potential v(U ).

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Eq. (3.4) is the final, exact result for the quantum Hamiltonian 8 of the theory. It is valid for all values of soliton moduli U M and conjugate momenta p N .

Covariance
Given the form of (3.4), it appears that the Hamiltonian is not invariant under arbitrary reparameterizations of the moduli U . This is not the case and the manifestly invariant form of the Hamiltonian can be recovered once we properly order the kinetic term operators.
The canonical change of variables in configuration space (2.9) from Φ to U, χ -plus constraints -effectively maps a Cartesian coordinate system to a curvilinear one. This map describes how the curved moduli space is embedded in the total infinite-dimensional space of modes. The orthogonal directions to M, within the constraint surface, are parameterized by the massive oscillator modes of χ. It is possible to use the theory of embedded surfaces in order to construct the exact metric for the infinite-dimensional Cartesian space in the new curvilinear coordinate system. This was explicitly done by Fujii et al. in [28], who found that the Hamiltonian can be expressed in terms of the Laplace-Beltrami operator in the curvilinear coordinate frame. Hence the full theory, in the new set of variables, is reparameterization invariant as expected.
It is interesting to see how covariance becomes manifest in the subsector of the theory with all fluctuations switched off. We have, from (3.4), where the last two terms in the first line are obtained from commutators upon appropriately ordering the momentum operators in the kinetic term. This expression can be manipulated as follows. First note that where R is the scalar curvature on moduli space

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and [28] Using this definition we can re-express (4.3) as Substituting (4.2) and (4.6) into (4.1), one finds that all bilinears in the Γ's as well as the terms involving derivatives of Γ's mutually cancel to leave In the last step we noted that The l.h.s. of (4.8) is in fact a covariant quantity when understood as a Hamiltonian acting on a wavefunction Ψ with canonical normalization M d d U Ψ * Ψ = 1. The timeindependent Schrödinger equation takes the form Redefining Ψ = G 1/4Ψ leads to the correct curved space normalization M d d U √ GΨ * Ψ = 1 and modifies (4.9) to where the l.h.s. is now the Laplace-Beltrami operator on the soliton moduli space. Hence we have arrived at the explicitly covariant expression In fact, the quantity ∫ (∇ 2 φ) 2 has a nice geometric interpretation. Using the results of [28] one finds that 9 where H is the extrinsic mean curvature and d = dim R M. This curvature invariant encodes information about how the moduli space M is embedded as a submanifold into the infinitedimensional flat configuration space. 9 The notation of [28] is rather different from the one used here, so it is useful to describe the precise map: We have ∂P φ a (x) → B Ax a . Then ∇ 2 φ a (x) → g ab ∇aB Ax b = nH Ax and the integration over x is performed by contracting the H's with ηAx,By.

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Our covariant result is then simply This Hamiltonian defines a quantum mechanics with target M; we will refer to (4.13) as the 'truncated Hamiltonian' in the following. The curvature terms are O( 2 ) effects and may be viewed as intrinsic and extrinsic 'quantum potentials.' The appearance of the Ricci scalar is well documented in background-independent approaches to quantum mechanics on curved spaces. In that context, it has been observed that the coefficient of the Ricci scalar term is ambiguous, depending on the operator ordering prescription [34]. Here there is no ambiguity because the correct ordering prescription is inherited from the parent theory, which is defined on a flat configuration space. It is also interesting to observe that even in the limit where the fluctuations have been completely decoupled, the Hamiltonian for the collective coordinates encodes information about the extrinsic geometry [28].

Semiclassical analysis
Up to this point we have not explicitly kept track of powers of coupling constants. As is typical in the soliton literature [4,10,11], we will assume that there is effectively a single coupling g such that, in terms of the canonically normalized fieldΦ, the potentialṼ (Φ; g) has the scaling propertyṼ (Φ; g) = 1 g 2Ṽ (gΦ; 1) =: Thus, if we define the rescaled field Φ = gΦ, then then the entire coupling dependence of the Lagrangian (2.1) becomes L(Φ,Φ; g) = 1 g 2 L(Φ,Φ; 1) .

(5.2)
We will assume that we have been working with the rescaled field Φ all along and that we previously set the coefficient of g −2 in front of (2.1) to one. Note that if φ is the rescaled classical solution, it will be independent of g and hence the canonically normalized classical solutionφ will go as g −1 , which is the usual behavior we expect from a soliton configuration. Under the assumption (5.2), it is clear from the path integral point of view that g 2 plays the role of and the semiclassical expansion is a g expansion. Once the factor of g −2 is restored in front of the Lagrangian, the Hamiltonian, (2.2), becomes Meanwhile, the definitions of the metric and potential on moduli space read

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and v(U ) := 1 g 2 The canonical transformations are given by In the above we have rescaled the fluctuations χ, π so that they are canonically normalized fields, while the power of g −2 in (5.7) originates from the definitions (5.4). Setting A M = 1 2 (a M +ā M ) as before, our full quantum Hamiltonian (3.4) can now be re-written as We can then expand the A M G M N A N term in powers of the coupling as follows: Notice also that the first line contains a term linear in the fluctuations χ through Ξ. The presence of this tadpole is due to the fact that φ(x; U (t)) is not an exact solution to the time-dependent equations of motion, irrespective of whether or not φ(x; U ) is an exact solution to the time-independent ones. This is what motivates the small velocity assumption: As it stands, (5.9) is valid for all values of soliton momenta but makes little JHEP06(2014)003 sense in perturbation theory, since the scalar propagator would be higher order in the coupling compared to the tadpole. However, if one considers appropriately slowly-moving solitons, p 2 χ can be viewed as a legitimate interaction term.
In a similar vein, since we do not solve the time-dependent equations of motion exactly, there is no need to insist on an exact solution to the time-independent equations. We merely require an approximate solution so that the tadpole term, s(x; U ) · χ, coming from the potential may also be viewed as an interaction term.
Thus we will continue by making the assumptionṡ so that we are expanding around an approximate solution to the time-dependent equations of motion. Note that the latter condition implies that In other words, the integral of the potential evaluated on the classical solution is constant up to O(g 2 )-suppressed corrections, which may be moduli dependent. The constant M cl is interpreted as the classical -or leading order -contribution to the soliton mass, while the corrections give a U -dependent potential on the moduli space.
In this small-velocity and small-potential approximation, the semiclassical expansion of the full Hamiltonian becomes where H (−2) = M cl , Here, H (n) is O(g n ) provided that (5.10) and (5.11) hold, and we recall that G M N ∼ O(g 2 ).
V (3,4) (φ) denote the third and fourth derivatives of the potential, evaluated on the soliton solution φ. Finally, we have used the results of section 4 to simplify the terms in H (2) that are zeroth order in fluctuations.

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Let us briefly discuss the issue of Lorentz invariance. Eq. (5.12) is in principle a double expansion: a quantum expansion in the coupling, as well as an expansion in small soliton velocities. A subset of the collective coordinates, {U i } D−1 i=1 ⊂ {U M }, correspond to the center-of-mass position of the soliton solution φ. The conjugate variables, p i , correspond to the center-of-mass momentum. One expects that any observable computed exactly in the quantum theory should be covariant under Lorentz transformations. On the one hand, expanding around slowly-moving solitons -in particular p i ∼ O(1/g) -naturally breaks the Lorentz symmetry of the original theory. On the other, the scaling (5.10) suggests that relativistic corrections should appear as quantum effects associated with the p 2 i χ tadpoles. In fact, it can be explicitly seen for the case of kink solitons in two-dimensional φ 4 theory that re-summing all the tree-level diagrams obtained by gluing together the p 2 i χ tadpole interactions restores Lorentz invariance for the soliton energy [8,35]. This computation should be extendable to the class of theories we are studying, but we will not explicitly consider it here.

Reduction to QM on the soliton moduli space
It is straightforward to use our results for the semiclassical expansion of the Hamiltonian to determine the behavior of the leading-order dynamics. Keeping terms in H through O(1), we have Let us focus on the fluctuation terms. We make a mode expansion where the ζ's are eigenfunctions of the operator ∆(U ) with strictly positive eigenvalues ω 2 k : ∆(U )ζ k = ω 2 k (U )ζ k . They are orthonormal and satisfy the completeness relation The modified completeness relation is due to the fact that we have excluded the zeroeigenvalue modes from the expansion. The ζ k (x, U ) form a basis for the subspace of configuration space orthogonal to the tangent space T U M. Using (6.4), one can show that the commutator [χ, π], (2.20), is equivalent to the standard creation and annihilation commutators

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We have written the mode expansions (6.2) as though the non-zero spectrum of ∆ is purely continuous. While the spectrum of ∆ is guaranteed to have a continuous component, 10 there could additionally be a discrete component beyond the zero-modes. Strictly positive discrete eigenvalues correspond to breather-like modes, and the mode expansion should include a sum over them. We will understand ' dk' in the above and following expressions as representing the integral over the continuous spectrum plus the sum over the breather-like modes, if present.
Using (6.2), (6.3), and (6.5), it is then easy to see that so that the full Hamiltonian is In particular, when acting on a state which does not involve massive fluctuations, the last term above vanishes and one is left with the zero-point energy of the fluctuation fields. In a renormalizable theory, the divergent part of this quantity can be removed, after vacuum energy subtraction, by mass renormalization; see footnote 8. The finite piece then generates a one-loop correction to the potential M cl + δv(U ) → M 1-loop + δv(U ) 1-loop [1][2][3]. Hence, the final result for the leading contribution in the semiclassical approximation and when restricting to incoming and outgoing states that do not contain perturbative excitations is which is a quantum mechanics on the soliton moduli space. We will refer to (6.8) as the 'semiclassical Hamiltonian'. This quantum mechanics, as written, is not covariant with respect to general coordinate transformations on M. However, following the discussion around (4.9), it can be trivially made covariant by replacing p M G M N p N → G −1/4 p M G 1/2 G M N p N G −1/4 . These two quantities differ by Y , which is higher order in the g-expansion and hence can be neglected in (6.8).
It is interesting to note that, even after this replacement, the two quantum mechanical systems on M defined by the truncated Hamiltonian (4.13) and the semiclassical Hamiltonian (6.8) are different. Although the intrinsic and extrinsic quantum potentials of the truncated Hamiltonian are present in the semiclassical expansion (5.12), it would be inconsistent to include them in the semiclassical Hamiltonian (6.8), without first accounting JHEP06(2014)003 for all O(g) and O(g 2 ) corrections from integrating out the fluctuations. Furthermore, the semiclassical approximation demands that the O(1), 'one-loop' corrections from χ, π be accounted for in the semiclassical Hamiltonian: They are of the same order as the kinetic term and moduli-dependent classical potential δv(U ), due to the necessity of imposing (5.10) and (5.11).

Acknowledgments
We would like to thank Sebastian Guttenberg for helpful discussions and comments. CP is a Royal Society Research Fellow and partly supported by the U.S. Department of Energy under grants DOE-SC0010008, DOE-ARRA-SC0003883 and DOE-DE-SC0007897. ABR is supported by the Mitchell Family Foundation. We would like to thank the Mitchell Institute at Texas A&M and the NHETC at Rutgers University respectively for hospitality during the course of this work. We would also like to acknowledge the Aspen Center for Physics and NSF grant 1066293 for a stimulating research environment which led to questions addressed in this paper.

A Mode expansions for χ, π
For a fixed value of the moduli, χ and π are simply n-tuples of scalar fields on R D−1 ; they can be expanded in any complete basis for the Hilbert space L 2 [R D−1 , R n ]. A particular basis that is naturally adapted to the problem is the basis of eigenfunctions of the Hermitian operator ∆(U ), defined in (2.8). Since the form of this operator depends on the moduli, so will its eigenfunctions; we denote the complete set of eigenfunctions by {ζ I (x; U )}, where I runs over an indexing set. This set will include both the continuous part of the spectrum, as well as the discrete part of the spectrum, which includes the zero-modes and may additionally contain other massive breather-like modes. We write schematically where χ I (t), π I (t) comprise the complete set of degrees of freedom in χ(x; U ), π(x; U ). The ζ I satisfy where by 'δ IJ ' and ' I ' we mean (2π) D−1 δ(k − k ′ ) and and similarly ψ From here we can explicitly see that the constraint surface corresponds to χ A = π A = 0. Meanwhile, Another identity that follows trivially from (A.1) and (A.2) and will be useful below is Finally, if I, J index the non-zero modes, which include the continuous spectrum and any possible breather-like modes, then the commutator of the fields χ, π is equivalent to These can be used to show, for example, that More generally, the commutator of any U -derivative of χ with another U -derivative of χ is zero, and similarly for π.

B Some details on the canonical transformation
First let us consider {Φ(x), Π(y)} ′ D in order to derive the classical form of Π M 0 as given in (2.23). Substituting in (2.9), (2.13) for Φ, Π and using (2.20), we can write the result as The first term is what we want; thus we must choose the functional Π M 0 so that the remaining terms vanish. Consider the x dependence of these remaining terms. The term in the first line is tangential to T U M ⊂ L 2 [R D−1 (x) ] since it is proportional to the zero-mode ∂ M φ(x), while the term in the last line is in the orthogonal complement (T U M) ⊥ since it JHEP06(2014)003 involves the projection operator δ(x−z)−∂ M φ(x)G M Q ∂ Q φ(z). The term involving ∂ Q χ(x) can be decomposed into a piece along T U M and a piece orthogonal to it. Substituting this into (B.1), we find that {Φ(x), Π(y)} ′ D = δ(x − y) if and only if both of the following hold: Note that we can write where Ξ P Q was defined in (2.24). The first of (B.2) implies whence the second equation implies that Here we have omitted the possible addition of a term depending only on the coordinates (U, χ). Consideration of {Π(x), Π(y)} ′ D shows that it is consistent to set this term to zero. We observe that if we set χ, π = 0, then the momentum transformation (2.13) with (B.5) reduces to Π(x) = p M ∂ M φ(x; U ). This is exactly what one would expect for the classical momentum density of the moving soliton.
At the quantum level we take the change of momentum variables to be (2.25). The basic commutators are the right-hand sides of (2.20), multiplied by a factor of i. Using these we have  can be used to show where we have defined S QRS := ∂ Q G RS + Γ QRS − ∫ χ · ∂ Q ∂ R ∂ S φ , (B.14) which is totally symmetric, S QRS = S (QRS) . Making note of the comment below (A.9) and using (A.7), one also finds that [(p P − ∫ π · ∂ P χ), (p Q − ∫ π · ∂ Q χ)] ≈ − 2iΘ [P |R G RS (C −1 ) S|Q] , [(p P − ∫ ∂ P χ · π), (p Q − ∫ ∂ Q χ · π)] ≈ 2i(C −1 ) [P |R G RS Θ S|Q] ,  Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.