= Solution
A <collective coordinate> describes a position, orientation, or another parameter of a family of static solitons. For a family $\Phi(\mathbf x;q)$, promote $q$ to a slowly varying $q(t)$ and substitute into the field action. In a scalar theory with unit kinetic coefficient, this gives the <collective-coordinate effective Lagrangian>
$$
L_{\rm eff}=\frac12g_{ab}(q)\dot q^a\dot q^b-V(q),\qquad
g_{ab}=\int\frac{\partial\Phi}{\partial q^a}\cdot\frac{\partial\Phi}{\partial q^b}\,d^dx.
$$
For a <gauge-theory soliton>, one also solves the <Gauss law constraint in gauge theory> constraint and projects out pure <gauge transformations>; arbitrary variations of gauge representatives do not define the physical metric. Tangent vectors to an exactly equal-energy family are <zero modes in field theory>. If $V=E_0$ is constant, the <Euler-Lagrange equations> of this <moduli-space approximation> are
$$
\ddot q^a+\Gamma^a_{bc}\dot q^b\dot q^c=0,
$$
the <geodesic> equations of its <Riemannian metric>. The approximation neglects radiation and deformation modes; it describes motion sufficiently slow that these omitted degrees of freedom remain unexcited to the required accuracy.
In <collective-coordinate quantization>, take the wavefunction measure $\sqrt{\det g}\,d^nq$ and the minimal scalar <Hamiltonian operator>
$$
\boxed{\widehat H=-\frac{\hbar^2}{2}\Delta_g+V(q),\qquad
\Delta_g=\frac1{\sqrt{\det g}}\partial_a\left(\sqrt{\det g}\,g^{ab}\partial_b\right).}
$$
The <Laplace-Beltrami operator> supplies coordinate-invariant kinetic energy. Global identifications and statistics must be imposed on the wavefunctions; the classical metric alone does not choose them. Curvature-ordering terms and loop corrections are additional quantum input.
For the <phi-four kink>, use the normalization and profile of Question 1. Substituting $\phi(x,t)=\phi_K(x-X(t))$ gives
$$
L_{
m eff}=-M+\frac12\dot X^2\int(\phi_K')^2dx=-M+\frac12M\dot X^2,
\qquad M=\frac{4c^3}{3}.
$$
The metric is constant because of <translation invariance>. This proves the <translational dynamics of a phi-four kink>: classically the centre moves at constant velocity, and quantum mechanically
$$
\boxed{\widehat H=M-\frac{\hbar^2}{2M}\partial_X^2,\qquad E(p)=M+\frac{p^2}{2M}.}
$$
<Plane waves> label the continuous translational <momentum>, with no position-dependent potential. Uniform-motion Lorentz invariance upgrades the dispersion to $\sqrt{M^2+p^2}$; the displayed collective Lagrangian is its small-velocity expansion. Small perturbations also include an internal shape mode and continuum radiation, which this single <collective coordinate> omits. The <fluctuation operator of a phi-four kink> in this normalization is
$$
-\partial_x^2+4c^2-6c^2\operatorname{sech}^2(c(x-X));
$$
the translational eigenfunction has $\omega^2=0$, the shape mode has $\omega^2=3c^2$, and continuum modes in the <spectrum> start at $\omega^2=4c^2$. Thus the free-coordinate states describe the kink's low translational energies, not its full excitation spectrum or quantum mass correction.
For two <Abelian Higgs vortices> at <critical coupling>, the static energy is $2M_v$ and the <Abelian Higgs vortex moduli space> has four real dimensions. Let $z_1,z_2$ be their positions, $Z=(z_1+z_2)/2$, and $q=z_1-z_2$. The centre of mass decouples; the relative metric is rotationally symmetric and can be written
$$
ds^2=2M_v|dZ|^2+F(\rho)(d\rho^2+\rho^2d\theta^2),\qquad q=\rho e^{i\theta},\quad\theta\sim\theta+\pi.
$$
At large separation, $F(\rho)\to M_v/2$, recovering two free particles. Although there is no static separation potential, the nonconstant metric produces velocity-dependent interaction. Coincidence is smooth in the <relative coordinate for two identical vortices> $w=q^2$, not in the double-valued $q$. Smoothness gives $F(\rho)\sim C\rho^2$ for $C>0$ near zero. A head-on <geodesic> continues through $w=0$ to the opposite real ray, so $q$ changes its line by $\pi/2$: \b[the vortices scatter through a right angle]. This geometric argument does not require an explicit formula for $F$.
With ordinary bosonic exchange statistics, relative wavefunctions are single-valued in $w$ and smooth at coincidence. In the separated polar coordinate they obey $\Psi(\rho,\theta+\pi)=\Psi(\rho,\theta)$, with even integer angular labels. Their kinetic operator is
$$
\widehat H_{
m rel}=-\frac{\hbar^2}{2F(\rho)}\left[\partial_\rho^2+\frac1\rho\partial_\rho+\frac1{\rho^2}\partial_\theta^2\right].
$$
The apparent singularity at $\rho=0$ must be resolved with the smooth $w$ coordinate and regularity, rather than arbitrary boundary conditions on a punctured cone. The free centre-of-mass motion and the asymptotically free relative geometry give quantum scattering states; a flat static energy does not imply that the metric is flat or that scattering is absent. This is not a prediction of a discrete family of static two-vortex bound separations. The smooth collision geometry is developed in David Tong's https://arxiv.org/abs/hep-th/0509216.
For a <Skyrmion> of <baryon number> one, the <Skyrmion hedgehog ansatz> is
$$
U_0(\mathbf x)=\cos f(r)+i\sin f(r)\widehat{\mathbf x}\cdot\boldsymbol\sigma,\qquad f(0)=\pi,\quad f(\infty)=0,
$$
where $\boldsymbol\sigma$ are the <Pauli matrices>. Include a centre $\mathbf X$ and an orientation $A\in SU(2)$ through $U=A U_0(\mathbf x-\mathbf X)A^{-1}$. Hedgehog symmetry identifies spatial rotations with opposite internal rotations, so there are three independent orientation coordinates, not six. Since $A$ and $-A$ give the same field, the physical orientation space is $SO(3)$, with <SU(2) group> as its double cover. Write $A^{-1}\dot A=i\boldsymbol\omega\cdot\boldsymbol\sigma/2$. The leading collective Lagrangian has the form
$$
L_{
m eff}=-M+\frac12M|\dot{\mathbf X}|^2+\frac12\Lambda|\boldsymbol\omega|^2,
$$
where $\Lambda>0$ is the rotational <moment of inertia> obtained by integrating the profile's field kinetic energy.
For the fermionic quantization appropriate to baryons, the <Finkelstein-Rubinstein constraints> on the double cover impose $\Psi(-A)=-\Psi(A)$. In <SU(2) representations>, the central element $-1$ acts by $(-1)^{2j}$, so $j$ must be half-integer. Left and right group actions supply <isospin> and <spin angular momentum>; hedgehog symmetry makes their magnitudes equal. The <rotational quantization of a unit Skyrmion> therefore gives
$$
\boxed{I=J=j=\frac12,\frac32,\ldots,\qquad E_j=M+\frac{j(j+1)\hbar^2}{2\Lambda}.}
$$
The $j=1/2$ level has four spin-isospin states and models the <nucleon> doublet, <proton> and <neutron>, each with two spin states. The $j=3/2$ level has sixteen states and models the <Delta baryon> quartet, each with four spin states. The rotor predicts a splitting $3\hbar^2/(2\Lambda)$. Without the fermionic sign, single-valued functions on $SO(3)$ would instead allow integer $j$, which is a different quantization. High rotor levels can couple to deformation and pion radiation; this semiclassical approximation does not establish that its entire formal tower consists of stable particles.
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