Collective coordinate of a soliton ($q^a(t)$)

A collective coordinate parametrizes a family of static classical field-theory solitons, for example their position or internal orientation. Derivatives with respect to these parameters are zero modes in field theory when the family has constant energy. Slowly varying coordinates describe low-energy motion, provided the excluded modes and radiation are negligible; gauge variations must first be removed in a gauge-theory soliton.

The conjugate momentum to a periodic internal soliton angle generates shifts of that angle. Single-valued wavefunctions, or Bohr-Sommerfeld quantization with zero Maslov index, quantize this canonical momentum. A theta term can shift the relation between canonical momentum and mechanical charge without changing integer canonical labels.

A collective-coordinate approximation substitutes a finite-parameter soliton ansatz into the field action and integrates over space. Its Euler-Lagrange equations approximate the slow motion of the soliton parameters.

Substitute the instantaneous Sine-Gordon kink profile ${\theta }_K(\gamma (x-X))$ and prescribed velocity $-\gamma \dot{X}{\theta }_K^{\prime }$ with $\gamma =(1-{\dot{X}}^2{)}^{-1/2}$ into the field density. For $A_1=0$, $A_0=xf(t)$, the integrals $\int ({\theta }_K^{\prime }{)}^2=8$ and $\int {\theta }_K^{\prime }=2\pi$ give the displayed collective-coordinate effective Lagrangian. Its equation is $d(8\gamma \dot{X})/dt=-2\pi f$. This is relativistic particle motion with mass $8$ and topological coupling charge $2\pi$. When $\gamma$ varies, the prescribed velocity omits a width-change contribution; acceleration correction for an instantaneous boosted kink describes the exact chain-rule distinction.

The actual derivative of ${\theta }_K(\gamma (t)(x-X(t)))$ is $[\dot{\gamma }(x-X)-\gamma \dot{X}]{\theta }_K^{\prime }$. The omitted contribution is first order in acceleration. Its cross term with the translation velocity integrates to zero because $\int y({\theta }_K^{\prime }(y){)}^{2}dy=0$. The square adds ${\dot{\gamma }}^2\int y^2({\theta }_K^{\prime }{)}^{2}dy/(2{\gamma }^3)={\pi }^2{\dot{\gamma }}^2/(3{\gamma }^3)$ to the spatially integrated density, since ${\theta }_K^{\prime }=2\mathrm{sech}y$ and $\int y^2({\theta }_K^{\prime }{)}^{2}dy=2{\pi }^2/3$. Thus the instantaneous-boost effective action for a sine-Gordon kink is an adiabatic truncation rather than the exact action restricted to that accelerating configuration-space profile.

Substitute a static soliton family with slowly time-dependent collective coordinates into the action. The kinetic energy induces a Riemannian metric $g$ on the family, after imposing any Gauss law constraint in gauge theory constraint and projecting out gauge directions. If its static energy is constant, the Euler-Lagrange equations are the geodesic equations of $g$. An approximately flat family instead carries a potential $V(q)$. This approximation neglects excitations of the other field modes and is justified only when their effects are small on the time and energy scales studied.

Quantizing a moduli-space approximation gives wavefunctions with measure $\sqrt{\det g}{d}^{d}q$ and, under the minimal scalar ordering, the Hamiltonian operator $-{\hslash }^2{\Delta }_g/2+V$, where ${\Delta }_g$ is the Laplace-Beltrami operator. Global identifications, statistics and regularity supply additional restrictions on wavefunctions. Curvature-dependent ordering terms or quantum corrections are extra choices, not determined by the classical kinetic energy alone.