Scalar kinks and topological charge

Use one spatial coordinate and a real dimensionless field phi. Let xi be distance divided by a fixed length scale, and measure static energy in a positive unit E zero. The potential has degenerate minima at plus and minus one. A finite-energy configuration on the infinite line must approach a minimum at each end, with its derivative tending to zero sufficiently fast. These asymptotic requirements are part of the problem, not optional decorations added after finding a profile.

For a static field, the Euler–Lagrange equation is a second-order ordinary differential equation. Multiplying it by the spatial derivative gives a conserved mechanical first integral. The integration constant vanishes because both derivative and potential vanish at spatial infinity. The resulting equality of gradient and potential energy densities reduces this particular boundary problem to a first-order equation. The mechanical analogy uses an inverted potential and spatial coordinate as the evolution parameter; it is not literal time evolution of the original field.

Choose the increasing branch between minus one and plus one. Separating variables integrates an inverse hyperbolic tangent and produces a translated hyperbolic-tangent profile. The arbitrary center X reflects spatial translation symmetry. Reversing the profile produces the antikink. The width is fixed by the potential and gradient normalization; changing the width arbitrarily generally produces a trial field rather than another static solution of the same theory.

The topological charge is the endpoint difference divided by two. Under continuous finite-energy deformations that keep those endpoints fixed, this charge cannot change. It is a boundary charge, rather than a Noether charge generated by a continuous internal symmetry. A kink–antikink pair can have net charge zero and may annihilate while conserving that charge. Thus a nontrivial sector obstructs relaxation to one uniform vacuum, but charge alone does not tell us the energy of every configuration within that sector.

Translating the center costs no energy. Differentiating the family of static solutions with respect to X therefore produces a zero-frequency fluctuation proportional to the kink’s spatial derivative. It represents a collective position coordinate. A zero eigenvalue associated with a symmetry differs from a negative fluctuation eigenvalue, which would indicate a direction of decreasing energy under the stated constraints. The boundary conditions and function space matter when making either statement.

Small oscillations, radiation, relativistic motion, and kink collisions add dynamical questions beyond the static solution. In the Lorentz-invariant parent theory a boosted kink is a moving solution, but a superposition of two kink profiles is generally not an exact nonlinear solution. Quantum corrections require a renormalization and fluctuation prescription rather than merely replacing the classical energy by a diagram count. The calculation here establishes a classical finite-energy configuration and its boundary sector, leaving those additional questions explicitly separate.