Bogomolny bounds and first-order equations

Use a real dimensionless scalar field, dimensionless coordinate xi, and positive energy unit E zero. Define a real function W of the field and choose potential one half the square of its field derivative. This is a deliberately specified model class. For this lesson take W equal to phi minus one third phi cubed. Its potential is one half of one minus phi squared, squared: the coefficient differs from the preceding lesson’s potential, so its kink width and energy will also differ.

Expanding the square of the spatial derivative minus or plus the derivative of W reproduces the gradient and potential terms. The remaining cross term integrates to the endpoint difference of W by the chain rule. The square is nonnegative for real fields and positive spatial metric, giving the energy bound. This argument uses a static energy, not the oscillatory Lorentzian action; an arbitrary action saddle need not be an energy minimum.

For an increasing field from minus one to plus one, delta W is positive and the saturating equation is phi prime equal to W phi. With the chosen W, this gives phi prime equal to one minus phi squared and solution tanh of xi minus X. Substitution into the second-order Euler–Lagrange equation verifies that the first-order solution is a static solution. The converse is not automatic for an arbitrary stationary configuration or a different boundary problem.

The boundary difference is four thirds, so the minimum energy is four thirds of E zero if the saturating solution exists. Reversing the endpoints changes the first-order sign and leaves the energy bound positive. A bound can remain valid even when no field satisfying the chosen first-order equations exists. In higher-dimensional systems, compatibility of all first-order equations, flux constraints, and boundary data makes this existence question particularly important.

Consider the trial family tanh of k xi for positive k. All members share the same endpoints, but only the unit-width member satisfies this model’s first-order equation. Increasing k raises gradient energy while decreasing integrated potential energy, and decreasing k reverses the trade. Integrating the fourth power of the hyperbolic secant yields an energy proportional to k plus its inverse, whose minimum occurs at one. This explicitly separates topological sector from being the sector’s minimum.

Square completion can extend to vortices, monopoles, and Euclidean gauge instantons, but each theory has its own normalization and special coupling conditions. A generic deformation of the potential can destroy the simple first-order equations while leaving a stable soliton. Quantum corrections may shift a classical mass unless additional symmetry and a properly defined quantum charge constrain them. One should identify which conclusion follows from a classical inequality, which from an existence theorem, and which requires a separate quantum argument.