Classical field-theory soliton

A classical field-theory soliton is a smooth, localized finite-energy solution that behaves as a persistent object. Stability may follow from a topological charge or from a balance among energy terms with different scaling behavior.

A non-topological soliton is a localized finite-energy field configuration whose existence or stability is not protected by a nontrivial topological charge. A conserved ordinary Noether charge can instead obstruct its decay. A Q-ball is a standard complex-scalar example: its profile is localized while its phase rotates in time.

For a complex scalar field with global $U(1)$ invariance and Lagrangian $\mathcal{L}=\frac{1}{2}|{\partial }_t\varphi {|}^2-\frac{1}{2}|\nabla \varphi {|}^2-U(|\varphi |)$, a Q-ball is a localized phase-rotating configuration with conserved Noether charge $Q=\int \mathrm{Im}(\bar{\varphi }{\varphi }_t)$. At fixed charge its real profile can minimizeSuch solutions require a suitable potential and charge range; the complex nature of a field alone does not ensure existence. A common condition is ${\inf }_{f>0}2U(f)/f^2<U^{\prime \prime }(0)$.

This nonnegative potential has its only vacuum at $s=0$, since $U(s)=s[(s-1/2{)}^2+3/4]$. For the complex kinetic normalization $\frac{1}{2}|\partial \varphi {|}^2$, the small-amplitude mass squared is $2U^{\prime }(0)=2$, whereas ${\min }_{s>0}2U(s)/s=3/2$. The radial Q-ball equation has a positive localized profile for frequencies $3/2<{\omega }^2<2$, for example in three spatial dimensions. It is a standard non-topological soliton potential: the effective profile potential $U(f^2)-{\omega }^2{f}^2/2$ is negative somewhere, while its quadratic coefficient near zero is positive. The conserved Noether charge, not vacuum winding, supplies the stabilization mechanism.

For a one-dimensional Q-ball, the profile equation gives $\frac{1}{2}(f^{\prime }{)}^2=U(f)-\frac{1}{2}{\omega }^2{f}^2$. Its rest energy is therefore $M=\int [(f^{\prime }{)}^2+{\omega }^2{f}^2]$. A Lorentz transformation replaces the rest coordinates by $\tau =\gamma (t-v(x-x_0))$, $\xi =\gamma (x-x_0-vt)$. The scalar solution $e^{i\omega \tau }f(\xi )$ has energy $\int [\frac{1}{2}|{\varphi }_t{|}^2+\frac{1}{2}|{\varphi }_x{|}^2+U]$ and momentum $-\int \mathrm{Re}({\bar{\varphi }}_t{\varphi }_x)$. Substitution and $dx=d\xi /\gamma$ give the displayed values and $E^2-P^2=M^2$.

For $f_{\lambda }(x)=f(\lambda x)$ in $d$ space dimensions, write $T=\frac{1}{2}\int |\nabla f{|}^2$, $W=\int U(f)$ and $I=\int f^2$. Eliminating the rotation frequency by $\omega =Q/I$ gives the displayed fixed-charge scaling. The positive-power charge term can balance the gradient and potential terms even for $d\ge 3$. This evades the static Derrick theorem because the field is time-dependent and the relevant variations preserve its Noether charge.

A breather is a spatially localized nonlinear-wave solution periodic in time. It does not separate into independently traveling lumps while it breathes. A Sine-Gordon breather arises from a complex-conjugate soliton pair. In the quantum sine-Gordon model, the associated neutral kink-antikink bound particles are also called breathers.

A weak-coupling expansion of a vacuum-subtracted soliton mass begins with the classical energy and adds the Gaussian fluctuation approximation. For Sine-Gordon theory, $M=8m/{\beta }^2+\Delta M_{1\mathrm{loop}}+O(m{\beta }^2)$. The term of order $m$ includes both the vacuum-subtracted fluctuation frequencies and the appropriate mass counterterm.

In a common finite-volume regularization in quantum field theory, the oscillator contribution is $\Delta M_{\mathrm{osc}}=\frac{1}{2}{\sum }_n({\Omega }_n-{\Omega }_n^{(0)})$, including every discrete and continuum-derived mode and treating translation as a collective coordinate. Physical frequencies are ${\Omega }_n=m{\omega }_n$ when coordinates are measured in units of $m^{-1}$. A mass counterterm is added only afterwards. The vacuum subtraction removes the extensive vacuum energy but can leave an ultraviolet divergence.

Compare the same total number of finite-box modes in the soliton and vacuum sectors before removing the ultraviolet cutoff. A localized bound state replaces a continuum mode, and its contribution must be retained. For the Sine-Gordon kink fluctuation operator and the odd phase branch $\delta (k)=2\arctan (1/k)$, the bound translation mode replaces the free $k=0$ oscillator. Thus the complete oscillator correction is $-m/2-(m/2)\int dk\delta (k)k/[2\pi \sqrt{k^2+1}]$ in this convention. The continuum integral alone misses the finite $-m/2$ contribution.

The scattering phase shift $\delta (k)=2\arctan (m/k)$ of the Sine-Gordon kink fluctuation operator gives a continuum frequency sum. In mode-number regularization of soliton masses, the zero-frequency translation mode replaces the vacuum oscillator at $k=0$. The bare difference is $-m/2-(2\pi {)}^{-1}{\int }_0^{\Lambda }\delta (k)k/\sqrt{k^2+m^2}dk$. Integration by parts cancels the lower endpoint against $-m/2$ and produces the displayed upper surface term. Although $\delta (\Lambda )$ vanishes, its product with $\omega (\Lambda )$ tends to $2m$. Vacuum normal ordering cancels the logarithmic bulk ultraviolet divergence, leaving this finite one-loop soliton mass correction. Parameters and finite counterterms must be fixed by a stated renormalization condition.

The rest mass of a stable soliton is the lowest energy in its topological sector minus the vacuum energy, in the infinite-volume limit. If Euclidean kernels $K_Q(\tau )$ have nonzero overlap with the sector's lowest-energy states, then $M=-{\lim }_{\tau \to \infty }{\tau }^{-1}\log [K_1(\tau )/K_0(\tau )]$, followed by the infinite-volume limit. A Euclidean path integral represents these kernels, with boundary wavefunctionals included if necessary.

If a labeled soliton approaches $x=vt+a_{-}$ before a collision and $x=vt+a_{+}$ afterwards, its spatial shift is $\Delta x=a_{+}-a_{-}$. For $v\ne 0$, define its time delay at a fixed spatial point by $\Delta t=-\Delta x/v$. Thus a positive forward shift of a right-moving soliton is a negative time delay. For a stationary soliton, the spatial shift is well defined but this time-delay definition is not.

A narrow outgoing energy packet has phase $-ET/\hslash +\delta (E)$, so stationary phase gives the arrival shift $\Delta T=\hslash {\delta }^{\prime }(E)$. For two equal-mass solitons with relative rapidity $\theta$, $E=2\mathcal{M}\cosh (\theta /2)$. The convention $S=e^{2i{\delta }_{\mathrm{pw}}}$ instead gives $2\hslash {\delta }_{\mathrm{pw}}^{\prime }=\Delta T$. Classical time delays determine only the energy-dependent phase difference, not its constant branch.

A many-soliton spatial shift is pairwise additive when $\Delta x_i={\sum }_{j\ne i}\Delta x_{i|j}$. In the all-kink sector of Sine-Gordon theory, $\Delta x_{i|j}=-\mathrm{sgn}(v_i-v_j)\log {\tanh }^2[({\theta }_i-{\theta }_j)/2]/\cosh {\theta }_i$. Dominant spectator exponentials in the Sine-Gordon multisoliton tau representation multiply the effective exponential of kink $i$, so their logarithms add. This is a classical signature of factorized scattering, with no independent many-body contribution to the asymptotic shift.

A finite-energy field configuration has integrable energy density after the vacuum energy has been subtracted. The boundary conditions used for a localized classical field-theory soliton require approaching a vacuum at spatial infinity. In one dimension the two ends may approach distinct scalar-field vacua; in a gauge theory the asymptotic field can have nontrivial winding even when its local energy density tends to zero.

For arbitrary maps $\varphi :{\mathbb{R}}^2\to S^2$, integrability of the Dirichlet energy does not imply continuous extension to the one-point compactification. For large $r$, takeand interpolate smoothly to a constant on a bounded disk. Its exterior energy isThe field has no limit because $a(r)$ keeps winding. A continuous extension at the point at infinity instead requires $\varphi (x)\to {\varphi }_{\infty }$ uniformly outside sufficiently large disks. This fixed-limit boundary condition is standard for a sigma-model lump, but is extra information beyond finite energy for arbitrary fields.

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.

A topological charge is an integer or other discrete invariant determined by the homotopy class or characteristic class of a field configuration. Continuous finite-energy deformations cannot change it without crossing a singular or forbidden configuration.

A normalized vacuum field on a sphere surrounding a core can define $S^{d-1}\to S^{d-1}$ with integer topological degree. Nonzero degree obstructs its extension through the enclosed ball while staying in the vacuum manifold. This accounts for the winding number of a vortex and the spherical degree of a magnetic monopole, with the appropriate vacuum targets.

A nonzero topological degree obstructs smooth unwinding with fixed boundary data, but does not by itself give a stable finite size. In three dimensions a two-derivative energy scales as $R$ and can decrease as a configuration shrinks at fixed degree. The four-derivative Skyrme term scales as $R^{-1}$ and can balance this tendency. Derrick theorem addresses the energetic issue; a singular limiting configuration can evade smooth homotopy conservation.

A local current whose conservation follows from field geometry rather than requiring the field equation. For Skyrmions, pulling back the closed volume form on the target three-sphere gives the current associated with topological baryon number in the Skyrme model. Its spatial integral is a topological charge, unchanged by smooth evolution with the prescribed boundary behaviour.

A closed target $d$-form $\omega$ pulls back under a spacetime field $\varphi$ to a closed form $\alpha ={\varphi }^{\ast }\omega$. Its dual defines an identically conserved topological current. For a compact oriented spatial $d$-manifold and an equally dimensional oriented target with $\int \omega =1$, the spatial integral is the topological degree. Conservation follows without the field equations; changing boundary flux can change the charge.

A connected component, or homotopy class, of admissible field configurations with fixed asymptotic conditions. Distinct topological sectors cannot be joined by a continuous deformation within that configuration space. A nonzero topological charge labels a sector, but a stable finite-size energy minimum also depends on the energy functional.

For negative frequency $E$, the one-dimensional focusing nonlinear Schrödinger equation has the localized stationary profileGalilean boosts generate moving profiles with a translational collective coordinate and constant velocity.

Let a real localized profile satisfy $Ef=-f^{\prime \prime }/2-f^3+y^{2}f/2$. Then $\psi (x,t)=e^{i\Theta (x,t)}f(x-X(t))$ solves the cubic focusing nonlinear Schrodinger equation in the potential $x^2/2$ with the displayed phase and any harmonic-oscillator trajectory $X$. Imaginary terms require ${\Theta }_x=\dot{X}$; real terms require $\ddot{X}=-X$ and the remaining time-dependent phase. The translation is exact and causes no profile deformation, unlike a generic collective-coordinate effective Lagrangian approximation.

For $i{\psi }_t=-{\psi }_{xx}/2-|\psi {|}^2\psi$, translation of a solution at velocity $u$ must be accompanied by the displayed phase. Direct differentiation cancels the terms proportional to ${\partial }_x{\psi }_0$, leaving the original equation. A standing profile of frequency $E=-{\kappa }^2/2$ thus gives $\kappa \mathrm{sech}(\kappa (x-ut-X_0))e^{iux-i(u^2/2+E)t+i{\theta }_0}$. Its centre travels uniformly, while its L2 norm is unchanged.

Derrick's scaling argument tests a static field configuration by rescaling space. A soliton can be stationary only if energy terms with opposite scaling powers balance at the original scale.

For smooth fields into ${\mathbb{R}}^l$ with nonnegative differentiable potential and finite energy, Derrick scaling in two spatial dimensions forces the potential integral to vanish. Thus the field takes values where $U=0$ and $\nabla U=0$, so its static field equation makes every component harmonic. Each first derivative is an entire harmonic function in L2 space; the mean-value inequality on arbitrarily large balls forces it to vanish. Hence the field is constant. The scale identity alone does not exclude a pure-gradient model; this additional argument uses the flat unconstrained target. Curved-target harmonic maps, for example the O3 nonlinear sigma model, can instead have finite-energy nonconstant solutions.

Under Derrick scaling, an energy term containing $m$ spatial derivatives and homogeneous of degree $m$ in those derivatives scales as ${\lambda }^{m-D}$ in $D$ spatial dimensions.

For a smooth gauge field on ${\mathbb{R}}^3$ with finite positive magnetic energy, dilate its connection one-form by $A_i^{(s)}(x)=s{A}_i(sx)$. Its gauge curvature scales as $F_{ij}^{(s)}(x)=s^2{F}_{ij}(sx)$, so the energy scales as $s^4{s}^{-3}E=sE$. A static stationary configuration must have zero derivative at $s=1$, forcing $E=0$ and hence $F=0$. This Derrick scaling argument assumes the dilation preserves the boundary conditions and that boundary terms vanish. It does not exclude four-dimensional Yang-Mills instantons, whose Euclidean action is scale invariant.

In three space dimensions, rescale $A_i(x)$ to $\lambda A_i(\lambda x)$ and $\Phi (x)$ to $\Phi (\lambda x)$. The gauge curvature then scales with two derivatives, and the adjoint covariant derivative of the Higgs field with one. Magnetic, Higgs-gradient and potential energies give the displayed formula. Stationarity requires $V_B=V_D+3V_U$, which permits nonzero energies. In the Bogomolny-Prasad-Sommerfield monopole limit $V_U=0$, the Bogomolny equations give $V_B=V_D$.

Stationarity of a static solution under Derrick scaling requires $dE(\lambda )/d\lambda {|}_{\lambda =1}=0$. This necessary relation among the separately scaling energy terms is the Derrick virial identity.

A scalar-field kink is a one-dimensional finite-energy static solution approaching different vacua as $x\to -\infty$ and $x\to +\infty$. Its boundary values define a topological sector.

An antikink is a scalar-field kink with the opposite orientation of its vacuum boundary values. If ${\varphi }_K(x)$ joins vacua ${\varphi }_{-}$ to ${\varphi }_{+}$, then ${\varphi }_K(-x)$ joins ${\varphi }_{+}$ to ${\varphi }_{-}$. For an even double-well potential and an odd centered phi-four kink, the centered antikink is $-{\varphi }_K(x)$. With $Q=[\varphi (+\infty )-\varphi (-\infty )]/({\varphi }_{+}-{\varphi }_{-})$, a kink has $Q=1$ and an antikink has $Q=-1$. Their localized energies coincide by spatial reflection.

For the static energythe kink joining the vacua $-c$ and $c$ isIt saturates the Bogomolny bound $E\ge 4c^3/3$ by satisfying the first-order equation ${\varphi }^{\prime }=c^2-{\varphi }^2$.

For the real scalar field with metric $(+,-)$ and potential $U={\kappa }^2({\varphi }^2-1{)}^2/2$, the static phi-four kink is ${\varphi }_0(x)=\tanh (\kappa x)$ and satisfies ${\varphi }_0^{\prime 2}=2U({\varphi }_0)$. Its mass is $M=\int {\varphi }_0^{\prime 2}dx=4\kappa /3$. A Lorentz boost gives $\varphi (t,x)={\varphi }_0(\gamma (x-vt))$. The stress-energy tensor gives $P=-\int {\varphi }_t{\varphi }_{x}dx=\gamma v\int {\varphi }_0^{\prime 2}dx=\gamma Mv$. Similarly, $E=\frac{1}{2}[\gamma (1+v^2)+{\gamma }^{-1}]M=\gamma M$, proving the energy–momentum relation $E^2-P^2=M^2$.

For $V(\varphi )=\lambda (m^2-{\varphi }^2{)}^2$ with $\lambda ,m>0$, set $\kappa =\sqrt{2\lambda }m$. A well-separated kink–antikink pair at $-d/2,d/2$ is approximated by $\varphi =m\tanh \kappa (x+d/2)-m\tanh \kappa (x-d/2)-m$. At its midpoint, ${\varphi }_x=0$ and $\varphi =m-4m{e}^{-\kappa d}+O(e^{-2\kappa d})$. The static stress-energy tensor component $T^{11}={\varphi }_x^2/2-V$ is therefore negative there. Momentum conservation gives the force on the left half-line as $T^{11}(-\infty )-T^{11}(0)$, which is positive. Its leading magnitude is $F\sim 32m^2{\kappa }^2{e}^{-2\kappa d}$. Thus the pair attracts. This is a controlled leading-tail estimate at large separation, not an exact superposed solution or a theorem about all subsequent collision outcomes.

For the initial field $\varphi (x)=\tanh (x+c)-\tanh (x-c)-1$ and zero initial time derivative, evaluate the scalar-field momentum flux at the midpoint: ${\varphi }_x(0)=0$ and $\varphi (0)=2\tanh c-1$. The force on the left kink is the displayed positive value, attractive toward the right antikink. This exact initial expression has the large-separation asymptotic $32e^{-2d}$ for $d=2c$. If the initial velocity at the cut is $\psi (0)$, subtract $\psi (0{)}^2/2$.

Linearizing about the phi-four kink gives the displayed one-dimensional Schrödinger operator in unit kinetic normalization. Its translational zero mode in field theory is proportional to ${\mathrm{sech}}^2(cx)$. The localized shape eigenfunction $\mathrm{sech}(cx)\tanh (cx)$ has squared frequency $3c^2$, while the continuum starts at $4c^2$. Substitution verifies both bound-state eigenfunctions. The Pöschl-Teller potential permits a short completeness argument for the bound modes. In $y=cx$ let $A_{\ell }={\partial }_y+\ell \tanh y$. Then $H_K/c^2=A_2^{†}{A}_2$, its partner is $A_2{A}_2^{†}=A_1^{†}{A}_1+3$, and $A_1{A}_1^{†}=-{\partial }_y^2+1$. The last operator has no bound states. The kernels of $A_1$ and $A_2$ give the two stated modes, while the partner spectra exclude any further normalizable bound modes.

For ${\varphi }_K(x-X)=c\tanh (c(x-X))$ in the unit kinetic normalization, $\int ({\varphi }_K^{\prime }{)}^{2}dx=M=4c^3/3$. Substitution of $X(t)$ therefore gives the free-particle collective-coordinate effective Lagrangian $-M+M{\dot{X}}^2/2$. Its quantization has momentum $p$ and energy $M+p^2/(2M)$ to this order. The exact uniformly moving classical kink is a Lorentz boost of the static one, with energy $M/\sqrt{1-v^2}$.

Kinks in a phi-six model commonly arise from a nonnegative degree-six potential with three degenerate vacua. For the unit-vacuum normalizationthe kink joining $0$ to $1$ isand has mass $1/4$. A rescaled model with $U(\varphi )={\varphi }^2({\varphi }^2-4{)}^2$ has the elementary kinkand its mass is $4\sqrt{2}$ in the normalization $E=\int [({\varphi }^{\prime }{)}^2/2+U]dx$.

For $U={\varphi }^2({\varphi }^2-b^2{)}^2/2$ and $b>0$, the three scalar-field vacua are $-b,0,b$. The static first integral is ${\varphi }_x^2=2U$. In either adjacent interval, $y={\varphi }^2/b^2$ obeys $y_x=\pm 2b^{2}y(1-y)$, giving the displayed four oriented kink profiles with $\sigma ,\eta \in \{1,-1\}$. Each has rest energy $b^4/4$. The intermediate-vacuum obstruction to a kink excludes a direct static connection from $-b$ to $b$: reaching zero would give zero derivative, and uniqueness of the smooth second-order field equation forces the zero solution. A Lorentz boost replaces $x-x_0$ by $(x-vt-x_0)/\sqrt{1-v^2}$.

// Destination: relativistic-quantum-field.bigb

If a proposed kink must cross a third degenerate vacuum, its first integral makes both the field derivative and potential vanish there. ODE uniqueness prevents it from crossing at finite distance, so the two elementary kink sectors can be joined only at infinite separation.

A gauge-theory soliton is a finite-energy localized classical field configuration stabilized by topology or a balance among differently scaling energy terms.

A smooth finite-energy monopole in an $SU(2)$ Yang-Mills theory with an adjoint Higgs field uses the normalized Higgs direction at infinity to define a map $S_{\infty }^2\to S^2$. Its nonzero degree supplies the magnetic topological sector. Unlike a singular Abelian Dirac monopole, the non-Abelian gauge and Higgs fields can have a regular core. In the zero-potential limit the Bogomolny-Prasad-Sommerfield monopole equations are $B_i=\pm D_i\Phi$.

A Bogomolny-Prasad-Sommerfield monopole is a finite-energy Yang-Mills-Higgs configuration satisfying $B_i=\pm D_i\Phi$. Its magnetic charge is the degree of the normalized Higgs field at spatial infinity.

For the trace inner product and adjoint covariant derivative, compatibility gives $\Delta |\Phi {|}^2=2|D\Phi {|}^2+2⟨\Phi ,D_i{D}_i\Phi ⟩$. The gauge-theory Bianchi identity and $B_i=\pm D_i\Phi$ imply $D_i{D}_i\Phi =0$. Therefore $\Delta |\Phi {|}^2=2|D\Phi {|}^2$, equal to the energy density $|B{|}^2+|D\Phi {|}^2$ on a Bogomolny-Prasad-Sommerfield monopole. If the energy density is instead defined with an overall factor $1/2$, this Laplacian is twice that density.

For zero scalar potential, the static field equations are $D_i{D}_i\Phi =0$ and $D_i{F}_{ij}=[\Phi ,D_j\Phi ]$. If $B_i=-D_i\Phi$, the gauge-theory Bianchi identity gives the first equation. For the second,using $F_{ik}={ϵ}_{ik\ell }{B}_{\ell }$ and ${ϵ}_{ijk}{ϵ}_{ik\ell }=-2{\delta }_{j\ell }$. Thus the first-order equations imply the full second-order equations locally, independently of the radial ansatz.

A spherically symmetric SU(2) monopole can be written with the internal Higgs direction aligned with the spatial radial direction and the gauge field built from the invariant tensor ${ϵ}_{aij}$.

For a Bogomolny-Prasad-Sommerfield monopole, the dimensionless radial Bogomolny equations are $dK/d\rho =-KH$ and $dH/d\rho =(1-K^2)/{\rho }^2$. The displayed profiles solve both equations: differentiating $\rho /\sinh \rho$ gives $K(1/\rho -\coth \rho )=-KH$, while differentiating $H$ gives $1/{\rho }^2-{\mathrm{csch}}^2\rho =(1-K^2)/{\rho }^2$. At the origin $K=1-{\rho }^2/6+O({\rho }^4)$ and $H=\rho /3+O({\rho }^3)$, giving a regular core. At infinity $K$ decays exponentially and $H=1-1/\rho +O(e^{-2\rho })$, giving the vacuum Higgs magnitude and a long-range Abelian magnetic field.

For ${\Phi }^a=f(r)x^a/r$ and $A_i^a={ϵ}_{iaj}{x}^j\alpha (r)$ with $[e_a,e_b]={ϵ}_{abc}{e}_c$, set $K=1+r^2\alpha$. The equation $B=-D\Phi$ equates radial coefficients to give $f^{\prime }=-2\alpha -r^2{\alpha }^2$ and transverse coefficients to give $r{\alpha }^{\prime }+2\alpha =-f/r-r\alpha f$. These are exactly the displayed two radial equations. The sign depends on the chosen orientation and on the order of indices in the gauge ansatz.

The Abelian Higgs model couples a complex scalar field to a $U(1)$ gauge field and permits magnetic vortex solitons in two spatial dimensions.

For $|D\Phi {|}^2-\lambda (|\Phi {|}^2-{\eta }^2{)}^2/4$, write $\Phi =(\eta +h/\sqrt{2})e^{i\theta }$ and absorb $\partial \theta /e$ into the vector field. The quadratic Lagrangian has a real scalar of mass $m_h=\sqrt{\lambda }\eta$ and a vector of mass $m_A=\sqrt{2}e\eta$. The two scalar plus two massless-vector polarizations in the symmetric phase become one radial scalar plus three massive-vector polarizations. The removed phase supplies the longitudinal mode; local vortices retain global winding and quantized flux.

// Target: cosmology.bigb

In two dimensions, scale ${\Phi }_{\lambda }(x)=\Phi (\lambda x)$ and $A_{j,\lambda }(x)=\lambda A_j(\lambda x)$. The magnetic field has scaling weight two, the gauge covariant derivative of the Higgs field has weight one, and the potential has weight zero. Their integrated energies therefore have the powers shown. Stationarity gives $E_B=E_P$, a Derrick virial identity. This balance allows a planar Abelian Higgs vortex to have finite size even though a scalar-gradient-plus-nonnegative-potential model has no comparable magnetic term.

This planar Abelian Higgs model replaces the usual magnetic energy by $B^2/h$. Use $D_i={\partial }_i-i{A}_i$ and $B={\partial }_1{A}_2-{\partial }_2{A}_1$. The magnetic term is singular at a zero of the Higgs field, so the admissible class must have finite energy; formal field equations are first derived where $h>0$. Its positive-flux Bogomolny equations are $(D_1+i{D}_2)\Phi =0$ and $B=h(1-h)/2$. On those solutions $B/h=(1-h)/2$ extends regularly across zeros.

Write $\Phi =e^{u/2+i\chi }$ away from its zeros and use the oriented planar Hodge star with $\ast d{x}^1=d{x}^2$. The first Bogomolny equation gives $A=d\chi -\frac{1}{2}\ast du$. A zero of multiplicity $n_j$ contributes $u=2n_j\log |x-p_j|+\text{smooth}$ and phase winding $2\pi n_j$. Combining the smooth magnetic field with these delta sources gives the displayed equation and the topological boundary condition $u\to 0$. Prescribed zero positions and multiplicities reduce multi-vortex construction to this scalar elliptic problem; a regular solution reconstructs both smooth fields.

Let $j_i=\mathrm{Im}(\bar{\Phi }{D}_i\Phi )$. The identityand completing the square give, when the current boundary integral vanishes,For flux $2\pi N>0$, the lower bound is $\pi N$, saturated by the two first-order Bogomolny equations. Division by $h$ is interpreted in the finite-energy sense at its zeros.