Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 49 / 2

For the usual scalar field theory with a nonnegative potential $U$, shifted to zero in the vacuum, write its static energy in $d$ dimensions asApply Derrick scaling ${\varphi }_{\lambda }(x)=\varphi (\lambda x)$. The gradient acquires a factor $\lambda$ and the volume a factor ${\lambda }^{-d}$, givingA smooth finite-energy static solution is a stationary point of this energy, so its Derrick virial identity isFor $d\ge 3$, both terms are nonnegative, forcing $T=P=0$. The field is a constant vacuum: there is no nontrivial finite-energy static soliton in this scalar-gradient-plus-nonnegative-potential model. The assumptions matter: extra derivative terms, gauge fields, time dependence or a potential not bounded below by its vacuum value can change the argument.

For $d=2$, the relation becomes $P=0$ and says nothing further about $T$. A gradient-only energy is scale invariant, so Derrick theorem alone does not rule out nonconstant solutions; sigma-model lumps are examples. For a scalar field with isolated potential minima, $P=0$ constrains the continuous field to one vacuum and hence makes it constant. If the vacuum values form a nontrivial manifold, the scaling relation alone leaves open a nonconstant map into it. In neither case does the two-dimensional scaling argument itself prove existence.

The Derrick scaling of the Abelian Higgs energy must rescale the gauge potential as well as the Higgs field. SetThen $B_{\lambda }(x)={\lambda }^{2}B(\lambda x)$ andLet $E_B=\int B^2$, $E_D=\int |D\Phi {|}^2$ and $E_P=\frac{1}{4}\int (1-|\Phi {|}^2{)}^2$, with all integrals over the plane. Changing variables givesAt a static solution, $V^{\prime }(1)=2E_B-2E_P=0$. ConsequentlyThe covariant-gradient contribution is scale invariant, while the magnetic and potential terms balance. This is why the scalar-only nonexistence conclusion does not extend to the Abelian Higgs model.

At this critical coupling, construct solutions of the Bogomolny vortex equations rather than superpose isolated vortices, which would not respect the nonlinear field equations. Define $D_j={\partial }_j-i{A}_j$ and $j_k=\mathrm{Im}(\overline{\Phi }{D}_k\Phi )$. Direct differentiation using $[D_1,D_2]\Phi =-iB\Phi$ givesIntegrating the last divergence away under the usual finite-energy vortex boundary conditions and completing the magnetic square yieldsThus positive vortices saturate this Bogomolny bound whenSuch fields solve the static second-order equations: under any compactly supported variation the flux is unchanged, and the first variation of both zero squares vanishes. The bound here is $2\pi N$, twice the bound in the frequently used half-normalized Abelian Higgs model energy.

For the prescribed-zero construction of planar Abelian Higgs vortices, choose arbitrary points $z_a\in \mathbb{C}$ and positive integer multiplicities $m_a$, permitting coincident vortices. Away from the zeros, write $\Phi =e^{h/2+i\chi }$, where $h=\log |\Phi {|}^2$. The first Bogomolny vortex equation givesChoose the phase winding $\chi ={\sum }_a{m}_a\arg (z-z_a)$ locally. Distributionally its curl contributes $2\pi {\sum }_a{m}_a{\delta }_{z_a}$, so the second Bogomolny vortex equation becomes the Taubes equationNear each zero, require $h=2m_a\log |z-z_a|+$ a smooth remainder. The planar existence theorem for this equation gives a solution for every finite multiset of prescribed points, unique in this decaying class; see the original theorem. This elliptic existence result is the nontrivial analytic ingredient of the construction, not a linear superposition assumption.

Reconstruct $\Phi =e^{h/2+i\chi }$ and $A$ from the formulas above. The phase factors have integer winding, so $\Phi$ is single-valued. Near a zero, the modulus and phase combine into $(z-z_a{)}^{m_a}$ times a smooth nonvanishing factor. The singular part of $\nabla \chi$ cancels the rotated logarithmic-gradient part of $A$, so the physical fields extend smoothly through the cores. This produces the desired Abelian Higgs vortices at the selected positions. Their vortex number is $N={\sum }_a{m}_a$. Replacing $(A,\Phi )$ by $(-A,\overline{\Phi })$ constructs the negative-sign antivortex family; the positive construction does not assert arbitrary mixtures of positive and negative cores solve the same first-order equations.

Since $|\Phi |\to 1$, the Higgs field is nonzero outside a sufficiently large disk. On a large positively oriented circle write $\Phi =\rho e^{i\chi }$ locally, where $\rho >0$. Its normalized phase is a map from the circle to the unit circle. The phase need not be a globally single-valued real function, but its total change is $2\pi N$ for an integer winding number $N$. This integer is the same on all sufficiently large circles because the intervening annulus contains no zeros.

The gauge covariant derivative givesIntegrate around the circle. The assumed rapid decay makes the integral of the second term tend to zero. For example, uniform $|D\Phi |=o(R^{-1})$ and $\rho \to 1$ on radius-$R$ circles suffice. HenceBy Stokes theorem, ${\oint }_{|x|=R}A={\int }_{|x|\le R}B{d}^{2}x$. Integrability of $B$ allows the limit, giving the magnetic flux quantization of an Abelian Higgs vortex:The topological charge $N$ is the degree of $\Phi /|\Phi |$ at infinity, counting net phase winding around the cores. For isolated zeros it is the sum of their signed local winding numbers. In the positive Abelian Higgs vortex construction, each core of multiplicity $m_a$ has local phase $m_a\arg (z-z_a)$, sothe number of vortices counted with multiplicity. Antivortices have the opposite sign. The integer is invariant under smooth gauge transformations defined throughout the disk, whose boundary winding is zero.