Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 47 / 2

Take $r>0$ and $m>0$, the positive-energy sigma-model regime, and keep $\theta =0$ for the energy calculation. For a static field the energy isFor either sign $s=\pm 1$, complete the square:Here $\int {\varphi }_x\sin \varphi dx=-[\cos \varphi {]}_{-\infty }^{+\infty }=2T$. Choose $s$ to have the sign of $T$. The Bogomolny bound is thereforeSaturation requires ${\varphi }_x=sm\sin \varphi$ and $\sin \varphi {\alpha }_x=0$. Finite-energy vacua have $\sin \varphi =0$, so their endpoint cosine values are $\pm 1$ and this particular topological charge is $T\in \{-1,0,1\}$. This is the static kink bound in a mass-deformed spherical sigma model. Negative $r$ would instead make the derivative energy unbounded below, so positivity is an essential physical convention.

For $T=1$, choose the positive Bogomolny equation and constant angular orientation $\alpha ={\alpha }_0$. Integrating $d\varphi /\sin \varphi =mdx$ givesThis interpolates from $\varphi (-\infty )=0$ to $\varphi (+\infty )=\pi$, so $T=-(-1-1)/2=1$. Its useful profiles are $\sin \varphi =\mathrm{sech}[m(x-x_0)]$ and $\cos \varphi =-\tanh [m(x-x_0)]$. It obeys the second-order field equation because differentiating ${\varphi }_x=m\sin \varphi$ gives ${\varphi }_{xx}=m^2\sin \varphi \cos \varphi$; the angular field equation holds since $\alpha$ is constant. ThusBoth $x_0$ and ${\alpha }_0$ are collective coordinates: translation and global angular rotation do not change the energy. This static solution is the zero-charge member of the rotating charged kink in a spherical sigma model.

Fix ${ϵ}^{01}=+1$ with $x^0=t$, $x^1=x$. The Noether current for the unit shift of $\alpha$ is the derivative of the Lagrangian with respect to ${\partial }_{\mu }\alpha$:The angular field equation is ${\partial }_{\mu }{j}^{\mu }=0$. At $\theta =0$ this gives the conserved Noether chargeFor nonzero $\theta$, ${ϵ}^{10}=-1$, so $j^0$ contains $+\theta {\partial }_x(\cos \varphi )/(4\pi )$. Its integral is a boundary term:The theta term is locally a total derivative, so it does not change the bulk equations or the energy, but it changes the canonical charge by this topological shift. This is the theta-angle shift of a sigma-model kink charge. Reversing the convention for ${ϵ}^{01}$ reverses the displayed theta shifts. Distinguishing the mechanical charge $Q_0$ from the canonical symmetry charge $Q_{\theta }$ will be important for quantization.

At $\theta =0$, the field equations areWith $\alpha =\omega t+{\alpha }_0$ and static $\varphi$, the second equation holds automatically and the first reduces toFor $|\omega |<m$, put $\kappa =\sqrt{m^2-{\omega }^2}$. The finite-energy $T=1$ solution isThis is a rotating charged kink in a spherical sigma model. Its orientation moves around a circle while its energy profile stays at rest. The limiting value $|\omega |=m$ gives no localized finite-width kink; larger $|\omega |$ does not give this finite-energy interpolation.

Using ${\varphi }_x=\kappa \sin \varphi$ and $\int {\sin }^2\varphi dx=2/\kappa$, its rest energy and mechanical charge areEliminating $\omega$ givesFor this part $Q=Q_0$ since $\theta =0$. The static mass is recovered at $Q_0=0$, and the mass grows with the magnitude of the global charge. If theta is restored, the energy remains the same function of $Q_0$, but $Q_0=Q_{\theta }+\theta T/(2\pi )$.

The internal orientation ${\alpha }_0$ is a periodic collective coordinate with period $2\pi$. Its conjugate momentum is the integrated canonical charge. The Bohr-Sommerfeld quantization for a positively oriented circuit of this angle is, in units $\hslash =1$,There is no oscillator turning point and hence no half-integer Maslov correction for this cyclic coordinate. Equivalently, a wavefunction $e^{i{Q}_{\theta }{\alpha }_0}$ must be single valued. At $\theta =0$, the allowed signed charges and leading semiclassical masses in the $T=1$ family are thereforeThe signed integers include both rotation directions and the nonrotating $\ell =0$ state. Every finite integer gives $|\omega |<m$.

For nonzero theta it is still the canonical charge that is integral, while the mechanical charge is shifted:For the stated family set $T=1$; the reversed kink gives the corresponding $T=-1$ family. This is the semiclassical charged-kink spectrum with a theta angle. One may equivalently describe the spectrum by fractional mechanical charges instead of integral canonical charges, but the two conventions must not be mixed. The theta periodicity follows by relabeling the integer when $\theta \mapsto \theta +2\pi$. These are leading semiclassical energies; the quantization argument does not assert that quantum fluctuation corrections vanish.