Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 47 (Classical and Quantum Solitons)

Use the Minkowski metric with signature $(+,-)$, so the Sine-Gordon equation is ${\varphi }_{tt}-{\varphi }_{xx}+\sin \varphi =0$. The light-cone coordinates here satisfyIn particular, there is no extra factor of four with this coordinate normalization. The Bäcklund transformation requires $a\ne 0$, because one of its equations contains $a^{-1}$.

Put $F=(\psi +\varphi )/2$ and $G=(\psi -\varphi )/2$. In the parameter convention of this paper the Sine-Gordon Bäcklund transformation gives $G_{+}=a\sin F$ and $F_{-}=a^{-1}\sin G$. Differentiating, for a twice differentiable transformed field, givesSince $\psi =F+G$, the addition formula yields ${\psi }_{+-}=\sin (F+G)=\sin \psi$. Therefore the transformed field satisfies the same Sine-Gordon equation:Subtracting the two differentiated equations also gives ${\varphi }_{+-}=\sin \varphi$, showing the compatibility with the seed equation. This proves the unheaded request before the numbered parts, without assuming a particular soliton form.

With zero seed, both Bäcklund transformation equations concern ${\varphi }_a/2$. On a nonconstant branch, separation of variables uses $d\log |\tan ({\varphi }_a/4)|=d{\varphi }_a/[2\sin ({\varphi }_a/2)]$, givingFor $C>0$, absorb its magnitude into an additive constant $c=\log C$ in the exponent. A convenient smooth representative isThe constant-phase condition for this traveling profile determines its velocity:The profile is a traveling soliton with velocity $v_a=(1-a^2)/(1+a^2)$, strictly between $-1$ and $1$. The scalar-field vacua on the two sides differ by $2\pi$. With the topological charge convention $Q=[\varphi (+\infty )-\varphi (-\infty )]/(2\pi )$, this branch has $Q=\mathrm{sgn}a$: it is a Sine-Gordon kink for $a>0$ and an antikink for $a<0$. Its width is proportional to ${\gamma }_a^{-1}$, and its derivative decays exponentially away from its center, giving a finite-energy field configuration.

For completeness, the classical rest mass in this paper's coupling convention is $8m/\beta$, as derived in Question 2; a Lorentz boost gives energy $8m{\gamma }_a/\beta$. This localized, topologically protected traveling field is the required classical field-theory soliton. A negative $C$ reverses the field and hence the topological charge; $C=0$ gives the vacuum rather than a soliton. Vacuum shifts by $4\pi$ can be made without changing the displayed Bäcklund transformation equations.

Choose the positive-exponential branches from part (i) and set their additive constants to zero. First take $0<b<1$, and defineThen $\tan ({\varphi }_b/4)=e^{\gamma (x-vt)}$ and $\tan ({\varphi }_a/4)=e^{-\gamma (x+vt)}$. The tangent subtraction formula givesConsequently the allowed Sine-Gordon superposition formula produces the smooth fieldThis is the negative of the Sine-Gordon two-kink solution, and hence a two-antikink configuration. The auxiliary seeds have opposite topological charges, but their charges cannot simply be added to infer the charge of the nonlinear two-step Bäcklund transformation. Indeed, the displayed final field tends to $2\pi$ at the left spatial end and $-2\pi$ at the right, so its total topological charge is $-2$.

Let $T=|t|$ become large. Near the right transition, $x=vT+O(1)$, the tangent argument has the asymptotic formso the local field is $-4\arctan e^{\gamma (x-vT)+\log v}$, a single antikink. Near the left transition, the local field is $4\arctan e^{-\gamma (x+vT)+\log v}+o(1)$, again a decreasing antikink. The resulting asymptotic center lines areThus two incoming antikinks with topological charges $(-1,-1)$ and velocities $(+v,-v)$ separate again with exactly the same topological charges and velocities. There is no radiative tail in these asymptotic profiles. Labeling the outgoing objects by their preserved rapidities makes this elastic soliton scattering; labeling the left and right lumps instead describes reflection with exchanged velocities.

For the right-moving soliton, its incoming intercept is ${\gamma }^{-1}\log v$ and its outgoing intercept is $-{\gamma }^{-1}\log v$. The spatial shifts are therefore $\Delta x_{+}=-2{\gamma }^{-1}\log v$ and $\Delta x_{-}=2{\gamma }^{-1}\log v$. Define the soliton time delay as the change in arrival time at a fixed distant spatial point relative to continuation of the incoming straight line, so $\Delta t=-\Delta x/v_{\mathrm{particle}}$. Both objects have the same signed soliton time delay, which is an advance:This is the Sine-Gordon two-kink time advance. In physical coordinates $T_{\mathrm{phys}}=t/m$, the time shift is $2\log v/(m\gamma v)$. The explicit intercepts fix the sign convention unambiguously.

The remaining real parameter choices are covered without changing the calculation. For any $b\ne 0$ with $b^2\ne 1$, put $v_b=(1-b^2)/(1+b^2)$, $u=|v_b|$, $\gamma =(|b|+|b{|}^{-1})/2$ and $ϵ=\mathrm{sgn}(b{v}_b)$. The same choice of zero additive constants givesEach scattered object's topological charge is $-ϵ$, the velocities are $\pm u$, and the signed soliton time delay is $2\log u/(\gamma u)$. If $b=\pm 1$, the superposition coefficient vanishes and this representative is the vacuum; there is no pair of separated moving solitons and no scattering delay to assign. Thus the scattering conclusion requires the nondegenerate case $0<u<1$.

To keep the paper's coupling convention explicit, write $g=\sqrt{\beta }$ and use physical coordinates $X=x/m$, $T=t/m$. Assume $\beta >0$ and weak coupling $g\ll 1$. The canonical real scalar field is $\phi =\varphi /g$, and the physical potential isThe symbol $g$ is the coupling usually appearing in canonical Sine-Gordon theory; the paper's $\beta$ is its square. The vacua have $g\phi =2\pi n$. A static Sine-Gordon kink has $g{\phi }_K=4\arctan e^{m(X-X_0)}$ and satisfies $\frac{1}{2}{\phi }_K^{\prime 2}=U({\phi }_K)$. Since $1-\cos (g{\phi }_K)=2{\mathrm{sech}}^2[m(X-X_0)]$, its classical mass isThe topological sector is important: this energy is measured relative to a vacuum, and the kink joins distinct vacua at the two spatial ends. Expanding around a spatially constant vacuum cannot construct this state by any finite-order perturbation in $g$.

For the general one-loop soliton mass correction, start with a canonical scalar field potential $U$ and a stable static soliton ${\phi }_K$. Write $\phi ={\phi }_K+\eta$. The term linear in $\eta$ vanishes by the classical Euler-Lagrange field equation. The quadratic fluctuation Hamiltonian isChoose a common large box and a common finite-mode regularization in quantum field theory. Expand the nonzero normal modes as $\eta ={\sum }_n{q}_n{f}_n$, with ${\mathcal{H}}_K{f}_n={\omega }_n^2{f}_n$ and normalized eigenfunctions. Each pair $(q_n,p_n)$ is a quantum harmonic oscillator contributing ground-state energy $\hslash {\omega }_n/2$. In the vacuum, replace ${\mathcal{H}}_K$ by ${\mathcal{H}}_0=-{\partial }_X^2+U^{\prime \prime }({\phi }_{\mathrm{vac}})$. Subtract the two ground-state energies and add the local counterterms evaluated on the soliton relative to the vacuum. This derives the general formulaThe prime excludes exact zero modes in field theory from oscillator quantization; their zero frequencies contribute no zero-point energy, but their role in mode counting must not be forgotten. The two sums mean a paired finite regulator, not separate subtractions of divergent answers. Equivalently the nonzero-mode term is the regulated difference of the square-root traces of the two fluctuation operators. Discrete bound states and continuum modes both contribute. This is a vacuum-subtracted soliton mass and the first term in the semiclassical soliton mass expansion.

Translation gives a zero mode in field theory because differentiating the static equation yields ${\mathcal{H}}_K{\phi }_K^{\prime }=0$. A Gaussian oscillator or an unprimed functional determinant is inappropriate along this flat direction. Replace its amplitude by the position collective coordinate $X_0(T)$ and require the residual fluctuation to obey $\int \eta {\phi }_K^{\prime }dX=0$, preventing double counting. The associated change-of-variables Jacobian supplies the zero-mode normalization. At low speed the collective-coordinate effective Lagrangian for a soliton is $-M_{\mathrm{cl}}+\frac{1}{2}{M}_{\mathrm{cl}}{\dot{X}}_0^2+\cdots$; quantizing the position gives the soliton momentum and its translational states, not an extra oscillator rest energy. More generally, every physical continuous modulus needs a collective coordinate; gauge directions require gauge fixing rather than additional physical states.

The Sine-Gordon kink fluctuation operator is particularly simple:while $D{D}^{†}=-{\partial }_X^2+m^2={\mathcal{H}}_0$. This supersymmetric factorization of the one-soliton potential shows stability and generates all nonzero eigenfunctions from vacuum plane waves. The sole normalizable bound state is the translational zero mode of a sine-Gordon kink, proportional to $\mathrm{sech}m(X-X_0)$; there is no positive-frequency internal bound oscillator. The continuum has $\omega (k)=\sqrt{k^2+m^2}$ and no reflection. Applying $D^{†}$ to $e^{ikX}$ gives a transmission amplitudeThis scattering phase shift changes the density of continuum modes. The high-frequency vacuum subtraction cancels the extensive vacuum contribution but still leaves a logarithmic ultraviolet divergence.

The finite part also requires consistent mode-number regularization of soliton masses. The periodic-box phase-shift quantization condition is $k_{n}L+\delta (k_n)=2\pi n$. Match $2N+1$ modes: the kink has its translation mode plus the two continuum modes at each $n=1,\dots ,N$, while the vacuum has the $k=0$ oscillator of frequency $m$ and the corresponding continuum pairs. With $\hslash =1$ and $\Lambda =2\pi N/L$, expanding $k_n-k_n^{(0)}=-\delta (k_n^{(0)})/L+o(L^{-1})$ givesIntegration by parts uses $\delta (0)=\pi$ and yieldsThe cutoff surface term for a Sine-Gordon kink tends to $-m/\pi$. It cannot be dropped merely because $\delta (\Lambda )\to 0$: $\omega (\Lambda )$ grows at the same time.

A renormalization condition must specify which mass and coupling are held fixed. For vacuum normal ordering, or cancellation of the vacuum tadpole diagram with the elementary mass fixed at $m$, the quartic interaction gives the mass countertermEvaluating this Sine-Gordon vacuum tadpole counterterm on the kink givesThe logarithmic ultraviolet divergence cancels, leaving the renormalized one-loop Sine-Gordon kink mass in the stated vacuum scheme:This example illustrates why a zero mode in field theory must be treated as a collective coordinate, why vacuum subtraction alone need not remove ultraviolet divergences, and why the finite relation between the two regulators matters. Different finite counterterms amount to different definitions of the renormalized parameters; an unregulated frequency difference without a renormalization condition is not a physical mass prediction.

In elastic diagonal factorized scattering, the Faddeev-Zamolodchikov algebra describes an exchange of two particle operators. Exchanging the pair twice must restore the original state. This is analytic unitarity:Keeping the reversed species in this identity avoids an unstated parity assumption. For the particular amplitudes derived below, crossing and analytic unitarity also imply $S_{\bar{A}A}=S_{A\bar{A}}$. Hermitian analyticity of a two-particle S-matrix states $S_{ab}(\theta {)}^{\ast }=S_{ba}(-{\theta }^{\ast })$ and, together with analytic unitarity, gives $|S_{ab}(\theta )|=1$ for real rapidity differences. Crossing symmetry analytically turns an incoming particle into an outgoing antiparticle, relating the two channels by $S_{A\bar{A}}(\theta )=S_{AA}(i\pi -\theta )$, with the reverse relation obtained by crossing again. The shift $i\pi$ follows from the sign reversal of the two-momentum $p(\theta )=(m\cosh \theta ,m\sinh \theta )$.

Put $u=\lambda /2$. The given amplitude is the unit-modulus hyperbolic scattering block $S_u(\theta )=\sinh [(\theta +iu)/2]/\sinh [(\theta -iu)/2]$. Its numerator and denominator interchange under $\theta \mapsto -\theta$, proving analytic unitarity. For real $u$ it also obeys Hermitian analyticity of a two-particle S-matrix. Applying crossing symmetry givesThis crossed amplitude also has unit modulus on the real axis. Thus both requested channel constraints hold.

The bound-state conclusion needs a coupling range. In the fundamental attractive range $0<u<\pi$, equivalently $0<\lambda <2\pi$, the denominator has a simple pole at $\theta =iu$ inside the physical rapidity strip. Its residue iswith positive imaginary coefficient. Under the usual one-particle interpretation of this direct-channel bound-state pole, it couples two charge-$+1$ particles to a new charge-$+2$ particle $B$. In the center-of-mass frame its constituents have analytically continued rapidities $\pm iu/2$, and their total two-momentum is $(2m\cos (u/2),0)$. Thus the relativistic bound-state mass from a rapidity pole givesIt is positive and less than the two-particle threshold $2m$. Without the coupling qualification the requested deduction is false: at $\lambda =0$ the amplitude is identically one after removing the apparent $0/0$ at the origin. A free massive complex scalar field has this diagonal amplitude, charge-$\pm 1$ particles and no isolated charge-$+2$ bound state. At $u=\pi$ the amplitude similarly becomes constant $-1$ and the apparent boundary pole cancels. Neither endpoint supplies the claimed particle.

For bound-state fusion of factorized S-matrices, represent $B$ as the residue of $A({\theta }_B+iu/2)A({\theta }_B-iu/2)$ at its bound-state separation. Move a third $A({\theta }_A)$ through both constituents using the Faddeev-Zamolodchikov algebra, then take the same residue. The bound-state normalization occurs on both sides and cancels. Consequently bootstrap fusion giveswhere $\theta ={\theta }_B-{\theta }_A$. This has analytic unitarity as a product of two shifted blocks. Its poles occur at $\theta =iu/2$ and $\theta =3iu/2$, modulo $2\pi i$; numerator zeroes occur at the corresponding negative positions, subject to cancellations at special couplings.

The nearer pole, $\theta =iu/2$, has residue $-2i\sin u$. It is a crossed-channel pole in diagonal factorized scattering, not a new direct-channel charge-$+3$ state. Indeed, the exchanged momentum has invariantusing $m_B=2m\cos (u/2)$. The exchanged particle is therefore the already present $A$, with the appropriate charge flow at the crossed vertex. Equivalently, crossing symmetry puts a positive-residue direct pole of $S_{B\bar{A}}$ at $i(\pi -u/2)$, corresponding to $B+\bar{A}⟶A$. This distinction avoids assigning an extra mass by applying the direct-channel formula to every pole.

The farther pole, $\theta =3iu/2$, lies in the physical rapidity strip only for $0<u<2\pi /3$, equivalently $0<\lambda <4\pi /3$. Its residue iswhich has positive imaginary coefficient in that range. It gives a charge-$+3$ bound state $C$. The relativistic bound-state mass from a rapidity pole now yieldsThe positive root in the admitted range isThis is the three-particle bound state from equal-mass fusion; in its own rest frame the constituent rapidities are $iu,0,-iu$. At $u=2\pi /3$ the farther apparent pole cancels because its numerator also vanishes; for $2\pi /3<u<\pi$ it lies outside the physical rapidity strip and does not require a new charge-$+3$ particle. Charge-conjugate partners carry charges $-2$ and $-3$ where the corresponding states exist. Fusion distinguishes an existing crossed-channel particle from a genuinely new direct-channel bound state, and the latter requires the stated smaller coupling range.