Class Of Soliton Solutions Research Articles

Solitons and vertex operators in twisted affine Toda field theories

Affine Toda field theories in two dimensions constitute families of integrable, relativistically invariant field theories in correspondence with the affine Kac-Moody algebras. The particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling and which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary. Here these results are extended from the untwisted to the twisted algebras. The new soliton solutions and their masses are found by a folding procedure which can be applied to the affine Kac-Moody algebras themselves to provide new insights into their structures. The relevant foldings are related to inner automorphisms of the associated finite dimensional Lie group which are calculated explicitly and related to what is known as the twisted Coxeter element. The fact that the twisted affine Kac-Moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions.

Spin content of quantum solitons

The classical soliton solution, quantized by means of suitable translational and rotational collective coordinates, is embedded into the one-particle irreducible representation of the Poincaré group corresponding to a definite spin. It is shown that within the conventional quasiclassical expansion such embedding leads to a set of nontrivial consistency conditions imposed on the classical solution. The validity of these relations is considered for a number of soliton models in 2 + 1 and 3 + 1 dimensions.

Quantum reduplication of classical solitons

The possible existence of a series of quantum copies of classical soliton solutions is discussed, which do not exist when the effective Planck constant of the theory γ tends to zero. Within the conventional weak-coupling expansion in √ γ such non-classical solitons are O(√ γ) in energy and therefore lie in between the true classical solutions and elementary quantum excitations. Analytic results concerning the shape functions, masses and characteristic scales of such quantum excitations are given for soliton models of a self-interacting scalar field. Stability properties and quantization of fluctuations in the neighborhood of these configurations are also discussed in detail.

Nonclassical analogs of solitons in quantum field theory

The possible existence of quantum copies of classical soliton solutions that cannot exist if the effective Planck constant γ of the theory tends to zero is discussed. In the framework of the standard expansion in √γ, such nonclassical solitons have energy of orderO(√γ) and thus occupy an intermediate position between purely classical configurations and second-quantized degrees of freedom. Analytic calculations are made for the profile functions, masses, and stability of such configurations in theϕ4 kink model.

On the physical content of the SU(2) Skyrme model

The authors study the solutions of the Euler-Lagrange equation for the Skyrme model in the SU(2) hedgehog representation. The authors derive sum rules and apply Derrick's theorem to characterize the solitonic solutions with baryon number B=1,2,3. They are seen to correspond to definite single values of a dimensionless control parameter, which can be expressed as the ratio of a dimensional 'size' parameter for the solution, classically arbitrary, and the product of the Skyrme parameter and the pion decay constant. They discuss the stability of the classical soliton solutions, and show that they have a minimum in the quantum energy after introducing collective coordinates. They find the masses at the minima for different baryon number and spin and obtain the values for the physical quantities of the B=1 baryon at its minimum quantum energy.

Legendre condition and the stabilization problem for classical soliton solutions in generalized Skyrme models

Legendre condition and the stabilization problem for classical soliton solutions in generalized Skyrme models

On SSQM and the U(N) non-linear Schrodinger equation

The method for obtaining the superpartner potential in supersymmetric quantum mechanics (SSQM) is discussed in connection with non-linear equations and reflectionless potentials. The correspondence between a new class of soliton solutions to the U(N) non-linear Schrodinger equation, obtained via application of SSQM, and previously known soliton solutions is also discussed.

Colliding gravitational waves with noncollinear polarization: A class of soliton solutions

We construct a general class of exact solutions of the vacuum Einstein field equations describing the interaction of colliding gravitational plane waves with noncollinear polarization by using the inverse scattering method.

On the collision of gravitational plane waves: a class of soliton solutions

The inverse scattering method is used to obtain a class of solutions of the vacuum Einstein equations describing the space-time following the collision of two gravitational plane waves. The general features of these solutions are analyzed in terms of the behavior of the Weyl scalars, and some degenerate cases are discussed.

Pion Fluctuations about the Skyrmion

We consider a chiral perturbation about the classical soliton solution (Skyrmion) in the Skyrme model. It is shown that there appear spurious zero-mode solutions associated with the rotation and translation degrees of freedom. This derives constraint conditions among the canonical conjugate momenta, when the collective coordinate method is applied. We then quantize the pion-Skyrmion system by the use of the method of Gervais-Jevicki-Sakita, the zero-mode solutions being properly removed. As an application the Green’s function for pion-nucleon scattering is derived, where the recoil of the nucleon and the transition to the Δ isobar are both taken into account.

Pion Fluctuations about the Skyrmion

Recently, there is an increasing interest in the Skyrme modeP) as an effective field theory of baryons underlied by the QeD. By means of a collective coordinate method) static properties of nucleons have been shown to be reproduced well in the model. Another interesting feature of the model is to predict an inner repulsion of about 1 Ge V in the overlapping two nucleons, in addition to a prediction of asymptotic one-pion exchange interaction at longer distances.) The model now enjoys several applications and extentions. Fluctuations about a classical soliton solution in the Skyrme model (Skyrmion) are considered to describe excitation modes of baryons and mesonic interactions. Several authors have attempted to calculate breathing modes of nucleons:)-8) An interesting point is that the excitation energies are suggested to be considerably small compared with experiment, while a simple quark model predicts too high excitations. Such investigations have been recently extended to general partial waves of the fluctuation modes, and interesting features of the Skyrme model have been derived.7).B) In Ref. 7), the authors noted that coupling terms involving spin and isospin are lacked in the method, and that a quantum number A = I + I is obtained as a conserved quantity, where I is the orbital angular momentum and I the isospin of the pion field. They afterwards proposed a recoupling scheme to connect the former result with that of physical channels.8) In this paper we want to show that a chiral perturbation method of Schnitzer) for the fluctuating field resolves this problem naturally. The Skyrme model assumes the lagrangian of the field as + l2 e-zTr{[(oU) U t , (ovU)U t ),}

Theoretical aspects of hybrid chiral bag models

In hybrid chiral bag models (HCBM's) the quarks are the source for the pion field outside the bag. If we want to solve this model with a classical external soliton solution and quantized fermions, it is necessary to evaluate the vacuum expectation values (VEV's) of those operators that contain fermion fields and appear in the boundary conditions. When the external solution is the so-called hedgehog solution, $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\pi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)={f}_{\ensuremath{\pi}}\ensuremath{\theta}(r)\stackrel{^}{r}$, the relevant VEV is $\frac{i}{16\ensuremath{\pi}}〈0|\ensuremath{\oint}\ensuremath{\int} {d}^{2}s[\stackrel{-}{\ensuremath{\psi}},(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\tau}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{r}){\ensuremath{\gamma}}_{5}\mathrm{exp}(i\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\tau}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{r}{\ensuremath{\gamma}}_{5}\ensuremath{\theta})\ensuremath{\psi}]|0〉=\frac{sin2\ensuremath{\theta}}{16\ensuremath{\pi}\ensuremath{\eta}}+\frac{{C}_{0}(\ensuremath{\theta})}{R},$ where $\ensuremath{\eta}$ is a cutoff parameter ($\ensuremath{\eta}\ensuremath{\rightarrow}0$). To obtain this result we have used a multiple-reflection expansion of the Green's function, while ${C}_{0}(\ensuremath{\theta})$ is evaluated numerically. We discuss the infinite contribution in the above VEV, and show that $\frac{4\ensuremath{\pi}{C}_{0}(\ensuremath{\theta})}{R}$ is precisely the derivative of the Casimir energy with respect to $\ensuremath{\theta}$. We also discuss some solutions of the HCBM for bag radii varying from $0 \mathrm{to} \ensuremath{\infty}$.

Lorentz invariance of the extended object

It has been known for a few years that the Heisenberg field of the extended object can be obtained in expanded form as a power series of quantum operators and creation and annihilation operators by solving the Yang–Feldman equation. Such an expression is called a dynamical map (mapping of the infield into the Heisenberg field). We will show that the Heisenberg field thus expressed (the dynamical map) is Lorentz covariant if it satisfies the equal time canonical commutation relation. In this paper we limit ourselves to the invariance of the first order term. Also, our Heisenberg field is (1+1) dimensional and is of the tree approximation. In the course of the calculation, we find that the quantum mechanical operator and the quantized field may be mixed by the Lorentz transformation if the space derivative of the classical field soliton solution is assumed to decrease not faster than 1/x2. It indicates that the Hilbert space may not be the direct product of two sub-Hilbert spaces, even though it is a product of two subspaces, one of which is a Fock space of the quantized field and the other is of the quantum mechanical operator.

Solitons and magnon bound states in ferromagnetic Heisenberg chains

We investigate the relation between magnon, magnon bound states, and the classical soliton solutions in the isotropic, the anisotropic-exchange, and the easy-axis ferromagnetic Heisenberg chains. The Dyson-Maleev boson representation is used to study the interaction between magnons, and bound states are investigated in terms of two-spin Green's functions. For easy-axis and anisotropic-exchange magnets, a mapping to the Bose gas with attractive $\ensuremath{\delta}$-function interactions is established, and it yields the eigenvalues of the magnon and the $m$th bound state in the weak-coupling and continuum limits. The classical limit of the two-spin Green's function reveals that, to leading order in temperature, the bound-state resonance and the associated effects on the two-magnon continuum survives. An important result is a ${T}^{2}$ dependence of the "binding" energy of the bound-state resonance. As a consequence, in a classical description the bound states enter only in order ${T}^{2}$. Finally, we quantize the soliton solutions according to the Bohr-Sommerfeld and de Broglie rules. This approach is found to be exact for the $s=\frac{1}{2}$ isotropic Heisenberg chain and for sufficiently small wave numbers for all $s$ values. In the anisotropic-exchange and easy-axis models, it agrees with the results obtained from the mapping on the Bose gas, for large quantum numbers and in the easy-axis case for $s\ensuremath{\gg}\frac{1}{2}$ in addition. On this basis, we conclude that the envelope solitons considered here lead to bound-state resonances and associated effects in a classical treatment. Moreover, their semiclassical quantization gives remarkably accurate energy levels.

Renormalization and Quantum Correction of Soliton Solutions

The renormalization procedure for soliton solutions in quantum systems in the framework of canonical formalism of quantum field theory is formulated. The steps of calculations are put in a systematic form and a detailed account of each step is clarified. It is shown that further renormalization is not needed in the presence of solitons; one can always obtain finite Yalues by the use of the renormalizecl quantities of the homogeneous case. § l. Introduction l\1athematical scientists have known for a long time that some non-linear clas­ sical field equations have a variety of solutions which behave as certain extended objects.]) It is clue to their particle-like behaviour that these solutions are called soliton solutions or solitary waves. In solid state physics the appearance of ex­ tended objects in various states is a common phenomenon. Due to their obvious quantum origin, these extended objects demand a quantum theory. The vortex-type solution in the Higgs mocleFl and the monopole solution in non-Abelian gauge the­ ory3l have attracted a great deal of attention among high energy particle physicists to extended objects created in field theoretical models. Though most of these analy­ ses were made at the classical level, they were aimed at a quantum theory for extended objects. There have been several attempts to study the quantum effects in soliton solutions. In Ref. 5), semiclassical quantization methods were developed using the \VKB approximation in the functional integral formalism. There the quantum. correction of the one-soliton energy was calculated in the (1 + 1) -dimensional J4/ model. The analysis began with the classical soliton solution and considered cor­ rections of order tz. A divergence appearing in the quantum correction of the energy is assumed to be cancelled by a mass counterterm. In Ref. 6) the method of collective coordinates was applied to th,e quantiza­ tion of the soliton sol uti on using the path integral formalism. The position of the soliton was treated as a collective coordinate and thus separated from the internal degrees of freedom. In Ref. 7) a soliton sector perturbation theory was developed. It was pointed out in Ref. 7a) that the ultraviolet divergence which arose in the one-loop computation of the soliton energy is removed, if one chooses the same

Local field theory for solitons

A method is developed for constructing the Lagrangian that describes the interaction of classical soliton solutions. We apply it to the Abelian Higgs model in 2 + 1 dimensions and the Georgi-Glashow model in 3 + 1 dimensions, and various features of the relevant Lagrangians are investigated.

Manifestly covariant approach for the one-soliton sector

The one-loop contributions to the meson propagator and to the meson-soliton scattering amplitude in the one-soliton sector are analyzed using ordinary Feynman rules. It is found that, at the level of the propagator, the most severe infrared divergences cancel out while others remain. The structure of the remaining divergences is studied. We conclude that, among other things, the coefficient function of the leading divergence only depends upon the classical soliton solution. An extended Lehmann-Symanzik-Zimmerman reduction technique, to work in the one-soliton sector, is proposed. Then, it is shown that, under certain mathematical assumptions, the above-mentioned S-matrix element turns out to be free of zero-mode singularities.

Semiclassical fields in soliton sectors

The quantum expansion of a local field around a classical soliton solution is reconsidered from a strictly field theoretical point of view. Relativistically covariant soliton representations of the canonical commutation relations are found for two-dimensional models in the lowest quantum approximation, and their properties are discussed.

Hadronic temperature and black solitons

Hawking has shown that event-horizon-containing classical soliton (black-hole) solutions of Einstein's equation radiate all species of particles with a thermal spectrum, the temperature being defined in terms of surface gracity. For a spinless soliton, the temperature is inversely proportional to the radius of its event-horizon. Assuming that there exists a fundamental strongly-interacting (massive) spin-2 field satisfying an Einstein-like equation with a strong coupling parameter, we propose to identify temperature in hadronic physics with strong surface gravity effects. The existence of black-body solitonic solutions for such an equation may then explain the thermal spectrum in E T observed in high-energy collisions.

Nonlinear Schrödinger equation: A testing ground for the quantization of nonlinear waves

Quantization of the nonlinear Schr\"odinger equation is carried out by the method due to Kerman and Klein. A viable procedure is inferred from the quantum interpretation of the classical (soliton) solution. The groundstate energy for a system with $n$ particles is calculated to an accuracy which includes the first quantum correction to the semiclassical result. It is demonstrated that the exact answer can be obtained systematically only at the next level of approximation. For the calculation of the first quantum correction, the quantum theory of the stability of periodic orbits in field theory is developed and discussed. Since one is dealing with a finite many-body problem, the field theory can be written so that no infinite terms are encountered, but the Hamiltonian can also be artificially rearranged so as to destory this feature. For learning purposes the calculations are carried out with the various alternatives, and our methods prove capable of providing a uniform final result.