In this chapter we discuss two laboratory systems where kinks are known to exist. The first system is trans polyacetylene which has a broken Z2 symmetry as in the λφ4 model. The second system is a Josephson junction transmission line, which is a laboratory realization of the sine-Gordon system. Helium-3 is another laboratory system that contains a wide variety of topological defects and the reader is referred to [174] for a discussion. In the third section of this chapter we describe Scott Russell's solitons in water. These solitons are not topological like the others discussed in this book but we include the discussion anyway since the reader's curiosity may have been aroused by the story in the Preface.

Polyacetylene

Polyacetylene consists of a linear chain of CH bonds. A sequence of x units is written as (CH)x. In the ground state of polyacetylene, the carbon atom forms three σ bonds, one of them is to the H in the CH unit, one to the unit on the left and one to the right. In addition, there is one more electron orbital that can cause bonding. This is called the π electron, and the π bond can form to the left or to the right. Then there are two possible sequences – first when the double (σ and π) bond is to the carbon on the right and the single to the left, the second when the double bond is to the left and the single to the right. These two possibilities are illustrated in Fig. 9.1 [149] in the trans configuration of polyacetylene.

In this chapter we study the effect of a kink on other bosonic or fermionic fields that may be present in the system. Under certain circumstances, it might be energetically favorable for a bosonic field, denoted by χ, to become non-trivial within the kink. Then we say that there is a “bosonic condensate” which is trapped on the kink. On a domain wall, the condensate has dynamics that are restricted to lie on the world-sheet of the wall.

The situation is similar for a fermionic field though there are subtleties. For a fermionic field, denoted by ψ, the Dirac equation is solved in the presence of a kink background made up of bosonic fields. This determines the various quantum modes that the fermionic excitations can occupy. In several cases, there can be “zero modes” of fermions in the background of a kink and this leads to several new considerations. (Fermionic zero modes were first discovered in [27, 84] in the context of strings.) In addition to the zero mode, there may be fermionic bound states. The high energy states that are not bound to the wall are called “scattering (or continuum) states.”

A difference between bosonic and fermionic condensates is that bosonic solutions can be treated classically but fermionic solutions can only be interpreted in quantum theory. For example, while there may be a bosonic solution with χ = 0, the solution ψ = 0 of the Dirac equation has no meaning because this solution is not normalizable. Solutions of the Dirac equation are only meant to supply us with the modes that fermionic particles or antiparticles can occupy, and as such are required to be normalizable.

The Z2 and sine-Gordon kinks discussed in the last chapter are not representative of kinks in models where non-Abelian symmetries are present. Kinks in such models have more degrees of freedom and this introduces degeneracies when imposing boundary conditions, leading to many kink solutions with different internal structures (but the same topology). Indeed, kink-like solutions may exist even when the topological charge is zero. The interactions of kinks in these more complicated models, their formation and evolution, plus their interactions with other particles are very distinct from the kinks of the last chapter.

We choose to focus on kinks in a model that is an example relevant to particle physics and cosmology. The model is the first of many Grand Unified Theories of particle physics that have been proposed [63]. The idea behind grand unification is that Nature really has only one gauge-coupling constant at high energies, and that the disparate values of the strong, weak, and electromagnetic coupling constants observed today are due to symmetry breaking and the renormalization-group running of coupling constants down to low energies. Since there is only one gauge-coupling constant in these models, there is a simple grand unified symmetry group G that is valid at high energies, for example, at the high temperatures present in the very early universe. At lower energies, G is spontaneously broken in stages, eventually leaving only the presently known quantum chromo dynamics (QCD) and electromagnetic symmetries SU(3)c × U(1)em of particle physics, with its two different coupling constants. It can be shown [63] that the minimal possibility for G is SU(5).

Solitons were first discovered by a Scottish engineer, J. Scott Russell, in 1834 while riding his horse by a water channel when a boat suddenly stopped. A hump of water rolled off the prow of the boat and moved rapidly down the channel for several miles, preserving its shape and speed. The observation was surprising because the hump did not rise and fall, or spread and die out, as ordinary water waves do.

In the 150 years or so since the discovery of Scott Russell, solitons have been discovered in numerous systems besides hydrodynamics. Probably the most important application of these is in the context of optics where they can propagate in optical fibers without distortion: they are being studied for high-data-rate (terabits) communication. Particle physicists have realized that solitons may also exist in their models of fundamental particles, and cosmologists have realized that such humps of energy may be propagating in the far reaches of outer space. There is even speculation that all the fundamental particles (electrons, quarks etc.) may be viewed as solitons owing to their quantum properties, leading to a “dual” description of fundamental matter.

In this book I describe the simplest kinds of solitons, called “kinks” in one spatial dimension and “domain walls” in three dimensions. These are also humps of energy as in Scott Russell's solitons. However, they also have a topological basis that is absent in hydrodynamical solitons. However, they also have a topological basis that is absent in hydrodynamical solitons. This leads to several differences e.g. water solitons cannot stand still and have to propagate with a certain velocity, while domain walls can propagate with any velocity. Another important point in this regard is that strict solitons, such as those encountered in hydrodynamics, preserve their identity after scattering.

A particle in a classical harmonic oscillator potential, mω2x2/2, has minimum energy when it sits at rest at the bottom of the potential. Then the particle's energy vanishes. The Heisenberg uncertainty principle however modifies this picture for the quantum harmonic oscillator. The particle cannot sit at rest (with definite momentum) at the bottom of the potential (a definite location). Indeed, the quantum zero point motion lifts the ground state energy to ω/2. Further, the excited states of the simple harmonic oscillator are discrete and occur at energies (n + 1/2)ω, n = 0, 1, 2, …

Just as the classical harmonic oscillator is modified by quantum effects, any classical solution to a field theory is also modified by quantum effects. Quantum effects give corrections to the classical kink energy owing to zero point quantum field fluctuations. These quantum corrections are small provided the coupling constant in the model is weak. To “quantize the kink” means to evaluate all the energy levels of the kink (first quantization) and to develop a framework for doing quantum field theory in a kink background. This involves identifying all excitations in the presence of the kink and their interactions. The field theory of the excitations in the non-trivial background of the kink is akin to second quantization. Finally, one would also like to describe the creation and annihilation of kinks themselves by suitable kink creation and annihilation operators. This would be the elusive third quantization.

Initially we calculate the leading order quantum corrections to the energy of the Z2 and sine-Gordon kinks. As these two examples illustrate, the precise value of the quantum correction depends on the exact model and kink under consideration.

We will work in natural units in which ħ = c = 1. In these units, all dimensionful quantities have dimensions of mass to some power. One way to convert from mass (g) to length (cm) and time (s), is to remember the values for the Planck mass, time, and length: mP = 1.2 × 1019 GeV, tP = 5.4 × 10−43 s, lP = 1.6 × 10−33 cm. Also, mPtP = 1 = mPlP in natural units. It is also useful to remember mP = 2.2 × 10−5 g and, when dealing with magnetic fields, the conversion: 1 Gauss = 1.95 × 10−20 GeV2. In addition, for cosmological estimates it is convenient to know that 1 pc = 3.1 × 1018 cm.

The metric signature is taken to be (+,−,−,−).

The quark model of hadrons, developed by Gell-Mann and Zweig, began to be taken seriously in the mid to late 1960s. The discovery of scaling in deep inelastic electron–nucleon reactions in the late 1960s seemed to imply that at very short distances, or very high momentum transfers, the nucleon constituents (valence quarks) behaved like weakly interacting point particles. However, the interactions between quarks had to be very strong at long distances, or small momentum transfers, to confine them in hadrons and thus explain the non-observation of isolated quarks. Politzer and Gross and Wilczek, who received the Nobel prize in 2004, showed that the only renormalizable field theory of quarks that had the property of an increasing force at long distance and a decreasing force at short distance was of the type discovered by Yang and Mills. Quarks must be spin-1/2 fermions, with fractional electric charge, and must come in three colors (a new quantum charge akin to electric charge) in order to explain the systematics of hadron spectroscopy. Interactions between quarks are mediated by gluons (the glue which holds them together). Gluons are massless spin-1 bosons, as are photons, but unlike photons they interact among themselves directly (via point interactions) because they also carry a color charge. Such theories are called nonabelian gauge theories. This theory of quarks and gluons, quantum chromodynamics (QCD), is the accepted theory of the strong interactions. Unfortunately, it has been very difficult to make quantitative predictions with QCD, owing to its complexity and peculiar properties. For a more thorough discussion of the history of QCD and its experimental support see Close.