III Classical and Quantum Solitons

0Introduction

0 Introduction

Given a classical field theory, if we want to “quantize” it, then we find the

vacuum of the theory, and then do perturbation theory around this vacuum. If

there are multiple vacua, then what we did was that we arbitrarily picked a

vacuum, and then expanded around that vacuum.

However, these field theories with multiple vacua often contain soliton so-

lutions. These are localized, smooth solutions of the classical field equations,

and they “connect multiple vacuums”. To quantize these solitons solutions, we

fix such a soliton, and use it as the “background”. We then do perturbation

theory around these solutions, but this is rather tricky to do. Thus, in a lot of

the course, we will just look at the classical part of the theory.

Recall that when quantizing our field theories in perturbation theory, we

obtain particles in the quantum theory, despite the classical theory being com-

pletely about fields. It turns out solitons also behave like particles, and they are

a new type of particles. These are non-perturbative phenomena. If we want to

do the quantum field theory properly, we have to include these solitons in the

quantum field theory. In general this is hard, and so we are not going to develop

this a lot.

What does it mean to say that solitons are like particles? In relativistic field

theories, we find these solitons have a classical energy. We define the “mass”

of the soliton to be the energy in the “rest frame”. Since this is relativistic, we

can do a Lorentz boost, and we obtain a moving soliton. Then we obtain an

energy-momentum relation of the form

− P · P = M

This is a Lorentz-invariant property of the soliton. Together with the fact that

the soliton is localized, this is a justification for thinking of them as particles.

These particles differ from the particles of perturbative quantum fields, as

they have rather different properties. Interesting solitons have a topological

character different from the classical vacuum. Thus, at least naively, they cannot

be thought of perturbatively.

There are also non-relativistic solitons, but they usually don’t have interpre-

tations as particles. These appear, for example, as defects in solids. We will not

be interested in these much.

What kinds of theories have solitons? To obtain solitons, we definitely need

a non-linear field structure and/or non-linear equations. Thus, free field theories

with quadratic Lagrangians such as Maxwell theory do not have solitons. We

need interaction terms.

Note that in QFT, we dealt with interactions using the interaction picture.

We split the Hamiltonian into a “free field” part, which we solve exactly, and

the “interaction” part. However, to quantize solitons, we need to solve the full

interacting field equations exactly.

Having interactions is not enough for solitons to appear. To obtain solitons,

we also need some non-trivial vacuum topology. In other words, we need more

than one vacuum. This usually comes from symmetry breaking, and often gauge

symmetries are involved.

In this course, we will focus on three types of solitons.

–

In one (space) dimension, we have kinks. We will spend 4 lectures on this.

– In two dimensions, we have vortices. We will spend 6 lectures on this.

–

In three dimensions, there are monopoles and Skyrmions. We will only

study Skyrmions, and will spend 6 lectures on these.

These examples are all relativistic. Non-relativistic solitons include domain walls,

which occur in ferromagnets, and two-dimensional “baby” Skyrmions, which are

seen in exotic magnets, but we will not study these.

In general, solitons appear in all sorts of different actual, physical scenarios

such as in condensed matter physics, optical fibers, superconductors and exotic

magnets. “Cosmic strings” have also been studied. Since we are mathematicians,

we probably will not put much focus on these actual applications. However, we

can talk a bit more about Skyrmions.

Skyrmions are solitons in an effective field theory of interacting pions, which

are thought to be the most important hadrons because they are the lightest.

This happens in spite of the lack of a gauge symmetry. While pions have no

baryon number, the associated solitons have a topological charge identified with

baryon number. This baryon number is conserved for topological reasons.

Note that in QCD, baryon number is conserved because the quark number

is conserved. Experiments tried extremely hard to find proton decay, which

would be a process that involves baryon number change, but we cannot find

such examples. We have very high experimental certainty that baryon number is

conserved. And if baryon number is topological, then this is a very good reason

for the conservation of baryon number.

Skyrmions give a model of low-energy interactions of baryons. This leads

to an (approximate) theory of nucleons (proton and neutron) and larger nuclei,

which are bound states of any number of protons and neutrons.

For these ideas to work out well, we need to eventually do quantization.

For example, Skyrmions by themselves do not have spin. We need to quantize

the theory before spins come out. Also, Skyrmions cannot distinguish between

protons and neutrons. These differences only come up after we quantize.