Is algebraic topology used in physics?

Light is a particular type of excitation of the electromagnetic field, one in which the electric and magnetic fields oscillate in phase with each other. Recall that, from a mathematical point of view, vector fields (including the electric or magnetic fields) are simply spatially varying collections of vectors, with one vector assigned to each point in space. These fields normally vary continuously and differentiably as functions of position. But in three dimensions, there can be isolated points or curves on which this well-behaved nature breaks down. These singular points or singular curves are subsets of space on which some variable associated with the field (such as phase or polarization) is undefined; since the field properties must be well-defined at each point, this forces the amplitude of the field to vanish at that point. In analogy to fluid mechanics, such singular points or curves are often called vortices or vortex lines, and they appear in many optical contexts, such as laser speckle, 'donut' or 'corkscrew' beams that carry orbital angular momentum, and light beams with a variety of other sorts of nontrivial spatially dependent structure.

It has long been known that topology is useful for studying and classifying systems with singularities. The set of possible field configurations can be viewed as an abstract space, and the singularities amount to holes in this space. The normal strategy for studying a space with holes is to look at collections of spheres of different dimensions and see whether or not they can be contracted to a point without crossing any holes. This technique is called homotopy theory, and is one of the major pillars of algebraic topology. Homotopy theory will be discussed in some detail in chapter 3.

We will see later that quite complex singularity structures can appear in optics, including multiple optical vortex lines that can form complicated knots and that can become linked or braided with each other.

Topology can enter optics in many other ways, beyond the study of vortices. For example:

• A different type of singularity may occur when wavefronts 'pile-up' on top of each other, leading to a sudden amplitude change when some curve is crossed. These ray-optical discontinuities are called caustics [1].
• Topology also enters optics via the study of non-dynamical phases (Berry phases, Pancharatnam phases, etc). These are phase factors acquired by a quantum mechanical wavefunction that are not due to the usual Hamiltonian time evolution factor
  . They are instead a result of changes in the parameters that define the Hamiltonian. In the optical case these parameters might include, for example, the index of refraction or the birefringence of a material. Non-dynamical phases of this type have made appearances in nearly every area of physics over the past 30 years and can be of either geometrical or topological origin. It will be seen in chapter 8 that such phases arise in optical systems in a variety of ways.
• When the fields at different boundary regions of a space are in states with different topological properties, the field solution can give rise to localized, particle-like wave pulses called solitons. These solitons occur in fields ranging from fluid dynamics to elementary particle physics, and optical solitons have become an important topic in fiber optic systems.
• One further area where topology has entered optics in just the past few years is in the construction of optical systems that can mimic the behavior of unusual solid-state systems known as topological insulators. Topological insulators are periodic systems that are insulators in their interior (their bulk), but which act as conductors on their boundaries. Such materials have associated topological invariants, and are governed by Hamiltonians that wrap in nontrivial ways around some compact space as a set of parameters is varied. Such systems have unusual properties that will be discussed in chapter 9, and several types of optical systems have been constructed or proposed in which photons simulate the unusual topological behavior of their solid-state analogs.

All of the areas discussed above contain some features in common. In particular, each of the systems has solutions that are characterized by various integer-valued quantities, generically called topological quantum numbers. The inability of these discrete numbers to vary continuously leads to greater stability of the system's properties than might otherwise be expected. The essence of such topological quantum numbers is that they measure global properties that belong to the system as a whole, rather than local properties like temperature or pressure that can be determined by measurements at individual points.

Topology has become ubiquitous in physics, making prominent appearances in subjects ranging from solid-state physics to superstring theory. In the remainder of this chapter, we give a brief historical overview of the rise of topological methods in physics in general.

Although earlier moments, such as Euler's use of graph theory to investigate the Konigsberg bridges problem (1736), could be singled out as the beginning of the study of topology, the subject only really became an important area of mathematics with the work of Poincaré around 1890. While investigating problems in celestial mechanics, he was led to the study of smooth mappings between surfaces, fixed point theorems, singularities of vector fields, and other topics that would now be considered topological. Poincaré's topological studies of solutions to differential equations as curves on manifolds were continued in the early 20th century by Birkhoff and others, with the results eventually being systematically applied to mechanical systems by Kolmogorov, Arnold, and Moser.

Aspects of topology, then known as analysis situs or geometria situs, had made appearances in physics before this, of course. For example, Gauss' law and Ampère's law in electrodynamics are both topological in nature: they involve line or surface integrals that remain invariant under continuous deformations of the underlying curve or surface; in modern terminology we would say that these integrals (the electric and magnetic fluxes) are topological invariants. In fact, integer linking numbers (chapter 5) made their first appearance in a study by Gauss of Ampère's law.

Similarly, in fluid mechanics the study of vortices has a long history. Then, starting in the 1860s, Peter Tait and William Thomson (Lord Kelvin) tried to model atoms as knotted vortex lines in the ether. The motivations included the fact that the multiplicity of different atoms could be explained by the variety of different ways a vortex line could be knotted, and the fact that the stability of atoms could be attributed to the inability to untie a knot without cutting it open; in other words, atomic stability follows from the topological stability of the knots. Different spectral lines could also be explained by different vibrational modes of the structure. The work of Tait and Kelvin led to knot theory becoming a major branch of topology, but after the idea of a space-filling ether was abandoned, knots disappeared from physics for almost a century, until they re-emerged in superstring theory and statistical mechanics, and then in other areas such as optics.

Despite occasional appearances of topological invariants in classical mechanics and electrodynamics, it was not until the advent of quantum mechanics that topology began to gain the central role in theoretical physics that it enjoys today. This key role is largely the result of three developments in quantum mechanics. The first two are Dirac's analysis of magnetic monopoles (1931) and the discovery of the AharonovBohm effect (1959). These are briefly discussed in the next two sections. Much later, the discovery of the quantum Hall effect in 1980 led to an explosive expansion of the role of topology in condensed matter physics. The quantum Hall effect will be discussed in chapter 9.

1.2.1.Dirac monopoles

Although never seen experimentally, the possibility of isolated magnetic charges or monopoles has long been studied theoretically, starting with the work of Paul Dirac in the 1930s [2]. In analogy to electric charges, a point-like magnetic monopole should produce a magnetic field (in Gaussian units)

where g is the magnetic charge and

is the magnetic scalar potential. Because of the identity

the magnetic analog of Gauss' law is

where

is the three-dimensional Dirac delta function.

Recall that when a particle of momentum

propagates with displacement

, the wavefunction picks up a phase factor,

The phase of a single wavefunction at a given point has no physical relevance, but the phase difference between points is meaningful, since it is measurable through interference effects. When there is a field present, the minimal coupling procedure of electromagnetism leads (for a particle of charge e) to an effective shifting of the momentum,

The phase difference between points A and B therefore gains a path-dependent contribution

where

is the length element tangent to the integration path.

Now consider a closed loop C. The electric and magnetic fluxes through the loops are related to the electric and magnetic charges e and g by Gauss' and Ampère's laws:

where S is any surface bounded by C. The phase change when the particle is transported around this loop is proportional to the magnetic flux enclosed by C,

:

where

is the area element, S is a smooth, oriented surface enclosed by C, and Stokes' theorem was used in the second equality. But the wavefunction at the initial point must be single valued, which implies that the phase shift must be an integer multiple of 2π:

This in turn implies that the magnetic flux is quantized:

In fact, Dirac showed that the product of the electric and magnetic charges must also be quantized:

for integer n. Thus the existence of magnetic monopoles would explain the quantization of electric charge.

The integer n is found to be a topological quantum number, counting the windings of mappings around the singular point where the monopole resides. In modern terms, the Dirac quantization condition states that the first Chern class of the electromagnetic field over a two-dimensional sphere S2 enclosing the monopole is an integer:

where

is the curvature of the field. This integer is called a Chern number. Chern numbers will be discussed in chapter 5.

The origin of this quantization is the singular point in the field at r = 0. Dirac in fact found a singular vortex line along the negative z-axis. This vortex line is unphysical and can be moved around by gauge transformations, but there is no single gauge choice defined at all points in space that will eliminate the vortex everywhere. However, Wu and Yang [3, 4] showed that the vortex line could be eliminated by defining two overlapping coordinate regions; the gauge fields on the overlap region are related by an appropriate gauge transformation. The work of Wu and Yang illustrated the fact that the proper mathematical setting for the description of electromagnetism (and, more generally, of all gauge theories) is the theory of fiber bundles (chapter 4).

Studies of topological structures in field theories took off in the 1970s, beginning with the discovery of quantized magnetic flux lines in superconductors [5], which (taken together with Dirac's work), led to the realization that magnetic monopoles should exist in non-Abelian field theories [6, 7].

It has now been found that monopoles occur in many quantum field theory models, and are ubiquitous in grand unified field theories in particle physics. Magnetic monopole-like structures can also occur in other types of physical systems (see chapter 8).

1.2.2.AharonovBohm effect

A second demonstration of the importance of topology in physics came with the AharonovBohm effect [8]. Consider a charged particle moving in the vicinity of a current-carrying solenoid. There is a magnetic field

inside the solenoid, but the field vanishes outside. The vector potential,

, however is nonzero everywhere, inside and out. Prior to the rise of the AharonovBohm effect, it was believed that the field

was the physically important variable and that

was simply a mathematical convenience of no physical significance. However, Aharonov and Bohm showed that when the particle circles the solenoid in a closed loop

, staying entirely in the

region, there is nevertheless a phase shift given by

and that this shift is an integer multiple of 2π. The existence of the AharonovBohm effect was verified in an experiment by Chambers in 1960 [9].

The solenoid contains a singularity in the vector potential. One can therefore view the solenoid as a hole in the space of allowed field configurations. The quantization arises from the topological fact that curves in

-space that enclose the solenoid are non-contractible. The integer n here counts the number of times the loop encloses the singularity: it is a winding number. This winding number characterizes the distinct homotopy classes (see chapters 3 and 5) of the field. The AharonovBohm phase accumulated as the electron circles the solenoid is an example of the geometric Berry phase to be discussed in chapter 8.

1.2.3.Topology in optics