Dirac

Mathematical Details



Topology of SO(3)

A topological space is described as being simply connected if every loop it contains can be shrunk to a single point; a loop with this property is called contractible[1]. For example, ordinary Euclidean space is simply connected, and so is the surface of a sphere, but the surface of a doughnut isn’t. The group of rotations in three dimensions, SO(3), is not simply connected, because the set of rotations around any fixed direction by angles ranging from –π to π forms a loop that is not contractible. This becomes clear if you picture SO(3) as a solid sphere of radius π, with a rotation by the angle θ around an axis pointing in the direction of a unit vector u being represented by the vector θu. Antipodal points on the surface of the sphere correspond to identical rotations, so any continuous path that crosses the surface must re-appear on the opposite side of the sphere. Having formed a loop from any diameter of the sphere, the two endpoints of the diameter will necessarily remain on opposite sides. Gathering up portions of the loop and poking them through the surface won’t make the loop contractible, because doing this to any segment that lies between two points on the surface breaks the original segment in three, and only pairs of such segments could be contracted down to points.

A belt in a fixed configuration defines a path through SO(3): starting from one end, the cumulative rotation that determines the orientation of each portion of the belt will vary continuously along its length. So the belt trick sheds some light on the topology of SO(3): it shows that a loop that crosses SO(3) twice (the path corresponding to a doubly-twisted belt with identically-oriented ends) can be contracted down to a single point (the path corresponding to a perfectly flat belt).

Two paths between points A and B in a topological space are called homotopic if they can be continuously deformed into each other, or equivalently, if the loop you get by travelling along the first path from A to B and then along the second one from B back to A is contractible. A homotopy class consists of all the paths between two chosen points that are homotopic to each other. It turns out that for every rotation R in SO(3) there are two homotopy classes of paths from the identity of the group, I, to R. This means that, for a given buckle orientation, there are only two distinct classes of configuration for a belt which cannot be deformed into each other while keeping the ends of the belt fixed. Belts with any even number of twists (including zero) lie in one class; belts with any odd number of twists lie in the other.

The set of homotopy classes of paths from I to R, for all rotations R, can be made into a group, known as the universal covering group of the original group, SO(3). To multiply two homotopy classes of paths, c and d, just pick a representative path in each class, p(t) and q(t), which give rotations in SO(3) for every value of the parameter t between 0 and 1. The product of c and d is then the homotopy class of the path p(t)q(t), multiplied point by point as elements of SO(3).

Because there are two classes of paths for every rotation, the universal covering group of SO(3) has twice as many elements as SO(3) itself, and is referred to as its double cover. Topologically, the double cover is equivalent to two copies of SO(3) glued together along the surface of the sphere described earlier. Just as two disks joined only along their border are topologically equivalent to the surface of a three-dimensional sphere, two solid spheres glued together this way are equivalent to the hypersurface of a four-dimensional sphere, and this space is simply connected.

Spinors and the belt trick

Algebraically, the double cover of SO(3) is equivalent to the group SU(2) of 2×2 complex unitary matrices with determinant 1. Particles with spins of (1/2), such as electrons, are described at each point by two-component complex vectors, known as spinors, which transform under rotation by being multiplied by an element of SU(2). So the belt trick ultimately reveals something about the behaviour of electrons: when it comes to SU(2), a single, complete rotation is not equivalent to no rotation at all, and an electron rotated through an angle of 2π ends up with its spinor wave function multiplied by –1. It takes two complete rotations to return the wave function to its original value, just as it takes two complete twists to produce a belt that can be made flat.

This connection can be made quite precise, as follows. Associated with any normalised spinor (z,w) are two real spatial vectors, and their cross product:

n = (zw*+z*w, i(zw*–z*w), zz*–ww*)  
f = ((w*2+w2z2z*2)/2, i(w*2w2z2+z*2)/2, zw+z*w*)  
n × f = (i(w*2w2+z2z*2)/2, –(w*2+w2+z2+z*2)/2, i(z*w*–zw))  

These are unit vectors, and they are orthogonal to each other. The vector n can be thought of as the direction of the electron’s spin axis, in the sense that (z,w) is an eigenvector, with eigenvalue 1, of the angular momentum measured along that axis:

σn = nxσx + nyσy + nzσz  
σn(z,w) = (z,w)  

If we rewrite z and w in polar form, and factor out the phase of z, the spinor can be written:

(z,w) = eiψ (a, b eiθ)  

The two vectors, and their cross product n × f, become:

n = (2ab cos(θ), 2ab sin(θ), a2b2)  
f = (b2cos(2(θ+ψ)) – a2cos(2ψ),  
    b2sin(2(θ+ψ)) + a2sin(2ψ),  
    2ab cos(θ+2ψ))  
n × f = (b2sin(2(θ+ψ)) – a2sin(2ψ),  
    b2cos(2(θ+ψ)) – a2cos(2ψ),  
    2ab sin(θ+2ψ))  

If we multiply the spinor by a phase eiφ, transforming ψ to ψ+φ, this has no effect on n, but changes f to f ', where:

f ' = cos(2φ) f – sin(2φ) n × f  

In other words, f rotates around n, by an angle of 2φ. So f almost keeps track of the spinor’s phase. However, some information can’t be recovered from n and f alone: multiplication by eiπ=–1 rotates f by 2π, so these vectors can’t discriminate between (z,w) and (–z,–w).

Now, imagine the electron with its spin axis n perpendicular to a belt buckle, and the vector f lying in the plane of the buckle; the precise orientation of f amounts to an arbitrary choice of overall phase. Rotating the buckle then alters the direction of n and f in a way that’s consistent with the SU(2) transformation of the electron’s spinor, and the additional information about the spinor’s change of sign after a 2π rotation is captured by the non-removable twist in the belt. After a 4π rotation, as the belt trick shows, the twist becomes removable, in accord with the fact that the spinor has been restored to its original value.

References: Spinors are discussed in Chapter 41 of Gravitation by C.W. Misner, K.S. Thorne and J.A. Wheeler (W.H. Freeman, 1973) and Chapter 13 of General Relativity by Robert M. Wald (University of Chicago Press, 1984). More about the belt trick can be found in Knots and Physics by Louis H. Kauffman (World Scientific, Singapore, 1993).

[1] The term contractible loop should not be confused with the definition of a contractible space. A loop in the space X is contractible to a point if there is a continuous map f:S1 × [0,1] → X, where f(S1,0) is the loop, while f(S1,1)=p is a single point in X. As we vary t from 0 to 1, f(S1,t) gives a series of closed curves in X which converge on the point p. This definition does not imply that the loop f(S1,0) is a contractible space. My thanks to Terry Padden for pointing out the potential for confusion here.



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