Separation of Topological Singularities

left spin

-

3

2

-1

-

1

2

0

1

2

1

3

2

right spin

-

3

2

-1

-

1

2

0

1

2

1

3

2

distance

phase shift

arrows

yes

no

automatic

x range

y range

gluing direction

When considering nearly continuous fields of nonzero vectors in a 2D plane, there are possibly some nontrivial topological situations with enforced discontinuities at some discrete set of points. If we make a loop around one of these so-called critical points, the phase makes some integer number of rotations. Such a number has good conservation properties—the number of rotations while going through some loop is the sum of such numbers for critical points inside it. In complex analysis this is called the argument principle, in differential equations theory this number is called the Conley (or Morse) index, and in physics it is very similar to the so-called spin—while rotating around the spin axis of a particle, the quantum phase rotates

s

times, where

s

is the spin.

This Demonstration helps you to imagine the behavior of the phase while separating two critical points, as in the case of the spontaneous creation of a particle-antiparticle pair. It uses a very simple complex function, showing only a qualitative picture. To handle spins having a multiplicity of 1/2 as in physics, we have to identify vectors with their opposites—forget about the arrows— by using a field of directions

(S(1)/{1,-1})

.