Srednicki — Quantum Field Theory · Problem 93.6
Canonical Quantization of Scalar Fields — Mark Srednicki
Physics — QFT · instanton-winding-number-power
Problem
Use eq.\,(93.16) to compute the winding number for , where is the map given in eq.\,(93.29). Hint: first show that can be written in the form , where is a three-vector that you should specify. Is your result in accord with the theorem of problem 93.4?Intuitive explanation
The winding number measures how many times a map winds around the target space as you traverse the domain—here, you're checking whether applying multiple times () scales the winding number linearly. The key insight is to recognize that has an exponential structure in the Pauli matrices, which makes the winding number calculation transparent: when you raise to the -th power, the exponent scales by , and the winding number should scale accordingly. This tests whether winding is additive under composition, a fundamental topological property.
Intuitive explanation
The winding number measures how many times a map winds around the target space as you traverse the domain—here, you're checking whether applying multiple times () scales the winding number linearly. The key insight is to recognize that has an exponential structure in the Pauli matrices, which makes the winding number calculation transparent: when you raise to the -th power, the exponent scales by , and the winding number should scale accordingly. This tests whether winding is additive under composition, a fundamental topological property.