Srednicki — Quantum Field Theory · Problem 93.4

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · winding-number-product-maps-s3

Problem

Show that if

U_n(\mathbf{x})

and

U_k(\mathbf{x})

are maps from

S^3 \to S^3

with winding numbers

n

and

k

, then

U_n(\mathbf{x})U_k(\mathbf{x})

is a map with winding number

n + k

. Hint: consider smoothly deforming

U_n(\mathbf{x})

to equal one for

x_3 < 0

. How should

U_k(\mathbf{x})

be deformed?