Srednicki — Quantum Field Theory · Problem 92.3

Canonical Quantization of Scalar Fields — Mark Srednicki

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

Problem

Show that if

U_n(\phi)

and

U_k(\phi)

are maps from

S^1 \to S^1

with winding numbers

n

and

k

, then

U_n(\phi)U_k(\phi)

is a map with winding number

n + k

. Hint: consider smoothly deforming

U_n(\phi)

to equal one for

0 \leq \phi \leq \pi

. How should

U_k(\phi)

be deformed?