  It is shown how the characteristic thermal effects that observers experience
in space-times possessing an event horizon can manifest already in a simple
quantum system with affine symmetry living on the real line. The derivation
presented is essentially group theoretic in nature: a thermal state emerges
naturally when comparing different representations of the group of affine
transformations of the real line. The freedom in the choice of different
notions of translation generators is the key to the Unruh effect "on a line" we
describe.
