  We present a quantum theory of distances along a curve, based on a linear
line element that is equal to the operator square root of the quadratic metric
of Riemannian geometry. Since the linear line element is an operator, we treat
it according to the rules of quantum mechanics and interpret its eigenvalues as
physically observable distances; the distance eigenvalues are naturally
quantized. There are both positive and negative eigenvalues, which requires
interpretation. Multi-element curves are defined as direct sums of line
elements, and behave much like systems of spin half electrons in a magnetic
field. For a curve of many elements an entropy and energy and temperature are
quite naturally defined, leading via standard statistical thermodynamics to a
relation between the most probable curve length and temperature. That relation
may be viewed as a universal heat-shrinking property of curves. At this stage
of the theory we do not include bodies or particles in the mix, do not suggest
field equations for the quantum geometry, and questions of interpretation
remain. The theory might conceivably be testable using observations of the
early Universe, when the temperature of space was presumably quite high. In
particular cosmogenesis may be thought of as time stopping at an infinite
temperature as we go backwards in time to the beginning.
