Some considerations on thermodynamics of quantum mechanics on a circle and Kałuża-Klein models

Introduction/purpose: Quantum mechanics on a circle has been investigated and applied to a Kałuża-Klein model with a compactified dimension. Methods: Methods of quantum mechanics and statistical mechanics were employed. Additionally, a Kałuża-Klein toy model with compactified dimension was considered. Results: The resulting partition function can be evaluated in a closed form, giving a special function. It presents a phase transition depending on the geometry. When used in a Kałuża-Klein model, it showed a phase transition regulated by the radius of the circle, and the transition disappears when the radius is infinite, that is in the flat space. Conclusion: Quantum mechanics on a circle exhibits many peculiar characteristics. Energy levels are discrete because of the geometry, in contrast to the configuration space of the real line. It presents a phase transition depending on the circle radius, also when embedded in a Kałuża-Klein model. This characteristic disappears when the radius becomes infinite, in the flat space.