Sean Carroll on Hilbert Space, Entropy and Infinity

Physicist Sean Carroll explains that mathematicians use “space” for abstract collections, not only the three-dimensional space of everyday experience.

He describes Hilbert space as the mathematical setting for possible quantum states of a system. Its dimensions are not ordinary directions in physical space.

Carroll uses a classical example: position and velocity each require three components, so describing both gives a six-dimensional phase-space representation.

He explains dimensions as a way to represent information needed to specify a state, while more complicated systems may require additional variables.

Asked about entropy, Carroll offers an information-based intuition: one may know a container’s temperature and pressure without knowing every molecule’s detailed state.

He says mathematical models can have finite- or infinite-dimensional Hilbert spaces and that the physical answer depends on the system and theory.

Fridman asks how to think about infinity. Carroll says mathematics can define and manipulate infinite structures precisely even when intuition struggles.

They discuss whether infinity is a feature of physical reality or a convenient mathematical description, acknowledging uncertainty.

Carroll notes that infinite behavior can arise naturally in simple mathematical examples, such as a program that never halts.