Non-normalizable quantum states are usually discarded as mathematical artefacts in quantum mechanics. However, such states naturally occur in quantum gravity as solutions to physical constraints. This suggests reconsidering the interpretation of such states. Some of the existing approaches to this question seek to redefine the inner product, but this arguably leads to further challenges.

In this talk, I will propose an alternative interpretation of non-normalizable states using pilot-wave theory. First, I will argue that the basic conceptual structure of the theory contains a straightforward interpretation of these states. Second, to better understand such states, I will discuss non-normalizable states of the quantum harmonic oscillator from a pilot-wave perspective. I will show that, contrary to intuitions from orthodox quantum mechanics, the non-normalizable eigenstates and their superpositions are bound states in the sense that the pilot-wave velocity field vy→0 at large ±y. Third, I will introduce a new notion of equilibrium, named pilot-wave equilibrium, and use it to define physically-meaningful equilibrium densities for such states. I will show, via an H-theorem, that an arbitrary initial density with compact support relaxes to pilot-wave equilibrium at a coarse-grained level, under assumptions similar to those for relaxation to quantum equilibrium. I will conclude by discussing the implications for pilot-wave theory, quantum gravity and quantum foundations in general.

Based on:

I. Sen. "Physical interpretation of non-normalizable harmonic oscillator states and relaxation to pilot-wave equilibrium" arXiv:2208.08945 (2022)

Zoom link: https://pitp.zoom.us/j/93736627504?pwd=VGtxZE5rTFdnT1dqZlFRWTFvWlFQUT09