Species Quantum Mechanics

TL;DR

The paper develops a framework called Species Quantum Mechanics by promoting swampland tower data to quantum operators and identifying conjugate pairs between the species number $N_s$ and the tower mass scale $m_t$, yielding an uncertainty relation that underpins the CRV pattern. It connects these quantum structures to ${\cal N}=2$ Calabi–Yau moduli and their periods, showing that dualities arise as symplectic transformations of the canonical variables and that CY data encode the same conjugacy relations that give CRV. A species wave function is proposed, satisfying a Schrödinger equation on moduli space, with plane-wave solutions in the canonical field $\phi$ that relate to the logarithm of black-hole entropy and to OVV-type quantization in quantum gravity. The analysis distinguishes three asymptotic limits (Type IV, III, II) corresponding to decompactifications and emergent strings, establishes universal relations $m_t(t)\sim(\Im\mathcal F_0)^{-1/2}$ and $N_s(t)\sim\Im\mathcal F_1(t)$, and suggests instanton-corrected CRV generalizations and future extensions to interior moduli space and additional swampland constraints.

Abstract

In this note we introduce some concepts of Species Quantum Mechanics. Specifically, we consider quantum operators that correspond to the species number $N_s$ and the tower mass scale $m_t$ in the context of the swampland distance conjecture. We discuss the commutation relations, a possible wave function, and symplectic duality transformations on the conjugate variables. Furthermore, we argue that the Castellano-Ruiz-Valenzuela (CRV) pattern is a consequence of the canonical commutation rules of moduli space quantum mechanics. We also connect the canonical quantization to the periods of ${\cal N}=2$ Calabi-Yau compactifications to explore other aspects of the CRV pattern, including its possible connection to the Ooguri-Vafa-Verlinde black hole quantization procedure.