Species quantum mechanics

In this paper we take the initial steps toward a formulation of species quantum mechanics. Specifically, we consider quantum operators that correspond to the species number <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>N</a:mi> <a:mi>s</a:mi> </a:msub> </a:math> and the tower mass scale <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>m</c:mi> <c:mi>t</c:mi> </c:msub> </c:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi mathvariant="script">N</e:mi> <e:mo>=</e:mo> <e:mn>2</e:mn> </e:math> Calabi-Yau compactifications to explore other aspects of the CRV pattern, including its possible connection to the Ooguri-Vafa-Verlinde black hole quantization procedure.