How the Geometric Calculus Resolves the Ordering Ambiguity of Quantum Theory in Curved Space

TL;DR

This paper demonstrates that geometric calculus based on Clifford Algebra naturally resolves the ordering ambiguity in defining the Hamiltonian for a quantum particle in curved space, ensuring Hermiticity and classical correspondence.

Contribution

It introduces a geometric calculus approach that unambiguously defines the Hamiltonian and momentum operators in curved space quantum mechanics, eliminating ordering ambiguities.

Findings

Hamiltonian is the D'Alembert operator in curved space

Momentum operator is Hermitian and self-adjoint

Expectation values follow classical geodesics

Abstract

The long standing problem of the ordering ambiguity in the definition of the Hamilton operator for a point particle in curved space is naturally resolved by using the powerful geometric calculus based on Clifford Algebra. The momentum operator is defined to be the vector derivative (the gradient) multiplied by $-i$; it can be expanded in terms of basis vectors $\gamma_\mu$ as $p = -i \gamma^\mu \p_\mu$. The product of two such operators is unambiguous, and such is the Hamiltonian which is just the D'Alambert operator in curved space; the curvature scalar term is not present in the Hamiltonian if we confine our consideration to scalar wave functions only. It is also shown that $p$ is Hermitian and self-adjoint operator: the presence of the basis vectors $\gamma^\mu$ compensates the presence of $\sqrt{|g|}$ in the matrix elements and in the scalar product. The expectation value of such operator follows the classical geodetic line.