Infinite Divisibility in Euclidean Quantum Mechanics

Abstract

An interesting and well studied class of quantum mechanical Hamiltonians consists of examples of the form

$$\mathcal{H} = - \tfrac{1}{2}{\partial ^2}/\partial {x^2} + V(x),$$

expressed in units where m = ћ = 1^l, where the real potential V(x) is chosen so that the spectrum of $\mathcal{H}$ is nonnegative and the real, normalizable, nowhere vanishing ground state φ _o (x) has zero energy eigenvalue; such systems are referred to as “simple systems” in this article. In this case, the ground state itself determines the Hamiltonian completely since

$$\mathcal{H} = - \tfrac{1}{2}[{\partial ^2}/\partial {x^2} - {\phi ''_o}(x)/{\phi _o}(x)],$$

, and therefore φ _o (x) — just as much as $\mathcal{H}$ itself — can be said to determine all the properties of the system.

In simple — but selected — quantum systems, the probability distribution determined by the ground state wave function is infinitely divisible. Like all simple quantum systems, the Euclidean temporal extension leads to a system that involves a stochastic variable and which can be characterized by a probability distribution on continuous paths. The restriction of the latter distribution to sharp time expectations recovers the infinitely divisible behavior of the ground state probability distribution, and the question is raised whether or not the temporally extended probability distribution retains the property of being infinitely divisible. A similar question extended to a quantum field theory relates to whether or not such systems would have nontrivial scattering behavior.

Dedicated to Gérard G. Emch on the occasion of his 70th birthday