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<p>This article bootstraps finite dimensional state spaces to construct infinite dimensional state spaces. As is well-known, basic quantum mechanics may be defined over the three fields: the reals, the complexes, and the quaternions. Of course, standard quantum physics is over the complex numbers. In finite dimensions, the state spaces are well understood as the three families of projective spaces, over the respective three fields. We show that state space of any dimension, including non-separable, is the projectivization of a <i>basic</i> inner product space, not necessarily complete. By “basic” is meant that the inner product space contains an orthonormal set whose <i>algebraic</i> linear span is dense in the inner product space. Viewing the inner product space as a dense linear subspace of a Hilbert space, the orthonormal set can be equivalently described as a Hilbert basis for the Hilbert space. We axiomatize infinite dimensional state spaces in a way that originates with Hardy: <i>how state subspaces are defined and how measurements restrict to state subspaces</i>. It turns out that the structure of state space is determined even when restricting attention solely to discretely-valued measurements. Consequently, for this article, there is no need to consider continuous probability distributions for state transitions. The von Neumann–Lüders projection postulate is derived for projective observables.</p>

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State spaces: infinite dimensions from finite

  • 期刊论文

  • 发表日期：

    2026年2月17日

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  • Norman J. Goldstein

摘要

This article bootstraps finite dimensional state spaces to construct infinite dimensional state spaces. As is well-known, basic quantum mechanics may be defined over the three fields: the reals, the complexes, and the quaternions. Of course, standard quantum physics is over the complex numbers. In finite dimensions, the state spaces are well understood as the three families of projective spaces, over the respective three fields. We show that state space of any dimension, including non-separable, is the projectivization of a basic inner product space, not necessarily complete. By “basic” is meant that the inner product space contains an orthonormal set whose algebraic linear span is dense in the inner product space. Viewing the inner product space as a dense linear subspace of a Hilbert space, the orthonormal set can be equivalently described as a Hilbert basis for the Hilbert space. We axiomatize infinite dimensional state spaces in a way that originates with Hardy: how state subspaces are defined and how measurements restrict to state subspaces. It turns out that the structure of state space is determined even when restricting attention solely to discretely-valued measurements. Consequently, for this article, there is no need to consider continuous probability distributions for state transitions. The von Neumann–Lüders projection postulate is derived for projective observables.