Banach space formalism of quantum mechanics

Abstract

This paper presents a generalization of quantum mechanics from conventional Hilbert space formalism to Banach space one. We construct quantum theory starting with any complex Banach space beyond a complex Hilbert space, through using a basic fact that a complex Banach space always admits a semi-inner product. Precisely, in a complex Banach space $\mathbb{X}$ with a given semi-inner product, a pure state is defined by Lumer \cite{Lumer1961} to be a bounded linear functional on the space of bounded operators determined by a normalized element of $\mathbb{X}$ under the semi-inner product, and then the state space $\mathcal{S} (\mathbb{X})$ of the system is the weakly closed convex set spanned by all pure states. Based on Lumer's notion of the state, we associate a quantum system with a complex Banach space $\mathbb{X}$ equipped with a fixed semi-inner product, and then define a physical event at a quantum state $\omega \in \mathcal{S}(\mathbb{X})$ to be a projection $P$ (bounded operator such that $P^2 =P$) in $\mathbb{X}$ satisfying the positivity condition $0 \le \omega (P) \le 1,$ and a physical quantity at a quantum state $\omega$ to be a spectral operator of scalar type with real spectrum so that the associated spectral projections are all physical events at $\omega.$ The Born formula for measurement of a physical quantity is the natural pairing of operators with linear functionals satisfying the probability conservation law. A time evolution of the system is governed by a one-parameter group of invertible spectral operators determined by a scalar type operator with the real spectrum, which satisfies the Schr\"{o}dinger equation. Our formulation is just a generalization of the Dirac-von Neumann formalism of quantum mechanics to the Banach space setting. We include some examples for illustration.