Mathematical structures of quantum theories

Abstract: We argue that Hilbert spaces are not suitable to represent quantum states mathematically, in the sense that they require properties that are untenable by physical entities. We first demonstrate that the requirements posited by complex inner product spaces are physically justified. We then show that completeness in the infinite-dimensional case requires the inclusion of states with infinite expectations, coordinate transformations that take finite expectations to infinite ones and vice versa, and time evolutions that transform finite expectations to infinite ones in finite time. This means we should be wary of using Hilbert spaces to represent quantum states as they turn a potential infinity into an actual infinity. The main point of the paper, then, is to raise awareness that results that rely on the completeness of Hilbert spaces may not be physically significant. While we do not claim to know what a physically more appropriate closure should be, we note that Schwartz spaces, among other things, guarantee that the expectation of all polynomials of position and momentum are finite, their elements are uniquely identified by these expectations, and they are closed under Fourier transform.

Abstract: The observables associated with a quantum system S form a non-commutative algebra  $\mathcal{A}_S$. It is assumed that a density matrix ρ can be determined from the expectation values of observables. But $\mathcal{A}_S$ admits inner automorphisms $a\mapsto uau^{-1},\; a,u\in \mathcal{A}_S$, u*u=uu*=𝟙, so that its individual elements can be identified only up to unitary transformations. So since Tr ρ(uau*) = Tr(u*ρu)a, only the spectrum of ρ, or its characteristic polynomial, can be determined in quantum mechanics. In local quantum field theory, ρ cannot be determined at all, as we shall explain. However, abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables in abelian algebras $\mathcal{A}_M\subset \mathcal{A}_S$ (M for measurement, S for system). We study the uncertainties in extending  $\rho\mid_{\mathcal{A}_M}$ to $\rho\mid_{\mathcal{A}_S}$ (the determination of which means measurement of $\mathcal{A}_S$) and devise a protocol to determine $\rho\mid_{\mathcal{A}_S}\equiv \rho$ by determining  $\rho\mid_{\mathcal{A}_M}$. The problem we formulate and study is a generalization of the Kadison–Singer theorem. We give an example where the system S is a particle on a circle and the experiment measures the abelian algebra of a magnetic field B coupled to S. The measurement of B gives information about the state ρ of the system S due to operator mixing. Associated uncertainty principles for von Neumann entropy are discussed in the appendix, adapting the earlier work by Białynicki-Birula and Mycielski (1975 Commun. Math. Phys. 44 129) to the present case.