Abstract: We show that Hilbert spaces should not be considered the ``correct'' spaces to represent quantum states mathematically. We first prove 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 makes Hilbert spaces physically unsound as they model a potential infinity as an actual infinity. We suspect that at least some problems in quantum theory related to infinities may be ultimately caused by the wrong space being used. We strongly believe a better solution can be found, and we look at Schwartz spaces for inspiration, as, among other things, they guarantee that the expectation of all polynomials of position and momentum are finite, their elements are uniquely identified by these expectations, and they are the only space 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.