Quantum mechanics from an epistemic state space

TL;DR

This paper derives the formal structure of quantum mechanics, including Hilbert spaces and Born's rule, from epistemic principles emphasizing knowledge limitations and the role of potential knowledge in physical states.

Contribution

It introduces an epistemic formalism where quantum states are based on collective potential knowledge, deriving Hilbert space structure and measurement probabilities from epistemic assumptions.

Findings

Derives Hilbert space formalism from epistemic principles.

Establishes Born's rule as the unique probability rule.

Connects commutativity of operators with simultaneous knowability.

Abstract

We derive the Hilbert space formalism of quantum mechanics from epistemic principles. A key assumption is that a physical theory that relies on entities or distinctions that are unknowable in principle gives rise to wrong predictions. An epistemic formalism is developed, where concepts like individual and collective knowledge are used, and knowledge may be actual or potential. The physical state $S$ corresponds to the collective potential knowledge. The state $S$ is a subset of a state space $\mathcal{S}=\{Z\}$, such that $S$ always contains several elements $Z$, which correspond to unattainable states of complete potential knowledge of the world. The evolution of $S$ cannot be determined in terms of the individual evolution of the elements $Z$, unlike the evolution of an ensemble in classical phase space. The evolution of $S$ is described in terms of sequential time $n\in \mathbf{\mathbb{N}}$, which is updated according to $n\rightarrow n+1$ each time potential knowledge changes. In certain experimental contexts $C$, there is initial knowledge at time $n$ that a given series of properties $P,P',\ldots$ will be observed within a given time frame, meaning that a series of values $p,p',\ldots$ of these properties will become known. At time $n$, it is just known that these values belong to predefined, finite sets $\{p\},\{p'\},\ldots$. In such a context $C$, it is possible to define a complex Hilbert space $\mathcal{H}_{C}$ on top of $\mathcal{S}$, in which the elements are contextual state vectors $\bar{S}_{C}$. Born's rule to calculate the probabilities to find the values $p,p',\ldots$ is derived as the only generally applicable such rule. Also, we can associate a self-adjoint operator $\bar{P}$ with eigenvalues $\{p\}$ to each property $P$ observed within $C$. These operators obey $[\bar{P},\bar{P}']=0$ if and only if the precise values of $P$ and $P'$ are simultaneoulsy knowable.