Hidden Variable Theory

 

Abstract

This notebook page presents a conceptual formulation of Hidden Variable Theory in quantum mechanics. Hidden variables ($hv$) are defined as the components of a hypothetical complete physical state that are absent from the standard quantum state description.

The notes depict the Universe ($U$) as containing both the quantum description and these hidden variables, formally expressed as

$$Q\,U\,hv \subseteq \text{Universe},$$

where $Q\,U\,hv$ denotes the quantum-plus-hidden-variable description. A complementary relation is given:

$$(Q\,U\,hv) \cup Q_s \subseteq U,$$

with $Q_s$ identified as quantum space.

The left-hand diagram and accompanying statements emphasize that while quantum hidden variables belong to the Universe, the Universe itself is not exhausted by any finite quantum-hidden-variable description ($Universe \not\subseteq (Q\,Uhr)$). Consequently the Universe is represented as an infinite sum

$$U = \sum_{i=0}^{N} (Q\,Uhv_i)\dots$$

in the limiting regime $n\to N$, $\pi\to\infty$.

Taken together, the page articulates the core claim of hidden-variable approaches: the quantum state is incomplete, and a complete ontological description of physical reality requires additional variables that lie outside the quantum formalism yet remain part of the larger Universe.