From Heisenberg and Schr\"odinger to the P vs. NP Problem

TL;DR

This paper explores the philosophical and historical foundations of quantum mechanics and relates them to the P vs. NP problem, highlighting the contrast between procedural construction and recognitional verification in scientific reasoning.

Contribution

It offers an epistemological reinterpretation linking quantum mechanics' dual formalisms to the P versus NP problem, emphasizing the structural tension in knowledge production.

Findings

Quantum formalisms embody procedural and recognitional modes of knowing.

Historical analysis of quantum mechanics' development.

The contrast reflects a fundamental aspect of scientific reasoning.

Abstract

This essay offers an epistemological reinterpretation of the foundational divide between matrix mechanics and wave mechanics. Though formally equivalent, the two theories embody distinct modes of knowing: procedural construction and recognitional verification. These epistemic architectures anticipate, in philosophical form, the logical asymmetry expressed by the P versus NP problem in computational complexity. Here, the contrast between efficient generation and efficient recognition is treated not as a mathematical taxonomy but as a framework for understanding how knowledge is produced and validated across physics, computation, and cognition. The essay reconstructs the mathematical history of quantum mechanics through the original derivations of Werner Heisenberg, Max Born, Pascual Jordan, Paul Dirac, Erwin Schr\"odinger, Paul Ehrenfest, and Wolfgang Pauli, culminating in John von Neumann's unification of both approaches within the formalism of Hilbert space. By juxtaposing Heisenberg's algorithmic formalism with Schr\"odinger's representational one, it argues that their divergence reveals a structural feature of scientific reasoning itself-the enduring tension between what can be procedurally constructed and what can only be recognized.