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Review of Economic Analysis

John von Neumann between Physics and Economics: A Methodological Note

Luca Lambertini

Published Jul 1, 2013• 3 citations
Publisher

Abstract

A methodological discussion is proposed, aiming at illustrating an analogy between game theory in particular (and mathematical economics in general) and quantum mechanics. This analogy relies on the equivalence of the two fundamental operators employed in the two fields, namely, the expected value in economics and the density matrix in quantum physics. I conjecture that this coincidence can be traced back to the contributions of von Neumann in both disciplines.

DOI
10.15353/rea.v5i2.1407

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hep-thCross-listedarXiv:2610.40174•Oct 30, 2026
A methodological discussion is proposed, aiming at illustrating an analogy between game theory in particular (and mathematical economics in general) and quantum mechanics. This analogy relies on the equivalence of the two fundamental operators employed in the two fields, namely, the expected value in economics and the density matrix in quantum physics. I conjecture that this coincidence can be traced back to the contributions of von Neumann in both disciplines.
The relationships between game theory and quantum mechanics let us propose certain quantization relationships through which we could describe and understand not only quantum but also classical, evolutionary and the biological systems that were described before through the replicator dynamics. Quantum mechanics could be used to explain more correctly biological and economical processes and even it could encloses theories like games and evolutionary dynamics. This could make quantum mechanics a more general theory that we had thought. Although both systems analyzed are described through two apparently different theories (quantum mechanics and game theory) it is shown that both systems are analogous and thus exactly equivalents. So, we can take some concepts and definitions from quantum mechanics and physics for the best understanding of the behavior of economics and biology. Also, we could maybe understand nature like a game in where its players compete for a common welfare and the equilibrium of the system that they are members.
This paper attempts to explain both the basic concepts and the mathematical formalism of von Neumann's operator method in quantum statistics, in terms that should be understandable by anyone who has mastered first courses in quantum mechanics and statistical mechanics. A simple model is used throughout the paper to illustrate the various theorems and techniques. The discussion begins with the statistical interpretation of the Heisenberg uncertainty principle, the definition of mixed ensembles, and the meaning of measurement in quantum mechanics. The von Neumann projection operator is explained, and is used to set up the ensemble density matrix and the von Neumann entropy operator. In the second part of the paper, to be published later, the quantum-mechanical equation of change with time is applied to the von Neumann operators.
These notes provide an introduction to Haagerup noncommutative LpL^pLp spaces for quantum field theorists. The aim is to explain how the familiar calculus of density matrices, fractional powers, and Schatten norms extends to local observable algebras of type~III, where an intrinsic trace is unavailable. We develop the construction first for a matrix algebra, making the crossed product, its trace, and the dual action explicit, before passing to general von Neumann algebras. Particular attention is paid to the representation of normal states by Haagerup densities and to the distinction between the crossed-product trace and the canonical Haagerup functional used to compute expectation values. As a concrete application, we calculate the sandwiched Rényi divergence for a coherent excitation of a chiral current, verifying monotonicity under restriction and the Rényi quantum null energy condition at every real order α>1α>1α>1. The presentation assumes familiarity with quantum mechanics and elementary quantum field theory, while introducing the operator-algebraic concepts needed for the construction and calculation.