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Realism and the Inequivalence of the Two Quantum Pictures

Charles Alexandre Bédard

Abstract

The standard claim that the Schrödinger and Heisenberg pictures of quantum mechanics are equivalent rests on the fact that they yield identical empirical predictions. This equivalence therefore assumes the instrumentalist worldview in which theories serve only as tools for prediction. Under scientific realism, by contrast, theories aim to describe reality. Whereas the Schrödinger picture posits a time-evolving wave function, the Heisenberg picture posits so-called descriptors, time-evolving generators of the algebra of observables. These two structures are non-isomorphic: descriptors surject onto but do not reduce to the Schrödinger state. Hence, under realism, the pictures are inequivalent. I argue that this inequivalence marks an opening toward a richer, separable ontology for quantum theory. On explanatory grounds, descriptors provide genuinely local accounts of superdense coding, teleportation, branching, and Bell inequality violations -- phenomena that the Schrödinger framework does not explain fully locally.

Realism and the Inequivalence of the Two Quantum Pictures

Abstract

The standard claim that the Schrödinger and Heisenberg pictures of quantum mechanics are equivalent rests on the fact that they yield identical empirical predictions. This equivalence therefore assumes the instrumentalist worldview in which theories serve only as tools for prediction. Under scientific realism, by contrast, theories aim to describe reality. Whereas the Schrödinger picture posits a time-evolving wave function, the Heisenberg picture posits so-called descriptors, time-evolving generators of the algebra of observables. These two structures are non-isomorphic: descriptors surject onto but do not reduce to the Schrödinger state. Hence, under realism, the pictures are inequivalent. I argue that this inequivalence marks an opening toward a richer, separable ontology for quantum theory. On explanatory grounds, descriptors provide genuinely local accounts of superdense coding, teleportation, branching, and Bell inequality violations -- phenomena that the Schrödinger framework does not explain fully locally.

Paper Structure

This paper contains 19 sections, 1 theorem, 21 equations.

Table of Contents

  1. Introduction
  2. The Instrumentalist Equivalence
  3. Realism
  4. Heisenberg-Picture Descriptors
  5. Generators
  6. Separability
  7. Evolution of Descriptors and Observables
  8. Recovering the Density Operator
  9. Evolution of Evolutions
  10. No Action at a Distance
  11. The Realist Inequivalence
  12. Non-isomorphic State Spaces
  13. Larger Schrödinger-Picture Descriptions?
  14. Additional Explicans
  15. Local Superdense Coding
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\mathfrak{U}$ be the whole system considered, with Heisenberg reference vector $\rvert\boldsymbol 0\rangle \in \mathcal{H}^{\mathfrak{U}}$. Assume that the whole Hilbert space admits, for a suitable set of indices $I$, the following decomposition where $\mathcal{H}^{\mathfrak S_i}$ has dimension $d_i \in \mathbb N_{>1} \cup {\infty}$. For all possible pairs of evolution $U$ and $U'$ of $\mat

Theorems & Definitions (2)

  • Theorem 1
  • proof