Quantum Mechanics Based on Real Numbers: A Consistent Description

2025/03/21 by Pedro Barrios Hita, Anton Trushechkin, А. С. Трушечкин +8 · 2 voices · 9 citations
Physics and Astronomy · #Quantum Mechanics and Applications #Advanced Mathematical Theories and Applications #Quantum and Classical Electrodynamics

paper · pdf · doi:10.1103/4k13-sdjh

Abstract

Complex numbers play a crucial role in quantum mechanics. However, their necessity remains debated: whether they are fundamental or merely convenient. Recently, it was shown that any real-number quantum theory satisfying certain postulates can be falsified with multipartite experiments. In this Letter we show that a physically motivated postulate about composite quantum systems allows us to construct quantum mechanics based on real numbers that reproduces predictions for all multipartite quantum experiments. Thus, we argue that real-valued quantum mechanics cannot be falsified, and therefore the use of complex numbers is a matter of convenience.

Citations

• The Logic of Quantum Mechanics
• Quantum theory does not need complex numbers
• Gap between quantum theory based on real and complex numbers is arbitrarily large
• Quantum Theory From Five Reasonable Axioms

Cited by

• Schrödinger's real-valued equation revisited
• Re-examining the role of state texture in gate identification and fixed-point resource theories
• Quantum theory does not need complex numbers
• Quantum mechanics over real numbers fully reproduces standard quantum theory
• Formal equivalence between Maxwell equations and the de Broglie-Bohm theory for two-dimensional optical microcavities
• Phase-adjusted realification of a ℂ³ Kochen-Specker configuration into ℝ⁶
• Gleason's Theorem for a Qubit as Part of a Composite System
• Exploring Imaginary Coordinates: Disparity in the Shape of Quantum State Space in Even and Odd Dimensions
• Indefinite Causal Order Reverses the Real-Complex Hierarchy
• Expectation value dynamics within real Hilbert space quantum mechanics

Discussions

• Don't think fact that it's not loophole-free is the main issue here. See arxiv.org/abs/2503.17307 and arxiv.org/abs/2504.02808. Both point out the problem is the use of the tensor product in their rea [bsky, 1 points, 2 comments]
• Yes, there are an equivalent ways to re-formulate QM theory in the real space only. These authors came with such Hermitian operator mapping to reals, $$\operatorname{T}(\Pi)=\text{Re}(\Pi)\otimes I_F [stackexchange, 0 points]

Related

• Quantum theory does not need complex numbers
• Quantum Mechanics interpreted in Quantum Real Numbers
• In defense of real quantum theory
• Quantum mechanics over real numbers fully reproduces standard quantum theory
• On Complex numbers in Quantum Mechanics