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This work explores the role of the imaginary unit in quantum mechanics as a representation of action-phase geometry.

June 19, 2026Open Access

What i Really Represents: From Spacetime Action to Quantum Phase

Key Points

This work explores the role of the imaginary unit in quantum mechanics as a representation of action-phase geometry.
Developed a conceptual framework linking action and phase through mathematical constructs like eiS/ℏ.
Analyzed implications of phase gradients, action periodicity, and multi-photon interference within quantum context.
Proposed that observable probabilities arise from interference among phase-weighted histories.
Argues that the imaginary unit i is fundamental to understanding action-phase geometry in quantum systems.
Demonstrates how constants ℏ and h relate to action and phase interpretations.
Indicates that observed probabilities emerge after interference among histories, refining our understanding of quantum amplitudes.

Abstract

What i Really RepresentsFrom Spacetime Action to Quantum Phase: The Imaginary Unit as aMarker of Action-Phase GeometryPeter Davies, v1.0230th April, 2026AbstractQuantum mechanics differs from classical probability theory by assigning complex amplitudes rather than real probabilities to physical alternatives. The appearance of the imaginary unit i in the Schr¨odinger equation, unitary time evolution, path integrals, gauge transformations, and geometric formulations of quantum mechanics suggests that i is not merely a calculational artifact.This paper argues that i is best understood as the algebraic marker of action-phase geometry: the phase-rotation structure through which action, measured in units of ℏ, becomes a probability-preserving quantum amplitude. In this interpretation, spacetime geometry determines the action accumulated along possible histories, while quantum mechanics maps this action into phase through eiS/ℏ. The constants ℏ and h acquire complementary interpretations: ℏ is action per radian of phase, while h = 2πℏ is action per full phase cycle. Observable probabilities arise only after interference among phase-weighted histories, with the final quadratic product converting the surviving amplitude structure into measurable probability. The paper develops this interpretation through phase gradients, action periodicity, proper-time phase, gauge connections, gravitational phase modulation, light–matter phase matching, and multi-photon interference. The proposed contribution is not a replacement of quantum mechanics, quantum field theory, or general relativity, but a conceptual synthesis: the repeated appearance of i, ℏ, h, action, proper time, and interference may be understood as different aspects of a single action-phase structure linking spacetime and quantum mechanics. The proposal is interpretive rather than a Bell-local hidden-variable completion of quantum mechanics. It does not assign pre-existing local outcome values to measurement events, and it is intended to remaincompatible with the nonclassical correlation structure required by Bell-type results. - Preprint version v1.02. This manuscript presents a conceptual framework and has not yet undergone peer review. Later versions may incorporate corrections, expanded references, and feedback on framing.

What i Really Represents: From Spacetime Action to Quantum Phase

Key Points

Abstract

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