Quantum mechanics theory may work without imaginary numbers, new analysis suggests

Physicists from Heinrich Heine University Düsseldorf (HHU) have examined a fundamental property of quantum mechanics in collaboration with the German Aerospace Center (DLR). In an article published in the journal Physical Review Letters, they show that this theory does not necessarily need to be formulated with imaginary numbers—real numbers can, in fact, also be used.

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A. Lyoubi-Idrissi | I-Field Laboratory -

The Irreversibility Field (I-Field):A Classical Framework for Fundamental Irreversibility in Physics

 We present the I-field: a classical scalar field minimally coupled to matter whose equation of motion contains an explicit time-asymmetric dissipation term, derived from the Euler-Lagrange-Rayleigh (ELR) formalism [@rayleigh1877]. The field does not modify the gravitational sector: Einstein's field equations are unchanged, and the total stress-energy tensor of matter plus I-field is covariantly conserved. In the limit $\gamma \to 0$ the theory reduces exactly to standard classical field theory.  The dissipation term $\gamma u^\mu \partial_\mu \mathcal{I}$, where $u^\mu$ is the four-velocity of the cosmological rest frame and $\gamma > 0$ is a coupling constant, is odd under time reversal while every term derived from a Lagrangian is even. This explicit breaking of time-reversal symmetry at the level of the field equation --- rather than through boundary conditions or statistical postulates --- has three consequences derived as theorems within the framework:   1. The I-field carries a covariant entropy production density   $\sigma_{\mathcal{I}} = \gamma\dot{\mathcal{I}}^2 \geq 0$   pointwise, establishing the second law of thermodynamics as   a field-theoretic identity rather than a postulate.     2. The energy transferred from matter to the I-field is strictly   non-negative, providing a microscopic account of dissipation   without invoking a heat bath or environment.     3. The preferred time direction is globally well-defined,   identified with the cosmological rest frame in which the   cosmic microwave background is isotropic [@fixsen2009].     The theory is self-contained and makes no modifications to the gravitational sector. The framework provides a minimal, classical extension of standard field theory in which irreversibility is fundamental rather than