Abstract. We develop a formalism for the Schrödinger equation in an eight dimensional complex Minkowski space and discuss its relation to the Dirac equation, properties of nonlocality, remote connectedness, Young’s double slit experiment, Bell’s Theorem, the EPR paradox and anticipatory parameters of spacetime; and also identify an imaginary temporal component as a small nonlinear term and find soliton or solitary wave solutions. These coherent solutions can carry information over long distances, are consistent with Lorentz invariance and appear to provide a fundamental methodology for describing the issue of quantum measurement and a new context for the basis of quantum theory. In the Copenhagen view models of reality are not desirable. However our new approach may enable the redefinition of concepts of reality from a new nonlocal anticipatory quantum theory. Certainly the most desirable consequence of scientific discovery is the ability to redefine our concepts of reality.
Abstract In considering models of reality we must consider the fundamental nature of quantum mechanics, which has had great success in describing the spectrum of light from distant stars and galaxies, formulating the properties of the strong, electro-weak or electromagnetic and weak decay forces, and explaining atomic order formulated as the atomic periodic table and atomic spectra. A vast array of technologies has resulted from quantum theory, from semiconductors to NMR machines, nuclear reactors and possibly encryption secured, fast quantum computers. Quantum theory, however, has yet to fully engage with the gravitational force. With the advent of quantum theory, several major issues stand out, leaving us with major philosophical questions on the nature of physics and reality, and our ability to comprehend them. The issues involve the interpretation of quantum measurement, the Heisenberg uncertainty principle on the completeness of quantum theory and nonlocality. The frontiers of science continue to open new concepts based on a multidimensional reality which we use to address these issues, including solving the Schrödinger and Dirac equations in complex Minkowski eight space. We also formulate Bell’s theorem and the Young’s double slit experiment in Rauscher’s complex eight-space. We formulate an interpretation of the quantum that yields locality in a hyper-dimensional complex space and that maintains the completeness of quantum theory and nonlocality in usual four space, with local connectivity and the Lorentz invariant conditions, where subluminal and luminal signaling are maintained. Are we really forced to choose between two alternate views of reality? One view that external reality exists independent of the observer, and the other that the observer somehow is a participator or an integral part of all of nature. Must we choose between these or is there another way out? We find our way out and we satisfy both Bohr (Copenhagen) and Einstein (determinism) in bringing about a practical outcome of the debate. By developing and applying complex eight space, we can accommodate locality or local realism in terms of contiguous event connections in complex eight space. In four space, nonlocality dominates and also can be considered locality in eight space. The determination of the state of a system naturally collapses to its final state where quantum mechanics is functioning completely in four and in eight dimensional spaces.
We outline the relativistic formalism which gives a more comprehensive explanation of the complexification scheme. Such issues as considering the Higgs Boson as a soliton depends on Lorentz invariance and relativistic causality constraints. We relate the complexification of Maxwell's equations to models of nonlocal micro and macro phenomena. In this chapter we relate the electromagnetic fields, F and A , the potentials to the gravitational field, G . We examine, for example, the manner in which advanced potentials may explain the remote connectedness which is indicated by the Clauser test of Bell's theorem. Similar arguments apply to Young's double slit experiment. The collective coherent phenomena of superconductivity is also explainable by considering the relativistic field theoretic approach in which wave equations are solved in the complex Minkowski space.
The Schrödinger equation need not be postulated. This paper demonstrates that the imaginary unit
i in quantum mechanics and the Schrödinger equation itself are both derived consequences of the Dirac equation when formulated in the real Clifford algebra
Cl(1,3) - the Spacetime Algebra introduced by Hestenes (1966). No new physics is proposed; the result collects and organises existing mathematical results into a single self-contained derivation chain:
Cl(1,3)→Hestenes-Dirac→γ0-split→Pauli→Schroedinger The role of
i is played by the spin-plane bivector
γ21=γ2γ1, a real geometric object satisfying
γ212=−1 as a consequence of the Minkowski metric. Complex-valued wavefunctions are thus projection artifacts: the residue of real four-dimensional geometry viewed from a three-dimensional observer frame. Key Results - The imaginary unit
i in the Schrödinger equation is identified as the spin-plane bivector
γ21 - a directed plane in spacetime, not an abstract algebraic postulate.- The Schrödinger equation is derived as the non-relativistic limit of the Hestenes--Dirac equation via the
γ0-split (observer frame choice), with
i inherited from the spacetime bivector structure.- Two of the five standard axioms of quantum mechanics (complex Hilbert space and the Schrödinger equation) are demoted from postulates to derived theorems. The measurement postulate and tensor product rule are retained.- Full compatibility with the Renou et al. (2021) no-go theorem and the Chen et al. (2022) experimental confirmation: the formulation is isomorphic to complex quantum mechanics, not a real restriction of it.
Cl(1,3) contains the required complex structure via
γ21.- Convergence with Penrose's twistor programme: both approaches conclude that complex structure in quantum mechanics and Lorentzian spacetime geometry have a common origin. Document Contents - complex_wavefunctions.pdf - Main paper (13 pages). Self-contained derivation from
Cl(1,3) to Schrödinger, with compatibility analysis (Renou/Chen/Penrose), axiom restructuring, and pedagogical appendix.- complex_wavefunctions.tex - LaTeX source. Standalone, no external dependencies. Scope and Originality This paper presents no new mathematical results. The identification
i↔γ21 is due to Hestenes (1967); the non-relativistic reduction in geometric algebra is given in Doran & Lasenby (2003). The contribution is expository: collecting the derivation chain into a single path with the geometric identity of
i tracked at each step, and situating the result in the context of the Renou et al. no-go theorem and its experimental confirmation.
The issue of whether Bell's theorem and other remote connectedness phenomena, such as Young's double slit experiment, demands superluminal or space-like signals or prior generated luminal signals is an area of hot debate. This also relates to the existence of advanced vs. retarded potentials and annihilation creation operators which are of interest in this regard. Using the complex model of we will examine the issue of the nonlocality of Bell's theorem as quantum mechanical ‘transactions’ providing a microscopic communication path between detectors across space-like intervals, which violate the EPR locality postulate [1]. See Chap. 4. This picture appears consistent with the remote connectedness properties of complex Minkowski space. Also, there are implications for macroscopic communications channels; another area of debate. Detailed discussions of Bell's theorem are given in [2].
This paper proposes a physical interpretation of the imaginary unit i in the Schrödinger equation, arguing that its complex structure is not a mathematical artifact but a direct signature of the zero field’s internal phase dynamics. In standard quantum mechanics, i enables oscillatory, unitary time evolution—yet its physical meaning remains unaddressed. Within the framework of R-theory (Resonance Theory), the universe is understood as a coherent resonant field with two inseparable components: a real, space-time-manifested aspect and an imaginary, non-local phase domain. This paper identifies i as the mathematical projection of this hidden imaginal axis. The wavefunction’s complex nature—traditionally viewed as formal—is reinterpreted as a physical necessity. The real amplitude corresponds to manifested energy, while the imaginary component expresses the latent coherence through which all energy oscillates between potential and presence. In this view, the Schrödinger equation becomes a dynamical law describing rotations within the zero field, where i functions as the operator of coherence rather than an abstract algebraic device. The implications are significant: Quantum oscillation reflects the continuous exchange between real and imaginal components of the zero field. Complexity is elevated from a mathematical convenience to an ontological principle. Consciousness emerges as the experiential side of the same imaginal phase that maintains physical coherence. Reality itself is intrinsically complex—built from orthogonal dimensions of manifestation and potential. This paper reframes the imaginary dimension as the fundamental axis through which the universe sustains resonance. What physics calls “imaginary” is revealed as the inner structure of reality, the phase without space that allows energy, matter, and awareness to exist as coherent oscillations within the zero field.