Inspired by a similar, more general treatment by Kahler, we obtain the spin operator by pulling to the Cartesian coordinate system the azimuthal partial derivative of differential forms. At this point, no unit imaginary enters the picture, regardless of whether those forms are over the real or the complex field. Hence, the operator is to be viewed as a real operator. Also a view of Lie differentiation as a pullback emerges, thus avoiding conceps such as flows of vector fields for its definition. Enter Quantum Mechanics based on the Kahler calculus. Independently of the unit imaginary in the phase factor, the proper values of the spin part of angular momentum emerge as imaginary because of the idempotent defining the ideal associated with cylindrical symmetry. Thus the unit imaginary has to be introduced by hand as a factor in the angular momentum operator |and as a result also in its orbital part| for it to have real proper values. This is a concept of real operator opposite to that of the previous paragraph. Kahler stops short of stating the antithesis in this pair of concepts, both of them implicit in his work. A solution to this antithesis lies in viewing units imaginary in those idempotents as being the real quantities of square minus 1 in rotation operators of real tangent Clifford algebra. In so doing, one expands the calculus, and launches in principle a geometrization of quantum mechanics, whether by design or not.
Inspired by a similar, more general treatment by Kahler, we obtain the spin operator by pulling to the Cartesian coordinate system the azimuthal partial derivative of differential forms. At this point, no unit imaginary enters the picture, regardless of whether those forms are over the real or the complex field. Hence, the operator is to be viewed as a real operator. Also a view of Lie differentiation as a pullback emerges, thus avoiding conceps such as flows of vector fields for its definition. Enter Quantum Mechanics based on the Kahler calculus. Independently of the unit imaginary in the phase factor, the proper values of the spin part of angular momentum emerge as imaginary because of the idempotent defining the ideal associated with cylindrical symmetry. Thus the unit imaginary has to be introduced by hand as a factor in the angular momentum operator |and as a result also in its orbital part| for it to have real proper values. This is a concept of real operator opposite to that of the previous paragraph. Kahler stops short of stating the antithesis in this pair of concepts, both of them implicit in his work. A solution to this antithesis lies in viewing units imaginary in those idempotents as being the real quantities of square minus 1 in rotation operators of real tangent Clifford algebra. In so doing, one expands the calculus, and launches in principle a geometrization of quantum mechanics, whether by design or not.
While the complex formalism of quantum mechanics has been extraordinarily successful, the fundamental geometric origin of its core elements—the imaginary unit, operator non-commutativity, quantum phase, and the intrinsic geometry with quantum spin—has remained unresolved for more than a century. Here we resolve these foundational questions by deriving an exact and physically equivalent formulation of the Schrödinger equation as a Real Geometric Flow—a moving-frame evolution on a 2𝑁-dimensional real state space. In this restored framework, the global imaginary unit 𝑖 fully decomposes into a set of local skew-symmetric generators that drive planar rotations in real space, revealing that the traditional complex formalism is a compressed representation of an intrinsic rotational geometry.Within the real geometric flow, energy eigenvalues appear as the Curvatures of the state trajectory, energy gaps correspond to Torsions that couple internal planes—naturally generating the geometry of quantum spin—and quantum phase emerges naturally as the Holonomy of the real moving frame. This formulation preserves the full empirical content of quantum mechanics while exposing the geometric mechanisms hidden beneath its notation, providing the first coherent explanation of curvature, torsion, non-commutativity, spin behavior, and quantum phase within a single framework.By demonstrating that the algebraic postulates of quantum mechanics arise from a deeper geometric order—one that stands alongside curvature-based theories such as General Relativity—this work establishes a unified differential-geometric foundation for quantum dynamics.It opens new directions for quantum control, quantum information, and fundamental physics.
Mathematically, the imaginary unit 𝑖 has long been regarded as an algebraic generator extending the real numbers to the complex field. Yet a century after the birth of quantum mechanics (1925–2025), the answer to its oldest question is now clear: quantum mechanics is not merely a mathematical structure—it is a physical reality, and all quantities appearing in its non-commutative relations are physical. Therefore, 𝑖 is not a mere algebraic extension but the universal operator demanded by the geometric completeness of non-commutative reality between position and momentum. The imaginary unit is a physical reality. The canonical relation [𝑥, 𝑝] = 𝑖ℏ is not merely algebraic—it is a geometric law, showing that the complex structure of quantum mechanics arises inevitably from physical reality itself. It stands without contradiction, without complexity. This is nature. In the (𝑥, 𝑝) plane, 𝑖 represents a 90° rotation and ℏ quantizes the curvature of that rotation. Complex numbers thus arise as the natural geometric language of non-commutative physics, providing a geometric resolution to the century-old question of physical completeness first raised by Einstein, Podolsky and Rosen (1935).
In standard quantum mechanics, the imaginary unit (i) is introduced axiomatically within the Hilbert space formalism, leaving its physical ontology largely unexplained. This paper demonstrates that within the Theory of Global Irreducible Unfolding (TGIU), the complex structure of the Schrödinger equation emerges naturally from a purely real, two-dimensional informational geometry. By representing the physical state as a real vector composed of structural coherence (\Phi) and entropy (S), we show that the conservation of total informational magnitude necessitates a continuous phase rotation in the (\Phi, S) plane. This rotation is generated by the Curvature-State Transform (CST) operator (\mathcal{J}_{\mathrm{CST}}), an antisymmetric matrix mathematically isomorphic to the imaginary unit (\mathcal{J}_{\mathrm{CST}}^2 = -I). Consequently, the complex quantum wavefunction \psi = \Phi + iS is redefined as the compact mathematical representation of a real informational state. In this framework, the Born rule for probability conservation (|\psi|^2 = \Phi^2 + S^2) translates directly to the conservation of informational magnitude, while the angular velocity of the phase evolution is shown to be driven by the TGIU Equilibrium Force (TEF). Ultimately, we prove that quantum mechanics is not fundamentally complex, but rather the complex representation of a deeper, real-valued symplectic informational dynamics.
As in classical mechanics, rotation in quantum mechanics is a transformation which deals with angular momentum. The difference with classical mechanics comes from the fact that angular momentum is a vector operator and not a usual vector and its components do not commute. As for any transformation in quantum mechanics, to each rotation we can associate an operator which acts in state space. The expression of this operator depends on whether the rotation is passive, that is we do a rotation of the coordinate axes and the physical system is left unchanged, or active, in which case the coordinate axes are unchanged and the rotation is performed on the physical system. In the first part (Chaps. 1 and 2) of this book, details concerning both aspects are given. Following the definition of the geometrical transformation associated with the most general rotation, we give the expression of the rotation operator for specific cases. Transformation laws for scalar fields, vector fields and spinor fields are given as well as transformation laws for scalar operators, vector operators and more generally, for operators of any rank. The second part (Chaps. 3 and 4) deals with angular momentum algebra. We define the coupling coefficients of 2, 3 and 4 angular momenta as well as the recoupling coefficients. The definition of the irreductible tensor operator, which is a generalisation of scalar and vector operators, is given as well as the Wigner-Eckart theorem. The application of this theorem to more complex cases is studied.
We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit i is systematically replaced by the temporal Hilbert transform operator Ht, which satisfies the anti-involution property Ht2=−I and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field ψ and its Hilbert-transformed partner ϕ=Ht[ψ], whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation ψa=ψ−iϕ, under which the standard U(1) gauge symmetry emerges naturally as a global SO(2) rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density ρ=ψ†ψ+ϕ†ϕ=∣ψa∣2 is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.
We present a formulation of quantum field theory based entirely on real numbers, which we call real quantum field theory (RQFT). The construction is obtained from the standard complex-number formulation by replacing the imaginary unit i with a matrix J throughout all formulas. Here J is the real 2x2 matrix satisfying the condition J^2=-1. This extends the recently developed formulation of real quantum mechanics based on the real Kähler space to systems with infinitely many degrees of freedom. As the basic example, we construct the RQFT for a scalar field. We define a field operator acting on the real Kähler analogue of the bosonic Fock space. This operator satisfies the Klein-Gordon equation and the J-form of canonical commutation relation. To construct the RQFT we develop the corresponding J-calculus. In particular, we introduce the direct and inverse J-Fourier transforms and the associated J-valued distributions. In RQFT, the usual unitarity condition for the complex S-matrix is replaced by the statement that the real scattering operator is both orthogonal and symplectic. The physical observables in RQFT coincide with those of ordinary QFT. Thus RQFT does not change the physical predictions of scalar QFT, but provides an equivalent formulation. We explore the possibility of purely real formulations of the Standard Model of elementary particles. We show that it does admit this formulation and the resulting theory has ortho-symplectic symmetry. The choice of the Standard Model as the testbed for exploring the possibility of purely real formulations is related with the fact it provides a realistic description of all known fundamental particles. The real formulation of the Standard Model naturally suggests a possible exit beyond the Standard Model. In particular, we consider the implications of breaking the J-symmetry as a marker for new physics.