As shown in our work on spacetime structures of quantum particles, Schrödinger wavefunctions in quantum mechanics can be utilised to construct the geometric structures of quantum particles which are considered to be three-dimensional differentiable manifolds. In this work we will extend this kind of geometric formulation of quantum particles by showing that wavefunctions that are normally used to describe wave phenomena in classical physics can in fact also be utilised to represent three-dimensional differentiable manifolds which in turns are identified with quantum particles. We show that such identification can be achieved by using a three-dimensional wave equation to construct three-dimensional differentiable manifolds that are embedded in a four-dimensional Euclidean space. In particular, the dual character that is resulted from the identification of a wavefunction with a three-dimensional differentiable manifold may provide a classical basis to interpret the wave-particle duality in quantum mechanics.
In this work, by summarising our recent works on the differential geometric and topological structures of quantum particles and spacetime manifolds, we discuss the possibility to classify quantum particles according to their intrinsic geometric structures associated with differentiable manifolds that are solutions to wave equations of two and three dimensions. We show that fermions of half-integer spin can be identified with differentiable manifolds which are solutions to a general two-dimensional wave equation, in particular, a two dimensional wave equation that can be derived from Dirac equation. On the other hand, bosons of integer spin can be identified with differentiable manifolds which are solutions to a general three-dimensional wave equation, in particular, a three-dimensional wave equation that can be derived from Maxwell field equations of electromagnetism. We also discuss the possibility that being restricted to three-dimensional spatial dimensions we may not be able to observe the whole geometric structure of a quantum particle but rather only the cross-section of the manifold that represents the quantum particle and the space in which we are confined. Even though not in the same context, such view of physical existence may comply with the Copenhagen interpretation of quantum mechanics which states that the properties of a physical system are not definite but can only be determined by observations.
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.
In our previous works, we suggest that quantum particles are composite physical objects endowed with the geometric and topological structures of their corresponding differentiable manifolds that would allow them to imitate and adapt to physical environments. In this work, we show that Dirac equation in fact describes quantum particles as composite structures that are in a fluid state in which the components of the wavefunction can be identified with the stream function and the velocity potential of a potential flow formulated in the theory of classical fluids. We also show that Dirac quantum particles can manifest as standing waves which are the result of the superposition of two fluid flows moving in opposite directions. However, for a steady motion a Dirac quantum particle does not exhibit a wave motion even though it has the potential to establish a wave within its physical structure, therefore, without an external disturbance a Dirac quantum particle may be considered as a classical particle defined in classical physics. And furthermore, from the fact that there are two identical fluid flows in opposite directions within their physical structures, the fluid state model of Dirac quantum particles can be used to explain why fermions are spin-half particles.
Three out of four complex components of the Dirac spinor can be algebraically eliminated from the Dirac equation (if some linear combination of electromagnetic fields does not vanish), yielding a partial differential equation of the fourth order for the remaining complex component. This equation is generally equivalent to the Dirac equation. Furthermore, following Schrödinger [Nature (London), 169, 538 (1952)], the remaining component can be made real by a gauge transform, thus extending to the Dirac field the Schrödinger conclusion that charged fields do not necessarily require complex representation. One of the two resulting real equations for the real function describes current conservation and can be obtained from the Maxwell equations in spinor electrodynamics (the Dirac-Maxwell electrodynamics). As the Dirac equation is one of the most fundamental equations, these results both belong in textbooks and can be used for development of new efficient methods and algorithms of quantum chemistry.
A realistic interpretation of Schroedinger and Dirac equations for density matrices is proposed, in which the difference between the position arguments of the density matrix is considered as an objective extra space dimension. "Particle" solutions are found, which are perfectly localized both in position space and in momentum space (the position and momentum operators commute in the density matrix representation); definitions for all observable quantities are given and the values associated to the "particle" solutions are the correct ones, both for the non-relativistic and the relativistic case. Finally, a non-linear interaction (the electromagnetic one) is introduced in an attempt to single out the "particle" solutions of the Dirac equation from all other solutions; the dynamical evolution of the electromagnetic field is described by the classical (unquantized) Maxwell equations.