The hydrodynamic formulation of the Dirac equation has historically been hindered by the inability to close the system of physical variables without resorting to infinite moment hierarchies. We resolve this longstanding issue by developing a fully covariant chiral-hydrodynamic formulation of the Dirac field in curved spacetime. Working in the Weyl representation, we introduce two independent null vectors, $P_L^μ$ and $P_R^μ$, which decouple the left and right chiral components. This allows us to define chiral geodesic and stochastic velocities, yielding a closed system of exactly eight real equations that corresponds directly to the Dirac field degrees of freedom. Remarkably, this formulation naturally isolates the spin-orbit coupling $(q/2)σ^{μν}F_{μν}$ while demonstrating the vanishing of the spin-gravity coupling in torsion-free general relativity. To demonstrate the analytical power of this framework, we specialize to the Schwarzschild geometry. We obtain exact radial solutions in terms of confluent Heun functions and directly compute the quasi-bound state spectrum (fermionic resonances), quasinormal mode frequencies, and greybody factors. Furthermore, by establishing an exact energy balance equation -- representing the first law of thermodynamics for Dirac fields -- we derive the Hawking radiation flux purely from chiral flux conservation at the event horizon. This work not only provides a closed hydrodynamic theory for spin-1/2 fluids but also establishes a unified framework for analyzing quantum information and fermion dynamics in strong gravitational backgrounds.
The hydrodynamic formulation of the Dirac equation has historically been hindered by the inability to close the system of physical variables without resorting to infinite moment hierarchies. We resolve this longstanding issue by developing a fully covariant chiral-hydrodynamic formulation of the Dirac field in curved spacetime. Working in the Weyl representation, we introduce two independent null vectors, PLμ and PRμ, which decouple the left and right chiral components. This allows us to define chiral geodesic and stochastic velocities, yielding a closed system of exactly eight real equations that corresponds directly to the Dirac field degrees of freedom. Remarkably, this formulation naturally isolates the spin-orbit coupling (q/2)σμνFμν while demonstrating the vanishing of the spin-gravity coupling in torsion-free general relativity. To demonstrate the analytical power of this framework, we specialize to the Schwarzschild geometry. We obtain exact radial solutions in terms of confluent Heun functions and directly compute the quasi-bound state spectrum (fermionic resonances), quasinormal mode frequencies, and greybody factors. Furthermore, by establishing an exact energy balance equation -- representing the first law of thermodynamics for Dirac fields -- we derive the Hawking radiation flux purely from chiral flux conservation at the event horizon. This work not only provides a closed hydrodynamic theory for spin-1/2 fluids but also establishes a unified framework for analyzing quantum information and fermion dynamics in strong gravitational backgrounds.
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.
Inverted Relativistic Phase-Fluid Mechanics on Curved Spacetime introduces a covariant matter-phase model in which the phase field defines the local four-momentum, written in readable form as: p_mu = hbar * nabla_mu phi or equivalently: four-momentum = Planck constant × phase gradient The density n weights the conserved phase current and determines how strongly the phase flow contributes to the stress-energy tensor and therefore to spacetime curvature. In the minimal model, n has no independent gradient term. It acts as an algebraic density variable enforcing a relativistic Hamilton-Jacobi or mass-shell relation. This gives an inverted relativistic phase fluid: the phase determines motion, while the density weights its physical and gravitational effect. The model is then extended through the Madelung decomposition: psi = sqrt(n) * exp(i phi) This adds the amplitude-gradient term and produces the relativistic quantum-pressure contribution: Q_rel = - hbar^2 * Box(sqrt(n)) / sqrt(n) where Box is the covariant d’Alembert operator on curved spacetime. In this way, the minimal phase-fluid picture becomes connected to the full Klein-Gordon/Madelung quantum-fluid structure on curved spacetime. The paper provides a covariant bridge between: proper-time matter phase, de Broglie momentum-phase relations, relativistic fluid mechanics, Madelung quantum hydrodynamics, quantum pressure, stress-energy generation, and gravitational curvature. The model does not replace general relativity and does not claim to be a completed theory of quantum gravity. Instead, it formulates a compact theoretical framework in which matter phase, density, quantum amplitude structure and spacetime curvature are coupled without introducing a preferred reference frame. A compact summary of the construction is: The phase carries the motion, the density carries the weighting, the quantum pressure carries the wave nature, and the metric carries gravity. This preprint is intended as a theory note and starting point for further analytical and numerical work, including fixed-background simulations, weak gravitational backreaction, and full Einstein-Klein-Gordon/Madelung evolution.
We uncover the quantum geometric structure of graviton Wigner functions and stress-energy tensors via quantum field theory in curved spacetime, revealing the phase-space geometry of spacetime and relativistic quantum-state manifolds. We show that the polarization mode expansion of the underlying graviton field operator is fully captured by quantum geometry, as encoded in the spinor-helicity formalism, thereby establishing the geometric framework from the outset. Applying this to a weak gravitational background, we demonstrate the roles of the quantum metric and connection in describing graviton transport beyond the chiral vortical effect. We also clarify the role of the quantum metric in normalizing the perfect-fluid graviton stress tensor within this approach. This work paves the path for explorations of multistate Hilbert-space geometry in high-energy and gravitational physics. In addition, this framework naturally encompasses lower-spin excitations, allowing for a unified quantum geometric treatment of bosonic and fermionic many-body systems in condensed matter and particle physics.
Inverted Relativistic Phase-Fluid Mechanics on Curved Spacetime introduces a covariant matter-phase model in which the phase field defines the local four-momentum, written in readable form as: p_mu = hbar * nabla_mu phi or equivalently: four-momentum = Planck constant × phase gradient The density n weights the conserved phase current and determines how strongly the phase flow contributes to the stress-energy tensor and therefore to spacetime curvature. In the minimal model, n has no independent gradient term. It acts as an algebraic density variable enforcing a relativistic Hamilton-Jacobi or mass-shell relation. This gives an inverted relativistic phase fluid: the phase determines motion, while the density weights its physical and gravitational effect. The model is then extended through the Madelung decomposition: psi = sqrt(n) * exp(i phi) This adds the amplitude-gradient term and produces the relativistic quantum-pressure contribution: Q_rel = - hbar^2 * Box(sqrt(n)) / sqrt(n) where Box is the covariant d’Alembert operator on curved spacetime. In this way, the minimal phase-fluid picture becomes connected to the full Klein-Gordon/Madelung quantum-fluid structure on curved spacetime. The paper provides a covariant bridge between: proper-time matter phase, de Broglie momentum-phase relations, relativistic fluid mechanics, Madelung quantum hydrodynamics, quantum pressure, stress-energy generation, and gravitational curvature. The model does not replace general relativity and does not claim to be a completed theory of quantum gravity. Instead, it formulates a compact theoretical framework in which matter phase, density, quantum amplitude structure and spacetime curvature are coupled without introducing a preferred reference frame. A compact summary of the construction is: The phase carries the motion, the density carries the weighting, the quantum pressure carries the wave nature, and the metric carries gravity. This preprint is intended as a theory note and starting point for further analytical and numerical work, including fixed-background simulations, weak gravitational backreaction, and full Einstein-Klein-Gordon/Madelung evolution.
We demonstrate that light propagation in reciprocal chiral photonic media admits a unified description in terms of an emergent Dirac structure in helicity space. Starting from Maxwell's equations, we reformulate the electromagnetic field as a four-component spinor governed by an effective non-Hermitian optical Dirac equation. In this representation, the magnetoelectric response of the chiral medium appears as a helicity-dependent background that modifies the spectrum and eigenmodes, while the breaking of the spin-degenerate condition generates the intrinsic spin-orbit coupling between helicity and orbital degrees of freedom. After projection onto the positive-frequency sector, the theory reduces to an exact two-level helicity-orbital model. This model is found to have an analytical solution and describes coherent Rabi-like oscillations between spin-orbit-coupled vector modes. Chirality controls the helicity splitting and detuning, whereas the electromagnetic mismatch of the medium determines the coupling strength responsible for oscillatory spin-orbit conversion. The resulting dynamics is constrained by exact conservation of the total angular momentum, leading to reversible conversion between spin and orbital angular momentum with well-defined selection rules. Our work establishes an optical Dirac framework for structured light in chiral media, and provides experimentally accessible predictions for chirality-controlled oscillations, polarization dynamics, and orbital angular momentum conversion in structured optical fields.
We develop a many-particle quantum-hydrodynamical model of fermion matter interacting with the external classical electromagnetic and gravitational/inertial and torsion fields. The consistent hydrodynamical formulation is constructed for the many-particle quantum system of Dirac fermions on the basis of the nonrelativistic Pauli-like equation obtained via the Foldy–Wouthuysen transformation. With the help of the Madelung decomposition approach, the explicit relations between the microscopic and macroscopic fluid variables are derived. The closed system of equations of quantum hydrodynamics encompasses the continuity equation, and the dynamical equations of the momentum balance and the spin density evolution. The possible experimental manifestations of the torsion in the dynamics of spin waves is discussed.