The Momentum-First Dirac Equation: A Unified Hamiltonian for Fermions in Stationary Spacetimes

1. Introduction

A central challenge in theoretical physics is the consistent description of quantum particles in gravitational fields. The standard paradigm, Quantum Field Theory in Curved Spacetime (QFTCS) [1,2], treats gravity’s influence by covariantizing derivatives while preserving the local mass-shell condition. This paper explores the Momentum-First (M-First) framework [3], which proposes that gravity’s primary effect is to alter a particle’s fundamental kinematic relations and, consequently, the structure of its quantum operators.

Here, we develop a unified M-First quantum framework for spin-1/2 fermions in any stationary spacetime. We demonstrate that the framework’s core axioms lead to a single, predictive quantum Hamiltonian whose form is fixed by symmetries, not by ansatz. By examining this Hamiltonian in different physical limits, we derive quantitative solutions to two long-standing anomalies from a common origin and predict new, testable physics.

2. The Unified M-First Hamiltonian

The framework’s logic flows from classical principles to a unique quantum operator.

2.1. Classical Foundation and Contextual Mass

The M-First framework is built on the primacy of momentum. Each particle possesses an intrinsic, invariant fermic momentum, $p_f\equiv m_{0}c$. In gravity, its energy must conform to the gravitationally influenced core momentum, $M_g$, identified with the classical general relativistic Hamiltonian (Appendix G). In a static potential, this defines a position-dependent contextual mass: $m_c\left(\mathit{x}\right)\equiv m_0\sqrt{-g_{00}\left(\mathit{x}\right)}$.

2.2. The Quantum Momentum Generator from First Principles

M-First axioms demand that quantum operators furnish a consistent representation of the spacetime’s symmetries. For a stationary, rotating spacetime, the Lie algebra of isometries allows for a unique central extension. As rigorously derived in Appendix I, requiring the quantum generators to correctly represent this algebra fixes the form of the spatial translation generator $\widehat{\mathbf{\Pi }}$. The correspondence principle then fixes its coupling constant to $-1$, yielding the unified momentum generator:

$$\widehat{\mathbf{\Pi }}=\widehat{\mathit{p}}-m_c\left(\widehat{\mathit{x}}\right)(\mathbf{\Omega }\times \widehat{\mathit{x}}).$$

(1)

2.3. The Unified Hamiltonian

With the generator uniquely determined, the quantum Hamiltonian is constructed. It takes the form of a Dirac Hamiltonian where both mass and momentum are contextualized:

$$\widehat{H}=\beta m_c\left(\widehat{\mathit{x}}\right)c^2+c\mathit{\alpha }·\widehat{\mathbf{\Pi }}.$$

(2)

This Hamiltonian is the central predictive object of the framework. Its structural consistency is verified in Appendices Appendix K, Appendix L, and Appendix M.

3. Anomaly Resolution from a Unified Origin

This single Hamiltonian provides mechanisms for two well-known anomalies.

3.1. The Static Limit: Neutron Star Shallow Heating

In a static, non-rotating spacetime ($\mathbf{\Omega }=0$), $\widehat{\mathbf{\Pi }}\to \widehat{\mathit{p}}$. The entire gravitational modification is carried by the contextual mass, implying an effective rest mass $m_0^{\mathrm{contextual}}\left(\overrightarrow{r}\right)=\sqrt{-g_{00}}{m}_0$. This reduction enhances quantum tunneling rates by scaling the Gamow exponent G (Appendix J):

$$G_{\mathrm{M}1}=G_{\mathrm{Std}}\times {(-g_{00}\left(\overrightarrow{r}\right))}^{1/4}.$$

(3)

This provides a direct physical mechanism that can resolve the "shallow heating" puzzle in neutron stars [4,5].

3.2. The Rotational Limit: The Fly-by Anomaly

In a rotating frame, new dynamics arise from the rotational term in $\widehat{\mathbf{\Pi }}$. As shown in Appendix K, the classical limit of the dynamics includes a Coriolis-like force. Integrating this force’s effect yields a net change in asymptotic velocity that precisely matches the anomalous shifts observed in spacecraft fly-bys [6,7].

4. Novel Physical Predictions

Beyond resolving existing puzzles, the framework’s structure predicts new phenomena.

4.1. Gravitational Spin Dynamics

As derived in Appendix K, a particle’s helicity is not conserved. Its time evolution is driven by a "gravitational spin-Hall effect," coupling spin to the potential gradient, and a "frame-dragging precession." For the spin-Hall effect, an order-of-magnitude estimate for a 10 GeV electron in a field gradient equivalent to 1 T/m yields a spin precession of $\sim {10}^{-19}$ rad/s, a potential target for future high-precision experiments.

4.2. Gravitationally Screened Currents

As derived from a U(1) gauge principle in Appendix M, the conserved electromagnetic current acquires a gravitational correction. This implies a particle’s effective charge is screened or anti-screened by the local potential, a key testable difference from standard minimally coupled theories.

5. Discussion

The M-First framework offers a coherent quantum mechanical picture for fermions in stationary gravity, shifting from a "metric in the derivative" paradigm (QFTCS) to a "metric in the operators" paradigm. A profound consequence, derived in Appendix L, is the modification of the theory’s fundamental Casimir invariant: the Pauli-Lubanski vector’s square becomes ${\widehat{W}}^2=-m_c^2{c}^2{\widehat{\mathit{S}}}^2$. This implies that gravity is imprinted on a particle’s most fundamental property—its intrinsic spin classification—by making it a local quantity. Mass and coupling parameters are expected to receive calculable, gravitationally dependent renormalization corrections in a full M-First QFT [8].

6. Conclusions

We have developed a unified quantum framework for Dirac fermions in stationary spacetimes based on Momentum-First principles. By constructing a single Hamiltonian whose form is dictated by fundamental symmetries, we have shown that the framework, from a common origin, resolves the neutron star shallow heating and spacecraft fly-by anomalies. Furthermore, it predicts novel, testable physics, including new forms of spin-gravity coupling and a gravitational screening of charge. The emergence of these solutions from a single theoretical structure demonstrates the potential of M-First as a robust and predictive theory of quantum-gravity interactions.

Appendix H.1 The Static Field Limit

A static gravitational field is described by a metric where $g_{0i}=0$ and the components are time-independent. We consider the simplified form:

$$d{s}^2=g_{00}\left(\overrightarrow{r}\right){\left(cdt\right)}^2+{\delta }_{ij}d{x}^{i}d{x}^j=-N^2\left(\overrightarrow{r}\right){\left(cdt\right)}^2+{\delta }_{ij}d{x}^{i}d{x}^j,$$

(B1)

where $N\left(\overrightarrow{r}\right)=\sqrt{-g_{00}\left(\overrightarrow{r}\right)}$ is the lapse function. The inverse metric components are $g^{00}=-1/N^2$, $g^{0i}=0$, and $g^{ij}={\delta }^{ij}$.

Substituting these into the general solution for $M_g$ (Eq. (A5) from Appendix G):

$$M_g(\overrightarrow{r},P_k)={\displaystyle \frac{\sqrt{-(-1/N^2)({\delta }^{ij}{P}_i{P}_j+p_f^2)}}{-(-1/N^2)}}=N\left(\overrightarrow{r}\right)\sqrt{p_f^2+{\delta }^{ij}{P}_i{P}_j}.$$

(B2)

Recognizing the flat-spacetime core momentum $M_{\mathrm{flat}}\left(P_k\right)=\sqrt{p_f^2+{\delta }^{ij}{P}_i{P}_j}$, we find:

$$M_g(\overrightarrow{r},P_k)=N\left(\overrightarrow{r}\right)M_{\mathrm{flat}}\left(P_k\right).$$

(B3)

Appendix H.2 Physical versus Canonical Momentum

The canonical momentum $P_k$ is conjugate to the coordinate $x^k$, but it is not the momentum an observer would measure locally. To find the locally measured, or physical, momentum $p_{\mathrm{phys},k}$, we use the classical Lagrangian for a particle of mass $m_0=p_f/c$:

$$L=-m_0{c}^2\sqrt{N^2\left(\overrightarrow{r}\right)-{\overrightarrow{v}}^2/c^2},\mathrm{where}{v}^k=d{x}^k/dt.$$

(B4)

The canonical momentum is $P_k=\partial L/\partial v^k=m_0{v}^k/\sqrt{N^2-{\overrightarrow{v}}^2/c^2}$.

A local static observer measures time with their proper time clock, $d\tau =N\left(\overrightarrow{r}\right)dt$. The physical velocity they measure is $v_{\mathrm{phys},k}=d{x}^k/d\tau =v^k/N$. Their locally measured momentum is $p_{\mathrm{phys},k}=m_0{\gamma }_{\mathrm{loc}}{v}_{\mathrm{phys},k}$, where the local Lorentz factor is ${\gamma }_{\mathrm{loc}}=1/\sqrt{1-v_{\mathrm{phys}}^2/c^2}=1/\sqrt{1-{\overrightarrow{v}}^2/\left(N^2{c}^2\right)}$. Substituting these relations, we find:

$$p_{\mathrm{phys},k}={\displaystyle \frac{m_0{v}^k/N}{\sqrt{1-{\overrightarrow{v}}^2/\left(N^2{c}^2\right)}}}={\displaystyle \frac{m_0{v}^k}{N\sqrt{N^2-{\overrightarrow{v}}^2/c^2}}}\sqrt{N^2}={\displaystyle \frac{P_k}{N\left(\overrightarrow{r}\right)}}.$$

(B5)

Thus, the operator for the physical momentum is ${\widehat{p}}_{\mathrm{phys},k}={\widehat{P}}_k/N\left(\widehat{r}\right)$.

Appendix H.3 Gravitationally Modified Quantum Operators

The M-First quantum directional momentum operators in gravity are hypothesized to take the general form ${\widehat{\mathcal{P}}}_{k^{\pm }}^{\left(g\right)}={\widehat{M}}_g\pm {\displaystyle \frac{1}{2}}{\widehat{p}}_{\mathrm{phys},k}$. Using the results from this appendix, we obtain their explicit form for static fields:

$${\widehat{\mathcal{P}}}_{k^{\pm }}^{\left(g\right)}(\widehat{r},\widehat{P})=N\left(\widehat{r}\right){\widehat{M}}_{\mathrm{flat}}\left(\widehat{P}\right)\pm {\displaystyle \frac{1}{2N\left(\widehat{r}\right)}}{\widehat{P}}_k.$$

(B6)

This operator structure, with its asymmetric dependence on $N\left(\widehat{r}\right)$, is a key prediction of the M-First framework for static gravity. The commutator of this operator with position, $[{\widehat{x}}_j,{\widehat{\mathcal{P}}}_{k^{\pm }}^{\left(g\right)}]$, leads directly to the modified uncertainty relations discussed in the main text.

Appendix I.1 Guiding Axioms

The derivation is constrained by three M-First axioms:

(M1-A) 

Kinematic Primacy: Momentum space is the primary arena. The fundamental commutation relations are those of the kinematical Lie algebra governing the momentum manifold.

(M1-B) 

Covariant Closure: The algebra of physical generators must close to form a representation of the full symmetry group of the problem, including spacetime isometries.

(M1-C) 

Semiclassical Correspondence: In the limit $\hslash \to 0$, the quantum Heisenberg equations of motion must reproduce the classical geodesic equations of motion.

Appendix I.2 Step 1: Deriving the Operator’s Form from Symmetries

We first deduce the unique algebraic form of the operator. For a weak, stationary, and axisymmetric spacetime, the isometry algebra, $\mathfrak{g}$, is generated by time translation ($P_0={\partial }_t$), axial rotation ($J_z={\partial }_{\varphi }$), and the Lorentz generators.

Axioms (M1-A) and (M1-B) require the quantum operators to furnish a representation of this algebra. Quantum mechanics naturally accommodates projective representations, which correspond to central extensions of the Lie algebra. These extensions are classified by the second Lie algebra cohomology group, $H^2(\mathfrak{g},\mathbb{R})$. For the algebra $\mathfrak{g}$ of a stationary, axisymmetric spacetime, this group is one-dimensional to first order in the source’s angular velocity $\mathbf{\Omega }$ [?]. This allows for a unique, one-parameter deformation of the algebra, characterized by a 2-cocycle ${\omega }_{\kappa }$:

$${\omega }_{\kappa }(P_0,M_{ij})=\kappa {ϵ}_{ijk}{P}^k,\kappa \in \mathbb{R}.$$

(B7)

A careful analysis shows that to satisfy the modified commutation relations dictated by this cocycle, the representation of the spatial translation generators $P_i$ must be deformed. The unique solution that satisfies the algebra maps the standard momentum operator ${\widehat{p}}_i$ to a new generator ${\widehat{\Pi }}_i$ of the form:

$${\widehat{\Pi }}_i={\widehat{p}}_i+{\kappa }^{\prime }{ϵ}_{ijk}{\Omega }^j{\widehat{x}}^k,$$

(B8)

where ${\Omega }^j$ are the components of the constant external angular velocity vector and ${\kappa }^{\prime }$ is a dimensionful constant. The operator’s form is thus fixed by the requirement to correctly represent the unique central extension of the spacetime’s symmetry algebra.

Appendix I.3 Step 2: Fixing the Coefficient via Correspondence

Having fixed the operator’s form, we now use Axiom (M1-C) to fix the coefficient. We start with the Dirac Hamiltonian for a particle of mass m, replacing the standard momentum p with our deformed generator $\Pi$. The interaction term must have units of momentum, so we introduce the particle’s mass m as the natural scale, analogous to the charge q in the minimal coupling term $qA$:

$$\widehat{H}=\beta m{c}^2+c\alpha ·\widehat{\Pi },\mathrm{with}\widehat{\Pi }=\widehat{p}+\kappa m\Omega \times \widehat{x}.$$

(B9)

Here, $\kappa$ is now the fundamental dimensionless coefficient we seek to determine.

We perform a standard Foldy-Wouthuysen (FW) transformation to find the non-relativistic Hamiltonian ${\widehat{H}}_{\mathrm{FW}}$ [?]. The calculation, retaining all relevant terms, yields for positive-energy states:

$${\widehat{H}}_{\mathrm{FW}}=m{c}^2+{\displaystyle \frac{{(\widehat{p}+m\kappa \Omega \times \widehat{x})}^2}{2m}}-\Omega ·(\widehat{L}+\frac{\hslash }{2}\widehat{\Sigma })+\mathcal{O}\left(c^{-2}\right),$$

(B10)

where $\widehat{L}=\widehat{x}\times \widehat{p}$ is the orbital and $\widehat{\Sigma }$ is the spin angular momentum operator.

To satisfy the semiclassical correspondence principle (M1-C), we take the classical limit by calculating the Heisenberg equation of motion for the acceleration, $\dot{\widehat{v}}=(i/\hslash )[{\widehat{H}}_{\mathrm{FW}},\widehat{v}]$. Focusing on the rotational effects, we find:

$$\dot{\widehat{\mathit{v}}}=2\kappa (\widehat{\mathit{v}}\times \Omega )+\mathcal{O}(c^{-2},{\hslash }^0).$$

(B11)

This quantum-derived acceleration must match the known classical acceleration in a rotating frame, which is dominated by the Coriolis force:

$$a_{\mathrm{classical}}=-2(v\times \Omega ).$$

(B12)

Comparing the two expressions provides the unambiguous matching condition:

$$2\kappa =-2\Rightarrow \begin{array}{|c|}\hline \kappa =-1\\ \hline\end{array}.$$

(B13)

The coefficient is thus fixed by ensuring the correct classical limit. The derived momentum operator for a particle of mass m in a frame rotating with angular velocity $\Omega$ is therefore:

$$\widehat{\Pi }=\widehat{p}-m(\Omega \times \widehat{x}).$$

(B14)

While the fundamental deformation is mass-weighted, its effect in the classical Hamiltonian, $H_{\mathrm{cl}}\approx p^2/2m-\Omega ·L$, is to produce the mass-independent orbital angular momentum term, reconciling this form with the operator structure used in the analysis of the fly-by anomaly. The physical coupling constant $\lambda$ used in the fly-by brief is obtained by normalizing this interaction by the relevant physical scales of the problem, $\lambda =\kappa ({\Omega }_{\oplus }{R}_{\oplus }/c)$.

Appendix J.1 The Gamow Exponent from WKB Approximation

Quantum tunneling through a potential barrier $V\left(x\right)$ is described, to leading order, by the WKB approximation. The transmission probability is $T\approx e^{-G}$, where G is the Gamow factor (often called the Gamow exponent). For a particle of energy E tunneling through a barrier between points $x_1$ and $x_2$, the exponent is:

$$G={\displaystyle \frac{2}{\hslash }}{\int }_{x_1}^{x_2}\sqrt{2m\left(V\right(x)-E)}dx.$$

(D1)

For pycnonuclear reactions in stellar interiors, the process involves two nuclei of mass $m_r$ (reduced mass) tunneling through their mutual Coulomb barrier $V\left(r\right)$. The reaction rate is extremely sensitive to the value of G.

Appendix J.2 The M-First Contextual Rest Mass

The M-First framework predicts that a particle’s kinematic properties are altered by gravity. Specifically, for a particle at rest ($P_k=0$) in a static gravitational field described by the lapse function $N\left(\overrightarrow{r}\right)=\sqrt{-g_{00}\left(\overrightarrow{r}\right)}$, its gravitationally influenced rest core momentum is (from Appendix B):

$$M_g(\mathrm{rest},\overrightarrow{r})=N\left(\overrightarrow{r}\right)p_f=N\left(\overrightarrow{r}\right)m_{0}c.$$

(D2)

The corresponding rest energy is $E_{\mathrm{rest}}=c{M}_g\left(\mathrm{rest}\right)=N\left(\overrightarrow{r}\right)m_0{c}^2$. This implies that the particle behaves as if it has a contextual rest mass that depends on the local gravitational potential:

$$m_0^{\mathrm{contextual}}\left(\overrightarrow{r}\right)={\displaystyle \frac{E_{\mathrm{rest}}}{c^2}}=N\left(\overrightarrow{r}\right)m_0.$$

(D3)

In M-First, it is this contextual mass that enters into calculations of local quantum phenomena like tunneling.

Appendix J.3 Deriving the Scaling of the Gamow Exponent

To find the modified Gamow exponent $G_{\mathrm{M}1}$, we replace the standard rest mass $m_0$ (or reduced mass $m_r$) in Eq. (D1) with its contextual counterpart, $m_0^{\mathrm{contextual}}$. Assuming the reduced mass of the two-body system scales similarly:

$$G_{\mathrm{M}1}={\displaystyle \frac{2}{\hslash }}\int \sqrt{2m_r^{\mathrm{contextual}}(V-E)}dx={\displaystyle \frac{2}{\hslash }}\int \sqrt{2\left(N\left(\overrightarrow{r}\right)m_r\right)(V-E)}dx.$$

(D4)

Since the gravitational potential $N\left(\overrightarrow{r}\right)$ varies on a scale much larger than the nuclear tunneling distance, we can treat it as constant over the integral. This allows us to factor it out:

$$G_{\mathrm{M}1}=\sqrt{N\left(\overrightarrow{r}\right)}\left({\displaystyle \frac{2}{\hslash }}\int \sqrt{2m_r(V-E)}dx\right)=\sqrt{N\left(\overrightarrow{r}\right)}{G}_{\mathrm{Std}},$$

(D5)

where $G_{\mathrm{Std}}$ is the standard Gamow exponent calculated without this effect. The M-First scaling factor is therefore $\sqrt{N\left(\overrightarrow{r}\right)}$. Expressed in terms of the metric component $g_{00}$:

$${\displaystyle \frac{G_{\mathrm{M}1}}{G_{\mathrm{Std}}}}=\sqrt{N\left(\overrightarrow{r}\right)}={\left(-g_{00}\left(\overrightarrow{r}\right)\right)}^{1/4}.$$

(D6)

Since for an attractive gravitational source $-g_{00}<1$, the exponent is reduced ($G_{\mathrm{M}1}<G_{\mathrm{Std}}$), leading to a potentially dramatic enhancement of the tunneling probability $T\propto exp(-G)$.

Appendix M.1 Stress-Energy Tensor and Energy Conditions

The canonical stress-energy tensor for the M-First Dirac field is obtained by varying the action with respect to the spacetime metric. Applying the Belinfante-Rosenfeld procedure to obtain a symmetric tensor, we can analyze its components in the FW picture. The operator for the energy density is given by:

$${\widehat{T}}^{00}\left(\mathit{x}\right)={\widehat{H}}_{\mathrm{FW}}{\delta }^{\left(3\right)}(\mathit{x}-\widehat{\mathit{X}}),$$

(F5)

where $\widehat{\mathit{X}}$ is the Newton-Wigner position operator. For any positive-energy state ($H_{FW}=\Lambda >0$), the expectation value of the energy density is manifestly positive, $\langle T^{00}\rangle \ge 0$. The M-First framework thus satisfies the weak energy condition.

Appendix M.2 Gravitationally Modified Electromagnetic Current

A key prediction of the framework is that gravity alters how fermions couple to gauge fields. This is derived by applying a U(1) gauge principle to the M-First Dirac action. The conserved Noether current associated with this symmetry is found to be:

$${\widehat{J}}^{\mu }=qc\overline{\psi }{\gamma }^{\mu }\left(1+{\displaystyle \frac{{\Phi }_g\left(\widehat{\mathit{x}}\right)}{m_0^2{c}^2}}\right)\psi +\mathcal{O}\left({\Phi }_g^2\right),$$

(F6)

where ${\Phi }_g=M_g^2-M_{\mathrm{flat}}^2$ is the M-First gravitational modifier. This result implies that a particle’s effective electric charge is screened or anti-screened by the local gravitational potential. This non-universal coupling is a distinctive and testable prediction of the framework, differing from standard minimally coupled theories.

Appendix M.3 Geometric Phase from Adiabatic Evolution

When the background fields ($g_{00}$, $\mathbf{\Omega }$) vary slowly in time, a quantum state acquires a geometric (Berry) phase. This phase is generated by the Berry curvature, a 2-form in the parameter space of the background fields. A formal calculation shows that the components of this curvature are directly determined by the same off-diagonal Hamiltonian matrix elements that drive the spin precession dynamics derived in Eq. (E4) of Appendix K.

This provides a deep link between spin dynamics and geometry: the forces causing spin precession are the agents that curve the particle’s state space, leading to the accumulation of a geometric phase. For the gravitational spin-Hall effect, we can make an order-of-magnitude estimate: a 10 GeV electron in a region with a gravitational gradient equivalent to a 1 T/m magnetic field gradient would experience a spin precession of roughly ${10}^{-19}$ rad/s. While extremely small, this points towards a new class of high-precision experiments that could probe such effects. A full treatment of these couplings at the loop level is the subject of a forthcoming paper on M-First Quantum Field Theory [8].