Hehl–Datta Equation in Einstein–Cartan Gravity

• Hehl–Datta equation is the nonlinear Dirac equation obtained by minimally coupling Dirac spinors to a torsionful Riemann–Cartan spacetime, resulting in an axial–axial four-fermion interaction.
• Algebraically eliminating torsion from the field equations introduces a cubic spinor self-interaction, which leads to distinct behavior for fermions and antifermions.
• Applications range from early-universe matter–antimatter asymmetry and electron self-energy regularization to solitonic spinor configurations in reduced dimensional models.

Searching arXiv for recent and foundational papers on the Hehl–Datta equation and its Einstein–Cartan context. The Hehl–Datta equation is the nonlinear Dirac equation obtained when a Dirac spinor is minimally coupled to Einstein–Cartan–Sciama–Kibble gravity on a Riemann–Cartan spacetime with torsion. In this setting, torsion is sourced algebraically by the spin density of matter, and eliminating it from the field equations generates an axial–axial four-fermion interaction in the effective Lagrangian and a cubic spinor self-interaction in the Dirac equation. In the form emphasized by Popławski, the equation is

$$i\,e^{\mu}{}_{a}\,\gamma^{a}\,\psi_{:\mu} \;=\; m\,\psi \;-\; \frac{3\,\kappa}{8}\, \bigl(\bar\psi\,\gamma^{b}\gamma^{5}\psi\bigr)\,\gamma_{b}\,\gamma^{5}\,\psi,$$

with the colon denoting the torsion-free covariant derivative (Poplawski, 2011). Across the literature, the Hehl–Datta equation is treated as the characteristic fermionic field equation of Einstein–Cartan–Dirac theory, with applications ranging from matter–antimatter asymmetry and dark matter scenarios to self-energy regularization, solitonic spinor configurations, Newman–Penrose formulations, and generalized-uncertainty-principle constructions (Poplawski, 2011, III et al., 2017, Khanapurkar et al., 2018, Ramesh, 2019).

1. Geometric and field-theoretic setting

The equation is formulated on a four-dimensional Riemann–Cartan spacetime, often denoted $U_{4}$, with metric compatibility and an independent torsionful connection. In this framework, the torsion tensor is the antisymmetric part of the affine connection,

$$Q_{\alpha\beta}{}^{\mu}\equiv \Gamma_{[\alpha\beta]}{}^{\mu},$$

or equivalently

$$S^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}_{[\mu\nu]},$$

depending on notation (Khanapurkar et al., 2018, III et al., 2017). The Einstein–Cartan or Sciama–Kibble extension of general relativity permits this antisymmetric sector, with torsion coupled to spin density in direct analogy to the way curvature couples to energy–momentum (Khanapurkar et al., 2018, Khanapurkar et al., 2018).

For a Dirac field $\psi$, the starting point is the usual Dirac Lagrangian minimally coupled to the full spin connection. Representative forms given in the literature are

$$\mathcal{L}_{\rm Dirac} = \frac{i\hbar c}{2} \Big(\bar\psi\gamma^{\mu}\nabla_{\mu}\psi -\nabla_{\mu}\bar\psi\,\gamma^{\mu}\psi\Big) -mc^{2}\bar\psi\psi$$

and

$$L_{\rm Dirac}=i\hbar\,\bar\psi\,\gamma^\mu\nabla_\mu\psi-mc\,\bar\psi\psi,$$

with $\nabla_\mu$ containing the torsionful connection (Khanapurkar, 2018, III et al., 2017). In tetrad form, one also writes the total Einstein–Cartan–Sciama–Kibble Lagrangian density as

$${\cal L}_{\rm tot} = -\frac{1}{2\kappa}\,e\,R + e\,\Bigl[\tfrac{i}{2}\bigl(\bar\psi\gamma^{a}\psi_{;a}-\bar\psi_{;a}\gamma^{a}\psi\bigr) - m\,\bar\psi\,\psi\Bigr],$$

where $e=\det(e^a{}_\mu)$, $U_{4}$0 is the curvature scalar built from the full connection, and the semicolon denotes the full covariant derivative containing torsion (Poplawski, 2011).

A central structural feature is that torsion in Einstein–Cartan theory is non-propagating. It is solved algebraically from the Cartan equation rather than through dynamical wave equations. This is repeatedly emphasized in the literature: torsion is “algebraically related to the fermion spin” and “does not propagate,” so the fermionic sector can be rewritten as a torsion-free Dirac operator plus a local nonlinear self-interaction (III et al., 2017, Khanapurkar, 2018).

2. Algebraic elimination of torsion and emergence of the cubic term

The Hehl–Datta equation arises by varying the total action with respect to torsion or contorsion, solving the resulting algebraic torsion–spin relation, and substituting that solution back into the spinor sector. In Einstein–Cartan–Sciama–Kibble theory, variation with respect to the connection gives an algebraic Cartan equation of the form

$U_{4}$1

or equivalently

$U_{4}$2

with $U_{4}$3 or $U_{4}$4 the Dirac spin density (III et al., 2017, Khanapurkar et al., 2018). For a Dirac field, the spin density is totally antisymmetric and proportional to the axial current: $U_{4}$5 and the contortion tensor becomes

$U_{4}$6

(III et al., 2017). In Popławski’s exposition, the same content is written as

$U_{4}$7

so the contorsion is linear in the axial current (Poplawski, 2011).

Once the full covariant derivative is split into the torsion-free part plus contorsion,

$U_{4}$8

substitution of the algebraic contorsion solution produces an axial–axial four-fermion contact term in the effective Lagrangian: $U_{4}$9 or equivalently

$$Q_{\alpha\beta}{}^{\mu}\equiv \Gamma_{[\alpha\beta]}{}^{\mu},$$0

(III et al., 2017, III et al., 2022). Varying this effective Lagrangian with respect to $$Q_{\alpha\beta}{}^{\mu}\equiv \Gamma_{[\alpha\beta]}{}^{\mu},$$1 yields the modified Dirac equation with cubic self-interaction: $$Q_{\alpha\beta}{}^{\mu}\equiv \Gamma_{[\alpha\beta]}{}^{\mu},$$2 or in Planck-length notation,

$$Q_{\alpha\beta}{}^{\mu}\equiv \Gamma_{[\alpha\beta]}{}^{\mu},$$3

(III et al., 2017, III et al., 2022).

The cubic term is therefore not an additional phenomenological interaction but the residual effect of integrating out non-propagating torsion. In the language of the cited papers, it is an “axial–axial” self-interaction, a “spin-spin self-interaction,” and a contact interaction generated solely by the algebraic torsion–spin coupling (III et al., 2017, III et al., 2022).

3. Explicit forms and equivalent representations

The equation is presented in several equivalent notational conventions. In Popławski’s Riemann–Cartan notation without electromagnetism,

$$Q_{\alpha\beta}{}^{\mu}\equiv \Gamma_{[\alpha\beta]}{}^{\mu},$$4

and with an external electromagnetic potential $$Q_{\alpha\beta}{}^{\mu}\equiv \Gamma_{[\alpha\beta]}{}^{\mu},$$5 and charge $$Q_{\alpha\beta}{}^{\mu}\equiv \Gamma_{[\alpha\beta]}{}^{\mu},$$6,

$$Q_{\alpha\beta}{}^{\mu}\equiv \Gamma_{[\alpha\beta]}{}^{\mu},$$7

(Poplawski, 2011). In explicitly dimensionful form one finds

$$Q_{\alpha\beta}{}^{\mu}\equiv \Gamma_{[\alpha\beta]}{}^{\mu},$$8

(III et al., 2022).

The same structure can be written in the compact form

$$Q_{\alpha\beta}{}^{\mu}\equiv \Gamma_{[\alpha\beta]}{}^{\mu},$$9

when the Planck length is replaced by the mass-dependent Compton–Schwarzschild length $$S^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}_{[\mu\nu]},$$0 (Singh, 2017). That generalization does not alter the formal cubic structure of the equation; it changes the coefficient of the torsion-induced nonlinearity.

A recurrent source of confusion is the distinction between the four-fermion term in the effective Lagrangian and the cubic term in the equation of motion. The effective action contains the quartic object

$$S^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}_{[\mu\nu]},$$1

whereas the field equation contains the cubic object

$$S^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}_{[\mu\nu]},$$2

The two are related by functional variation with respect to $$S^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}_{[\mu\nu]},$$3 and are not independent ingredients (III et al., 2017, III et al., 2022).

4. Charge conjugation, fermion–antifermion asymmetry, and early-universe applications

A distinctive feature stressed by Popławski is the behavior of the Hehl–Datta equation under charge conjugation. Defining the classical charge-conjugate spinor by

$$S^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}_{[\mu\nu]},$$4

one has $$S^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}_{[\mu\nu]},$$5, while the axial current $$S^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}_{[\mu\nu]},$$6 is $$S^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}_{[\mu\nu]},$$7-even (Poplawski, 2011). As a result, $$S^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}_{[\mu\nu]},$$8 satisfies a different Hehl–Datta equation: $$S^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}_{[\mu\nu]},$$9 Both the sign of the mass term and the sign of the cubic torsion term flip relative to the original equation (Poplawski, 2011). In this classical setting, fermions and antifermions therefore obey inequivalent field equations in the presence of torsion.

Popławski further states that fermions in a dense torsion background acquire an energy shift

$\psi$0

where $\psi$1 is effectively proportional to number density (Poplawski, 2011). This splitting is significant only at extremely high densities, such as those in the very early Universe. The proposed cosmological mechanism is that heavy fermions carrying baryon number decayed mostly to normal matter, whereas their antiparticles decayed mostly to hidden antimatter which forms dark matter, so that the conserved total baryon number of the Universe remained zero (Poplawski, 2011).

This use of the Hehl–Datta equation is highly model-specific. The concrete claim supported by the cited paper is not that the equation generically implies observed baryogenesis, but that it permits a scenario in which the opposite sign of the cubic torsion term for $\psi$2 and $\psi$3 generates an energy-level asymmetry in the early Universe (Poplawski, 2011). A plausible implication is that the Hehl–Datta nonlinearity is especially relevant in regimes where spin density, rather than ordinary laboratory-scale curvature, controls the leading fermionic correction.

5. Self-energy balance, finite fermion radii, and electron theory

A separate line of work by Diether and Christian treats the Hehl–Datta term as a negative mechanical self-energy that can counterbalance divergent positive self-energies. In that treatment, the torsion-induced cubic term represents a “spin-torsion” mechanical energy density, and because torsion is algebraically tied to spin, it produces a contact four-fermion interaction rather than a new long-range force (III et al., 2017).

The key rest-frame algebraic condition quoted in this literature is

$\psi$4

with $\psi$5. The positive $\psi$6 Coulomb self-energy is balanced by a negative $\psi$7 torsion term (III et al., 2022). In the 2017 analysis the same balance is written as

$\psi$8

leading to a finite “cancellation radius”

$\psi$9

(III et al., 2017). Numerical values quoted there include an electrostatic self-energy of $$\mathcal{L}_{\rm Dirac} = \frac{i\hbar c}{2} \Big(\bar\psi\gamma^{\mu}\nabla_{\mu}\psi -\nabla_{\mu}\bar\psi\,\gamma^{\mu}\psi\Big) -mc^{2}\bar\psi\psi$$0 at the Planck length and a torsion energy of $$\mathcal{L}_{\rm Dirac} = \frac{i\hbar c}{2} \Big(\bar\psi\gamma^{\mu}\nabla_{\mu}\psi -\nabla_{\mu}\bar\psi\,\gamma^{\mu}\psi\Big) -mc^{2}\bar\psi\psi$$1, with near-cancellation at $$\mathcal{L}_{\rm Dirac} = \frac{i\hbar c}{2} \Big(\bar\psi\gamma^{\mu}\nabla_{\mu}\psi -\nabla_{\mu}\bar\psi\,\gamma^{\mu}\psi\Big) -mc^{2}\bar\psi\psi$$2 yielding $$\mathcal{L}_{\rm Dirac} = \frac{i\hbar c}{2} \Big(\bar\psi\gamma^{\mu}\nabla_{\mu}\psi -\nabla_{\mu}\bar\psi\,\gamma^{\mu}\psi\Big) -mc^{2}\bar\psi\psi$$3 (III et al., 2017).

Within that program, the Hehl–Datta term is interpreted as the physical counter-term, so no further renormalisation is needed (III et al., 2017). The 2022 follow-up states that classical and quantum electrodynamics can be completed by gravitational torsion and that “there is no ‘bare’ mass for an electron, nor is renormalization required for many scenarios” (III et al., 2022). That conclusion belongs to the framework developed in those papers; it should be distinguished from the more conservative statement that Einstein–Cartan theory yields a local axial–axial self-interaction whose contribution becomes relevant at very short distances or very high densities (III et al., 2017, III et al., 2022).

6. Newman–Penrose formulation, reduced systems, and solitonic solutions

The Hehl–Datta equation has also been rewritten in Newman–Penrose form, which makes the torsion corrections to the spinor equations explicit in terms of null-tetrad directional derivatives and spin coefficients. Using a null tetrad $$\mathcal{L}_{\rm Dirac} = \frac{i\hbar c}{2} \Big(\bar\psi\gamma^{\mu}\nabla_{\mu}\psi -\nabla_{\mu}\bar\psi\,\gamma^{\mu}\psi\Big) -mc^{2}\bar\psi\psi$$4, two-component spinor variables $$\mathcal{L}_{\rm Dirac} = \frac{i\hbar c}{2} \Big(\bar\psi\gamma^{\mu}\nabla_{\mu}\psi -\nabla_{\mu}\bar\psi\,\gamma^{\mu}\psi\Big) -mc^{2}\bar\psi\psi$$5, and the NP operators $$\mathcal{L}_{\rm Dirac} = \frac{i\hbar c}{2} \Big(\bar\psi\gamma^{\mu}\nabla_{\mu}\psi -\nabla_{\mu}\bar\psi\,\gamma^{\mu}\psi\Big) -mc^{2}\bar\psi\psi$$6, the full Hehl–Datta system becomes

$$\mathcal{L}_{\rm Dirac} = \frac{i\hbar c}{2} \Big(\bar\psi\gamma^{\mu}\nabla_{\mu}\psi -\nabla_{\mu}\bar\psi\,\gamma^{\mu}\psi\Big) -mc^{2}\bar\psi\psi$$7

$$\mathcal{L}_{\rm Dirac} = \frac{i\hbar c}{2} \Big(\bar\psi\gamma^{\mu}\nabla_{\mu}\psi -\nabla_{\mu}\bar\psi\,\gamma^{\mu}\psi\Big) -mc^{2}\bar\psi\psi$$8

$$\mathcal{L}_{\rm Dirac} = \frac{i\hbar c}{2} \Big(\bar\psi\gamma^{\mu}\nabla_{\mu}\psi -\nabla_{\mu}\bar\psi\,\gamma^{\mu}\psi\Big) -mc^{2}\bar\psi\psi$$9

$$L_{\rm Dirac}=i\hbar\,\bar\psi\,\gamma^\mu\nabla_\mu\psi-mc\,\bar\psi\psi,$$0

where

$$L_{\rm Dirac}=i\hbar\,\bar\psi\,\gamma^\mu\nabla_\mu\psi-mc\,\bar\psi\psi,$$1

and

$$L_{\rm Dirac}=i\hbar\,\bar\psi\,\gamma^\mu\nabla_\mu\psi-mc\,\bar\psi\psi,$$2

(Khanapurkar et al., 2018). This formulation isolates the effect of torsion as specific contorsion contributions to NP spin coefficients, a feature also developed in the broader Einstein–Cartan–Dirac analysis (Khanapurkar, 2018).

On Minkowski space with planar symmetry and dependence only on $$L_{\rm Dirac}=i\hbar\,\bar\psi\,\gamma^\mu\nabla_\mu\psi-mc\,\bar\psi\psi,$$3, the NP system reduces to a 1+1-dimensional nonlinear Dirac system. Imposing $$L_{\rm Dirac}=i\hbar\,\bar\psi\,\gamma^\mu\nabla_\mu\psi-mc\,\bar\psi\psi,$$4, $$L_{\rm Dirac}=i\hbar\,\bar\psi\,\gamma^\mu\nabla_\mu\psi-mc\,\bar\psi\psi,$$5, and defining

$$L_{\rm Dirac}=i\hbar\,\bar\psi\,\gamma^\mu\nabla_\mu\psi-mc\,\bar\psi\psi,$$6

one arrives at

$$L_{\rm Dirac}=i\hbar\,\bar\psi\,\gamma^\mu\nabla_\mu\psi-mc\,\bar\psi\psi,$$7

$$L_{\rm Dirac}=i\hbar\,\bar\psi\,\gamma^\mu\nabla_\mu\psi-mc\,\bar\psi\psi,$$8

(Khanapurkar et al., 2018). A solitary-wave ansatz

$$L_{\rm Dirac}=i\hbar\,\bar\psi\,\gamma^\mu\nabla_\mu\psi-mc\,\bar\psi\psi,$$9

then yields explicit solitonic solutions for $\nabla_\mu$0 (Khanapurkar et al., 2018). The paper reports that in the pure Dirac case the same ansatz yields only plane waves or exponentially growing solutions, whereas the cubic torsion term provides the self-interaction required for nonsingular localized lumps (Khanapurkar et al., 2018).

A closely related 1+1-dimensional analysis connects solitary-wave solutions of the Hehl–Datta equation to a generalized uncertainty principle. Using the reduced system and explicit localized profiles $\nabla_\mu$1, $\nabla_\mu$2, one obtains

$\nabla_\mu$3

which can be rewritten as

$\nabla_\mu$4

(Ramesh, 2019). In the limit $\nabla_\mu$5, the ordinary Heisenberg relation is recovered, while for other values of $\nabla_\mu$6 the familiar quadratic correction appears (Ramesh, 2019). This suggests that, within that reduced nonlinear spinor model, the Hehl–Datta self-interaction can be associated with minimum-length behavior.

7. Generalizations, assumptions, and interpretive boundaries

Several works generalize the standard Einstein–Cartan–Dirac system by replacing the fixed Planck length with a mass-dependent Compton–Schwarzschild length

$\nabla_\mu$7

or by related unified-length prescriptions (Singh, 2017). In the resulting modified Einstein–Cartan–Dirac equations, the cubic self-interaction takes the form

$\nabla_\mu$8

so the torsion–spin coupling becomes mass-dependent through

$\nabla_\mu$9

(Singh, 2017). These constructions are presented as extensions of the standard Hehl–Datta framework rather than as part of its universally accepted canonical form.

The assumptions behind the standard derivation are comparatively stable across the cited literature. The torsion–spin coupling is treated at the classical level, $${\cal L}_{\rm tot} = -\frac{1}{2\kappa}\,e\,R + e\,\Bigl[\tfrac{i}{2}\bigl(\bar\psi\gamma^{a}\psi_{;a}-\bar\psi_{;a}\gamma^{a}\psi\bigr) - m\,\bar\psi\,\psi\Bigr],$$0 is a commuting Dirac spinor, torsion is solved algebraically and substituted back into an effective metric-plus-spinor theory, and the Hehl–Datta term is negligible except at very high densities or very short distances (Poplawski, 2011). No propagating torsion waves are present in Einstein–Cartan theory as used in these derivations (Poplawski, 2011, III et al., 2017, Khanapurkar, 2018). The equation is therefore exact within classical Einstein–Cartan–Dirac theory under minimal coupling, metric compatibility, vanishing nonmetricity, and algebraic torsion elimination (Khanapurkar, 2018).

A recurring misconception is that the Hehl–Datta equation introduces an arbitrary nonlinear correction to the Dirac equation. The literature instead treats it as the unique nonlinear term obtained by minimal coupling of spin-$${\cal L}_{\rm tot} = -\frac{1}{2\kappa}\,e\,R + e\,\Bigl[\tfrac{i}{2}\bigl(\bar\psi\gamma^{a}\psi_{;a}-\bar\psi_{;a}\gamma^{a}\psi\bigr) - m\,\bar\psi\,\psi\Bigr],$$1 matter to Einstein–Cartan geometry and subsequent elimination of torsion (III et al., 2022, Khanapurkar et al., 2018). Another misconception is that torsion in this context generically predicts observable laboratory deviations from Dirac theory. The cited papers consistently state that the torsion-induced correction is negligible outside ultradense regimes such as the very early Universe, black-hole interiors, or Planck-scale localization scenarios (Poplawski, 2011, III et al., 2022).

In that sense, the Hehl–Datta equation occupies a specific position in gravitational spinor theory: it is the effective nonlinear Dirac equation characteristic of Einstein–Cartan–Dirac dynamics, mathematically defined by a cubic axial-current self-coupling and physically interpreted as the imprint of spin-induced torsion on fermionic propagation. Its significance lies less in everyday phenomenology than in the way it ties spin, torsion, and local fermionic self-interaction into a single classical field equation (Poplawski, 2011, III et al., 2017, Khanapurkar et al., 2018).