Eidic Geometry, Gravitational Embedding and Berry-Induced Uncertainty

Eidic Geometry, Gravitational Embedding and Berry-Induced Uncertainty Enrique García Salcinesa,* aRabanales Campus, Carretera Madrid-Córdoba Km. 397, 14014, University of Cordoba, Spain Note This technical note records how the orientational (eidic) field theory introduced in [1] can be extended so that its high–coherence sector reproduces a first–order gravitational theory of Cartan–Palatini type, while its Berry sector already contains a Heisenberg-type uncertainty structure. The aim is to fix priority on the idea that the same orientational medium that supports Hopf-labeled finite-energy configurations can (i) supply a tetrad and a spin connection and (ii) generate quantum commutation relations from a first-orderin-time term. 1. Idea In the eidic framework of [1] the fundamental object is not a positional field but an orientational section Φ on a gauge/associated bundle over 4D spacetime. From Φ we already built a gauge-invariant direction 𝐧 ∈ 𝑆! and an abelian composite with Hopf charge 𝑄 ∈ ℤ, stabilized by a Faddeev–Skyrme term [2,3]. Here we observe that, in a regime where the eidic configuration is smooth and highly coherent (covariant derivatives small, curvature mild), the orientational data can be “soldered’’ to the tangent bundle and read as a coframe. Concretely, pick a local frame in the internal space and define 𝑒"#(𝑥) = 𝒩 Ξ"$(Φ(𝑥)) (𝐷#Φ)$, 0 where Ξ"$(Φ) is an algebraic projector built from Φ(for instance using the CP% direction and its orthogonal complement) and 𝒩is a normalization chosen so that det0(𝑒"#) ≠ 0 on the coherent patch. When the eidic field varies slowly, (𝐷#Φ)$span four independent directions and the matrix (𝑒"#) can be interpreted as a tetrad. A compatible spin connection 𝜔"&# is then obtained either from metric-compatibility with 𝑒" or from the underlying gauge connection. Thus the same eidic medium supplies the pair (𝑒", 𝜔"&) that ordinary first-order gravity takes as fundamental. Scale of validity. This “geometrization” is meant for the long-wavelength / highcoherence regime of the eidic theory: schematically ∥ 𝐷Φ ∥≪ Λcoh and eidic curvature ∥ 𝐹[𝐧] ∥≪ Λcoh !, where Λcoh is the coherence scale that also suppresses the Faddeev– Skyrme term. At shorter scales the description reverts to the full eidic Lagrangian (with knots, links and VK scaling); at longer scales it is consistent to trade the orientational data for (𝑒", 𝜔"&). 2. Gravitational sector from induced data * [email protected] Once 𝑒" and 𝜔"& are available, the natural gravitational Lagrangian is the Cartan–Palatini one, possibly with the usual topological refinements: ℒgrav =1 16𝜋𝐺 𝜖"&'( 𝑒"∧ 𝑒&∧ 𝑅'( +1 16𝜋𝐺 𝛾 𝑒"∧ 𝑒&∧ 𝑅"& + 𝛽 ℒ)* + 𝜃+ Tr(𝑅 ∧ 𝑅) + 𝜃, 𝜖"&'(𝑅"& ∧ 𝑅'(, 0 with 𝑅"& = 𝑑𝜔"& + 𝜔"'∧ 𝜔'&. Here the Holst term (with Barbero–Immirzi parameter 𝛾) and the Nieh–Yan density appear on the same footing as in standard first-order gravity [4,5]; the point is that now they are functionals of the eidic configuration, not extra fields. In the smooth/coherent limit this reduces to Einstein–Cartan, and for vanishing torsion to ordinary GR. 3. Eidic torsion source The original eidic theory contains a topological/helicity current built from the CP% direction, e.g. 𝐽top #∝ 𝜖#-./𝐴0,-𝐻./, 0 which measures how much the orientational field is twisted or linked. Since Cartan– Palatini gravity admits torsion 𝑇"= 𝐷𝑒", the most natural local coupling between “angular twist’’ and geometry is ℒcpl = 𝜉 𝑇"∧ 𝒥top "[Φ], 0 where 𝒥top " is the tetrad-projected version of the topological current. This makes regions of large eidic helicity act as sources of torsion. In weakly twisted configurations the current vanishes and torsion is driven to zero, so the theory flows back to GR; near knotted, high-𝑄 configurations the same medium that supports Hopf solitons also twists the coframe. 4. Berry term, Heisenberg-type uncertainty, and its normalization Independently of the geometric limit, the eidic Lagrangian admits a first-order-in-time piece of the form ℒBerry = 𝜂2 𝜌eidic 𝜃U, 0 where 𝜃 is the local angular phase of the eido and 𝜌eidic its coherence/amplitude. This is the standard geometric/Berry structure: it tells us that 𝜃and 𝜌form a canonical pair. When one quantizes a system with such a term, the coefficient in front fixes the commutator: [𝜃 V(𝑥), 𝜌W(𝑥)] = 𝑖 𝜂2.0 In the usual quantum–mechanical normalization we identify 𝜂2= ℏ, 0 so that the eidic theory reproduces the familiar relation [𝜃 V(𝑥), 𝜌W(𝑥)] = 𝑖ℏ ⇒ Δ𝜃 Δ𝜌 ≥ ℏ 2.0 Written this way, the Heisenberg uncertainty is not an extra postulate but a consequence of having a Berry-type, first-order time term for the orientational field. Since 𝜂2 is, in the eídic picture, a property of the medium, this makes explicit how a single orientational field can underwrite both a gravitational (long-wavelength) sector and a quantum (firstorder/phase) sector. 5. Compatibility with knotted sectors Because the construction of 𝑒" and 𝜔"& is done in the coherent/slow sector, it does not destroy the finite-energy, Hopf-labeled configurations of the original eidic Lagrangian. Static configurations can still be compactified ℝ3→ 𝑆3, the map 𝐧: 𝑆3→ 𝑆!still defines 𝑄 ∈ ℤ, and the Faddeev–Skyrme rigidity still enforces the Vakulenko–Kapitanskii lower bound 𝐸 ≥ 𝐶 ∣ 𝑄 ∣3/5 [6]. The difference is that now these knotted eidic textures live on (and weakly backreact on) an induced Cartan–Palatini geometry. 6. Outlook This embedding opens the way to study eidic textures near horizons or domain walls (where coherence is cut and Berry phases become physical), to place electroweak-type holonomy effects on top of an induced geometry, and to revisit cosmological mechanisms in a setting where quantum, topological and gravitational pieces all come from one orientational medium. References [1] García Salcines, E. (2025). “Eidic Field Theory: Hopf sectors, Skyrme-type rigidity and the Vakulenko–Kapitanskii 3/4 bound”. SSRN. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5705578 [2] L. D. Faddeev, A. J. Niemi, “Stable knot-like structures in classical field theory”, Nature 387 (1997) 58–61. [3] L. D. Faddeev, “Quantization of solitons”, Princeton preprint IAS-75-QS70 (1975). [4] S. Holst, “Barbero’s Hamiltonian derived from a generalized Hilbert–Palatini action”, Phys. Rev. D 53 (1996) 5966–5969. [5] H. T. Nieh, M. L. Yan, “An identity in Riemann–Cartan geometry”, J. Math. Phys. 23 (1982) 373. [6] S. Vakulenko, L. Kapitanskii, “Stability of solitons in S^2 in the nonlinear sigma model”, Sov. Phys. Dokl. 24 (1979) 433.