Quantum geometry is conventionally understood in parameter or momentum space and need not be related to a geometry in configuration space. Here we show for Dirac fermions in a background electromagnetic field that an induced Lorentzian geometry is generated from the fermionic quantum metric. We study a Lorentz covariant phase space worldline path integral construction, whose application of the adiabatic theorem onto spin degrees of freedom leads to the appearance of the Berry phase at $O(\hbar)$, and the quantum metric at $O(\hbar^2)$ in the worldline action. Both the Feynman and retarded/advanced propagators are examined in 3+1 and 2+1 dimensions, whose different on-shell implementations lead to related quantum metrics; QED/QED${}_3$ and condensed matter Dirac theory are considered. An effective theory is found by integrating out the momenta degrees of freedom, which gives rise to an emergent Lorentzian metric in configuration space, whose perturbation to the flat metric is dependent on the quantum metric and the background field strength. Apart from the spin degrees of freedom encoded into the Berry phase, the effective theory describes a scalar particle moving in the induced quantum geometry. Phenomenological applications to the covariant geometry are further studied using spectral holonomy in a gapped 3-dimensional Dirac material.
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Can momentum-space quantum geometry induce a configuration-space spacetime metric? This paper studies massive Dirac fermions in background electromagnetic fields using phase-space worldline path integrals, treating Feynman and retarded/advanced propagators in $3+1$ and $2+1$ dimensions, with corresponding interpretations for gapped Dirac materials.
An adiabatic expansion of the spin dynamics produces the Berry connection at $O(\hbar)$ and a quantum-metric term quadratic in momentum derivatives at $O(\hbar^2)$. The covariant Feynman construction gives $\mathfrak g_{ab}=-(\eta_{ab}-\hat p_a\hat p_b)/(4p^2)$, a hyperbolic metric on normalized timelike momenta with a null radial direction. The Hamiltonian construction gives a related tensor with different normalization and complementary subspaces. Using leading Lorentz-force trajectories and semiclassical momentum integration, the quadratic term becomes an effective worldline kinetic term with $g^{\mathrm{ind}}_{\mu\nu}=\eta_{\mu\nu}+\hbar^2h_{\mu\nu}$, where $h$ is proportional to $F\mathfrak g F$ and retains dependence on the classical momentum.
For the covariant construction, the deformation has rank at most three and annihilates the rest-frame magnetic-field direction. A band-edge annulus example predicts a crossed-field contribution to counterpropagating-mode splitting, $\Delta\epsilon_n=n\hbar v_F\varepsilon_E\varepsilon_B/(2R)$, odd under reversal of either field.
The treatment is perturbative through $O(\hbar^2)$ and assumes slow, weak field-driven evolution suppressing interband transitions and vacuum non-persistence. Gap closing and strong-field Landau–Zener dynamics are excluded. The material models neglect lattice corrections, anisotropy, remote bands, disorder, and strong interactions; the spectral example additionally uses a near-band-edge approximation.
Little sign of AI-generated textOverclaimsUnclear noveltyStrong assumptionsPossible error
Central claim may not hold: The displayed induced metrics retain dependence on a chosen classical momentum history or local velocity, so the derivation does not generally establish the unique configuration-space geometry advertised.
The momentum elimination omits the second-order contribution from the Berry-induced first-order saddle displacement, which is nonzero already for the planar Dirac model and can cancel the retained quantum-metric correction at the band edge.
The curved-space conversion incorrectly treats a coordinate Jacobian as a vielbein capable of producing nonzero curvature from Minkowski space.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
Quantum geometry is conventionally understood in parameter or momentum space and need not be related to a geometry in configuration space. Here we show for Dirac fermions in a background electromagnetic field that an induced Lorentzian geometry is generated from the fermionic quantum metric. We study a Lorentz covariant phase space worldline path integral construction, whose application of the adiabatic theorem onto spin degrees of freedom leads to the appearance of the Berry phase at O(ℏ), and the quantum metric at O(ℏ2) in the worldline action. Both the Feynman and retarded/advanced propagators are examined in 3+1 and 2+1 dimensions, whose different on-shell implementations lead to related quantum metrics; QED/QED3 and condensed matter Dirac theory are considered. An effective theory is found by integrating out the momenta degrees of freedom, which gives rise to an emergent Lorentzian metric in configuration space, whose perturbation to the flat metric is dependent on the quantum metric and the background field strength. Apart from the spin degrees of freedom encoded into the Berry phase, the effective theory describes a scalar particle moving in the induced quantum geometry. Phenomenological applications to the covariant geometry are further studied using spectral holonomy in a gapped 3-dimensional Dirac material.
Can momentum-space quantum geometry induce a configuration-space spacetime metric? This paper studies massive Dirac fermions in background electromagnetic fields using phase-space worldline path integrals, treating Feynman and retarded/advanced propagators in 3+1 and 2+1 dimensions, with corresponding interpretations for gapped Dirac materials.
An adiabatic expansion of the spin dynamics produces the Berry connection at O(ℏ) and a quantum-metric term quadratic in momentum derivatives at O(ℏ2). The covariant Feynman construction gives gab=−(ηab−p^ap^b)/(4p2), a hyperbolic metric on normalized timelike momenta with a null radial direction. The Hamiltonian construction gives a related tensor with different normalization and complementary subspaces. Using leading Lorentz-force trajectories and semiclassical momentum integration, the quadratic term becomes an effective worldline kinetic term with gμνind=ημν+ℏ2hμν, where h is proportional to FgF and retains dependence on the classical momentum.
For the covariant construction, the deformation has rank at most three and annihilates the rest-frame magnetic-field direction. A band-edge annulus example predicts a crossed-field contribution to counterpropagating-mode splitting, Δϵn=nℏvFεEεB/(2R), odd under reversal of either field.
The treatment is perturbative through O(ℏ2) and assumes slow, weak field-driven evolution suppressing interband transitions and vacuum non-persistence. Gap closing and strong-field Landau–Zener dynamics are excluded. The material models neglect lattice corrections, anisotropy, remote bands, disorder, and strong interactions; the spectral example additionally uses a near-band-edge approximation.
This work resolves a foundational tension in Dirac's formulation of relativistic quantum mechanics: momentum appears in two distinct but undifferentiated roles—as a dispersion-locked parameter generating unitary evolution in the relativistic wave equation, and as a free phase variable subject to symplectic flow in the constraint formalism. We demonstrate that this duality reflects an intrinsic Lorentzian structure in energy-momentum space, characterized by the interval dΣ² = dE²/c² − dp². This geometry divides naturally into three sectors: symmetric (dispersion-locked), null (massless, E/|p| = c), and antisymmetric (free-momentum) regimes, each governed by different geometric constraints. Dispersion emerges as the global integral of an exact action one-form, while the (E,p) symplectic structure arises locally as the exterior derivative of the action's inexact form θ_E = E dp. Energy quantization appears as a topological stability condition requiring closed spectral loops to satisfy ∮ E dp = nh, defining "proto-eigenstates" whose Fourier encodings yield standard Hilbert-space eigenstates. We recover both the Bohr spectrum and Dirac fine-structure energies from spectral phase closure, revealing α = e²/(ℏc) as a geometric action-normalization ratio between Maxwell U(1) curvature and Lorentz spinor action—not a coupling parameter from QED perturbation theory. The framework synthesizes unitary flow and symplectic uncertainty into a single bilinear Lorentzian structure, showing quantum states as objective in their symmetric unitary existence but path-dependent in their antisymmetric observed manifestation. This geometric reconstruction bridges Dirac's wave mechanics and constraint formalism through native (E,p) phase space, offering a foundation for quantum theory that treats energy-momentum as primary coordinates rather than conjugate variables derived from spacetime position.
This Volume III develops a complete geometric formulation of quantum mechanics, demonstrating that quantum phenomena emerge from the very structure of spacetime. We demonstrate that Einstein's field equations—when interpreted in the spectral domain and taking phase coherence into account—already contain the entire structure of quantization. Quantum mechanics appears not as an axiom, but as a geometric consequence. Based on the experimental and mathematical results consolidated in Volumes~0, I, and II, the present work shows that quantization need not be postulated: it arises inevitably from three fundamental geometric elements—proper frequency, parallel transport, and phase coherence. The action appears as the phase accumulated along the worldline, and its discretization into coherence windows, derived without external hypotheses, naturally leads to the condition Sn=nh and to all associated quantum phenomenology. We derive the Schrödinger equation as the limit of short windows, obtain the commutator [x,p]=iℏ from the incompatibility between spatiotemporal synchronizations, and reinterpret interference, propagators, and decoherence as spectral properties of parallel transport. The Hilbert space arises as the spectral space of the metric, observables correspond to generators of isometric deformations, while the collapse is understood as the geometric selection of coherent subspaces. The formalism extends to relativistic particles, spin, quantum statistics, the Dirac equation, harmonic oscillators, and field quantization, where spectral modes of the metric assume the role of particles: photons as harmonic 1-forms, weak modes as connections of 2-forms, gluons as 3-forms, and the Higgs as the fundamental scalar curvature mode. The reinterpretation of the Standard Model thus becomes a direct consequence of spectral geometry. The Volume culminates in the Central Theorem —Geometry = Quantization— showing that the entire quantum structure is a manifestation of the spectral granularity of spacetime. We conclude with experimental predictions and falsification criteria, as well as the conceptual framework necessary for Volume~IV, dedicated to the complete quantization of the metric within the same paradigm.
A Geometric Origin for Fermi--Dirac Statistics: The Spin-Statistics Connection from Z2 Normal Holonomy Without Lorentz Invariance
The Pauli exclusion principle---the reason atoms have shell structure and matter is stable against collapse---is conventionally explained through the spin-statistics theorem, whose every standard proof relies on the machinery of relativistic quantum field theory (Lorentz invariance and microcausality). Yet the principle governs entirely non-relativistic physics. This paper asks whether its non-relativistic content admits a correspondingly non-relativistic explanation, and argues that it does: the Fermi--Dirac statistics of a half-integer-spin particle can be obtained from the topology of three-dimensional space together with the M\"obius geometry of the particle, with no appeal to Lorentz invariance or microcausality. The relativistic input of the standard proof is replaced by a geometric one. Concretely, the proof rests on two algebraic-topological lemmas about three-dimensional space (the configuration space of two identical particles is homotopy equivalent to ^2; real line bundles over ^2 are classified by H^1(^2;_2) _2), together with an explicit equivariant bundle map identifying the relevant exchange line bundle with the normal bundle of a non-orientable surface in ^3. The construction realises the spin-statistics correspondence as a topological identity in the spirit of Kuckert--Mund~KuckertMund2004 and Anastopoulos~Anastopoulos2004, but using only standard algebraic topology without recourse to QFT axioms or geometric quantisation. If the argument stands, the Pauli exclusion principle emerges as a statement about the topology of three-dimensional space and the geometry of the particle, rather than about the causal structure of spacetime. The required geometric input on the particle is encapsulated as a single postulate (Section~sec:setup): identical fermionic particles are modelled as compact embedded surfaces in ^3 equipped with light-speed-constrained flow fields generating _2 normal holonomy () = -1. In this setting we prove: enumerate[label=(*)] The two-particle configuration space is homotopy equivalent to the real projective plane: C_2 ^2, so _1(C_2) _2. Real line bundles over ^2 are classified by H^1(^2;_2) _2: there are exactly two, the trivial bundle (holonomy +1) and the tautological Möbius bundle (holonomy -1). The exchange holonomy of the fermionic vortex-knot wavefunction equals -1 (a unit normal section traversing undergoes a sign reversal). An explicit equivariant bundle map : N|_L_{ex}|_, (s,) = (s,), is constructed and proved to be a well-defined bundle isomorphism if and only if () = -1 (Section~sec:equivariant, Proposition~prop:chi). The argument rests on one geometric axiom---the Normal Transport Axiom (NTA), that is connection-preserving---which selects =-1 from the two phases the configuration-space topology admits, playing the role microcausality plays in the standard relativistic proof (Section~sec:equivariant). The contribution is to show this selecting input can be geometric rather than relativistic. Therefore the two-particle wavefunction is antisymmetric under exchange: Fermi--Dirac statistics follows. enumerate Points (i)--(iii) are rigorous theorems. Point~(iv) is also rigorous (the bundle map is explicit and proved); the NTA is the one remaining axiom---a precisely stated geometric assumption that supplies the selection input required by the Leinaas--Myrheim classification, with the same logical status as microcausality in the standard proof. Point~(v) follows under the NTA. At the heart of the result is a striking geometric correspondence: the non-orientability of the Möbius particle (in ^3) and the non-orientability of the two-particle configuration space ^2 are the same _2 at two different scales---the scale of one particle and the scale of a pair---connected by the explicit bundle map . : spin-statistics theorem; holonomy; Möbius surface; real projective plane; configuration space; line bundle classification; exchange statistics; normal holonomy; Fermi--Dirac statistics; equivariant b
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.
It is well-known that Dirac particles gain geometric phase, namely Berry phase, while moving in an electromagnetic field. Researchers have already shown covariant formalism for the Berry connection due to an electromagnetic field. A similar effect is expected to happen due to the presence of Gravity. We use WKB approximation to develop a covariant formalism of Berry-like connection in the presence of Einstein gravity, which can be further used to describe the Berry-like phase or simply Berry phase. We also extend this formalism for massless Dirac particles (Weyl particles).Then we further show that this connection can be split into two parts, one of which vanishes when the metric is spherically symmetric and thus can be linked to the Aharonov-Bohm-like effect in the 3 + 1 formalism. At the same time, the other term can be related to the Pancharatnam-Berry like effect.
Non-Hermitian systems exhibit spectral and topological phenomena absent in Hermitian physics; however, their geometric characterization remains subtle due to the intrinsic ambiguity of biorthogonal eigenspaces. Since left and right eigenvectors are not related by Hermitian conjugation, the associated Berry connection is generally nonunique, leading to complex geometric phases and ambiguously defined holonomies. Here we formulate a covariant geometric framework for non-Hermitian quantum systems based on the metric structure of the underlying Hilbert space. We show that, in the quantum regime with continuous state evolution, the conventional Berry connection and the associated Berry holonomy over closed parameter-space loops can be consistently defined only in the pseudo-Hermitian limit, where the spectrum is real. For generic non-Hermitian Hamiltonians with complex spectra, the relevant geometric object is instead the Aharonov--Anandan holonomy associated with cyclic evolution in projective Hilbert space. Within the pseudo-Hermitian regime, we construct a unique Hermitian Berry connection that is covariant under arbitrary GL(N,C) frame transformations and reduces to the standard Berry connection in the Hermitian limit. The resulting formalism separates the intrinsic geometry of the Hamiltonian eigenspace from contributions arising from the parameter dependence of the Hilbert-space metric, revealing that the conventional biorthogonal formulation generally mixes these distinct geometric effects. Consequently, geometric phases, synthetic gauge fields, and topological characteristics commonly attributed to non-Hermitian eigenspace geometry may, in part, originate from the underlying metric structure. Our framework therefore provides a consistent geometric foundation for Berry phases, non-Abelian holonomies, and topological invariants in non-Hermitian quantum systems.