⚛️ high-energy theory

Induced Lorentzian geometry from the fermionic quantum metric

This paper demonstrates that for Dirac fermions in a background electromagnetic field, integrating out momentum degrees of freedom within a Lorentz covariant path integral framework induces an emergent Lorentzian geometry in configuration space, where the metric perturbation is determined by the fermionic quantum metric and field strength.

Original authors: Patrick Copinger, Pablo Morales

Published 2026-11-02

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CC BY 4.0

Original authors: Patrick Copinger, Pablo Morales

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern physics, geometry is not merely the study of shapes and spaces we can see with our eyes. It is a fundamental language used to describe how the universe is put together, from the curvature of spacetime around a star to the invisible patterns that govern the behavior of electrons inside a computer chip. For decades, physicists have understood that quantum particles, such as electrons, carry their own internal "map" of geometry. This map, known as the quantum metric, usually exists in an abstract realm of momentum or parameters, describing how easily one quantum state can be distinguished from another nearby one. While this concept has successfully explained many phenomena in materials science, a lingering question has remained: can this abstract, internal geometry ever become a real, physical shape in the space where particles actually move?

A team of researchers has now provided a definitive answer to this question, showing that the internal geometry of fermions—particles like electrons—can indeed generate a new, physical geometry in the space they inhabit. By studying how these particles move through electromagnetic fields, the scientists discovered that the quantum metric acts as a blueprint, inducing a real, curved spacetime structure around the particle. This is not a metaphorical curvature but a tangible alteration of the path the particle takes, effectively creating a new landscape that depends on the strength of the surrounding fields and the particle's own quantum properties. The findings suggest that the geometry of the quantum world is not just a mathematical curiosity but a force that can actively reshape the physical stage upon which particles perform.

To reach this conclusion, the researchers focused on a specific type of particle called a Dirac fermion, which behaves like a relativistic electron. They employed a powerful mathematical tool known as the worldline formalism, which treats the history of a particle not as a single line but as a sum of all possible paths it could take. In this framework, the particle's journey is analyzed by looking at how its internal spin—a kind of intrinsic angular momentum—evolves as it moves. The team applied a principle called the adiabatic theorem, which describes how a system changes when its environment shifts very slowly. When this principle was applied to the spin of the fermion moving through an electromagnetic field, two distinct geometric features emerged from the mathematics: a phase shift known as the Berry phase, and the quantum metric itself.

The study revealed that these geometric features are not just side effects but are central to the particle's motion. As the researchers integrated out the complex momentum details of the particle's path, a surprising result appeared in the remaining equations: the particle began to move as if it were traveling through a curved space. This new space, or induced metric, was not the flat, empty stage of standard physics but a warped geometry directly shaped by the quantum metric and the strength of the external electromagnetic field. The researchers found that this effect occurs in both three-dimensional space and two-dimensional systems, and it holds true whether they looked at the particle's behavior using different mathematical descriptions of time and causality.

One of the most significant aspects of this work is that the induced geometry is Lorentzian, meaning it respects the rules of special relativity, where time and space are treated on an equal footing. The researchers showed that the curvature of this new space is proportional to the square of the electromagnetic field strength and the quantum metric. In simpler terms, the stronger the field and the more distinct the quantum states, the more the space around the particle curves. This curvature is not a random distortion; it follows a precise pattern determined by the interaction between the field and the particle's internal structure. The study also highlighted that the specific shape of this induced geometry depends on whether the particle is described as a Feynman propagator, which accounts for all possible time directions, or as an advanced or retarded propagator, which describes cause-and-effect evolution in a specific time direction.

The implications of this discovery extend beyond pure theory. The researchers demonstrated that this induced geometry has real, measurable consequences for how particles move in materials. For instance, in a gapped three-dimensional Dirac material, the induced metric creates a specific direction where the geometry remains flat, known as a magnetic null direction. This direction is fixed by the local magnetic field and the particle's velocity. Furthermore, the study showed that when particles travel in opposite directions around a loop in a crossed electric and magnetic field, they experience a difference in travel time. This delay, known as spectral holonomy, is a direct result of the mixed time-space components of the induced geometry, effectively splitting the energy of the counter-propagating modes.

This work bridges a gap between the abstract mathematics of quantum geometry and the concrete reality of spacetime curvature. By showing that the quantum metric can generate an emergent Lorentzian geometry, the researchers have provided a new way to understand how the microscopic world influences the macroscopic stage. The findings suggest that the geometry of space is not a fixed background but a dynamic entity that can be molded by the quantum properties of the matter within it. This insight opens new avenues for exploring how quantum effects might manifest in gravitational physics and condensed matter systems, offering a fresh perspective on the deep connection between the quantum world and the fabric of spacetime.

Technical Summary: Induced Lorentzian Geometry from the Fermionic Quantum Metric

Problem and Motivation
Quantum geometry, conventionally defined in parameter or momentum space via the Quantum Geometric Tensor (QGT), is typically distinct from the geometry of configuration space. The QGT comprises the Berry curvature (imaginary part), associated with topological phases, and the quantum metric (QM, real part), which quantifies the distinguishability of neighboring quantum states. While the QM is known to influence various condensed matter phenomena (e.g., Wannier localization, orbital susceptibility), its interpretation as a physical metric raises a fundamental question: can this internal quantum-state geometry induce an emergent spacetime geometry in configuration space?

This paper addresses this question by analyzing Dirac fermions in background electromagnetic fields. The authors aim to demonstrate that an induced Lorentzian geometry arises in configuration space directly from the fermionic quantum metric, moving beyond the standard view of quantum geometry as merely a feature of parameter space.

Methodology
The authors employ a Lorentz-covariant phase-space worldline path integral formalism. This approach allows for the application of the adiabatic theorem to the spin degrees of freedom within a first-quantized setting, which is necessary to handle the matrix-weighted actions of Quantum Field Theory (QFT) and Condensed Matter systems (specifically QED, QED3, and gapped Dirac materials).

The methodology proceeds through the following steps:

Formulation: The Feynman, retarded, and advanced propagators are expressed as path integrals over relativistic trajectories in phase space. The action includes a path-ordered exponential of the worldline Hamiltonian ( $\not{p}$ for Feynman, $H_k$ for retarded/advanced).
Adiabatic Expansion: The adiabatic theorem is applied to the spin degrees of freedom. This expansion generates a Berry phase at order $\mathcal{O}(\hbar)$ and the quantum metric at order $\mathcal{O}(\hbar^2)$ within the worldline action.
Diagonalization: The effective Hamiltonian is diagonalized using similarity transforms. For the Feynman propagator, the authors explore two diagonalization schemes: one using a $Spin(3,1)$ transform to $\gamma_0$ and another using a Clifford group transform to $\gamma_5$ . For retarded/advanced propagators, diagonalization is performed with respect to energy eigenvalues.
Integration of Momenta: The momentum degrees of freedom are integrated out from the path integral. The classical equations of motion (Lorentz force) are used to relate momentum changes to spacetime displacements. This step effectively pulls back the momentum-space quantum metric to configuration space.

Key Contributions and Results

Emergent Lorentzian Metric: The primary result is the derivation of an effective single-particle Lagrangian in configuration space that includes an induced metric $g^{\text{ind}}_{\mu\nu} = \eta_{\mu\nu} + \hbar^2 h_{\mu\nu}$ . The perturbation $h_{\mu\nu}$ is explicitly dependent on the quantum metric and the background electromagnetic field strength ( $F_{\mu\nu}$ ). Specifically, the perturbation scales as $h_{\mu\nu} \propto F_{\mu\alpha} g^{\alpha\beta} F_{\beta\nu}$ .
Distinction Between Propagators: The paper demonstrates that the induced geometry differs depending on the propagator used:
- Feynman Propagator: The adiabatic theorem is applied before enforcing the on-shell condition ( $p^2 = m^2$ ). The resulting quantum metric is covariant and defined on the future cone of momentum space. The induced metric perturbation is derived for both $\gamma_0$ and $\gamma_5$ diagonalizations, with the latter preserving manifest Lorentz invariance in the action.
- Retarded/Advanced Propagators: The adiabatic theorem is applied after enforcing the on-shell condition (or energy projection). The resulting quantum metric differs in form (e.g., lacking temporal components in the Hamiltonian formulation) and leads to a distinct, though related, induced metric perturbation.
Dimensional Analysis: The results are established for both $(3+1)$ -dimensional and $(2+1)$ -dimensional systems, covering both high-energy QED and condensed matter systems (gapped Dirac semimetals, gapped graphene, Chern insulators).
Phenomenological Application: The authors analyze the induced geometry in a gapped 3D Dirac material. They identify a "magnetic null direction" where the metric deformation vanishes, fixed by the local field and particle velocity. Furthermore, they show that crossed electric and magnetic fields generate a mixed time-space metric component ( $g^{\text{ind}}_{0i}$ ). This component leads to a spectral holonomy: a geometric splitting of counterpropagating modes in an annulus, resulting in an energy splitting proportional to $E_r B_z$ and a propagation delay dependent on the circulation of the fields.

Significance and Claims
The paper claims to affirmatively answer whether quantum-state geometry can relate to emergent spacetime geometry. By integrating out momentum degrees of freedom in the worldline formalism, the authors show that the quantum metric, traditionally a measure of state distinguishability in momentum space, directly generates a unique induced Lorentzian geometry in configuration space.

The significance lies in:

Unification: It provides a mechanism where the internal geometry of the Hilbert space (the QM) manifests as a physical metric in spacetime, distinct from the background metric.
Universality: The phenomenon is shown to apply across different propagator types (Feynman vs. retarded/advanced) and dimensions, suggesting a robust feature of fermionic systems in background fields.
New Geometric Effects: The induced geometry predicts specific phenomenological effects, such as the magnetic null direction and the geometric splitting of modes in crossed fields, which are distinct from standard Aharonov-Bohm effects.

The authors note that while the study focuses on fermionic systems, the techniques could potentially be extended to other systems with Berry phases, such as QCD, where color degrees of freedom might similarly induce geometry in configuration space. However, the paper confines its claims to the demonstrated results for Dirac fermions in electromagnetic backgrounds.

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