Electric Carroll Fermions

• Electric Carroll fermions are fermionic degrees of freedom in the ultra-relativistic Carroll regime, characterized by mass, ultralocal time dynamics, and suppressed spatial propagation.
• They arise as the c→0 limit of Dirac fermions where Carroll boosts act only on coordinates, leaving the spin representations invariant.
• Their various formulations, from flat-space actions to gravity coupling, offer insights into fracton models and flat-band holography.

Electric Carroll fermions are fermionic degrees of freedom in the ultra-relativistic Carroll regime in which time-derivative terms survive while spatial derivatives are suppressed or absent. In the recent literature, the term is used in two closely related senses: as the spin-$\tfrac12$ massive unitary irreducible representations of the Carroll group together with their ultralocal quantum fields, and as the $c\to0$ limits of relativistic Dirac fermions for which Carroll boosts act only through the coordinate part and not in the spin representation (Figueroa-O'Farrill et al., 2023, Bergshoeff et al., 2023).

1. Carrollian meaning of the electric sector

Carroll symmetry is obtained by contracting Poincaré symmetry in the limit of vanishing speed of light. Under Carroll boosts $\vec b$, one has

$$t' = t - \vec b\cdot\vec x,\qquad \vec x' = \vec x,$$

so space becomes absolute while time is shifted by a spatially dependent amount (Boer et al., 2021). In this regime, lightcones close up and spatial propagation is suppressed.

In the representation-theoretic language of the Carroll group, the massive sector is characterized by

$$E=E_0\neq 0,\qquad O_\tau \cong A^3=\{(\mathbf p,E_0)\mid \mathbf p\in\mathbb R^3\},$$

with little group $K_\tau\cong Spin(3)$. The corresponding massive spin-$s$ UIRs are denoted $\Romanbar{II}(s,E_0)$; for fermions the relevant case is $\Romanbar{II}(1/2,E_0)$ (Figueroa-O'Farrill et al., 2023).

A recurrent source of confusion is that “electric” and “magnetic” are not fundamental representation-theoretic labels. They arise at the field-theory level when one chooses Carroll-invariant actions and field equations. In the massive electric sector, the free equations are first order in time and contain no spatial derivatives; by contrast, magnetic Carroll theories retain spatial-derivative structures and, in the fermionic case, lead to a different boost representation (Figueroa-O'Farrill et al., 2023, Bergshoeff et al., 2023).

In the Dirac-limit approach, the electric sector is precisely the one for which the Carroll boost generators act only through the coordinate part, not in the spin representation. The spinor therefore transforms under spatial rotations in the standard way, but as a scalar under Carroll boosts. This contrasts with magnetic Carroll fermions, whose boost action is reducible but indecomposable (Bergshoeff et al., 2023).

2. Flat-space constructions and equations of motion

A central flat-space action for electric Carroll fermions is the minimal truncation obtained from the relativistic Dirac theory: $\boxed{ \mathcal{L}_{\text{electric Carroll} = i\,\bar\psi_+ \Gamma^0 \partial_t\psi_+ - \text{Re}(m)\,\bar\psi_+ \psi_+ . }$ This action follows from an off-diagonal parent Lagrangian for two Dirac spinors, an electric scaling of projected components, and the consistent truncation $c\to0$0. It contains no spatial derivative and preserves the electric Carroll transformation law in which only spatial rotations act on the spin indices (Bergshoeff et al., 2023).

The corresponding equation of motion is

$c\to0$1

or equivalently

$c\to0$2

The number of components is the same as for a relativistic Dirac spinor, but the dynamics is ultralocal in space (Bergshoeff et al., 2023).

A parallel derivation starts directly from the relativistic Dirac action

$c\to0$3

with the scalings

$c\to0$4

Taking $c\to0$5 suppresses the spatial derivative term and yields

$c\to0$6

with equation of motion

$c\to0$7

This makes explicit that electric Carroll Dirac theory is an infinite collection of identical $c\to0$8-dimensional systems labeled by $c\to0$9 (Ekiz et al., 8 Feb 2025).

A third formulation uses a Carrollian Clifford algebra adapted to the degenerate metric $\vec b$0, with Lagrangian

$\vec b$1

Here the adjoint is defined by $\vec b$2, and the equations of motion are

$\vec b$3

This model was introduced as the simplest consistent Carrollian fermion theory, rather than as a direct Lorentzian contraction (Sharma, 1 Feb 2025).

From the representation-theoretic side, the electric Carroll-Dirac equation is the ultralocal massive field equation

$\vec b$4

with $\vec b$5 a two-component spinor transforming in the spin-$\vec b$6 representation of $\vec b$7 (Figueroa-O'Farrill et al., 2023).

3. Representation theory, good and bad fermions, and coupling to gravity

For the massive UIR $\vec b$8, the one-particle Hilbert space is

$\vec b$9

and the generators act in momentum space as

$$t' = t - \vec b\cdot\vec x,\qquad \vec x' = \vec x,$$0

Thus the representation has constant energy $$t' = t - \vec b\cdot\vec x,\qquad \vec x' = \vec x,$$1 and unconstrained momentum $$t' = t - \vec b\cdot\vec x,\qquad \vec x' = \vec x,$$2; the dispersion relation is trivial,

$$t' = t - \vec b\cdot\vec x,\qquad \vec x' = \vec x,$$3

This is the group-theoretic origin of ultralocality and of the extreme degeneracy of the spectrum (Figueroa-O'Farrill et al., 2023).

A more geometric derivation uses null reduction from a Bargmann spacetime. In Lorentzian light-cone form, a Dirac spinor splits into “good” and “bad” components $$t' = t - \vec b\cdot\vec x,\qquad \vec x' = \vec x,$$4 and $$t' = t - \vec b\cdot\vec x,\qquad \vec x' = \vec x,$$5. In the parent Lorentzian theory, $$t' = t - \vec b\cdot\vec x,\qquad \vec x' = \vec x,$$6 is dynamical while $$t' = t - \vec b\cdot\vec x,\qquad \vec x' = \vec x,$$7 is constrained. After deforming to a Bargmann spacetime, the bad mode becomes dynamical, and null reduction yields the electric Carroll fermion. The resulting electric Carroll Lagrangian is

$$t' = t - \vec b\cdot\vec x,\qquad \vec x' = \vec x,$$8

In this formulation, the electric sector arises from the constrained modes of the parent theory, whereas the magnetic sector arises from the original dynamical modes (Majumdar et al., 6 May 2026).

The same work shows that the electric two-point function has the explicitly ultralocal form

$$t' = t - \vec b\cdot\vec x,\qquad \vec x' = \vec x,$$9

and that the electric sector admits a standard Fock-space quantization (Majumdar et al., 6 May 2026).

Electric Carroll fermions also admit a systematic coupling to magnetic Carroll gravity. In first-order form, the coupled action is

$$E=E_0\neq 0,\qquad O_\tau \cong A^3=\{(\mathbf p,E_0)\mid \mathbf p\in\mathbb R^3\},$$0

where only the spatial rotation connection $$E=E_0\neq 0,\qquad O_\tau \cong A^3=\{(\mathbf p,E_0)\mid \mathbf p\in\mathbb R^3\},$$1 enters the fermionic covariant derivative,

$$E=E_0\neq 0,\qquad O_\tau \cong A^3=\{(\mathbf p,E_0)\mid \mathbf p\in\mathbb R^3\},$$2

The boost connection does not couple directly, and the second-order formulation does not generate quartic fermion terms for the electric sector (Bergshoeff et al., 2023).

4. Quantization, propagators, and ultralocal dynamics

Canonical quantization of the electric Carroll Dirac theory proceeds in close analogy with the relativistic case, but with no spatial dispersion. The equal-time anti-commutator is

$$E=E_0\neq 0,\qquad O_\tau \cong A^3=\{(\mathbf p,E_0)\mid \mathbf p\in\mathbb R^3\},$$3

and the mode expansion uses spinors $$E=E_0\neq 0,\qquad O_\tau \cong A^3=\{(\mathbf p,E_0)\mid \mathbf p\in\mathbb R^3\},$$4 that are independent of $$E=E_0\neq 0,\qquad O_\tau \cong A^3=\{(\mathbf p,E_0)\mid \mathbf p\in\mathbb R^3\},$$5. The Hamiltonian is

$$E=E_0\neq 0,\qquad O_\tau \cong A^3=\{(\mathbf p,E_0)\mid \mathbf p\in\mathbb R^3\},$$6

so every one-particle excitation has energy $$E=E_0\neq 0,\qquad O_\tau \cong A^3=\{(\mathbf p,E_0)\mid \mathbf p\in\mathbb R^3\},$$7, independent of momentum (Ekiz et al., 8 Feb 2025).

The corresponding time-ordered propagator in position space is

$$E=E_0\neq 0,\qquad O_\tau \cong A^3=\{(\mathbf p,E_0)\mid \mathbf p\in\mathbb R^3\},$$8

while in energy space

$$E=E_0\neq 0,\qquad O_\tau \cong A^3=\{(\mathbf p,E_0)\mid \mathbf p\in\mathbb R^3\},$$9

The $K_\tau\cong Spin(3)$0 support makes the ultralocality manifest: fermions do not propagate spatially (Ekiz et al., 8 Feb 2025).

In the Carrollian-Clifford formulation, the system is first order in time and singular, so one uses Dirac brackets. The second-class constraints are

$K_\tau\cong Spin(3)$1

and the equal-time anticommutator becomes

$K_\tau\cong Spin(3)$2

The time-ordered two-point function is then

$K_\tau\cong Spin(3)$3

and its massless limit is finite,

$K_\tau\cong Spin(3)$4

This yields a completely time-independent massless two-point function, again reflecting extreme Carrollian ultra-locality (Sharma, 1 Feb 2025).

A useful correction to a common misconception is that all Carrollian fermionic excitations are equally immobile. The papers distinguish massive electric fermions, whose free field equations are ultralocal and whose spectrum is flat in momentum space, from other massless or magnetic sectors where different structures can arise (Figueroa-O'Farrill et al., 2023, Bergshoeff et al., 2023).

5. Interactions, discrete symmetries, and gauge-theoretic structures

The first explicit interacting quantized Carrollian Dirac fermion model is Carrollian Yukawa theory,

$K_\tau\cong Spin(3)$5

Its Feynman rules involve the electric Carroll fermion propagator

$K_\tau\cong Spin(3)$6

and the scalar propagator

$K_\tau\cong Spin(3)$7

At tree level, scalar exchange produces the ultralocal effective potential

$K_\tau\cong Spin(3)$8

This is a Dirac delta interaction with a time-dependent factor, rather than a spatially extended Yukawa tail (Ekiz et al., 8 Feb 2025).

The same work studies discrete symmetries. The Carrollian Dirac Lagrangian

$K_\tau\cong Spin(3)$9

is $s$0-, $s$1-, $s$2-, and $s$3-invariant, and the Yukawa extension preserves these symmetries as well. In the Carrollian scaling used there, the engineering dimensions in $s$4 dimensions are

$s$5

so the quartic scalar and Yukawa couplings are relevant under the Wilsonian flow (Ekiz et al., 8 Feb 2025).

Quantum-field-theoretic studies of Carrollian electrodynamics clarify the gauge sector that electric Carroll matter would couple to. The electric Carrollian gauge action is

$s$6

Its gauge transformations are

$s$7

and, after complete gauge fixing, the gauge-field two-point function becomes

$s$8

In the same study, the fermionic sector is constructed and quantized in parallel, but not yet coupled to the gauge field (Sharma, 1 Feb 2025).

The scalar Carrollian electrodynamics model in that paper also develops a BRST action and Nielsen identities, showing that the renormalized scalar mass is gauge independent once the theory is completely gauge fixed. A plausible implication is that analogous BRST control will be required for genuinely gauge-coupled electric Carroll fermions as well (Sharma, 1 Feb 2025).

6. Fractons, flat bands, holography, and open directions

Electric Carroll fermions are closely related to fractonic matter. The dipole-group construction maps every massive Carroll UIR $s$9 to a monopole fracton UIR $\Romanbar{II}(s,E_0)$0 with

$\Romanbar{II}(s,E_0)$1

In particular, electric Carroll fermions correspond to charged spinful fracton monopoles. Their immobility on the Carroll side matches the constrained mobility of charged fractons on the dipole-symmetry side (Figueroa-O'Farrill et al., 2023).

Condensed-matter realizations reinforce the “electric” interpretation. In one-dimensional flat-band fermion systems with compact localised states, the Hamiltonian can be written directly in terms of CLS operators,

$\Romanbar{II}(s,E_0)$2

and the infinite family of conserved charges

$\Romanbar{II}(s,E_0)$3

acts as a lattice version of Carroll supertranslations. The resulting correlators are ultra-local in space, and at gapless points the system exhibits emergent conformal Carroll symmetry (Ara et al., 2024).

A related line of work on spinless fermions near phase separation in a Tomonaga–Luttinger liquid finds that the effective velocity tends to zero, the spatial gradient term is suppressed, and the correct long-distance description is an electric Carroll scalar theory with density correlators scaling as

$\Romanbar{II}(s,E_0)$4

This provides a many-body realization of an electric Carroll fixed point, albeit without an explicit spinor construction (Biswas et al., 27 Jan 2025).

In flat-space holography, free massive electric Carrollian scalar theory already admits a regular Carroll-invariant vacuum state and a regular KMS state, while massless theories require more delicate nonregular sectors and zero-mode treatments. These results were obtained for Weyl algebras of scalar fields, but they strongly suggest that a parallel CAR-based analysis for electric Carroll fermions is both natural and technically necessary (Fredenhagen et al., 24 Apr 2026).

Several open problems recur across the literature. Interacting fermionic Carroll or fracton quantum field theories remain comparatively undeveloped; magnetic Carroll fermions are structurally subtler because of their indecomposable boost action; systematic Carrollian QED and QCD are proposed but not yet completed; and the role of electric Carroll fermions in flat-space holography, supergravity, and strongly correlated flat-band matter remains only partially understood (Figueroa-O'Farrill et al., 2023, Ekiz et al., 8 Feb 2025, Bergshoeff et al., 2023). These directions underscore that electric Carroll fermions are no longer just a formal contraction of Dirac theory, but a developing class of ultralocal quantum matter systems with links to representation theory, null reduction, non-Lorentzian geometry, condensed matter, and holography.