Worldline Formulations of Covariant Fracton Theories

Technical Summary: Worldline Formulations of Covariant Fracton Theories

Authors: Filippo Fecit and Davide Rovere
Subject Area: Theoretical High-Energy Physics (First-quantized/Worldline Formalism, Gauge Theory, Fracton Physics)

1. Problem Statement

The paper addresses the challenge of constructing first-quantized (worldline) formulations for covariant fracton gauge theories.

Fractons are exotic quasiparticles characterized by restricted mobility (e.g., they cannot move freely due to the conservation of dipole moments). While most fracton models are Lorentz-breaking and defined on lattices, "covariant fracton theories" are a family of Lorentz-invariant gauge theories of a rank-two symmetric tensor field $h_{\mu\nu}$ with a specific scalar gauge symmetry: $\delta_\Lambda h_{\mu\nu} = \partial_\mu \partial_\nu \Lambda$.

The fundamental problem is to bridge the gap between the standard second-quantized spacetime field theory (described via the Batalin-Vilkovisky [BV] formalism) and a first-quantized particle-like description (worldline formalism) that can reproduce the full spectrum of fields and gauge symmetries.

2. Methodology

The authors employ a systematic program to reproduce the BV spectrum of spacetime field theories from a worldline perspective using BRST quantization. Their approach follows these steps:

• Phase Space Construction: They augment the standard spacetime coordinates and momenta $(x^\mu, p^\mu)$ with additional bosonic oscillator variables ($\alpha_{\mu\nu}$ or $\alpha_\mu$). These oscillators are not used to describe spin (as in standard spinning particles) but are introduced to saturate the "double-momentum" constraints ($\sim p_\mu p_\nu$) necessitated by the fracton gauge symmetry.
• Constraint Algebra: They identify sets of constraints that form a first-class algebra. Because the gauge symmetry involves second derivatives, the resulting algebra often involves "structure functions" (coefficients depending on phase-space variables) rather than simple structure constants.
• BRST Quantization: They construct a nilpotent BRST charge $Q$ on an enlarged Hilbert space. By studying the cohomology of $Q$, they extract the spacetime field content.
• String Field Mapping: They represent the state in the BRST Hilbert space as a "string field" $\psi$, where the coefficients of the expansion in the oscillator basis correspond to the spacetime fields (gauge fields, ghosts, and antifields).
• Deformation Technique: To capture the full family of fracton theories, they treat the models as deformations of a base "vector model," adjusting the constraints to match different values of the theory's free parameters ($\alpha, \beta$).

3. Key Contributions and Results

The paper develops and analyzes three distinct worldline models:

1. The Tensor Model:

   • Uses rank-two symmetric tensor oscillators $(\alpha_{\mu\nu}, \bar{\alpha}_{\mu\nu})$.
   • Implements constraints $L = \alpha_{\mu\nu} p^\mu p^\nu$ and $\bar{L} = \bar{\alpha}_{\mu\nu} p^\mu p^\nu$.
   • Result: Successfully reproduces a specific subset of the fracton family where the parameters satisfy $2\alpha - \beta = 0$.

2. The Vector Model:

   • Uses vector oscillators $(\alpha_\mu, \bar{\alpha}_\mu)$.
   • Implements a more complex constraint $\ell = \alpha_\mu p^\mu \bar{\alpha}_\nu p^\nu$.
   • Result: Reproduces the specific fracton theory where $2\alpha + 3\beta = 0$ (for $D \neq 4$). It highlights that the specific theory captured is a consequence of the quantum operator ordering (Weyl ordering) required for nilpotency.

3. The Deformed Vector Model:

   • Deforms the vector model by introducing parameters $(\tilde{\alpha}, \tilde{\beta})$ into the $\ell$ constraint.
   • Result: This is the most significant contribution, as it reproduces almost the entire family of covariant fracton theories. It maps the worldline deformation parameters to the spacetime Lagrangian parameters $\alpha$ and $\beta$.

Gauge-Fixing Comparison:
The authors compare the BV-BRST spacetime gauge-fixing (using antighosts and Lagrange multipliers) with the worldline "Siegel gauge" (requiring the state to be in the kernel of the ghost operator $b$). They demonstrate that the Siegel gauge effectively reduces the field content to the physical sector and yields wave equations for the remaining components.

4. Significance

• Theoretical Unification: The work demonstrates that the worldline formalism is a robust alternative to second quantization, capable of handling higher-derivative gauge symmetries and complex constraint algebras.
• New Tool for Fracton Physics: It provides a new mathematical framework to study fracton excitations. While fractons are typically studied in condensed matter (Lorentz-breaking), this covariant approach allows for the use of powerful relativistic tools.
• Methodological Extension: The paper shows that the worldline approach can be extended to "non-standard" particle models where constraints are quadratic in both momenta and oscillators, suggesting that worldline methods could eventually serve as an equivalent formulation for a much broader class of quantum field theories.
• Future Directions: The authors suggest this could lead to efficient computations in condensed matter physics, particularly for topological and boundary-driven phenomena.