⚛️ lattice

Hamiltonian framework for Chiral Gauge Theories on a Disk Boundary

This paper proposes a novel Hamiltonian framework for chiral gauge theories on a disk boundary by introducing an alternative gauge field extension into the bulk that preserves non-perturbative topological features while enabling exact functional expressions suitable for future quantum simulations.

Original authors: Srimoyee Sen

Published 2026-10-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Srimoyee Sen

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

The universe at its smallest scale is governed by a set of rules known as the Standard Model, a framework that has successfully predicted the behavior of particles for decades. Yet, beneath this success lies a stubborn gap in our understanding. Physicists struggle to describe certain types of particles, specifically those that distinguish between left and right, known as chiral fermions, when they are confined to a grid-like structure used for computer simulations. This difficulty arises because the very act of discretizing space-time to make calculations possible seems to destroy the delicate handedness these particles possess. Without a way to simulate these theories on a computer, scientists cannot explore the most extreme conditions of the early universe or the deep, non-perturbative forces that hold matter together. To bridge this gap, researchers have turned to a clever geometric trick: imagining our four-dimensional world as the surface of a higher-dimensional object, much like a two-dimensional sheet of paper is the edge of a three-dimensional block. This approach allows the problematic particles to exist naturally on the boundary while the extra dimension handles the mathematical heavy lifting.

For years, this method worked well in the realm of static, time-averaged calculations, but it hit a wall when scientists tried to turn it into a dynamic model that could evolve over time. The previous method for filling the interior of this higher-dimensional object relied on a process that looked at the entire history of the system at once. It required the state of the interior at any single moment to depend on the state of the surface at every other moment in time, past and future. This created a logical knot for anyone trying to build a Hamiltonian framework, which is the standard language for describing how a physical system changes from one instant to the next. If the present depends on the future, the system cannot be simulated step-by-step, rendering the powerful tools of quantum simulation useless for these theories.

In a recent study, Srimoyee Sen proposes a new way to fill the interior of this geometric object that unties this knot. Instead of looking at the entire timeline at once, the new method solves the equations for the interior fields at each moment in time independently, using only the information available on the boundary at that exact instant. By treating the extra dimension as a series of snapshots rather than a continuous flow through time, the researcher constructed a prescription that allows the interior fields to be determined instantly from the boundary conditions. This change transforms the problem from one that is mathematically tangled into one that is compatible with the step-by-step logic required for a Hamiltonian formulation. The result is a framework that can, in principle, be used to simulate these complex theories on a quantum computer, opening the door to exploring phenomena that were previously out of reach.

The paper demonstrates that this new approach works perfectly for simple, linear versions of these theories, providing exact formulas for how the interior fields behave. When applied to more complex, non-linear theories, the method suggests that the interior remains smooth and free of mathematical singularities, provided the total number of topological twists on the boundary is zero. This is a crucial finding because it preserves the physical predictions of the original theory. Specifically, the study confirms that the new method prevents a specific particle, known as the eta-prime, from becoming a massless ghost particle, a fate that would have contradicted our observations of the real world. Furthermore, the method ensures that a mysterious parameter called the theta angle, which in other contexts could lead to a violation of time-reversal symmetry known as the strong CP problem, remains unphysical and harmless.

Perhaps most significantly, the research shows that when the boundary contains a net imbalance of these topological twists, the resulting mathematical singularities are pushed to the very edges of time, rather than appearing randomly in the middle of the system. This behavior mirrors the results of the older, time-averaged method, suggesting that the new, time-step approach captures the same deep physical truths without the computational bottleneck. The author argues that this construction does not just solve a technical hurdle but potentially clears the path for a full quantum simulation of chiral gauge theories. By allowing these theories to be treated with a Hamiltonian framework, the work offers a potential route to understanding the strong nuclear force and the origin of mass in a way that respects the fundamental asymmetry of the universe, all while avoiding the sign problems that have plagued previous attempts at simulation.

The study does not claim to have solved every problem or to have produced a working simulation of the entire Standard Model. Instead, it provides the necessary mathematical bridge to get there. It establishes that the interior of the geometric construction can be filled in a way that is both mathematically consistent and physically sound. The work suggests that the strange behavior of instantons and the absence of the strong CP problem, which were previously understood only through static, Euclidean calculations, can also be understood in a dynamic, real-time context. This opens a new avenue for future research, where scientists can potentially use quantum simulators to test these theories directly, moving from abstract mathematical consistency to concrete, observable predictions about the fundamental nature of reality.

Technical Summary: Hamiltonian Framework for Chiral Gauge Theories on a Disk Boundary

Problem Statement
The non-perturbative definition of Chiral Gauge Theories (CGTs), including the Standard Model (SM), remains elusive. While the "extra-dimension-approach" (EDA) offers a promising formulation where $2n$ -dimensional chiral fermions reside on the boundary of a $(2n+1)$ -dimensional disk, the current Euclidean implementation faces a critical bottleneck for Hamiltonian formulation. In the existing construction [22, 23], boundary gauge fields are extended into the bulk using the full $(2n+1)$ -dimensional equations of motion (EOM), a process known as "annealing flow." This method creates a non-local dependency in time: the bulk gauge field at a specific time slice depends on boundary gauge fields at all other time instants. This temporal non-locality prevents the construction of a Hamiltonian, which requires instantaneous functionals of the boundary fields. Furthermore, while the Euclidean formulation successfully addresses the $\eta'$ mass and the strong CP problem by pushing interior singularities to the temporal boundary when the net topological charge is zero, a Hamiltonian treatment is necessary to understand how these features connect to real-time physics and to enable quantum simulation, which avoids the sign problem inherent in Euclidean Monte Carlo methods.

Methodology
The paper proposes a new prescription for extending gauge fields from the $2n$ -dimensional boundary into the $(2n+1)$ -dimensional bulk. Instead of solving the full $(2n+1)$ -dimensional EOM, the author solves a spatial $(2n)$ -dimensional equation of motion at every instantaneous time slice $x_t$ .

The New Flow: The governing equation is $D_i \bar{F}_{iA} = 0$ (Eq. 11), where indices $i$ run only over spatial coordinates $(r, \phi, w, z)$ and $A$ runs over all five coordinates. This equation is solved subject to boundary conditions at the disk radius $R$ and a specific gauge condition $\Delta \partial_i \bar{A}_i = 0$ (Eq. 21).
Abelian Solution: For Abelian gauge fields, the author derives exact analytical expressions for the interior gauge fields $\bar{A}_\mu$ as functionals of the boundary fields $A_\mu$ . These solutions are expressed using modified Bessel functions and Green's functions, ensuring that the interior fields are instantaneous functionals of the boundary data at the same time $x_t$ .
Comparison with Slab: The paper notes that this prescription is specific to the disk geometry. It does not apply to slab manifolds (used in other EDA proposals [25]) because the slab requires decoupling mirror fermions on an anti-wall, a condition that conflicts with the instantaneous extension for certain time-dependent gauge configurations.

Key Results

Instantaneous Functionals: The proposed flow yields interior gauge fields that depend only on the boundary fields at the current time slice, thereby removing the temporal non-locality that obstructed Hamiltonian formulation.
Singularity Structure:
- Zero Net Topological Charge: For boundary configurations with zero net topological charge (e.g., instanton-anti-instanton pairs), the interior gauge fields are proven to be perfectly smooth and non-singular for all $r < R$ and all $x_t$ .
- Non-Zero Net Topological Charge: When the boundary carries a net topological charge (e.g., a single instanton), the interior solution remains smooth for all $x_t$ except at the temporal boundaries ( $x_t = 0, L_t$ ). The singularity is pushed to the temporal boundary of the manifold, consistent with the behavior observed in the original annealing flow [36].
Non-Abelian Extension: While analytical solutions are not available for non-Abelian fields, the paper argues that in the weak coupling regime, the non-linear terms are negligible, reducing the problem to the Abelian case. For $n=2$ (4D boundary), there are no topological obstructions to smooth extensions, and numerical solutions are expected to exist.
Preservation of Physics: The new prescription retains the crucial non-perturbative physics of the original construction:
- $\eta'$ Mass: The mechanism preventing the $\eta'$ from becoming a Goldstone Boson (GB) remains intact. The correlation between boundary and interior zero modes ensures the $\eta'$ acquires a mass gap for finite momentum modes, while the zero-momentum mode remains gapless.
- Strong CP Problem: The unphysical nature of the $\theta$ -angle is preserved. The gapless zero-momentum mode of the $\eta'$ allows the $\theta$ -angle to be rotated away, eliminating the strong CP problem within this framework.

Significance and Claims
The paper claims to provide a strategy to overcome the primary obstruction to a Hamiltonian treatment of the disk construction for Chiral Gauge Theories. By establishing a prescription that yields instantaneous bulk functionals, it opens a pathway for:

Hamiltonian Formulation: Enabling the explicit construction of a Hamiltonian for Abelian CGTs and the Abelian sector of non-Abelian CGTs.
Quantum Simulation: Facilitating the simulation of CGTs on quantum computers, which can circumvent the sign problem that plagues Euclidean lattice Monte Carlo methods at strong coupling.
Real-Time Physics: Clarifying the connection between the Euclidean results regarding the $\eta'$ and the strong CP problem and real-time dynamics.

The author emphasizes that while the analytical results are exact for Abelian fields, the framework is designed to be applicable to non-Abelian theories, where numerical methods would be required to obtain the functionals. The work does not claim to solve the full non-perturbative definition of the Standard Model immediately but provides the necessary structural foundation to attempt such a formulation via Hamiltonian methods and quantum simulation.

More like this