Dirac states from the 't Hooft model

The Big Picture: A Heavy Anchor and a Light Boat

Imagine a very heavy, immovable anchor (a heavy quark) sitting in the ocean. Tethered to it by a strong, stretchy rope is a tiny, fast-moving boat (a light quark).

In the world of particle physics, these two particles are bound together to form a "meson." The paper asks a fundamental question: How does the tiny boat move when the anchor is so heavy that it barely moves at all?

Usually, physicists use a complex set of rules called the Dirac equation to describe how fast-moving particles behave. However, this equation gets tricky when the particle is trapped inside a larger system. The author of this paper wanted to prove that if you take a heavy particle and make it infinitely heavy, the messy, complex rules of the whole system simplify perfectly into the standard Dirac equation for the light particle.

The Laboratory: A Flat, 2D Universe

To solve this without getting lost in mathematical chaos, the author uses a simplified version of our universe called QCD2.

The Analogy: Imagine our universe is a flat sheet of paper (2 dimensions) instead of a 3D room.
The Trick: In this flat world, the "glue" holding the particles together acts like a simple, straight line that gets stronger the further you pull it apart (a linear potential).
The Limit: The author also uses a mathematical trick called "large Nc," which essentially turns off the ability for new particle pairs to pop into existence. This keeps the system simple: just one heavy anchor and one light boat, with no extra noise.

The Discovery: The View Doesn't Change

One of the most surprising findings in the paper concerns perspective (or "frames of reference").

The Problem: In physics, if you watch a boat from a stationary dock, it looks different than if you watch it from a speeding train. Usually, the rules of how the boat moves change depending on how fast you are moving.
The Result: The author found that for this specific heavy-light system, the light boat's behavior is the same no matter how fast the whole system is moving.
The Metaphor: Imagine you are on a train looking at a fly buzzing inside a train car. Even if the train is speeding down the track, the fly's flight pattern relative to the car doesn't change just because the train is moving. The paper proves that the light quark behaves exactly like this fly: its internal dynamics are "frame-independent." The only thing that changes is a slight squishing of space (Lorentz contraction), which doesn't actually change the physics of the light particle itself.

The Puzzle of the "Infinite" Spectrum

The paper also tackles a weird quirk of the Dirac equation when the "rope" (the potential) is a straight line.

The Paradox: Normally, if you trap a particle in a box, it can only have specific, distinct energy levels (like rungs on a ladder). However, the math for a straight-line potential suggests the particle could have any energy level, like a slide where you can stop anywhere. This is called a continuous spectrum.
The Resolution: The author shows that because this light particle is actually part of a bound system with a heavy partner, nature forces it to pick only specific, discrete energy levels (the rungs on the ladder).
The Analogy: Think of a guitar string. Mathematically, a string could vibrate at any frequency. But because it is tied down at both ends, it can only vibrate at specific notes. The heavy quark acts like the "tie" that forces the light quark to pick specific, discrete notes, even though the math of the "rope" alone suggests it could play any note.

The Proof: Numbers Don't Lie

The author didn't just do the math on paper; they ran a computer simulation to check it.

They started with a heavy anchor that was "heavy" but not infinite.
They gradually made the anchor heavier and heavier.
The Result: As the anchor got heavier, the behavior of the light boat matched the predictions of the standard Dirac equation perfectly. The difference between the complex reality and the simple Dirac equation shrank to zero, proportional to how heavy the anchor became.

Summary

In short, this paper confirms a long-held intuition in physics: When a light particle is bound to an infinitely heavy one, it behaves exactly as if it were a free particle moving in a static field, described by the standard Dirac equation.

This holds true whether the system is sitting still or zooming through space. The complex, messy interactions of the full quantum world simplify down to the elegant, familiar rules of the Dirac equation when one partner is heavy enough to act as a fixed anchor.

Technical Summary: Dirac states from the 't Hooft model

Problem Statement
The dynamics of a light fermion bound to a heavy fermion are expected to be governed by the Dirac equation with an external potential. However, a conceptual tension exists: the Dirac equation with a static potential breaks spatial translation invariance, meaning its eigenstates possess definite energy but not definite spatial momentum. Conversely, a physical bound state of two particles possesses a well-defined center-of-mass (CM) momentum. While boosting a bound state should allow the study of Dirac dynamics in any reference frame, canonically quantized bound states involve interactions in their boost generators, making the dynamics frame-dependent. Furthermore, relativistic bound states in field theory are generally intractable analytically. The specific challenge addressed here is to rigorously derive the Dirac equation as the limit of a light-heavy quark bound state in Quantum Chromodynamics in 1+1 dimensions (QCD $_2$ ) and to clarify the resulting spectrum and frame dependence.

Methodology
The study utilizes QCD $_2$ in the limit of a large number of colors ( $N_c \to \infty$ ), a regime where quark-antiquark pair production is suppressed, allowing for an exact solution of the $q\bar{q}$ Fock states. The analysis employs canonical quantization in the temporal gauge ( $A^0_a = 0$ ) at equal time $t=0$ , avoiding the singularities associated with light-front quantization zero-modes or modified propagator prescriptions.

The methodology proceeds in three stages:

Derivation of the Bound State Equation (BSE): The $q\bar{q$ bound state is expressed using a gauge-invariant wave function involving a path-ordered gauge link. The BSE is derived for the wave function $\Phi(x)$ in the rest frame and subsequently for a state with non-zero CM momentum $P$ .
The Heavy Mass Limit: The limit $m_2 \to \infty$ (where $m_2$ is the heavy quark mass and $m_1$ is the light quark mass) is taken. The bound state mass is defined as $M = m_2 + M_D$ , where $M_D$ is the Dirac bound state energy. The analysis examines how the BSE reduces to the Dirac equation in both the rest frame and boosted frames, accounting for Lorentz contraction.
Analytic and Numerical Solutions: The paper presents analytic solutions for both the full BSE and the resulting Dirac equation using confluent hypergeometric functions ( $_1F_1$ ). A numerical study is conducted to verify the convergence of the $q\bar{q$ wave function to the Dirac wave function as $m_2$ increases, specifically checking the convergence of the energy spectrum and the wave function components.

Key Contributions and Results

Derivation of the Dirac Equation: The paper demonstrates that the QCD $_2$ bound state equation for a light quark ( $m_1$ ) and a heavy quark ( $m_2$ ) reduces exactly to the Dirac equation with a linear potential $V(x) = V'|x|$ in the limit $m_2 \to \infty$ (specifically when $V' \ll m_2$ ). This reduction holds in any reference frame.
Frame Independence: A significant finding is that the light quark wave function, once the heavy quark carries the CM momentum, is independent of the bound state's frame of reference, up to an irrelevant Lorentz contraction. This resolves the issue of frame dependence in boosted canonically quantized states for this specific limit.
Discrete vs. Continuous Spectrum: The Dirac equation with a linear potential in 1+1 dimensions admits a continuous spectrum of energy eigenvalues, similar to plane waves, because the potential is balanced by negative kinetic energy at large distances (describing positrons repelled by the potential). However, the paper shows that when the Dirac equation is obtained as the limit of the physical $q\bar{q$ bound states, the spectrum becomes discrete. The requirement that the full $q\bar{q$ wave function remains regular (avoiding singularities at $V'|x| = E - P$ ) imposes quantization conditions that select only specific discrete values of $M_D$ .
Analytic Solutions: Explicit analytic solutions for the bound state wave functions and the Dirac wave functions are provided in terms of confluent hypergeometric functions. The paper details the boundary conditions (smoothness at $x=0$ ) required to fix the discrete mass spectrum.
Numerical Verification: Numerical calculations confirm that as $m_2$ increases, the difference between the bound state energy and the heavy mass ( $M - m_2$ ) converges to a constant Dirac energy $M_D$ . Furthermore, the bound state wave functions converge to the Dirac wave functions with an error scaling as $O(1/m_2)$ .

Significance
The paper establishes a rigorous field-theoretic derivation of the Dirac equation for a light fermion in a linear potential from the exact solution of the 't Hooft model. It clarifies that while the Dirac equation with a linear potential generally allows a continuous spectrum, the physical context of a bound state imposes a discrete spectrum. This work provides a concrete example where relativistic bound state dynamics can be studied analytically, bridging the gap between the 't Hooft model and the standard Dirac formalism. The author notes that this framework could potentially be extended to $D=3+1$ dimensions given a suitable relativistic bound state framework.