Kummitus: a light-weight toolbox for counting DOF in perturbative QFT

The paper introduces \texttt{Kummitus}, an open-source Wolfram Mathematica toolbox designed to efficiently compute gauge-invariant propagators and determine the propagating degrees of freedom in perturbative quantum field theories for both research and pedagogical applications.

The Gist

Imagine you are an architect trying to design a new, incredibly complex building. You have blueprints (the mathematical equations) that look beautiful and symmetrical. But before you can start construction, you need to answer a very specific question: "How many actual rooms will this building have?"

In the world of theoretical physics, these "rooms" are called Degrees of Freedom (DOF). They represent the actual, physical particles that can move and interact.

However, there's a catch. The blueprints often include "ghost rooms" (mathematical artifacts that don't exist in reality) and "hallucinated corridors" (particles that move faster than light or have negative energy, which breaks the laws of physics). If you build on these, your entire theory collapses.

For decades, physicists have had to manually count these rooms, a process that is like trying to untangle a knot of 100 headphones while blindfolded. It's slow, prone to error, and often requires making shortcuts that might miss a hidden ghost.

Enter Kummitus.

What is Kummitus?

The name "Kummitus" comes from the Estonian word for "Ghost." It is a clever, lightweight software tool (a toolbox) written for a program called Wolfram Mathematica. Its job is simple but revolutionary: It counts the real rooms in your theoretical building and ignores the ghosts.

Here is how it works, using some everyday analogies:

1. The "Saturated Propagator" (The Ultimate Test)

In physics, to see what particles exist, you look at something called a "propagator." Think of this as a sound system in your building. If you shout in one room, how does the sound travel to another?

• If the sound travels clearly, you have a real particle.
• If the sound echoes weirdly or disappears, you have a "ghost" (a mathematical error).
• If the sound travels faster than the speed of light, you have a "tachyon" (a time-traveling nightmare).

Usually, physicists try to guess the answer by looking at parts of the blueprint. Kummitus, however, turns on the sound system and listens to the whole building at once. It calculates the "saturated propagator," which is the most direct, unfiltered way to hear exactly what is moving through the theory.

2. The "Ghost Hunter" (AI Assistance)

One of the hardest parts of this job is dealing with "gauge constraints." Imagine your building has secret doors that only open if you press three buttons in the exact right order. If you press them wrong, the door looks like it's open, but it's actually a trap (a "spurious pole").

Older methods often got confused by these traps and thought a healthy building was broken. Kummitus uses a modern trick: AI assistance.
The author used an AI (specifically "Claude Code") to help write a special module called SystemSolver. Think of this as a super-smart inspector who doesn't just look at the blueprints but walks through every single room, checks every door, and verifies that no "ghosts" are hiding in the corners. It ensures that the math doesn't accidentally flag a perfectly good theory as broken.

3. The "Universal Translator" (Spin-Parity)

The paper talks about "Spin-Parity projection operators." In plain English, this is like a sorting machine.
Your building blueprint might be a giant, messy pile of bricks (tensors). Kummitus takes that pile and sorts it into neat boxes: "Here are the spinning bricks," "Here are the stationary bricks," "Here are the heavy ones."
It translates the messy, high-level math into a simple list of particles, telling you exactly: "You have 2 particles with spin-2, 1 particle with spin-1, and 0 ghosts."

Why Does This Matter?

The paper tests Kummitus on famous and difficult theories, like:

• Maxwell's Theory (Light): It correctly identifies the two polarizations of light.
• Einstein's Gravity: It confirms that gravity has two "helicity" states (like left-handed and right-handed spins).
• Exotic High-Spin Theories: It tackles theories with "Spin-3" particles, which are so complex that previous methods often gave up or gave the wrong answer.

The Big Picture

Think of Kummitus as a universal "Truth Detector" for theoretical physics.

• Before: Physicists were like detectives trying to solve a crime by looking at a single clue. They often missed the bigger picture.
• Now: With Kummitus, they have a drone camera that flies over the entire crime scene, counts every person, and instantly tells them who is a suspect and who is just a bystander.

The author, Carlo Marzo, built this tool not just to solve hard problems, but to make the process transparent and teachable. He wants students and researchers to see exactly how the counting happens, rather than treating the answer as a "black box."

In short, Kummitus is the tool that helps physicists separate the real particles from the mathematical ghosts, ensuring that the theories they build to explain the universe are solid, stable, and real.

Technical Summary

1. Problem Statement

In perturbative Quantum Field Theory (QFT), constructing consistent theories beyond simple cases requires a precise characterization of the propagating degrees of freedom (DOF).

• The Challenge: Local fields often introduce an overabundance of particle content, including non-unitary representations (ghosts) and tachyons. While the physical spectrum is determined by the single-pole structure of the two-point function (propagator), explicitly computing the gauge-invariant saturated propagator is notoriously difficult.
• Limitations of Existing Methods:
  • High-level spectral analysis tools often rely on indirect methods (e.g., analyzing the determinant of kinetic matrices or trace residues) which fail for massless particles, degenerate spectra, or models where gauge symmetries cause cancellations between sectors (e.g., the Fierz-Pauli spin-2 model).
  • Direct computation is often stalled by the complexity of solving tensorial gauge constraints (Ward identities) and handling spurious divergences in the massless on-shell limit ($q^2 \to 0$).
  • Existing algorithms often involve "detours" or auxiliary constructions that obscure the direct link between the Lagrangian and the physical spectrum.

2. Methodology

The paper introduces Kummitus, an open-source Wolfram Mathematica toolbox designed to compute the gauge-invariant saturated propagator via the shortest possible algorithmic path. The methodology relies on the Spin-Parity Projection Operator (SPO) formalism.

Core Algorithmic Steps:

1. Decomposition via SPOs:

   • The quadratic Lagrangian is decomposed into irreducible spin-parity sectors ($J^P$) using projection operators $P_{\{S,p\}}^{i,j}$.
   • The kinetic term is rewritten in terms of a matrix $a_{i,j}^{\{S,p\}}$ acting on these sectors, effectively removing the complexity of high-rank tensor indices.

2. Gauge Symmetry Identification:

   • Gauge symmetries are identified by finding the null-vectors ($X_s^i$) of the kinetic matrix $a_{i,j}^{\{S,p\}}$.
   • These null-vectors define the gauge transformations $\delta \Phi = X_s^i P_{\{S,p\}}^{i,j} \Psi$.

3. Solving Gauge Constraints (The Critical Step):

   • To obtain the physical propagator, one must solve the source constraints $X_s^* P J = 0$.
   • Innovation: Instead of using naive symbolic solvers (which often fail due to spurious poles in the massless limit), Kummitus employs a covariant frame-specific approach:
     • Massive Poles: Tensors are expanded in the rest frame ($q = (\omega, \vec{0})$).
     • Massless Poles: Tensors are expanded in a light-like basis ($q = (\omega, 0, 0, \kappa)$), with the limit $\kappa \to \omega$ taken after the full solution is found.
   • This avoids spurious divergences and correctly handles the cancellation of unphysical modes.

4. Saturated Propagator Construction:

   • The saturated propagator $D_S(q)$ is constructed by inverting the non-degenerate sub-matrix of the kinetic term and contracting it with the constrained sources.
   • The result is expanded as a Laurent series around the poles ($q^2 \to M^2$).

5. Spectral Analysis:

   • Pole Order ($r$): If the pole order $r > 1$, the theory contains ghosts. If $r \neq 1$, the pole is spurious.
   • Residue Extraction: For simple poles ($r=1$), the numerator matrix is diagonalized. The eigenvalues determine the number of propagating DOF and the sign of their norm (unitarity).

6. AI-Assisted Implementation:

   • The paper notes the use of Large Language Models (specifically Claude Code) to develop a robust SystemSolver module. This module replaces naive Solve calls, systematically scanning source subsets to find regular solutions and ensuring the code is free of "black box" logic by allowing back-substitution verification.

3. Key Contributions

• Direct Propagator Construction: Kummitus provides a direct, algorithmic route to the gauge-invariant saturated propagator without relying on indirect spectral proxies.
• Handling of Massless and Degenerate Cases: The tool successfully resolves the interplay between spurious poles in different sectors (e.g., $2^+$ and $0^+$ in spin-2 theories) which previous indirect methods often missed.
• Open-Source Transparency: The code is publicly available on GitHub, designed to be inspectable, reproducible, and extensible. It includes built-in SPOs for fields up to rank-3.
• Pedagogical Utility: The tool serves as a transparent resource for the theoretical community to verify particle content and unitarity in complex models.

4. Results and Validation

The author validates Kummitus against a diverse set of models ranging from textbook examples to recent high-spin theories:

• Maxwell Theory (Spin-1): Correctly identifies 2 helicity states.
• Einstein Gravity (Massless Spin-2): Recovers the 2 helicity states and correctly handles the diffeomorphism constraints.
• TDIFF Gravity: Demonstrates that Transverse Diffeomorphism is sufficient for healthy massless spin-2 propagation, detecting an additional massive scalar if parameters are not tuned.
• Fierz-Pauli (Massive Spin-2): Correctly identifies the 5 DOF of a massive spin-2 particle without gauge constraints.
• Stueckelberg & Gauge-Invariant Massive Spin-2: Successfully analyzes complex multi-field models with intertwined gauge symmetries, confirming unitary propagation.
• Spin-3 Models:
  • Fronsdal (Massless): Identifies 2 propagating states.
  • Campoleoni-Francia: Identifies simultaneous propagation of helicity-3 and helicity-1.
  • Singh-Hagen (Massive): Recovers 7 propagating DOF using auxiliary fields.
  • Klishevich-Zinoviev (Gauge-Invariant Massive): Handles a highly constrained system with many gauge symmetries, confirming 7 DOF.
  • Percacci-Sezgin (Hook Symmetric): Validates models using 2-index symmetric tensors instead of totally symmetric ones.

5. Significance

• Bridging the Gap: Kummitus fills a critical gap in the literature by providing a tool that moves beyond "spectral estimation" to "exact propagator construction."
• Reliability: By explicitly solving gauge constraints in a controlled frame, it eliminates the risk of misidentifying healthy theories as pathological due to spurious mathematical artifacts.
• Future-Proofing: The modular design allows for the easy inclusion of higher-rank fields (spin > 3) and mixed-field scenarios, making it a vital tool for exploring non-linear completions of gravity and higher-spin theories.
• Collaborative Ecosystem: When used alongside other tools like PSALTer, Kummitus offers a multi-pronged approach to spectral analysis, allowing for cross-verification of results in complex theoretical models.

In summary, Kummitus represents a shift from heuristic spectral analysis to rigorous, algorithmic verification of the physical content of perturbative QFT models, ensuring that only consistent, unitary, and causal degrees of freedom are identified.