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The mass gap is a small statement with a vast mathematical burden: in the physical vacuum theory, the energy needed to create any excitation has a strictly positive lower bound. Establishing that statement requires first constructing the quantum theory in which it makes sense.
This site is designed for readers who have taken at most two semesters of quantum field theory, including QED and non-Abelian gauge theory. You do not need a prior course in constructive field theory, probability, or functional analysis. When one of those subjects enters, a short mathematical bridge introduces exactly what is needed.
Choose a route
Section titled “Choose a route”Use the worked prerequisite diagnostic to choose a short recovery detour. For a seminar, the reading-group plan pairs every lesson with a concrete discussion output.
| Your starting point | Begin with | Then explore |
|---|---|---|
| I know the QFT formalism and want to understand the problem. | The twelve-lesson course | The proof map |
| I understand the physical picture and want to understand rigor. | Defining quantum Yang–Mills | Mathematical bridges |
| I work in the subject and want to assess current results. | The research atlas | Current status and numerical benchmarks |
| I want to explore one idea before committing. | Read a correlator | What an operator can see |
A good first hour
Section titled “A good first hour”Read what the problem actually asks. Pay attention to the distinction between a formal Lagrangian and a constructed quantum field theory. Then open the correlator laboratory. Set the overlap with the lighter state to zero and watch what the measured effective mass does. This example explains why a beautiful exponential decay in one channel is only part of the evidence.
Finish with the proof map. Select “Continuum limit” and “The mass gap.” The gap is a property of the final physical theory; it cannot be supplied by a box size or a regulator that is later removed.
What we assume
Section titled “What we assume”You should recognize a Lagrangian, a path integral, a correlation function, a gauge transformation, and the idea of renormalization. Familiarity with the Yang–Mills field strength helps. We review the remaining conventions in the notation reference.
You can work through the course without previously studying measure theory or operator algebras. The spectral theorem bridge explains why exponential functions of an operator produce Euclidean correlators. Haar measure explains the finite-lattice integral. Uniform estimates explains why the order and compatibility of limits matter.
How to read a research entry
Section titled “How to read a research entry”Each dossier states its gauge group, dimension, regulator, model, and controlled regime before describing its conclusion. A result in three dimensions can be a major advance without being a construction in four dimensions. A theorem for a compactified or deformed theory can expose a mechanism while leaving an essential continuation argument open.
The atlas distinguishes theorems, numerical evidence, controlled models, and methods. These labels describe the kind of evidence. A separate publication field tells you whether the source is a journal article or a preprint. The editorial principles explain how sources are checked and how records should evolve.
Keep the target in view
Section titled “Keep the target in view”The official problem asks for a nontrivial quantum Yang–Mills theory on four-dimensional spacetime, with a positive mass gap, for every compact simple gauge group. The statement and its explanatory discussion are available from Arthur Jaffe and Edward Witten at the Clay Mathematics Institute.
The research collection is curated, with each source’s exact version, inspected passages and reading depth recorded separately. An editorial update is not a new proof verification or an exhaustive literature census. Start with lesson 1 →.