Physics:Quantum Yang-Mills mass gap

From HandWiki
Short description: Millennium Prize Problem in quantum field theory

← Back to Advanced and frontier topics


The Yang–Mills existence and mass gap problem is an open problem in mathematical physics and one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute. It asks for a mathematically rigorous construction of quantum Yang–Mills theory in four-dimensional spacetime and a proof that the theory has a positive mass gap.[1]

In physical terms, the problem concerns the mathematical foundations of non-abelian quantum gauge theory. Yang–Mills theories form the basis of the gauge-field description used in the Standard Model of particle physics. In particular, quantum chromodynamics, the theory of the strong interaction, is based on a non-abelian gauge theory.

The official problem statement asks one to prove that, for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ℝ4 and has a mass gap Δ>0.[1] Here ℝ4 denotes four-dimensional Euclidean space, and the mass gap is the positive energy difference between the vacuum and the lightest physical excitation of the theory.

The Yang–Mills mass gap problem asks why a quantum gauge field theory can produce massive excitations even when the classical gauge fields are massless.

Meaning of the problem

The problem has two closely related parts.

First, one must prove that quantum Yang–Mills theory exists as a mathematically well-defined quantum field theory. This means more than writing down the formal equations used in physics. The theory must satisfy rigorous axiomatic conditions of the kind used in constructive quantum field theory and axiomatic quantum field theory.[1][2][3][4]

Second, one must prove that the theory has a positive mass gap. This means that the vacuum is separated from the next possible energy state by a strictly positive amount. In particle language, the theory must not contain arbitrarily light physical particles.

For the gauge group SU(3), which is relevant to the strong interaction, the problem is closely connected with the expected existence of massive glueballs. A successful proof would show, in a mathematically rigorous way, why the quantum gauge theory produces massive color-neutral excitations rather than massless isolated gluons.

Yang–Mills theory

Yang–Mills theory is a generalization of gauge theory in which the gauge group is non-abelian. Unlike electromagnetism, whose gauge group is abelian, non-abelian gauge theories contain gauge fields that interact with one another.

This self-interaction is one reason Yang–Mills theory is difficult. It is also why it is physically important. Non-abelian gauge theories describe the weak and strong nuclear interactions in the Standard Model. In the strong interaction, the gauge group is SU(3), and the corresponding theory is quantum chromodynamics.

In the classical theory, the gauge fields can be massless. In the quantum theory, however, the physical spectrum is expected to contain a lowest nonzero mass. This is the mass gap.

Mass gap

In quantum field theory, a mass gap is the difference between the vacuum energy and the energy of the lightest excited state. If the vacuum has energy zero, then the mass gap is the smallest positive energy that can appear in the physical spectrum.

A theory with no mass gap can have excitations of arbitrarily small energy. A theory with a positive mass gap has a lowest nonzero energy scale. For Yang–Mills theory, the Clay problem requires proving that such a positive gap exists.[1]

One way of seeing the mass gap is through correlation functions. For a field ϕ, the long-time behavior of a two-point function may contain exponential decay of the form

⟨ϕ(0,t)ϕ(0,0)⟩∼∑nAnexp⁡(−Δnt).

The smallest positive value Δ0 corresponds to the mass gap. Lattice gauge theory calculations support the existence of such a gap in Yang–Mills theory and have been used to estimate glueball spectra.[5][6]

Relation to confinement

The mass gap problem is closely related to confinement in quantum chromodynamics. Confinement is the property that quarks and gluons are not observed as isolated particles under ordinary low-energy conditions. Instead, they appear inside color-neutral bound states such as hadrons.

In a pure Yang–Mills theory without quarks, the corresponding color-neutral bound states are glueballs. If glueballs exist as physical excitations, they should have positive mass. This is why the mass gap is physically connected with the absence of freely propagating massless gluons in the observed low-energy spectrum.

At the level of theoretical physics and lattice gauge theory, there is strong evidence for confinement and a mass gap. The Millennium problem, however, asks for a mathematically rigorous proof, not only a physical argument or numerical calculation.

Wightman and Osterwalder–Schrader axioms

The official Clay problem requires the constructed quantum Yang–Mills theory to satisfy rigorous axiomatic standards. The problem statement refers to the Wightman axioms and the Osterwalder–Schrader axioms.[1][7][8][9]

The Wightman framework describes quantum fields as operator-valued distributions acting on a Hilbert space. It imposes conditions such as relativistic covariance, a unique vacuum, positivity of energy, and local commutativity for spacelike separated fields.

The Osterwalder–Schrader framework gives related conditions for Euclidean quantum field theory. These axioms are important because Yang–Mills theory is often studied using Euclidean methods and lattice approximations. A rigorous construction must show that the Euclidean theory corresponds to a valid relativistic quantum field theory.

Importance

The problem is important because most interacting quantum field theories in four dimensions are difficult to define rigorously at all length scales. Many theories are best understood as effective field theories, valid below some cutoff scale.

Non-abelian Yang–Mills theory is different because it has asymptotic freedom. At high energies, the interaction becomes weaker, making it a natural candidate for a nontrivial four-dimensional quantum field theory that may exist at all scales.

A proof of existence and mass gap would therefore be a major result for both physics and mathematics. It would give a rigorous foundation to a central structure used in modern particle physics and would clarify the mathematical status of non-abelian gauge theory.

Relation to undecidability

The general problem of determining whether a quantum many-body system has a spectral gap is known to be undecidable in certain settings. This means that no universal algorithm can decide the answer for every possible system.[10][11]

This undecidability result does not by itself solve or refute the Yang–Mills mass gap problem. The Clay problem concerns a specific class of quantum gauge theories, not arbitrary quantum systems. It does, however, illustrate why spectral-gap questions can be mathematically subtle.

Status

The Yang–Mills existence and mass gap problem remains unsolved. Physicists have strong theoretical and numerical evidence that four-dimensional non-abelian Yang–Mills theory has a mass gap, but a complete proof satisfying the required axiomatic standards is not known.

The problem remains one of the most important open questions connecting quantum field theory, particle physics, geometry, analysis, and mathematical physics.

See also

Table of contents (184 articles)

Index

Full contents

1. Foundations (11)↑
  1. Physics:Quantum basics
  2. Physics:Quantum Postulates
  3. Physics:Quantum Hilbert space
  4. Physics:Quantum Observables and operators
  5. Physics:Quantum mechanics
  6. Physics:Quantum mechanics measurements
  7. Physics:Quantum state
  8. Physics:Quantum system
  9. Physics:Quantum superposition
  10. Physics:Quantum probability
  11. Physics:Quantum Mathematical Foundations of Quantum Theory
2. Conceptual and interpretations (13)↑
  1. Physics:Quantum Interpretations of quantum mechanics
  2. Physics:Quantum Wave–particle duality
  3. Physics:Quantum Complementarity principle
  4. Physics:Quantum Uncertainty principle
  5. Physics:Quantum Measurement problem
  6. Physics:Quantum Bell's theorem
  7. Physics:Quantum Hidden variable theory
  8. Physics:Quantum nonlocality
  9. Physics:Quantum contextuality
  10. Physics:Quantum Darwinism
  11. Physics:Quantum A Spooky Action at a Distance
  12. Physics:Quantum A Walk Through the Universe
  13. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together
  14. Physics:Quantum measurement problem
3. Mathematical structure and systems (13)↑
  1. Physics:Quantum Density matrix
  2. Physics:Quantum Exactly solvable quantum systems
  3. Physics:Quantum Formulas Collection
  4. Physics:Quantum A Matter Of Size
  5. Physics:Quantum Symmetry in quantum mechanics
  6. Physics:Quantum Angular momentum operator
  7. Physics:Quantum Runge–Lenz vector
  8. Physics:Quantum Approximation Methods
  9. Physics:Quantum Matter Elements and Particles
  10. Physics:Quantum Dirac equation
  11. Physics:Quantum Klein–Gordon equation
  12. Physics:Quantum pendulum
  13. Physics:Quantum configuration space
4. Atomic and spectroscopy (14)↑
    Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.
  1. Physics:Quantum Atomic structure and spectroscopy
  2. Physics:Quantum Hydrogen atom
  3. Physics:Quantum number
  4. Physics:Quantum Multi-electron atoms
  5. Physics:Quantum Fine structure
  6. Physics:Quantum Hyperfine structure
  7. Physics:Quantum Isotopic shift
  8. Physics:Quantum defect
  9. Physics:Quantum Zeeman effect
  10. Physics:Quantum Stark effect
  11. Physics:Quantum Spectral lines and series
  12. Physics:Quantum Selection rules
  13. Physics:Quantum Fermi's golden rule
  14. Physics:Quantum beats
5. Wavefunctions and modes (9)↑
    A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.
  1. Physics:Quantum Wavefunction
  2. Physics:Quantum Superposition principle
  3. Physics:Quantum Eigenstates and eigenvalues
  4. Physics:Quantum Boundary conditions and quantization
  5. Physics:Quantum Standing waves and modes
  6. Physics:Quantum Normal modes and field quantization
  7. Physics:Number of independent spatial modes in a spherical volume
  8. Physics:Quantum Density of states
  9. Physics:Quantum carpet
6. Quantum dynamics and evolution (17)↑
  1. Physics:Quantum Time evolution
  2. Physics:Quantum Schrödinger equation
  3. Physics:Quantum Time-dependent Schrödinger equation
  4. Physics:Quantum Stationary states
  5. Physics:Quantum Perturbation theory
  6. Physics:Quantum Time-dependent perturbation theory
  7. Physics:Quantum Adiabatic theorem
  8. Physics:Quantum Scattering theory
  9. Physics:Quantum S-matrix
  10. Physics:Quantum tunnelling
  11. Physics:Quantum speed limit
  12. Physics:Quantum revival
  13. Physics:Quantum reflection
  14. Physics:Quantum oscillations
  15. Physics:Quantum jump
  16. Physics:Quantum boomerang effect
  17. Physics:Quantum chaos
7. Measurement and information (9)↑
  1. Physics:Quantum Measurement theory
  2. Physics:Quantum Measurement operators
  3. Physics:Quantum Projective measurement
  4. Physics:Quantum POVM
  5. Physics:Quantum Weak measurement
  6. Physics:Quantum Measurement collapse
  7. Physics:Quantum entanglement
  8. Physics:Quantum Zeno effect
  9. Physics:Quantum limit
8. Quantum information and computing (10)↑
  1. Physics:Quantum information theory
  2. Physics:Quantum Qubit
  3. Physics:Quantum Entanglement
  4. Physics:Quantum Gates and circuits
  5. Physics:Quantum Computing Algorithms in the NISQ Era
  6. Physics:Quantum Noisy Qubits
  7. Physics:Quantum random access code
  8. Physics:Quantum pseudo-telepathy
  9. Physics:Quantum network
  10. Physics:Quantum money
9. Quantum optics and experiments (5)↑
    Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.
  1. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
  2. Physics:Quantum optics beam splitter experiments
  3. Physics:Quantum Ultra fast lasers
  4. Physics:Quantum Experimental quantum physics
  5. Physics:Quantum optics
    6. Template:Quantum optics operators
10. Open quantum systems (9)↑
  1. Physics:Quantum Open systems
  2. Physics:Quantum Master equation
  3. Physics:Quantum Lindblad equation
  4. Physics:Quantum Decoherence
  5. Physics:Quantum dissipation
  6. Physics:Quantum Markov semigroup
  7. Physics:Quantum Markovian dynamics
  8. Physics:Quantum Non-Markovian dynamics
  9. Physics:Quantum Trajectories
11. Quantum field theory (20)↑
    Structural dependency map of quantum field theory.
  1. Physics:Quantum field theory (QFT) basics
  2. Physics:Quantum field theory (QFT) core
  3. Physics:Quantum Fields and Particles
  4. Physics:Quantum Second quantization
  5. Physics:Quantum Harmonic Oscillator field modes
  6. Physics:Quantum Creation and annihilation operators
  7. Physics:Quantum vacuum fluctuations
  8. Physics:Quantum Propagators in quantum field theory
  9. Physics:Quantum Feynman diagrams
  10. Physics:Quantum Path integral formulation
  11. Physics:Quantum Renormalization in field theory
  12. Physics:Quantum Renormalization group
  13. Physics:Quantum Field Theory Gauge symmetry
  14. Physics:Quantum Non-Abelian gauge theory
  15. Physics:Quantum Electrodynamics (QED)
  16. Physics:Quantum chromodynamics (QCD)
  17. Physics:Quantum Electroweak theory
  18. Physics:Quantum Standard Model
  19. Physics:Quantum triviality
  20. Physics:Quantum confinement problem
12. Statistical mechanics and kinetic theory (9)↑
  1. Physics:Quantum Statistical mechanics
  2. Physics:Quantum Partition function
  3. Physics:Quantum Distribution functions
  4. Physics:Quantum Liouville equation
  5. Physics:Quantum Kinetic theory
  6. Physics:Quantum Boltzmann equation
  7. Physics:Quantum BBGKY hierarchy
  8. Physics:Quantum Relaxation and thermalization
  9. Physics:Quantum Thermodynamics
13. Condensed matter and solid-state physics (13)↑
  1. Physics:Quantum Band structure
  2. Physics:Quantum Fermi surfaces
  3. Physics:Quantum Semiconductor physics
  4. Physics:Quantum Phonons
  5. Physics:Quantum Electron-phonon interaction
  6. Physics:Quantum Superconductivity
  7. Physics:Quantum Topological phases of matter
  8. Physics:Quantum well
  9. Physics:Quantum spin liquid
  10. Physics:Quantum spin Hall effect
  11. Physics:Quantum phase transition
  12. Physics:Quantum critical point
  13. Physics:Quantum dot
14. Plasma and fusion physics (8)↑
    Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
  1. Physics:Quantum Fusion reactions and Lawson criterion
  2. Physics:Quantum Plasma (fusion context)
  3. Physics:Quantum Magnetic confinement fusion
  4. Physics:Quantum Inertial confinement fusion
  5. Physics:Quantum Plasma instabilities and turbulence
  6. Physics:Quantum Tokamak core plasma
  7. Physics:Quantum Tokamak edge physics and recycling asymmetries
  8. Physics:Quantum Stellarator
15. Timeline (8)↑
  1. Physics:Quantum mechanics/Timeline
  2. Physics:Quantum mechanics/Timeline/Pre-quantum era
  3. Physics:Quantum mechanics/Timeline/Old quantum theory
  4. Physics:Quantum mechanics/Timeline/Modern quantum mechanics
  5. Physics:Quantum mechanics/Timeline/Quantum field theory era
  6. Physics:Quantum mechanics/Timeline/Quantum information era
  7. Physics:Quantum mechanics/Timeline/Quantum technology era
  8. Physics:Quantum mechanics/Timeline/Quiz
16. Advanced and frontier topics (16)↑
  1. Physics:Quantum topology
  2. Physics:Quantum battery
  3. Physics:Quantum Supersymmetry
  4. Physics:Quantum Black hole thermodynamics
  5. Physics:Quantum Holographic principle
  6. Physics:Quantum gravity
  7. Physics:Quantum De Sitter invariant special relativity
  8. Physics:Quantum Doubly special relativity
  9. Physics:Quantum arithmetic geometry
  10. Physics:Quantum unsolved problems
  11. Physics:Quantum Yang-Mills mass gap
  12. Physics:Quantum gravity problem
  13. Physics:Quantum black hole information paradox
  14. Physics:Quantum dark matter problem
  15. Physics:Quantum neutrino mass problem
  16. Physics:Quantum matter-antimatter asymmetry problem

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 Jaffe, Arthur; Witten, Edward. "Quantum Yang-Mills theory". https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf. 
  2. ↑ Streater, R. F.; Wightman, A. S. (2000). PCT, spin and statistics, and all that. Princeton landmarks in physics (1st with rev ed.). Princeton, N.J: Princeton University Press. ISBN 978-0-691-07062-9. 
  3. ↑ Osterwalder, Konrad; Schrader, Robert (1973). "Axioms for Euclidean Green's functions". Communications in Mathematical Physics 31 (2): 83–112. doi:10.1007/BF01645738. Bibcode: 1973CMaPh..31...83O. 
  4. ↑ Osterwalder, Konrad; Schrader, Robert (1975). "Axioms for Euclidean Green's functions II". Communications in Mathematical Physics 42 (3): 281–305. doi:10.1007/BF01608978. Bibcode: 1975CMaPh..42..281O. 
  5. ↑ Lucini, Biagio; Teper, Michael; Wenger, Urs (2004). "Glueballs and k-strings in SU(N) gauge theories: calculations with improved operators". Journal of High Energy Physics 2004 (6): 012. doi:10.1088/1126-6708/2004/06/012. Bibcode: 2004JHEP...06..012L. 
  6. ↑ Chen, Y.; Alexandru, A.; Dong, S. J.; Draper, T.; Horváth, I.; Lee, F. X.; Liu, K. F.; Mathur, N. et al. (2006). "Glueball spectrum and matrix elements on anisotropic lattices". Physical Review D 73 (1). doi:10.1103/PhysRevD.73.014516. Bibcode: 2006PhRvD..73a4516C. 
  7. ↑ Streater, R. F.; Wightman, A. (1964). PCT, spin and statistics, and all that. New York, W.A. Benjamin. https://archive.org/details/pctspinstatistic0000stre. 
  8. ↑ Osterwalder, Konrad; Schrader, Robert (1973). "Axioms for Euclidean Green's functions". Communications in Mathematical Physics 31 (2): 83–112. doi:10.1007/BF01645738. Bibcode: 1973CMaPh..31...83O. 
  9. ↑ Osterwalder, Konrad; Schrader, Robert (1975). "Axioms for Euclidean Green's functions II". Communications in Mathematical Physics 42 (3): 281–305. doi:10.1007/BF01608978. Bibcode: 1975CMaPh..42..281O. 
  10. ↑ Cubitt, Toby S.; Pérez-García, David; Wolf, Michael (2018-10-01). "The Unsolvable Problem". Scientific American 319 (4): 28–37. doi:10.1038/scientificamerican1018-28. PMID 30273308. Bibcode: 2018SciAm.319d..28C. https://www.scientificamerican.com/article/the-unsolvable-problem/. 
  11. ↑ Castelvecchi, Davide (9 December 2015). "Paradox at the heart of mathematics makes physics problem unanswerable". Nature. doi:10.1038/nature.2015.18983. https://www.nature.com/articles/nature.2015.18983. 

Further reading

  • Bogoliubov, N.; Logunov, A.; Oksak; Todorov, I. (1990) (in en). General Principles of Quantum Field Theory. Dordrecht: Springer Netherlands. doi:10.1007/978-94-009-0491-0. ISBN 978-94-010-6707-2. 
  • Strocchi, Franco (1993). Selected topics on the general properties of quantum field theory: lecture notes. World Scientific lecture notes in physics. Singapore: World Scientific. ISBN 978-981-02-1149-3. 
  • Dynin, A. (2014). "Quantum Yang-Mills-Weyl Dynamics in the Schrödinger paradigm". Russian Journal of Mathematical Physics 21 (2): 169–188. doi:10.1134/S1061920814020046. Bibcode: 2014RJMP...21..169D. 
  • Dynin, A. (2014). "On the Yang-Mills mass gap problem". Russian Journal of Mathematical Physics 21 (3): 326–328. doi:10.1134/S1061920814030042. Bibcode: 2014RJMP...21..326D. 
  • Bushhorn, G.; Wess, J. (2004). Fundamental physics-- Heisenberg and beyond: Werner Heisenberg Centennial Symposium "Developments in Modern Physics". Berlin ; New York: Springer. ISBN 978-3-540-20201-1. 


Author: Harold Foppele





Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Quantum_Yang-Mills_mass_gap&oldid=4326241"