1. Gerard ’t Hooft Spinoza Institute Utrecht, the Netherlands Utrecht University

2. Contents First part: Absolute Quark Confinement in lattice QCD Second part: Absolute Quark Confinement as a topological Phenomenon Third part: The Gluon Chain Model (Greensite, Thorn) Try to do it better: compare Renormalization Procedure (infinite) Infrared Renormalization The renormalized Coulomb potential (in radiation gauge) Fourth part: Gauge invariant procedure Renormalized effective actions: an exercise in Legendre Transformations

3. What kinds of forces were holding them together?

4. ProtonLambda Antiproton Pi-plusPi-zero The hadronic particles …

5. Lattice QCD(K. Wilson, London, 1974) Using the expansion In the expansion, only terms where the energy increases linearly with inter- quark distance survive ! Part 1:

6. Part 2: Magnetic Confinement In case of spontaneous "breakdown" of N S H.B. Nielsen and P. Olesen, 1970.

7. Color Magnetic Super Conductivity N S + _ Electric Super Conductor Magnetic Super Conductor

8. G. ’t H (1974), A.M. Polyakov (1974) The Magnetic Monopole S. Mandelstam (1975), G. ’t H (1976)

9. Part 3: The gluon chain approach Anti- quark J. Greensite and C.B. Thorn hep-ph/0112326 Ansatz for the "Wave Function": Use variational principle, minimize then, improve Ansatz [ ? ]

10. This can be done better The gluon chain model gives reasonable – looking "stringlike“ structures for the mesons … but confinement is not built in … The chainlike states will surely not form a complete set of states.UNITARITY ? Describe a "modified" perturbative approach, where unitarity is guaranteed Infinite infrared renormalization

11. Lowest order Compare UV renormalization Combine this with the higher order terms

12. Perturbative Confinement now generates a potential V between charges obeying

13. Let V be a confining potential, typically: in space: then

14. should be treated exactly like a renormalization counter term. Compare our procedures in the renormalization group: the coefficients (here: ) must be adjusted in such a way that the higher order correction terms, together with the contributions from, should be as insignificant as possible. At lowest order, we should start with a Fock space of Eigen states of particles bound by the potential V. They are confined from the very beginning:

15. Part 4: A Classically Confining Theory: Stationary case: U ( D ) can become any monotonically increasing function of D

16. Q -Q D

17. Legendre Transformations: 1

18. The dual transformation

19. Quantum Chromodynamics is an extremely accurate theory. At short distances, the forces become weak, so that perturbative treatment there is possible. Calculating the QCD contributions to high-energy scattering processes has become routine. Interesting and important problems remain: - find a quark-gluon plasma - find more accurately converging calculation procedures... Utrecht University

20. Further References: Nucl. Phys. B 138 (1978) 1 Nucl. Phys. B 153 (1979) 141 Nucl. Phys. B 190 (1981) 455 Acta Phys. Austriaca Suppl. XXII (1980) 531 Physics Reports 142 (1986) #6, 357 hep-th / 9903189 Erice: hep-th / 9812204 Montpellier Proceedings (2002) The End