Report on the status of the yangmills millenium prize problem michael r. Having some experience in the matter, i decided to. A mathematically rigorous relativistic quantum yangmills theory with an arbitrary semisimple compact gauge lie group is set up in the hamiltonian canonical formalism. Fundamental particles,such as quarks and electrons,are described by fermionic. In a quantum eld theory, spacetime elds with relativistic eld equations are quantized and, in many calculations, the quanta of the elds are interpreted as particles. A quantum eld theory qtf such as yang mills theory is said to have a mass gap if the hamiltonian h has no spectrum in the interval 0. Faddeev 1 bulletin of the brazilian mathematical society volume 33, pages 201 212 2002 cite this article. Whats the distinctions between yangmills theory and qcd. By the 1950s, when yangmills theory was discovered, it was already known that the quantum version of maxwell theory known as quantum electrodynamics or qed gives an extremely accurate account of electromagnetic fields and forces. But the yangmills equations have certain properties in common with the maxwell equations. By the 1950s, when yangmills theory was discovered, it was already known that the quantum version of maxwell theoryknown as quantum.
Evidence for the mass gap has been demonstrated in physical experiments and computerbased mathematical models, and it is believed to be the reason that the strong force operates only. The groups of unitary symmetry are central to the formalism of quantum theory, starting with the pauli su2 group for spin, heisenbergs isospin, wigners irreducible representations, and ending with gellmanns quarks, yangmillss fields, strings and 11dimensional supergravity. A survey and problem description of the nonperturbative quantization of yangmills theory, in the context of the clay mathematics institutes list of millennium problems yangmills and mass gap is in. This paper shows that quantum theory describing particles in. Many physical theories have symmetries, which have important consequences. A quantization procedure for the yang mills equations for the minkowski space is carried out in such a way that field maps satisfying wightman axioms of constructive quantum field theory can be. Massdeformed yangmills theory in the covariant gauge and. A high energy hard transverse gluon, at the time t 0 of its emission, say in 3jet event, lies somewhere. When yangmills field theory was introduced, a problem is that the gauge invariance can not allow mass for the gauge field. I even the rst step in the probabilistic approach,namely, the construction of a random eld, remains open.
Quantum waves with mass disprove yang mills theory and the mass gap. Pdf from yangmills photon in curved spacetime to dark. The qcd analog of electric charge is a property called color. We show how su3 c chromodynamics, which is the theory of strong interactions, is a. In contrast, the classical lagrangian of yangmills field theory has no mass quadratic form, but only a selfinteraction quartic form. The energy difference between the lowest order and the vacuum state in yangmills quantum field theory, the mass gap is in principle responsible for the largescale structure of the qcd ground. Proving yang mills existence is important because the standard model produces such precise calculations in particle physics. Quantum field theory is little more than a field of holes, and. The basic starting point is that the axioms of special relativity on the one hand and those of quantum mechanics on the other, should be combined into one theory. In theoretical physics, quantum chromodynamics qcd is the theory of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion. One of them is called yangmills existence and mass gap.
In the process, one finds the prediction of antimatter. Becchirouetstoratyutin brst symmetry, physical states and unitarity. Yangmills theories, and especially quantum yangmills theories, have many subtle. The major breakthrough came with the observation 42 by r. We start from quantum field theory in curved spacetime to derive a new einsteinlike energy mass relation of the type e. Petersburg department of steklov mathematical institute. Among others, the quantum yangmills theory problem requires a mathematically complete proof of a positive mass gap in the yangmills spectrum on the fourdimensional minkowski vector space.
Lecture notes relativistic quantum field theory ii. Chromodynamic symmetries, confinement properties, and shortrange interactions of classical and quantum yangmills gauge theory jay r. In the conclusion of this paper we will analyze the yang mills theory and see how it has played a role in the development of modern gauge theories. Prove that for any compact simple gauge group g, quantum yang mills theory of r4 exists and has a mass gap. Two articles showing proof and disproof of yang mills. Qcd is a type of quantum field theory called a nonabelian gauge theory, with symmetry group su3. Relativistic quantum field theory is a mathematical scheme to describe the subatomic particles and forces. Yang mills theory modern particle theories, such as the standard model, are quantum yang mills theories.
Introduction and motivation taking into consideration both quantum and spacetime physics has shown that black holes are indeed thermodynamic systems. Gauge theory 1 gauge theory 1 notes and gauge theory 1 video nonableian. Pdf quantum energymass spectrum of yangmills bosons. Heisenbergs uncertainty principle shows the position and momentum of a particle to be uncertain. Quantum yangmills theory the physics of gauge theory. As an example, ill explain how maxwells equations can be regarded as a yangmills theory with gauge group u1. What all this means is that the mass gap was always the least of the problems of yangmills.
The mass lessness of classical yangmills waves was a serious obstacle to applying yangmills theory to the other forces, for the weak and nuclear forces are not. Riemann geometry and general relativity in various formalisms. Witten, gives both motivation and exposition of related mathematical results, known until now. Although quantum field in curved spacetime is not a complete quantum gravity theory, our prediction here of 95. Existence includes establishing axiomatic properties at least as strong as those cited in 45, 35. Yangmills theory exists and satisfies the standard of rigor that characterizes contemporary mathematical physics, in particular constructive quantum field theory, and the mass of the least massive particle of the force field predicted by the theory is strictly positive. It attempts to show that, to explain the massspectrum of qcd, one needs the mass scale parameter the mass gap instead of other massive particles. The pure yangmills theory is particularly interesting since it is a con.
Su3 yangmills theory in any detail, but in principle it is easy to construct using the recipe i will explain. In addition, with quantized gravitational mass, because it is a generic massless, semisimple yang mills theory, it is mathematically proved to be a perturbatively renormalizable quantum theory of. In addition, with quantized gravitational mass, because it is a generic massless, semisimple yangmills theory, it is mathematically proved to be a perturbatively renormalizable quantum theory of. Sep 06, 2016 research video based on the papers arxiv. In quantum mechanics, the position and velocity of a particle are noncommuting operators acting. Douglas april 2004 yangmills existence and mass gap. This is an archive of the quantum field theory ii course. The yangmills mass gap problem is solved by deriving su3 c chromodynamics as a corollary theory from yangmills gauge theory. Witten talk on the mass gap problem in 3d quantum yang.
Two articles showing proof and disproof of yang mills theory. The physics of mass gap problem in the general field theory. The yangmills theory relies on a quantum mechanical property called the mass gap. Quantum yang mills problem, navierstokes problem duration. Matrix logic offers a different language, which also has ramifications for cosmology. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous. The mass gap is filled from the finite nonzero eigenvalues of a configuration space inverse. Prove that for any compact simple gauge group g, quantum yangmills theory on r4 exists and has a mass gap. Mass in quantum yangmills theory comment on a clay millenium problem l. Quantum yangmills theory in two dimensions youtube. Mass in quantum yangmills theory comment on a clay. I however, the investigation was inconclusive and the question is still considered to be open. Mathematical quantum yangmills theory revisited springerlink. Arthur jaffe, edward witten, quantum yangmills theory.
Recently, gaugegravity duality has been extensively studied and it provides a method. Su2 as well as quantum chromodynamics, the theory of. The mass gap and its applications world scientific. The yangmills functional is analogous to the harmonic maps energy functional and the. Aug 01, 2012 witten talk on the mass gap problem in 3d quantum yang mills theory. Douglas april 2004 yang mills existence and mass gap. The physics of gauge theory since the early part of the 20th century, it has been understood that the description of nature at the subatomic scale requires quantum mechanics. Witten talk on the mass gap problem in 3d quantum yangmills. Yang mills theory is a gauge theory based on a special unitary group sun, or more generally any compact, reductive lie algebra. Yangmills as gravity theory, gravity as yangmills theory. Pdf four dimensional quantum yangmills theory and mass gap.
The yang mills theory is more of a pardon my french paradigm than a theory, in the sense yang mills sets up a framework for theories like qcd as opposed to giving an hypothesis. Minkowski space is not the same as four dimensional euclidean space but it can be generated from it. Duality transformation of yang mills theory on a lattice in 3d. Yangmills theory is a gauge theory based on a special unitary group sun, or more generally any compact, reductive lie algebra. Prove that for any compact simple gauge group g, a nontrivial quantum yangmills theory exists on and has a mass gap. Existence of yangmills theory with vacuum vector and mass gap igor hrnci. Witten talk on the mass gap problem in 3d quantum yangmills theory. Yangmills theory, in physics, a generalization of scottish physicist james clerk maxwells unified theory of electromagnetism, also known as maxwells equations, used to describe the weak force and the strong force in subatomic particles in terms of a geometric structure, or quantum field theory. For yang mills theory, one of the most important questions is to mathematically explain the mass gap, or nonzero mass, in quantum applications of the formulas. Yang mills theory seeks to describe the behavior of elementary particles using these nonabelian lie groups and is at the core of the unification of the electromagnetic force and weak forces i. Black holes radiate heat in a manner quite similar to black body radiation. Hofmann, institut fur theoretische physik, universitat heidelberg. Among seven problems, proposed for xxi century by clay mathematical institute, there are two stemming from physics. This equivalence is obtained as a consequence of the gaugeindependent broutenglerthiggs mechanism proposed recently by one of the authors.
For a simple compact yang mills gauge group and the antinormal quantization of yang mills energy mass functional of initial data in a box, the quantum energy mass spectrum is a sequence of non. One of them is called yang mills existence and mass gap. Yangmills theory is the nonabelian quantum field theory underlying the standard model of particle physics. Yangmills theory plays a central role in explaining fundamental interactions, because both the strong and weak interactions are described by yangmills theories.
Pdf due to the nonlinearity involved in quantum chromodynamics qcd, the required uncertainty in position of a transverse hard gluon. Prove that for any compact simple gauge group g, quantum yangmills theory on. Proof and disproof of yang mills theory and the mass gap. Yangmills theory seeks to describe the behavior of elementary particles using these nonabelian lie groups and is at the core of the unification of the electromagnetic force and weak forces i. Yang mills theory in 4d in a gauge invariant dual formulation. What is an intuitive explanation of yangmills theory. Existence includes establishing axiomatic properties at least as strong as those cited in. For a simple compact yangmills gauge group and the antinormal quantization of yangmills energymass functional of initial data in a box, the quantum energymass spectrum is a sequence of non. On the one hand, there is a detailed analogy between yangmills theory over 4manifolds and the geometry of maps from a riemann surface to a symplectic manifold. The theory is nonperturbative, without cutoffs, and agrees with the causality and stability principles. Report on the status of the yangmills millenium prize problem.
Among seven problems, proposed for xxi century by clay mathematical institute 1, there are two stemming from physics. Yang mills theory in 3d as quantum gravity with aether. The gauge principle quantum electrodynamics revisited 3. Later people invented spontaneous symmetry breaking and higgs mechanism t. Quantum yangmills problem, navierstokes problem duration. G, and construct the corresponding yangmills theory. As an input, yang mills requires some specified gauge group. Third, we show that the reflection positivity is violated for any value of the parameters in the massdeformed yangmills theory to. Nov 05, 2009 among seven problems, proposed for xxi century by clay mathematical institute, there are two stemming from physics. Su2 quantum yangmills theory, and a strongly perturbed bps monopole international conference on numerical analysis and applied mathematics 2017, thessaloniki r. Prove that for any compact simple gauge group g, quantum yangmills theory of r4 exists and has a mass gap. Existence of yangmills theory with vacuum vector and. The detailed statement of the problem, written by a.
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