Freedom and slavery in the subnuclear realm: quarks, gluons and confinement (draft)

Steven Weinberg once remarked that there is a tradition in theoretical physics that affected him, but ‘which by no means affected everyone’, namely that the strong (nuclear) interactions are ‘too complicated for the human mind’. Fortunately, he went on instead to clarify our understanding of the weak nuclear interactions and their connection with electromagnetism. (Whether he actually did say this I am not sure. Physics is full of folklore but I have read this statement at least twice over the years.) Those who did take up the challenge to break their heads on explaining the strong interactions included Murray Gell-Mann, who brought in powerful group-theoretic ideas; David Gross, David Politzer and Frank Wilczek, who demonstrated asymptotic freedom; and Kenneth Wilson, who provided important technical tools and demonstrated how the resulting theory of quantum chromodynamics (QCD) could be put on large computers and solved numerically. All were rightly awarded the famous Swedish prize. (As was Weinberg too of course, although Wilson was actually awarded the prize for his work on critical phenomena and phase transitions.)

It became clear in the late 60s and early 70s that quarks and gluons constituted the ‘machinary’ inside protons and neutrons, the nuclear building blocks of the familiar forms of matter in the universe; from ordinary hydrogen and helium to carbon, iron, gold and everything else in the periodic table. It also become clear that the strength of the gluon-exchanging color forces holding quarks together in subnuclear matter was immense; giving rise to both the stability of ordinary matter and–almost as a ‘feeble’ byproduct–the ferocious infernos in the cores of the sun and stars, powering them for billions of years.

Despite the successes of QCD, many crucial aspects of the strong or color interactions and the Yang-Mills gauge theories underpinning them, still remain elusive; indeed, one of the Clay Millennium prizes is for a rigorous formulation of quantum Yang-Mills theory on , which currently remains beyond reach. This does not imply that little progress has been made: Yang-Mills theories, with their deep geometric significance, have played a crucial role in mathematics, as have their supersymmetric extentions. Work in string theory has also revealed a deep and tantalizing connection between string theory on one space and a QFT without gravity defined on the boundary of that space, as embodied in the celebrated AdS/CFT conjecture. For example, supersymmetric Yang Mills theory living on the 4-dimensional boundary of a five-dimensional anti-deSitter space and Type IIB string theory on , where is the 5-dimensional sphere. This is a very promising development within string theory and the conjecture has certainly emphasised once again the central importance of gauge theories. (A more detailed discussion on the importance of the duality for studying QCD can found here.) One of the main problems has always been (and still is) the need for development of tractable quantitative analytical nonperturbative methods. However, for all practical purposes, the main approach to effectively dealing with strongly coupled Yang-Mills theories and QCD has been (and still is) via powerful lattice computer simulations using numerical methods.

A central conjecture and open problem in quantum field theory is the following, and from a mathematical physics perspective the one that intrigues and interests me the most

Let be a simple compact Lie group and let be a representation of . In (3+1) dimensions and in (2+1) dimensions , the pure gauge theory with connection or 1-form , curvature , gauge group and action exhibits confinement.

The specific case of the conjecture that is relevent for strong interactions is and , and is tantamount to confinement of quarks within subnuclear matter. A rigorous analytical proof of confinement is a classical and long-standing open problem within gauge theory and QCD, and essentially states that quarks cannot be isolated: color-charged quarks are confined together in pairs (mesons) or triplets (baryons) with an overall neutral net color. There is a linearly rising potential between quarks of the form , for a constant , and consequently no free quarks exist in nature.

Tentative topics for 2007

Blogging has to take a low priority I am afraid due to time constraints, but topics I would like to try and write about this year are tentatively listed below, and in no particular order. Some will be ‘lay person reviews’, some will be semi-technical, while others (denoted with *) will be technical articles at the text book or journal level perfused with some LaTex. I certainly don’t claim to have expertise in all of these areas (although quite a few I think I can discuss with some authority). I do however have a wide range of interests reflecting my background. Primarily I am writing about research topics and areas I have either worked in or have studied or read about that I find especially interesting; or simply summarizing something I am trying to understand. Other posts will include book reviews, primers on a few topics, and reviews of preprints or journal articles. Comments, input etc. is welcome as are corrections of the dumb errors I will invariably make. (My only request is that any discourse is kept civil.)

Despite difficulties, string theory remains a compelling area of mathematical research.

Statistical topology of polymer tangles: applications of Wilson loops and Chern-Simons theory (*)

Stochastic Analysis I: Markov processes, diffusions and Brownian motions. (*)

Stochastic Analysis II: some illustrative applications in finance and quantum mechanics. (*)

Topological  field theory and molecular biology: dna knots and superhelicity.

Brief primer: Lax pairs, integrability and the Bethe ansatz. (*)

Brief primer: characteristic functions and the linked cluster decomposition. (*)

Freedom and slavery in the subnuclear realm: quarks, gluons and confinement. (*)

Inferno: the thermonuclear processes that fire the sun and the stars.

Dissipative stochastic processes and the statistical mechanics of folding proteins. (*)

The power of statistical sampling: history and applications of the Monte Carlo method.

John Von Neumann and Subramanyan Chandrasekhar: great intellects of the 20th century.

Singular terminal indecomposable past sets: a class of conformal factors restoring geodesic completeness. (*)

Light scattering and radiative transfer in liquid suspensions of polymers and bioparticles: borrowing mathematical methods from nuclear and astrophysics. (*)

Strings, branes and black holes: a tantalizing connection. (*)

The Kerr solution of the vacuum Einstein equations–the most remarkable exact solution in mathematical physics? (*)

Possibility of pycnonuclear reactions in superdense matter, liquid metal hydrogen and collapsed stars. (*)

“I know how to make it work…and it will change the course of history…”: Mathematician Stanislaw Ulam and the hydrogen superbomb.

Wave propagation on stochastic geometries. (*)

The surface physics and chemistry of artificial polymer-blood interations.

Notes on differential geometry, gravitation and black hole mechanics (*)

Classical Book Review: The Mathematical Theory of Black Holes. S. Chandrasekhar.

Book review: String Theory and M-theory–a Modern Introduction. Becker, Becker and Schwarz.

Book review:Inventing Money. The Story of LTCM and the legends behind it. N Dunbar.

Stochastic games and Markov games: are casinos beatable?

Submissions to J. Math. Physics for 2007.