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 \mathbb{R}^{(3,1)}, 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, N=4 supersymmetric Yang Mills theory living on the 4-dimensional boundary of a five-dimensional anti-deSitter space AdS_{5} and Type IIB string theory on AdS_{5}\times S^{5}, where S^{5} 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 \mathbb{G} be a simple compact Lie group and let \mathcal{R} be a representation of \mathbb{G}. In (3+1) dimensions \mathbb{R}^{(3,1)}=\mathbb{R}^{3}\times\mathbb{R} and in (2+1) dimensions \mathbb{R}^{(2,1)}=\mathbb{R}^{2}\times\mathbb{R}, the pure gauge theory with connection or 1-form {A}=A_{\mu}(x)dx^{\mu}, curvature {F}=d{A}+{A}\wedge {A}, gauge group \mathbb{G} and action \int tr({F}\wedge *{F}) exhibits confinement.

The specific case of the conjecture that is relevent for strong interactions is \mathbb{G}=su(3) and \mathcal{R}=\mathbb{C}^{3}, 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 (q\overline{q}) pairs (mesons) or (qqq) triplets (baryons) with an overall neutral net color. There is a linearly rising potential between quarks of the form V(r)\sim k r, for a constant k, and consequently no free quarks exist in nature.

Confinement can be sought within pure Yang-Mills gauge theory on \mathbb{R}^{(3,1)} or \mathbb{R}^{(2,1)} with the quarks as external sources, and it is essentially the properties of the non-abelian quantum Yang-Mills vacuum or gluon fields that will determine confinement. The confining phase is generally defined in terms of Wilson loop functionals. In gauge theory, a Wilson loop is a gauge-invariant observable derived from the holonomy \mathit{h}(A(x),\gamma) of the gauge connection {A}(x) around a loop \gamma on a manifold \mathbb{R}^{(3,1)}=\mathbb{R}^{3}\times \mathbb{R}. The underlying non-abelian gauge group is \mathbb{G}. To demonstrate confinement, the vacuum expectation value of the path-ordered Wilson loop should exhibit an exponential area-law behaviour

\langle W({\gamma})\rangle =\langle tr_{R} {p}~exp(i \oint_{{\gamma}}{A}_{\mu}(x)dx^{\mu})\rangle \sim exp(-k\mathit{area}({\gamma}))

where \mathit{area}({\gamma}) is the area enclosed by the curve or loop {\gamma}\subset\mathbb{R}^{(3,1)} and k\in\mathbb{R} is a constant called the string tension. p is the path-ordering operator and the trace tr gaurantees invariance under gauge transformations. The area law is equivalent to the quark-antiquark linear confining potential V(r)\sim kr with the quarks connected by a string with (constant) tension k. A confining theory gives an area law but a non-confining theory gives a perimeter law exp(-k\ell), where \ell is the distance around the loop. The vacuum expectation of the Wilson loop functional \langle W(\gamma)\rangle is formally given in the quantum theory by the Feynman path integral or partition function

Z = \langle W(\gamma)\rangle = \int d\mu[{A}]exp(iS) W({\gamma})

where \int d\mu[{A}] is a functional integration measure over the gauge orbits that is generally not well defined, and S is the classical Yang-Mills action \int tr {F} \wedge *{F}, and {F}=d{A} + {A} \wedge {A} or F=F_{\mu\nu}(x)dx^{\mu}\wedge dx^{\nu} is the Yang-Mills curvature 2-form or field strength. The fundamental quantity which defines the vacuum structure is the n-point field correlator

D^{n}(x_{1}...x_{n}) =\langle F_{\mu_{1}\nu_{1}}(x_{1})\phi(x_{1},x_{2})...F_{\mu_{n}\nu_{n}}(x_{n})\phi(x_{n},x_{1})\rangle

where \phi(x,y)=Pexp(i \int_{y}^{x}A_{\mu}(z)dz) is a parallel transport operator.

However, on \mathbb{R}^{(3,1)} it is impossible to calculate the expectation value of the Wilson loop in a closed analytical form using path integrals, and standard perturbation theory is invalid at large distances or strong couplings where confinement occurs. Even on \mathbb{R}^{(2,1)} the problem is still very hard and you also have to bear in mind Richard Feynman’s remark that, ‘ there is a serious risk that working in (2+1) dimensions you are wasting your time or even that you are getting a false impression of how things work in (3+1)”. On \mathbb{R}^{(2,1)} even a U(1) abelian Maxwell theory with curvature F=dA confines in the sense that the potential between two point charges on a disc or plane is logarithmic since the force between them varies as 1/r and so an infinite energy is required to seperate them an infinite distance. But this is very different from the sought-after linear potential that characterizes Yang-Mills confinement, which must necessarily arise purely from the nonlinear or ‘nonabelian piece’ A \wedge A.

Despite a lack of rigorous proof within QCD or quantum field theory that Yang-Mills theory on \mathbb{R}^{(3,1)} confines quarks and gluons, powerful renormalization group arguments (from Gross, Wilczek and Politzer) have provided compelling theoretical justification that it does; the coupling constant g is indeed large enough at large seperations to produce such an expected effect. Confinement also explains why the massless gluon gauge fields do not produce long-range forces like the massless gauge particles associated with gravitation or electromagnetism: although the gluons are massless they ‘condense’ into string-like structures binding quarks together at their ends. If we had enough energy to try and pull and break the string two more quarks would appear at the ends, much like the breaking of a bar magnet does not produce two isolated monopoles. The ‘chromo- electric-magnetic’ flux tubes or ’strings’ that form at strong coupling are also somewhat analogous to the Abrikosov-Gorkov vortex lines in superconductors that arise from expulsion of magnetic flux. String-like structure is clearly an established feature of strong interactions at strong coupling. Indeed, string theory began as a theory of strong interactions before QCD became dominant and before the spin-2 mode in the spectrum suggested it might instead be a theory of gravitation.

Gluons can also self-interact forming states called ‘glueballs’ which have a mass, even though the gluons themselves are massless. (This is also related to the mass gap problem, a proof of the existance of which is also part of the Millennium problem. ) In a sense the self interaction of gluons–ie, that the carriers of the color force can feel the color force– make the Yang-Mills equations less like Maxwell’s equations and more like the Einstein equations for the gravitational field: both the Yang-Mills and Einstein equations are nonlinear and so the gauge particles in the corresponding quantum theories–respectively, gluons and gravitons–are self-interacting as a result.

At very high temperatures there is a phase transition whereby baryons ‘melt’ and the constituent quarks and gluons then form a free plasma. (Although recent work suggests it may be more like a liquid.) Recent experiments in relativistic heavy ion collisions (RHICs) have studied this. String theory has even provided a useful theoretical approach via the AdS/CFT duality, and is in a sense here returning to its origins in the realm of strong interaction physics. In the early universe quarks and gluons would have formed such a free plasma before condensing into baryons as the universe expanded and cooled. Frank Wilczek in his Nobel lecture gave a nice paraphrasing of Jean Jacques Rosseau when he stated that: “quarks are born free but everywhere they are in chains”.

As mentioned, confinement, as well as other quantities of interest within QCD such as particle masses etc., have been predominantly established from a lattice formulation of the theory using numerical supercomputer calculations. In lattice QCD, spacetime is approximated as a lattice and the quark and gluon fields are placed at lattice sites. Given a gauge group \mathbb{G}=su(3), each element U_{ij}\in \mathbb{G} is associated with each nearest-neighbour pair of lattice sites latex (i,j). A gluon link field is given by U_{ij}=exp(igA_{\mu}a), where \mu is the direction of a given bond and a is the lattice spacing. For gauge theory without fermions, Wilson’s formulation emphasises the analogies with magnetism in statistical mechanics in that the link variables U_{ij} are much like spins located at crystal bonds. The Yang Mills action is then given by summing over all elementary squares or ‘plaquettes’ \lbrace \Box\rbrace of the lattice so that

S(U)=\sum_{\Box}S_{\Box}(U_{\Box})

where S_{\Box}(U_{\Box})\sim Re(tr(U_{ij}U_{jk}U_{kl}U_{li})). The path integral is then Z=\sum_{U\in\mathbb{G}}exp(S(U)) and so observables \langle O(U)\rangle can be computed. The Wilson loop is now given as the trace of a product of links around a closed loop \gamma and is gauge invariant

W(\gamma)=\langle tr\prod_{ij}U_{ij}\rangle

If the loop \gamma is a rectangle of dimensions \mathit{area}(\gamma)=R\times T then for large and long rectangular loops it is found that W(\gamma)=W(R,T)\sim exp(-kRT). Lattice QCD therefore establishes confinement from brute-force numerical evaluation.

In order to try and establish the confining area-law behaviour from an analytical perspective, a number of prescriptions have been established to evaluate the vacuum expectation value of the Wilson loop functional within some reasonable approximation. Recently, Nair and others have considered a Hamiltonian approach with nonperturbative calculations leading to estimates for mass gap, string tension and confinement in (2+1) dimensions. Other approaches include (i) the stochastic vacuum models (SVM), and (ii) the centre vortex models (CVM) of confinement initiated by ‘t Hooft. An alternative approach has also been proposed in recent years again based on the AdS/CFT conjecture, as already discussed. However, there is no known string dual to pure QCD, although in principle the duality does provide a strategy to compute the area law.

Within the stochastic vacuum model the QCD vacuum is interpreted as a random or stochastic medium that is colour neutral essentially representing a quantum noise. Since asymptotic freedom basically states that the quantum theory behaves like the classical theory at short distances (weak coupling) there may be scales where a classical stochastic description is viable. The field correlators D^{(n)}(x_{1}...x_{n}) and the non-abelian Stokes theorem allow the Wilson loop to be written as a cluster expansion

\langle W(\gamma)\rangle = exp\bigg(\sum_{n}\frac{{ig}^{2}}{n!}\int \widetilde{D}^{(n)}(x_{1}...x_{n})d\sigma_{\mu_{1}\nu_{1}}(x_{1})...d\sigma_{\mu_{n}\nu_{n}}(x_{n})\bigg)

This is truncated at n=2 in a Gaussian dominance approximation, and an area law can be established. Integration is performed over the minimal surface S_{min} bounded by the curve \gamma, and \widetilde{D}^{(n)} is the cumulant or connected correlator obtained by subtracting out all disconnected averages. The result is \langle W(\gamma)\rangle \sim exp(-kS_{min}). (More details are in hep-ph/0007223). The approach has found a number of useful applications in QCD.

Within the centre vortex theory of confinement, first proposed over 25 years ago, the QCD vacuum is considered a condensate of color magnetic vortices with a flux quantised in terms of the centre group \mathbb{Z}_{N}. It is based on the notion that a random distribution of vortex color flux is sufficient to produce a confining area law. A considerable body of evidence including lattice simulations, now supports the claim that the infra-red properties of Yang-Mills theory are indeed well accounted for in terms of vortex effects and that vortices are the correct gluonic degrees of freedom in the infrared. ‘t Hooft initially proposed criteria for confinement within a vortex model: [i] there is a percolating cluster of vortices, [ii] the Wilson loops measure the linking with the vortex lines and [iii] vortices at different locations are weakly correlated.

The centre vortex model gives a well-known heuristic derivation of the area law. Consider an ensemble or ‘gas’ of centre vortices such that the vortices are distributed randomly and such that the intersection points of vortices within a 2-dimensional plane or disc {D}\subset \mathbb{R}^{3} are found at random uncorrelated locations. Let a Wilson loop {\gamma}\subset {D} with radius \ell be embedded within a larger circle or loop of radius L and on this slice or plane distribute m vortex intersection points at random. Each of these contributes a factor (-1) to the value of the Wilson loop if it falls within the area \mathit{area}({\gamma}) spanned by the loop. The probability for this to occur for any given point is \mathit{area(}\gamma)/\ell^{2}. The probability that n of the m vortex intersection points falls within \mathit{area}({\gamma}) is binomial. Since the Wilson loop takes the value of (-1)^{n} in the presence of n intersection points within the area \mathit{area(}{\gamma}) its expectation value is

\langle W(\gamma)\rangle=\sum_{n=o}^{m}(-1)^{n}(\frac{m}{n})(\frac{area({\gamma})}{\pi \ell^{2}})^{n}(1-\frac{area({\gamma})}{\pi \ell^{2}})^{m-n}=(1-2\rho \frac{area({\gamma})}{m})^{n}

Taking the limit as m\rightarrow\infty then gives

\langle W(\gamma)\rangle \longrightarrow exp(-2\rho~ \mathit{area}(\gamma))

The planar density of vortex loops is \rho =m/\pi \ell^{2} and the string tension is k=2\rho. Lattice calculations suggest that the vortex area density \rho represents a physical observable. So again, the confinement behavior does seem to emerge.

It can be demonstrated with some confidence that gauge theories do indeed exhibit confinement, a crucial property of strong interactions, but a rigorous analytical proof on \mathbb{R}^{3,1}, in the mathematical physics sense, remains beyond our grasp. Some might argue it is not necessary and that computer simulations and approximations are sufficient. But while there have certainly been interesting analytical developments, as well as important connections to mathematics, the lack of such a proof –and the inability to rigorously construct and define the quantum Yang Mills theory on a 4-manifold–demonstrates that we don’t really understand strongly coupled non-perturbative gauge theories as well as we should, or would like; even though they are at the very heart of the Standard Model. Until we more fully understand these crucial mathematical underpinnings of the Standard model it may never be possible to go beyond it (at least from the perspective of pure theory).

To conclude, it remains to be seen whether the mysteries of the strong interactions will ultimately prove to be ‘too complicated for the human mind’. At any rate, QCD, gauge theories and the ongoing quest to learn more about the strong interactions and their connections with the other forces, will probably keep people very busy and fascinated for a long time to come.