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From: Stephen Wolfram, A New Kind of Science
Notes for Chapter 9: Fundamental Physics
Section: Quantum Phenomena
Page 1061

Quantum field theory. In standard approaches to quantum field theory one tends to think of particles as some kind of small perturbations in a field. Normally for calculations these perturbations are on their own taken to be plane waves of definite frequency, and indeed in many ways they are direct analogs of waves in classical field theories like those of electromagnetism or fluid mechanics. To investigate collisions between particles, one thus looks at what happens with multiple waves. In a system described by linear equations, there is always a simple superposition principle, and waves just pass through each other unchanged. But what in effect leads to non-trivial interactions between particles is the presence of nonlinearities. If these are small enough then it makes sense to do a perturbation expansion in which one approximates field configurations in terms of a succession of arrangements of ordinary waves - as in Feynman diagrams. But just as one cannot expect to capture fully turbulent fluid flow in terms of a few simple waves, so in general as soon as there is substantial nonlinearity it will no longer be sufficient just to do perturbation expansions. And indeed for example in QCD there are presumably many cases in which it is necessary to look at something closer to actual complete field configurations - and correlations in them.

The way the path integral for a quantum field theory works, each possible configuration of the field is in effect taken to make a contribution Exp[I s/h], where s is the so-called action for the field configuration (given by the integral of the Lagrangian density - essentially a modified energy density), and h is a basic scale factor for quantum effects (Planck’s constant divided by 2 Pi). In most places in the space of all possible field configurations, the value of s will vary quite quickly between nearby configurations. And assuming this variation is somehow random, the contributions of these nearby configurations will tend to cancel out. But inevitably there will be some places in the space where s is stationary (has zero variational derivative) with respect to changes in fields. And in some approximation the field configurations in these places can be expected to dominate the path integral. But it turns out that these field configurations are exactly the ones that satisfy the partial differential equations for the classical version of the field theory. (This is analogous to what happens for example in classical diffraction theory, where there is an analog of the path integral - with h replaced by inverse frequency - whose stationary points correspond through the so-called eikonal approximation to rays in geometrical optics.) In cases like QED and QCD the most obvious solutions to the classical equations are ones in which all fields are zero. And indeed standard perturbation theory is based on starting from these and then looking at the expansion of Exp[I s/h] in powers of the coupling constant. But while this works for QED, it is only adequate for QCD in situations where the effective coupling is small. And indeed in other situations it seems likely that there will be all sorts of other solutions to the classical equations that become important. But apart from a few special cases with high symmetry, remarkably little is known about solutions to the classical equations even for pure gluon fields. No doubt the analog of turbulence can occur, and certainly there is sensitive dependence on initial conditions (even non-Abelian plane waves involve iterated maps that show this). Presumably much like in fluids there are various coherent structures such as color flux tubes and glueballs. But I doubt that states involving organized arrangements of these are common. And in general when there is strong coupling the path integral will potentially be dominated by large numbers of configurations not close to classical solutions.

In studying quantum field theories it has been common to consider effectively replacing time coordinates t by I t to go from ordinary Minkowski space to Euclidean space (see page 1047). But while there is no problem in doing this at a formal mathematical level - and indeed the expressions one gets from Feynman diagrams can always be analytically continued in this way - what general correspondence there is for actual physical processes is far from clear. Formally continuing to Euclidean space makes path integrals easier to define with traditional mathematics, and gives them weights of the form Exp[-β s] - analogous to constant temperature systems in statistical mechanics. Discretizing yields lattice gauge theories with energy functions involving for example Cos[Subscript[θ,i]-Subscript[θ,j]] for color directions at adjacent sites. And Monte Carlo studies of such theories suggest all sorts of complex behavior, often similar in outline from what appears to occur in the corresponding classical field theories. (It seems conceivable that asymptotic freedom could lead to an analog of damping at small scales roughly like viscosity in turbulent fluids.)

One of the apparent implications of QCD is the confinement of quarks and gluons inside color-neutral hadrons. And at some level this is presumably a reflection of the fact that QCD forces get stronger rather than weaker with increasing distance. The beginnings of this are visible in perturbation theory in the increase of the effective coupling with distance associated with asymptotic freedom. (In QED effective couplings decrease slightly with distance because fields get screened by virtual electron-positron pairs. The same happens with virtual quarks in QCD, but a larger effect is virtual gluon pairs whose color magnetic moments line up with a color field and serve to increase it.) At larger distances something like color flux tubes that act like elastic strings may form. But no detailed way to get confinement with purely classical gluon fields is known. In the quantum case, a sign of confinement would be exponential decrease with spacetime area of the average phase of color flux through so-called Wilson loops - and this is achieved if there is in a sense maximal randomness in field configurations. (Note that it is not inconceivable that the formal problem of whether quarks and gluons can ever escape to infinity starting from some given class of field configurations may in general be undecidable.)

Stephen Wolfram, A New Kind of Science (Wolfram Media, 2002), page 1061.
© 2002, Stephen Wolfram, LLC