arXiv subjects
Publications and source records attributed to Frank Wilczek.
We develop techniques for calculating the ground state wave functional and the geometric entropy for some simple field theories. Special attention is devoted to fermions, which present special technical difficulties in this regard. Explicit calculations are carried through for free mass bosons and fermions in two dimensions, using an adaptation of Unruh's technique to treat black hole radiance.
We expand upon on an earlier renormalization group analysis of a non-Fermi liquid fixed point that plausibly govers the two dimensional electron liquid in a magnetic field near filling fraction $ν=1/2$. We give a more complete description of our somewhat unorthodox renormalization group transformation by relating both our field-theoretic approach to a direct mode elimination and our anisotropic scaling to the general problem of incorporating curvature of the Fermi surface. We derive physical consequences of the fixed point by showing how they follow from renormalization group equations for finite-size scaling, where the size may be set by the temperature or by the frequency of interest. In order fully to exploit this approach, it is necessary to take into account composite operators, including in some cases dangerous ``irrelevant'' operators. We devote special attention to gauge invariance, both as a formal requirement and in its positive role providing Ward identities constraining the renormalization of composite operators. We emphasize that new considerations arise in describing properties of the physical electrons (as opposed to the quasiparticles.) We propose an experiment which, if feasible, will allow the most characteristic feature of our results, that is
We consider the modification of the formulas for black hole radiation, due to the self-gravitation of the radiation. This is done by truncating the coupled particle-hole system to a small set of modes, that are plausibly the most significant ones, and quantizing the reduced system. In this way we find that the particles no longer move along geodesics, nor is the action along the rays zero for a massless particle. The radiation is no longer thermal, but is corrected in a definite way that we calculate. Our methods can be extended in a straightforward manner to discuss correlations in the radiation, or between incoming particles and the radiation.
Guided by a Hamiltonian treatment of spherically symmetric geometry, we find a remarkably simple -- stationary, but not static -- form for the line element of Schwarzschild (and Reissner-Nordstrom) geometry. The line element continues smoothly through the horizon; by exploiting this feature we are able to give a very simple and physically transparent derivation of the Hawking radiance. We construct the complete Penrose diagram by enforcing time-reversal symmetry. Finally we outline how an improved treatment of the radiance, including effects of self-gravitation, can be obtained.
We derive some physical properties of ideal assemblies of identical particles obeying generalized exclusion statistics. We discuss fluctuations, and in this connection point out a fundamental contrast to conventional quantum statistics. We demonstrate a duality relating the distribution of particles at statistics $g$ to the distribution of holes at statistics $1/g$. We suggest applications to Mott insulators.
We construct models of excitations about a Fermi surface that display calculable deviations from Fermi liquid behavior in the low-energy limit. They arise as a consequence of coupling to a Chern-Simons gauge field, whose fluctations are controlled through a ${1\over{k^x}}$ interaction. The Fermi liquid fixed point is shown to be unstable in the infrared for $x<1$, and an infrared-stable fixed point is found in a $(1-x)$-expansion, analogous to the $ε$-expansion of critical phenomena. $x=1$ corresponds to Coulomb interactions, and in this case we find a logarithmic approach to zero coupling. We describe the low-energy behavior of metals in the universality class of the new fixed point, and discuss its possible application to the compressible $ν={1\over2}$ quantum Hall state and to the normal state of copper-oxide superconductors.
We show that a geometrical notion of entropy, definable in flat space, governs the first quantum correction to the Bekenstein-Hawking black hole entropy. We describe two methods for calculating this entropy -- a straightforward Hamiltonian approach, and a less direct but more powerful Euclidean (heat kernel) method. The entropy diverges in quantum field theory in the absence of an ultraviolet cutoff. Various related finite quantities can be extracted with further work. We briefly discuss the corresponding question in string theory.
I have been asked to discuss the status of QCD. It seems to me that there are three main points to be made about the present status of QCD: $\bullet$ QCD is right, and we can do many beautiful things with it. $\bullet$ There are several important concrete problems that lie just beyond the edge of our current understanding. $\bullet$ There are some foundational issues in QCD, and some recent developments, that may point toward entirely new directions. These points will, I believe, emerge quite clearly from the following more detailed discussion. The discussion will be in three parts. I'll first discuss elementary processes, then more complicated processes, and then finally foundational issues.
After briefly reviewing work I've done recently with Krishna Rajagopal on the nature of the equilibrium phase transition in QCD at finite temperature and zero baryon number and on the possibility of forming misaligned vacuum by a quench, I discuss -- also briefly -- two new items. These are extension of the arguments concerning the order of the transition to to 2+1 dimensions, relating in this case gauge to conformal field theories, that could be explored in numerical experiments; and the energy budget at the chiral transition and its relevance to the nature of the transition and to quenching.
The standard model of particle physics is marvelously successful. However, it is obviously not a complete or final theory. I shall argue here that the structure of the standard model gives some quite concrete, compelling hints regarding what lies beyond.
To model the dynamics of the chiral order parameter in a far from equilibrium phase transition, we consider quenching in the O(4) linear sigma model. We argue, and present numerical evidence, that in the period immediately following the quench long wavelength modes of the pion field are amplified. This results in large regions of coherent pion oscillations, and could lead to dramatic phenomenological consequences in heavy ion collisions.
Following Hawking, it is usual to mimic the effect of collapse space-time geometry on quantum fields in a semi-classical approximation by imposing suitable boundary conditions at the origin of coordinates, which effectively becomes a moving mirror. Suitable mirror trajectories induces a close analogue to the radiance of black holes, including a flux of outgoing radiation that appears accurately thermal. If the acceleration of the mirror eventually ceases the complete state of the radiation field is a pure quantum state, even though it is indistinguishable from an accurately thermal state for an arbitrarily long period of time and in a precise sense differs little from ``pure thermal'' closely followed by ``vacuum''. Suspicions that the semiclassical calculation of black hole radiance gives evidence for the evolution of pure into mixed states are criticized on this basis. Possible extensions of the model to mimic black holes more accurately (including the effects of back reaction and partial transparency), while remaining within the realm of tractable models, are suggested.
QCD is now a mature theory, and it is possible to begin to view its place in the conceptual universe of physics with appropriate perspective. There is a certain irony in the achievements of QCD. For the problems which initially drove its development only limited insight has been achieved. However I shall argue that QCD is actually {\it more\/} special and important a theory than one had any right to anticipate. After elaborating these quasi-philosophical remarks, I discuss two current frontiers of physics that illustrate the continuing vitality of the ideas. The recent wealth of beautiful precision experiments measuring the parameters of the standard model has made it possible to consider the unification of couplings in unprecedented quantitative detail. One central result emerging from these developments is a tantalizing hint of virtual supersymmetry. The possibility of phase transitions in matter at temperatures of order $\sim 10^2 Mev$, governed by QCD dynamics, is of interest from several points of view. There is the interesting possibility of long-range correlations in heavy ion collisions due to the creation of large regions of misaligned chiral condensate. Opening Talk at the Aachen conference, ``20 Years of QCD'', June 1992.
A black hole may carry quantum numbers that are {\it not} associated with massless gauge fields, contrary to the spirit of the ``no-hair'' theorems. We describe in detail two different types of black hole hair that decay exponentially at long range. The first type is associated with discrete gauge charge and the screening is due to the Higgs mechanism. The second type is associated with color magnetic charge, and the screening is due to color confinement. In both cases, we perform semi-classical calculations of the effect of the hair on local observables outside the horizon, and on black hole thermodynamics. These effects are generated by virtual cosmic strings, or virtual electric flux tubes, that sweep around the event horizon. The effects of discrete gauge charge are non-perturbative in $\hbar$, but the effects of color magnetic charge become $\hbar$-independent in a suitable limit. We present an alternative treatment of discrete gauge charge using dual variables, and examine the possibility of black hole hair associated with discrete {\it global} symmetry. We draw the distinction between {\it primary} hair, which endows a black hole with new quantum numbers, and {\it secondary} hair, which does not, and we point out some varieties of secondary hair that occur in the standard model of particle physics.
It is shown that the breakdown of a {\it global} symmetry group to a discrete subgroup can lead to analogues of the Aharonov-Bohm effect. At sufficiently low momentum, the cross-section for scattering of a particle with nontrivial $\Z_2$ charge off a global vortex is almost equal to (but definitely different from) maximal Aharonov-Bohm scattering; the effect goes away at large momentum. The scattering of a spin-1/2 particle off a magnetic vortex provides an amusing experimentally realizable example.