On The Phase Transition in D=3 Yang-Mills Chern-Simons Gauge Theory

$SU(N)$ Yang-Mills theory in three dimensions, with a Chern-Simons term of level $k$ (an integer) added, has two dimensionful coupling constants, $g^2 k$ and $g^2 N$; its possible phases depend on the size of $k$ relative to $N$. For $k \gg N$, this theory approaches topological Chern-Simons theory with no Yang-Mills term, and expectation values of multiple Wilson loops yield Jones polynomials, as Witten has shown; it can be treated semiclassically. For $k=0$, the theory is badly infrared singular in perturbation theory, a non-perturbative mass and subsequent quantum solitons are generated, and Wilson loops show an area law. We argue that there is a phase transition between these two behaviors at a critical value of $k$, called $k_c$, with $k_c/N \approx 2 \pm .7$. Three lines of evidence are given: First, a gauge-invariant one-loop calculation shows that the perturbative theory has tachyonic problems if $k \leq 29N/12$.The theory becomes sensible only if there is an additional dynamic source of gauge-boson mass, just as in the $k=0$ case. Second, we study in a rough approximation the free energy and show that for $k \leq k_c$ there is a non-trivial vacuum condensate driven by soliton entropy and driving a gauge-boson dynamical mass $M$, while both the condensate and $M$ vanish for $k \geq k_c$. Third, we study possible quantum solitons stemming from an effective action having both a Chern-Simons mass $m$ and a (gauge-invariant) dynamical mass $M$. We show that if M \gsim 0.5 m, there are finite-action quantum sphalerons, while none survive in the classical limit $M=0$, as shown earlier by D'Hoker and Vinet. There are also quantum topological vortices smoothly vanishing as $M \rightarrow 0$.Comment: 36 pages, latex, two .eps and three .ps figures in a gzipped uuencoded fil

On One-Loop Gap Equations for the Magnetic Mass in d=3 Gauge Theory

Recently several workers have attempted determinations of the so-called magnetic mass of d=3 non-Abelian gauge theories through a one-loop gap equation, using a free massive propagator as input. Self-consistency is attained only on-shell, because the usual Feynman-graph construction is gauge-dependent off-shell. We examine two previous studies of the pinch technique proper self-energy, which is gauge-invariant at all momenta, using a free propagator as input, and show that it leads to inconsistent and unphysical result. In one case the residue of the pole has the wrong sign (necessarily implying the presence of a tachyonic pole); in the second case the residue is positive, but two orders of magnitude larger than the input residue, which shows that the residue is on the verge of becoming ghostlike. This happens because of the infrared instability of d=3 gauge theory. A possible alternative one-loop determination via the effective action also fails. The lesson is that gap equations must be considered at least at two-loop level.Comment: 21 pages, LaTex, 2 .eps figure

Seeking an Even-Parity Mass Term for 3-D Gauge Theory

Mass-gap calculations in three-dimensional gauge theories are discussed. Also we present a Chern--Simons-like mass-generating mechanism which preserves parity and is realized non-perturbatively.Comment: 11 pages, revte

Land and politics in the Transvaal in the 1880s

African Studies Seminar series. Paper presented October, 1972It is evident that the population of the South African Republic did not constitute a single homogenous group, devoid of variation in wealth, education and life-style. The diversity of Afrikaner society has frequently been overlooked by historians however, and the courses of social differentation completely ignored. The State Archives at Pretoria do contain some enormous collections of largely unused material however, which for all their superficial dullness embody a vast amount of detailed and valuable data relating to this very problem HAD one the time and equipment to analyse in detail the information which is to be found in the land-registers, estates, death notices and wills, there can be no question but that the result would represent a most notable contribution to the social and economic history of South Africa. If in addition it were possible to collect the petitions, ballot papers and voting list, many of them still extant then another dimension could be superimposed. These would however, be projects of vast size, and although they will, hopefully, be attempted in the future, for the present we shall have to be satisfied with a more modest harvest of information. There is no expectation that however complete, the bare bones of statistical data will answer all of our questions. They can only be used to suggest the patterns of life extending beyond the range of the livelier facts called from volksraad minutes, newspapers, petitions, private correspondence and reminiscences. The major part of this paper will be concerned with the social and economic diversity reflected in landownership and in particular with its causes, general and local. No claim is made to completeness in dealing with the origins of the phenomenon, and attention is paid largely to that material which illustrates aspects of the problem which have been previously neglected. Practical considerations obviously restrict the scope of all research, and for the purposes of this paper most of the evidence relates to the district of Wakkerstroom

Center Vortices, Nexuses, and Fractional Topological Charge

It has been remarked in several previous works that the combination of center vortices and nexuses (a nexus is a monopole-like soliton whose world line mediates certain allowed changes of field strengths on vortex surfaces) carry topological charge quantized in units of 1/N for gauge group SU(N). These fractional charges arise from the interpretation of the standard topological charge integral as a sum of (integral) intersection numbers weighted by certain (fractional) traces. We show that without nexuses the sum of intersection numbers gives vanishing topological charge (since vortex surfaces are closed and compact). With nexuses living as world lines on vortices, the contributions to the total intersection number are weighted by different trace factors, and yield a picture of the total topological charge as a linking of a closed nexus world line with a vortex surface; this linking gives rise to a non-vanishing but integral topological charge. This reflects the standard 2\pi periodicity of the theta angle. We argue that the Witten-Veneziano relation, naively violating 2\pi periodicity, scales properly with N at large N without requiring 2\pi N periodicity. This reflects the underlying composition of localized fractional topological charge, which are in general widely separated. Some simple models are given of this behavior. Nexuses lead to non-standard vortex surfaces for all SU(N) and to surfaces which are not manifolds for N>2. We generalize previously-introduced nexuses to all SU(N) in terms of a set of fundamental nexuses, which can be distorted into a configuration resembling the 't Hooft-Polyakov monopole with no strings. The existence of localized but widely-separated fractional topological charges, adding to integers only on long distance scales, has implications for chiral symmetry breakdown.Comment: 15 pages, revtex, 6 .eps figure

Center vortices and confinement vs. screening

We study adjoint and fundamental Wilson loops in the center-vortex picture of confinement, for gauge group SU(N) with general N. There are N-1 distinct vortices, whose properties, including collective coordinates and actions, we study. In d=2 we construct a center-vortex model by hand so that it has a smooth large-N limit of fundamental-representation Wilson loops and find, as expected, confinement. Extending an earlier work by the author, we construct the adjoint Wilson-loop potential in this d=2 model for all N, as an expansion in powers of $\rho/M^2$, where $\rho$ is the vortex density per unit area and M is the vortex inverse size, and find, as expected, screening. The leading term of the adjoint potential shows a roughly linear regime followed by string breaking when the potential energy is about 2M. This leading potential is a universal (N-independent at fixed fundamental string tension $K_F$) of the form $(K_F/M)U(MR)$, where R is the spacelike dimension of a rectangular Wilson loop. The linear-regime slope is not necessarily related to $K_F$ by Casimir scaling. We show that in d=2 the dilute vortex model is essentially equivalent to true d=2 QCD, but that this is not so for adjoint representations; arguments to the contrary are based on illegal cumulant expansions which fail to represent the necessary periodicity of the Wilson loop in the vortex flux. Most of our arguments are expected to hold in d=3,4 also.Comment: 29 pages, LaTex, 1 figure. Minor changes; references added; discussion of factorization sharpened. Major conclusions unchange

The Effects of a Low Back Pain Vibration Modality on Trunk Postural Control

Abstract Introduction: Low back pain (LBP) is prevalent worldwide and is affecting even more individuals as the population ages. There has recently been an increase in production of low back pain (LBP) vibration modality belts that apply localized vibration to the lumbar region of the spine as it has shown to reduce pain. However, vibration is also known to perturb muscle spindles and thus interfere with proprioception. If a LBP vibration modality causes a proprioceptive deficit in the trunk lumbar region it could potentially increase an individual’s risk of injury due to poor postural control. Therefore, the effects of a LBP vibration modality on trunk motor control needed to be investigated further. Methods: 15 participants who had not experienced LBP for longer than 3 days in the previous year were recruited to partake in a control and experimental day approximately one week apart from each other. Each day consisted of 3 conditions including; pre vibration, post-vibration, and vibration ON. Between each condition participants sat on a standard chair for 15 minutes while wearing the vibration belt and the belt was also worn during the last vibration ON condition. On the control day the vibration belt was worn but not turned on between conditions or during the last vibration ON condition. Each condition consisted of 4 sudden unexpected trunk perturbations following by 3 half- and 3 full-trunk flexion repositioning tasks. Electromyography (EMG) was collected from several trunk muscles to analyze changes in trunk postural control and motion capture was collected to analyze changes in lumbar spine movement. Results: The magnitude of lumbar flexion caused from the sudden trunk perturbation decreased after sitting for 15-minutes and was exacerbated by vibration. No other significant differences in the variables measured after wearing the vibration belt for 15-minutes were found. However, profound differences were found during the vibration ON condition. Bilaterally the LES (p\u3c0.0001), LEO (p=0.03), and REO (p=0.002) displayed significantly delayed muscle activation onset latencies. Bilaterally the LES (p\u3c0.0001), LTES (p=0.01), RTES (p\u3c0.0001), EO (p\u3c0.0001), LRA (p\u3c0.0001), and RRA (p=0.0001) all showed significant increases in resting muscle activation pre-perturbation. The LLES (p=0.0002), RLES (p=0.0003), LTES (p=0.0008), RTES (p=0.03), LRA (p=0.02), and RRA (p=0.01) also all displayed significantly reduced muscle activation magnitudes post-perturbation. Discussion and Conclusion: The increased resting muscle activation pre-perturbation caused from the vibration can be explained as the tonic vibration reflex (TVR) because of vibrations stimulatory effect on Ia afferents. The delayed muscle activation onset latencies that were observed while wearing the vibration belt most likely occurred because of the ability for vibration to create a ‘busy line’ or vibration-locked discharge of the muscle spindles. Proper muscle spindle function is essential for providing critical proprioceptive information on body awareness and functions as a protective mechanism against injury. Additionally, the TVR can also lead to fatigue and subsequent altered trunk motor control. The findings of the current study reveal that impaired trunk motor control when wearing a LBP vibration modality belt needs to be considered due to the potentially greater risk for experiencing an injury to the low back or LBP