On the problem of mass-dependence of the two-point function of the real scalar free massive field on the light cone

We investigate the generally assumed inconsistency in light cone quantum field theory that the restriction of a massive, real, scalar, free field to the nullplane $\Sigma=\{x^0+x^3=0\}$ is independent of mass \cite{LKS}, but the restriction of the two-point function depends on it (see, e.g., \cite{NakYam77, Yam97}). We resolve this inconsistency by showing that the two-point function has no canonical restriction to $\Sigma$ in the sense of distribution theory. Only the so-called tame restriction of the two-point function exists which we have introduced in \cite{Ull04sub}. Furthermore, we show that this tame restriction is indeed independent of mass. Hence the inconsistency appears only by the erroneous assumption that the two-point function would have a (canonical) restriction to $\Sigma$.Comment: 10 pages, 2 figure

Restriction categories III: colimits, partial limits, and extensivity

A restriction category is an abstract formulation for a category of partial maps, defined in terms of certain specified idempotents called the restriction idempotents. All categories of partial maps are restriction categories; conversely, a restriction category is a category of partial maps if and only if the restriction idempotents split. Restriction categories facilitate reasoning about partial maps as they have a purely algebraic formulation. In this paper we consider colimits and limits in restriction categories. As the notion of restriction category is not self-dual, we should not expect colimits and limits in restriction categories to behave in the same manner. The notion of colimit in the restriction context is quite straightforward, but limits are more delicate. The suitable notion of limit turns out to be a kind of lax limit, satisfying certain extra properties. Of particular interest is the behaviour of the coproduct both by itself and with respect to partial products. We explore various conditions under which the coproducts are ``extensive'' in the sense that the total category (of the related partial map category) becomes an extensive category. When partial limits are present, they become ordinary limits in the total category. Thus, when the coproducts are extensive we obtain as the total category a lextensive category. This provides, in particular, a description of the extensive completion of a distributive category.Comment: 39 page

Deconfinement from Action Restriction

The effect of restricting the plaquette to be greater than a certain cutoff value is studied. The action considered is the standard Wilson action with the addition of a plaquette restriction, which should not affect the continuum limit of the theory. In this investigation, the strong coupling limit is also taken. It is found that a deconfining phase transition occurs as the cutoff is increased, on all lattices studied (up to $20^4$). The critical cutoff on the infinite lattice appears to be around 0.55. For cutoffs above this, a fixed point behavior is observed in the normalized fourth cumulant of the Polyakov loop, suggesting the existence of a line of critical points corresponding to a massless gluon phase, not unlike the situation in compact U(1). The Polyakov loop susceptibility also appears to be diverging with lattice size at these cutoffs. A strong finite volume behavior is observed in the pseudo-specific heat. It is discussed whether these results could still be consistent with the standard crossover picture which precludes the existence of a deconfining phase transition on an infinite symmetric lattice.Comment: 4 pages latex, 6 ps figures, uses espcrc2.sty (included). Poster presented at LATTICE96(topology

Varieties of \u3cem\u3eP\u3c/em\u3e-Restriction Semigroups

The restriction semigroups, in both their one-sided and two-sided versions, have arisen in various fashions, meriting study for their own sake. From one historical perspective, as “weakly E-ample” semigroups, the definition revolves around a “designated set” of commuting idempotents, better thought of as projections. This class includes the inverse semigroups in a natural fashion. In a recent paper, the author introduced P-restriction semigroups in order to broaden the notion of “projection” (thereby encompassing the regular *-semigroups). That study is continued here from the varietal perspective introduced for restriction semigroups by V. Gould. The relationship between varieties of regular *-semigroups and varieties of P-restriction semigroups is studied. In particular, a tight relationship exists between varieties of orthodox *-semigroups and varieties of “orthodox” P-restriction semigroups, leading to concrete descriptions of the free orthodox P-restriction semigroups and related structures. Specializing further, new, elementary paths are found for descriptions of the free restriction semigroups, in both the two-sided and one-sided cases