The chiral condensate in a constant electromagnetic field

We study the shift of the chiral condensate in a constant electromagnetic field in the context of chiral perturbation theory. Using the Schwinger proper-time formalism, we derive a one-loop expression correct to all orders in $m_{\pi}^2 / eH$. Our result correctly reproduces a previously derived ``low-energy theorem'' for $m_\pi = 0$. We show that it is essential to include corrections due to non-vanishing $m_\pi$ in order for a low energy theorem to have any approximate regime of validity in the physical universe. We generalize these results to systems containing electric fields, and discuss the regime of validity for the results. In particular, we discuss the circumstances in which the method formally breaks down due to pair creation in an electric field.Comment: 9 pages, 6 figures, LaTeX; removed extraneous section + minor revision

Tangle and Brauer Diagram Algebras of Type Dn

A generalization of the Kauffman tangle algebra is given for Coxeter type Dn. The tangles involve a pole or order 2. The algebra is shown to be isomorphic to the Birman-Murakami-Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which in our set-up, occurs when the Coxeter type is of type A with index n-1. The proof involves a diagrammatic version of the Brauer algebra of type Dn in which the Temperley-Lieb algebra of type Dn is a subalgebra.Comment: 33 page

The United States-Canadian Automotive Trading Relationship and the Legality of the Canadian Duty Remission Program

The automotive industry influences the entire U.S. economy. This huge industrial structure directly affects over four million jobs in the United States. In terms of foreign trade in 1985, the United States exported approximately $6 billion worth of new passenger cars to foreign nations; 96 percent of those automobiles went to Canada. This is an illustration of the U.S. automotive industry\u27s direct interest in the U.S.-Canadian trading relationship, and the desire to obtain the potential benefits of a broadened trading relationship between the two countries. Presently, the United States and Canada have a unique dutyfree automotive trading arrangement which is distinguishable from those of other automotive trading nations