Point-splitting regularization for gauge theories: Quantum electrodynamics

As a method of regularization, point splitting has played an essential role in the recent theoretical determination of the masses of the Higgs boson and the top quark. It is the purpose of this paper to put this pointsplitting regularization on a firm basis. The result turns out to be extremely simple: replace the usual vertex factor-ieγ ^µ in quantum electrodynamics by $$ - ie(\gamma _\mu - \frac{{\not p}}{{p \cdot \delta }}\delta _\mu ),$$ wherep is the momentum of the photon line, andδ ^µ is the distance for point splitting. No additional vertices are needed.     

Magnetic penetration depth in V3Si and LiTi2O4 measured by μSR

Muon spin relaxation (SR) studies have been performed in the normal spinel LiTi₂O₄ and the A-15 superconductor V₃Si to measure the magnetic penetration depth . The relaxation rate(T) 1/² in field-cooled measurements shows a sharp increase belowT _c followed by saturation at low temperatures in both systems. This feature implies an isotropic energy gap without anomalous zeros, and most likelys-wave pairing. The low temperature penetration depth (T 0) is determined to be 2100Å for LiTi₂O₄ and 1300Å for V₃Si respectively. Assuming a clean limit relation ^–2 n _s /m ^*, we derive the Fermi temperatureT _F n _s/ ^2/3 m ^* from the relaxation rate and the Sommerfeld constant asT _F ^3/4–1/4. Unlike conventional superconductors, both LiTi₂O₄ and V₃Si have a large ratio ofT _c /T _F 0.01, only slightly smaller than those ratios in more exotic superconductors.We thank C. Ballard and K. Hoyle for technical assistance. Work at Columbia University is supported by NSF Grant No. DMR-89-13784 and Packard Foundation (YJU). Ames Laboratory is operated for the U. S. Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. Work at Ames was supported by the Director for Energy Research, Office of Basic Energy Sciences.     

Volumes of restricted Minkowski sums and the free analogue of the entropy power inequality

In noncommutative probability theory independence can be based on free products instead of tensor products. This yields a highly noncommutative theory: free probability theory (for an introduction see [9]). The analogue of entropy in the free context was introduced by the second named author in [8]. Here we show that Shannon's entropy power inequality ([6, 1]) has an analogue for the free entropy (X) (Theorem 2.1).The free entropy, consistent with Boltzmann's formulaS=klogW, was defined via volumes of matricial microstates. Proving the free entropy power inequality naturally becomes a geometric question.Restricting the Minkowski sum of two sets means to specify the set of pairs of points which will be added. The relevant inequality, which holds when the set of addable points is sufficiently large, differs from the Brunn-Minkowski inequality by having the exponent 1/n replaced by 2/n. Its proof uses the rearrangement inequality of Brascamp-Lieb-Lüttinger ([2]). Besides the free entropy power inequality, note that the inequality for restricted Minkowski sums may also underlie the classical Shannon entropy power inequality (see 3.2 below).Research supported in part by grants from the National Science Foundation.