由力学变分原理到定态电磁场变分原理

本文以力学变分原理为基础,引出时空算子对偶概念,据以导出定态电磁场变分原理;进而,论述了时空淡化和磁单极子问题。     

Of connections and fields - I

We describe some instances of the appearance of Chern’s mathematical ideas in physics. By means of simple examples, we bring out the geometric and topological ideas which have found application in describing the physical world. These applications range from magnetic monopoles in electrodynamics to instantons in quantum chromodynamics to the geometric phase of quantum mechanics. The first part of this article is elementary and addressed to a general reader. The second part is somewhat more demanding and is addressed to advanced students of mathematics and physics.     

Of connections and fields—II

In the first part of this article¹ we gave an elementary introduction to Chern’s ideas and their impact on modern physics. In this concluding article we describe some more advanced applications of Chern’s ideas. This second part is some-what more demanding than the first part and is addressed to students with some background in mathematics and physics. Joseph Samuel is a theoretical physicist and by natural inclination a classical mechanic. Over the years he has strayed into other fields like optics, general relativity and very recently DNA elasticity. A unifying theme in his work is differential geometry and topology in physics. He keeps moderately fit by raising and lowering indices and relaxes by playing semiclassical guitar.     

Dirac’s conception of the magnetic monopole, and its modern avatars

Conclusions Dirac’s philosophy of mathematical elegance led to his pioneering work on magnetic monopoles and inspired a large number of subsequent developments. It remains true that magnetic monopoles have never been detected, despite numerous attempts, but after all it was Dirac who once remarked, “It is more important to have beauty in one’s equations than to have them fit experiment”. One might therefore assume that he was more than satisfied with the impact of this work. This is not quite the case. Towards the end of his life, the complete lack of experimental evidence for magnetic monopoles began to weigh on him. In 1981, a year short of his 80th birthday, he was invited to Trieste for a conference to commemorate the 50th anniversary of his monopole paper. He declined the invitation due to the strain of travelling, but sent a letter of thanks to Abdus Salam, the Nobel Laureate and Director of the Centre at Trieste. In this letter, Dirac wrote: “I am inclined now to believe that monopoles do not exist. So many years have gone by without any encouragement from the experimental side.” Dirac passed away nearly two decades ago, but it remains to be determined who was right about monopoles: Dirac in 1931, or Dirac in 1981.