现代物理的数学进展

Mathematical Progress of Modern Physics(1900-1988) Qu Chaochun (Department of Physics. Department of Mathematics) (Yunnan Univ

现代物理学中涌进过大量的数学分支与数学对象:有的被淘汰,有些被固定下来成为量子场论的组成部分,如李群与李代数(Lorentz)群,同位空间中的SU(n)等等。但这些仍属于古典数学,现代数学中的广义函数,点集拓扑学,现代维度理论(Dimensiongrad Theory)与强相互作用过程的亚群对称性(Demigroup Symmetey四维空时中所包含的点同Euclid直线一样多,如果不考虑这个定理。人们不能严格刻画量子场论的原理,现代物理,对物理学家是困难的,对数学家也是困难的。 在不明僚事物的实质时,用数学方法学掌握事物的可能性本身是惊人的。

Ever since the fifties of this century.new branches and new objects of mod ern mathematics are constantly emerging from the physics.some branchs may be obsolete or already obsolete.e.g. the non-standard analysis may be obsolete.Many objects have been being investigated for long(1954-1988).but no progress has been made.e.g.the quaternion field.lie grorp and lie algebra are accepted by modern Physics and arc the intrinsic constituent of the quantum field theory(Lorentz group.SU(n) in the isotopic spin space ) The year of 1970 was a turing point of the development for quantum field theory when the anomalous dimensionslity of quantum fields( 1970.K.G.Wilson),the equation of renomalization semi-group(1970.C.G.Callan.K.Symanzik) and the Demigroup symmetry of the strong interaction dynamics( 1985)compelled us to use the following branches of modern mathematics:Generalized Functions( 1950. L.Schwartz. Distribution Theory); To pological theory of G. Cantor' s point-set; The theory of semi-group and Demigroup (.1957, E.Hille et al); Dimension Theory (1941. W.Hurewiz and H.Wallman;1978.R.Engel king). Space time with 4-dimension has the points as many as Euclid line. If we take no account of this theorem,we can not give a rigorous statement of the fundamental principle in quantum field theory. Modern physics is not only difficult for physicists but also for mathematicians.