Classical 6j-symbols and the tetrahedron
Classical 6j-symbols and the tetrahedron
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经典 6j 符号和四面体
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发表时间：
1998
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通讯作者：
J. Roberts
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文献类型：
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作者：
J. Roberts
一个经典的6 j-符号是一个真实的数，它可以通过SU（2）的不可约表示与四面体的六条边的标号相关联。这种抽象的联系传统上只是用来表达6 j符号的对称性，这是一个纯粹的代数对象;然而，它有更深层次的几何意义。庞扎诺和雷奇，扩大工作的维格纳，给了一个惊人的（但未经证明）渐近公式有关的价值6 j-符号，当尺寸的代表是大的，体积诚实的欧几里得四面体的边长是这些尺寸。本文的目的是用几何量子化的方法证明和解释这个公式。一个令人惊讶的副产品是，一个普通的欧几里得四面体产生了一个家庭的12个剪刀全等，但不全等四面体。
A classical 6j-symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). This abstract association is traditionally used simply to express the symmetry of the 6j-symbol, which is a purely algebraic object; however, it has a deeper geometric significance. Ponzano and Regge, expanding on work of Wigner, gave a striking (but unproved) asymptotic formula relating the value of the 6j-symbol, when the dimensions of the representations are large, to the volume of an honest Euclidean tetrahedron whose edge lengths are these dimensions. The goal of this paper is to prove and explain this formula by using geometric quantization. A surprising spin-off is that a generic Euclidean tetrahedron gives rise to a family of twelve scissors-congruent but non-congruent tetrahedra.