从Kontsevich图到Feynman图：标量场星乘积的一个视角
From Kontsevich Graphs to Feynman graphs, a Viewpoint from the Star Products of Scalar Fields
AI总结：
本文提出新方法，在函数、场和泛函三个层次构造标量场的星乘积，证明Kontsevich图与Feynman图一一对应，并推广了微扰代数量子场论中的星乘积。
摘要：
在本文中，我们通过一种新方法构造了协变情形下标量场的星乘积。我们在三个层次上构造星乘积，分别是Rd上的函数层次、场层次和泛函层次。我们强调函数层次的星乘积是本质且是我们设定的起点。首先，函数的星乘积几乎包含了关于标量场和泛函的星乘积的所有代数和组合信息。其次，更有趣的一点是，函数的星乘积仅涉及有限维问题，它是由具有抽象系数的双向量场在Rd上生成的类似Moyal的星乘积。因此，Kontsevich图自然地发挥了一些作用。实际上，我们证明了某一类Kontsevich图与Feynman图之间存在一一对应关系。此外，我们还用函数层次的星乘积讨论了Wick定理、Wick幂以及Wick单项式的期望。我们的构造可以被视为微扰代数量子场论中星乘积以及[1]、[2]中引入的扭曲乘积的推广。
英文摘要：
In the present paper we construct the star products concerning scalar fields in the covariant case from a new approach. We construct the star products at three levels, which are levels of functions on Rd, fields and functionals respectively. We emphases that the star product at level of functions is essence and starting point for our setting. Firstly the star product of functions includes all algebraic and combinatorial information of the star products concerning the scalar fields and functionals almost. Secondly, a more interesting point is that the star product of functions concerns only finite dimensional issue, which is a Moyal-like star product on Rd generated by a bi-vector field with abstract coefficients. Thus the Kontsevich graphs play some roles naturally. Actually we prove that there is an ono-one correspondence between a class of Kontsevich graphs and the Feynman graphs. Additionally the Wick theorem, Wick power and the expectation of Wick-monomial are discussed in terms of the star product at level of functions. Our construction can be considered as the generalisation of the star products in perturbative algebraic quantum fields theory and twist product introduced in [1],[2].
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