The large-N limit of the Segal–Bargmann transform on UN
The large-N limit of the Segal–Bargmann transform on UN
复制标题
UN 上 Segal-Bargmann 变换的大 N 极限
DOI：
10.1016/j.jfa.2013.07.020
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发表时间：
2013
影响因子：
1.7
通讯作者：
Todd Kemp
中科院分区：
文献类型：
--
作者：
B. Driver;B. Hall;Todd Kemp
研究了酉群UN上的（双参数）Segal-Bargmann变换B s，tN，其中N很大.作用于在群的伴随作用下等变的矩阵值函数，变换具有有意义的极限Gs，t，当N→∞时，它可以被识别为复洛朗多项式空间上的算子。我们引入了迹多项式空间，并使用它来给出有效的计算方法来确定热算子的作用，从而确定Segal-Bargmann变换。证明了几个测度集中定理和极限定理，给出了有限维变换B s，t N到其极限Gs，t的直接联系.我们通过其在标准多项式基上的逆作用来刻画算子Gs，t。最后，证明了在s= t的情况下，极限变换Gt，t是Biane提出的“自由Hall变换”Gt.
We study the (two-parameter) Segal–Bargmann transform B s, t N on the unitary group U N, for large N. Acting on matrix-valued functions that are equivariant under the adjoint action of the group, the transform has a meaningful limit G s, t as N→∞, which can be identified as an operator on the space of complex Laurent polynomials. We introduce the space of trace polynomials, and use it to give effective computational methods to determine the action of the heat operator, and thus the Segal–Bargmann transform. We prove several concentration of measure and limit theorems, giving a direct connection from the finite-dimensional transform B s, t N to its limit G s, t. We characterize the operator G s, t through its inverse action on the standard polynomial basis. Finally, we show that, in the case s= t, the limit transform G t, t is the “free Hall transform” G t introduced by Biane.