Classical Segal–Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on \(\mathbb {R}^n\) with respect to a Gaussian measure, the space of monogenic square-integrable functions on \(\mathbb {R}^{n+1}\) with respect to another Gaussian measure and the space of Clifford algebra-valued linear functionals on symmetric tensor elements of \(\mathbb {R}^n\).

经典 Segal-Bargmann 理论研究了三个 Hilbert 空间酉同构，它们描述了波粒二象性和构型空间相空间。在这项工作中，我们将这些概念推广到 Clifford 代数值函数。我们在\(\mathbb {R}^n\)上的 Clifford 代数值平方可积函数空间之间建立了关于高斯测度的酉同构，即\(\mathbb { R}^{n+1}\)相对于另一个高斯测度和\(\mathbb {R}^n\) 的对称张量元素上的 Clifford 代数值线性泛函的空间。