复参数空间上概率振幅函数的几何相位和模关系
Geometric Phase and Modulo Relations for Probability Amplitudes as Functions on Complex Parameter Spaces
摘要：
本文研究了$\amp{ψ_f}ψ$形式的概率幅的相位和模的相应行为之间的一般微分关系，其中$\ket{ψ_f}$是Hilbert空间中的固定态，$\ketψ$是复参数空间上全纯线丛的一部分.这种丛上的振幅函数，虽然不是严格全纯的，但仍然满足涉及参数空间上U（1）Berry-Simon联络的广义Cauchy-Riemann条件。这些条件需要相位和模的梯度之间的可逆关系，因此允许从模重构相位（或反之亦然）以及关于幅度的任一极性分量的行为的其他条件。作为一种特殊情况，我们认为振幅函数值的空间上的纯态，射线空间${\cal R} = {\mathbb C}P^n$，其中转移概率有一个几何解释的测地距离与Fubini研究度量。结合广义柯西-黎曼条件，这种测地线解释导致额外的关系，特别是一个新的连接之间的模量的振幅和相位梯度，有点让人想起WKB公式。最后，建立了与几何相位的联系。
英文摘要：
We investigate general differential relations connecting the respective behavior s of the phase and modulo of probability amplitudes of the form $\amp{ψ_f}ψ$, where $\ket{ψ_f}$ is a fixed state in Hilbert space and $\ketψ$ is a section of a holomorphic line bundle over some complex parameter space. Amplitude functions on such bundles, while not strictly holomorphic, nevertheless satisfy generalized Cauchy-Riemann conditions involving the U(1) Berry-Simon connection on the parameter space. These conditions entail invertible relations between the gradients of the phase and modulo, therefore allowing for the reconstruction of the phase from the modulo (or vice-versa) and other conditions on the behavior of either polar component of the amplitude. As a special case, we consider amplitude functions valued on the space of pure states, the ray space ${\cal R} = {\mathbb C}P^n$, where transition probabilities have a geometric interpretation in terms of geodesic distances as measured with the Fubini-Study metric. In conjunction with the generalized Cauchy-Riemann conditions, this geodesic interpretation leads to additional relations, in particular a novel connection between the modulus of the amplitude and the phase gradient, somewhat reminiscent of the WKB formula. Finally, a connection with geometric phases is established.
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