N-bein formalism for the parameter space of quantum geometry

IF 2 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL
Jorge Romero, Carlos A Velasquez, J David Vergara
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Abstract

This work introduces a geometrical object that generalizes the quantum geometric tensor; we call it N-bein. Analogous to the vielbein (orthonormal frame) used in the Cartan formalism, the N-bein behaves like a ‘square root’ of the quantum geometric tensor. Using it, we present a quantum geometric tensor of two states that measures the possibility of moving from one state to another after two consecutive parameter variations. This new tensor determines the commutativity of such variations through its anti-symmetric part. In addition, we define a connection different from the Berry connection, and combining it with the N-bein allows us to introduce a notion of torsion and curvature à la Cartan that satisfies the Bianchi identities. Moreover, the torsion coincides with the anti-symmetric part of the two-state quantum geometric tensor previously mentioned, and thus, it is related to the commutativity of the parameter variations. We also describe our formalism using differential forms and discuss the possible physical interpretations of the new geometrical objects. Furthermore, we define different gauge invariants constructed from the geometrical quantities introduced in this work, resulting in new physical observables. Finally, we present two examples to illustrate these concepts: a harmonic oscillator and a generalized oscillator, both immersed in an electric field. We found that the new tensors quantify correlations between quantum states that were unavailable by other methods.
量子几何参数空间的 N-贝因形式主义
这项工作引入了一个几何对象,它概括了量子几何张量;我们称之为 N-bein。与 Cartan 形式论中使用的 vielbein(正交框架)类似,N-bein 的行为类似于量子几何张量的 "平方根"。利用它,我们提出了两种状态的量子几何张量,可以测量在连续两次参数变化后从一种状态移动到另一种状态的可能性。这个新张量通过其反对称部分确定了这种变化的交换性。此外,我们还定义了一种不同于贝里连接的连接,并将其与 N-贝因相结合,从而引入了一种满足比安奇等式的卡坦扭转和曲率概念。此外,扭转与前面提到的双态量子几何张量的反对称部分相吻合,因此,它与参数变化的交换性有关。我们还用微分形式描述了我们的形式主义,并讨论了新几何对象可能的物理解释。此外,我们还定义了由本研究中引入的几何量构建的不同量规不变式,从而产生了新的物理观测值。最后,我们举了两个例子来说明这些概念:一个谐波振荡器和一个广义振荡器,两者都浸没在电场中。我们发现,新的张量可以量化量子态之间的相关性,而其他方法却无法实现这一点。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
4.10
自引率
14.30%
发文量
542
审稿时长
1.9 months
期刊介绍: Publishing 50 issues a year, Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.
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