二维离散时间量子行走中的电磁格点规范不变性
Electromagnetic lattice gauge invariance in two-dimensional discrete-time quantum walks
AI总结：
本文提出一种在二维离散时间量子行走中引入电磁规范场的新方法，使离散导数平等处理时空，并定义规范不变且守恒的密度流，连续极限下对应狄拉克方程。
摘要：
规范不变性是物理学中较为重要的概念之一。我们在一维和二维空间维度下，结合离散时间量子行走的幺正演化来讨论这一概念，此时量子行走包含与合成外部电磁场的相互作用。这种相互作用以额外相位的形式引入，这些相位扮演规范场的角色。在此，我们提出一种纳入这些相位的方法，该方法不同于以往的工作。我们的提议使得规范变换下出现的离散导数能够以与标准格点规范理论类似的方式，平等地处理时间与空间。通过考虑演化的两步，我们定义了一个规范不变且守恒的密度流。在连续极限下，在适当选择参数的情况下，粒子的动力学变为狄拉克方程，且守恒流满足相应的守恒方程。
英文摘要：
Gauge invariance is one of the more important concepts in physics. We discuss this concept in connection with the unitary evolution of discrete-time quantum walks in one and two spatial dimensions, when they include the interaction with synthetic, external electromagnetic fields. One introduces this interaction as additional phases that play the role of gauge fields. Here, we present a way to incorporate those phases, which differs from previous works. Our proposal allows the discrete derivatives, that appear under a gauge transformation, to treat time and space on the same footing, in a way which is similar to standard lattice gauge theories. By considering two steps of the evolution, we define a density current which is gauge invariant and conserved. In the continuum limit, the dynamics of the particle, under a suitable choice of the parameters, becomes the Dirac equation, and the conserved current satisfies the corresponding conservation equation.
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