论文标题
一般协方差的离散类似物 - 第2部分：尽管您听说过，但完全是洛伦兹晶格理论
A Discrete Analog of General Covariance -- Part 2: Despite what you've heard, a perfectly Lorentzian lattice theory
论文作者
论文摘要
一般相对历史上的关键步骤是爱因斯坦采用了一般协方差原则，该原则要求为我们的时空理论进行协调独立的配方。一般协方差有助于我们将理论的实质性内容从仅仅代表性的文物中解脱出来。它是对时空理论的现代理解必不可少的工具。受量子重力的驱动，人们可能希望将这些概念扩展到量子时空理论（无论是什么）。相关的是，人们可能希望将这些概念扩展到离散的时空理论（即晶格理论）。本文提供了带有令人惊讶的后果的这种扩展，将第1部分（Arxiv：2204.02276）扩展到Lorentzian设置。 这种一般协方差的离散类似物表明，晶格结构不像固定的背景结构，而更像是坐标系，即仅仅是代表性伪像。这种离散的类似物建立在我们离散的时空理论中出现的晶格结构与出现在我们的连续时空理论中的坐标系之间的丰富类比。我认为，正确理解没有晶格基础理论之类的东西，而只有晶格表现出的理论。因果集理论社区充分说明，没有关于固定时空晶格的理论是洛伦兹不变的，但是我将讨论这最终是代表能力的问题，而不是物理学的问题。我们的代表性工具的对称性不需要锁定在所代表的事物的对称性上。没有什么可以阻止我们使用笛卡尔坐标来描述旋转不变的状态/动力学。正如本文所示，在洛伦兹（Lorentzian）环境中的晶格也是如此：没有什么可以阻止我们定义完美的洛伦兹晶格（ - 表述）理论。
A crucial step in the history of General Relativity was Einstein's adoption of the principle of general covariance which demands a coordinate independent formulation for our spacetime theories. General covariance helps us to disentangle a theory's substantive content from its merely representational artifacts. It is an indispensable tool for a modern understanding of spacetime theories. Motivated by quantum gravity, one may wish to extend these notions to quantum spacetime theories (whatever those are). Relatedly, one might want to extend these notions to discrete spacetime theories (i.e., lattice theories). This paper delivers such an extension with surprising consequences, extending Part 1 (arXiv:2204.02276) to a Lorentzian setting. This discrete analog of general covariance reveals that lattice structure is rather less like a fixed background structure and rather more like a coordinate system, i.e., merely a representational artifact. This discrete analog is built upon a rich analogy between the lattice structures appearing in our discrete spacetime theories and the coordinate systems appearing in our continuum spacetime theories. I argue that properly understood there are no such things as lattice-fundamental theories, rather there are only lattice-representable theories. It is well-noted by the causal set theory community that no theory on a fixed spacetime lattice is Lorentz invariant, however as I will discuss this is ultimately a problem of representational capacity, not of physics. There is no need for the symmetries of our representational tools to latch onto the symmetries of the thing being represented. Nothing prevents us from using Cartesian coordinates to describe rotationally invariant states/dynamics. As this paper shows, the same is true of lattices in a Lorentzian setting: nothing prevents us from defining a perfectly Lorentzian lattice(-representable) theory.
