Matter couplings in theories of gravity beyond General Relativity

在爱因斯坦的广义相对论中，人们很清楚如何将物质场与重力相耦合。标准方法通常被称为最小耦合原则，它包含两个步骤。首先，描述物质的方程是在Minkowski空间中建立的，这意味着时间和空间不再是不同的概念，而是成为最初平坦的四维洛伦兹流形的一部分。其次，用协变导数代替偏导数，用任意时空度规代替Minkowski度规。然而，当一个人把度规和联系作为自变量时，这个过程可能会产生不同的引力理论。当应用到狄拉克领域时，人们自然会得出爱因斯坦-卡尔坦理论，这一理论自20世纪20年代以来就广为人知，从20世纪60年代起又重新引起了人们的兴趣。众所周知，在爱因斯坦-卡坦理论中，电磁场不可能是最小的耦合，因为扭转场会与电磁场耦合，从而打破其规范不变性。这也适用于非阿贝尔规范场。该项目将研究各种修正的引力理论中的最小耦合过程，并研究由此产生的场方程。最近提出的各种引力理论都具有不同寻常的性质，例如，一些模型不再是局部洛伦兹不变的，而是在整体洛伦兹变换下不变的。众所周知，这种模型中的物质耦合过程是微妙的，但有可能规定一个一致的耦合过程，使一般相对论方程在适当的限度内返回。关于可以在小尺度上放松微分同态不变性的模型，人们所知较少。除了最小耦合外，人们还可以考虑非最小耦合。这里指的是任何涉及物质场和几何量的相互作用项，如曲率标量。然而，人们也可以考虑各种其他非最小耦合，其中物质与边界项相互作用，例如，边界项不一定具有几何解释。这是一个相对较新的研究领域，允许研究大量可能的模型。研究领域：数学科学--数学物理

In Einstein's theory of General Relativity it is well understood how to couple matter fields to gravity. The standard approach is often referred to as the principle of minimal coupling which contains two steps. First, the equations describing the matter are formulated in Minkowski space which means that time and space are no longer distinct concepts but become part of a 4-dimensional Lorentzian manifold, flat initially. Second, one replaces partial derivatives with covariant derivatives and the Minkowski metric with an arbitrary spacetime metric. This process, however, can yield different gravitational theories when one considers the metric and connection as independent variables. When applied to Dirac fields one naturally arrives at Einstein-Cartan theory, a theory well known since the 1920s with much renewed interesting from the 1960s onwards. It is well known that the electromagnetic field cannot be coupled minimally in Einstein-Cartan theory as the torsion field would couple to the electromagnetic potential, thereby breaking its gauge invariance. The also applied to non-Abelian gauge fields.The project will investigate the minimal coupling procedure in various modified theories of gravity and study the resulting field equations. Various recently proposed theories of gravity have unusual properties, for instance, some models are no longer locally Lorentz invariant but are invariant under global Lorentz transformations. The matter coupling procedure in such models is known to be subtle but it is possible to prescribe a consistent coupling procedure which return the general relativistic equations in the appropriate limit. Less is know about models which can loose diffeomorphism invariance on small scales.In addition to minimal couplings one can also consider non-minimal couplings. By this one means any interaction term involving the matter field and a geometrical quantity, like the curvature scalar. However, one can also consider a variety of other non-minimal couplings where the matter interacts with a boundary term, for example, which may not necessarily have a geometrical interpretation. This is a relatively new field of research allowing for a large amount of possible models to be investigated.Research Areas: mathematical sciences -> mathematical physics