According to the algebraic approach to spacetime, a thoroughgoing dynamicism, physical fields exist without an underlying manifold. This view is usually implemented by postulating an algebraic structure (e.g., commutative ring) of scalar-valued functions, which can be interpreted as representing a scalar field, and deriving other structures from it. In this work, we point out that this leads to the unjustified primacy of an undetermined scalar field. Instead, we propose to consider algebraic structures in which all (and only) physical fields are primitive. We explain how the theory of natural operations in differential geometry—the modern formalism behind classifying diffeomorphism-invariant constructions—can be used to obtain concrete implementations of this idea for any given collection of fields. For concrete examples, we illustrate how our approach applies to a number of particular physical fields, including electrodynamics coupled to a Weyl spinor.

根据时空的代数方法，一种彻底的动力学，物理场的存在没有潜在的流形。这种观点通常是通过假设一个标量值函数的代数结构（例如，交换环）来实现的，它可以被解释为表示一个标量域，并从中推导出其他结构。在这项工作中，我们指出这导致了未定标量场的不合理首要性。相反，我们建议考虑代数结构，其中所有（并且只有）物理场都是原始的。我们解释了自然运算的理论如何在微分几何中——分类微分同胚不变结构背后的现代形式主义——可以用来为任何给定的域集合获得这个想法的具体实现。对于具体的例子，我们说明了我们的方法如何应用于许多特定的物理领域，包括耦合到外尔旋量的电动力学。