An algebraic approach to physical fields
Section snippets
Spacetime algebraicism
Is spacetime a substance? One reason for answering yes is that the curvature of spacetime can explain phenomena such as light bending near massive objects according to general relativity. The view that spacetime is a substance independent of the things and processes in it is called (spacetime) substantivalism. The structure of spacetime is standardly represented by a smooth manifold equipped with a metric in manifold-based differential geometry. But there are also many reasons for thinking that
Spacetime algebraicism based on a scalar field
In this section, we provide some background by outlining our perspective on the algebraic approach to spacetime proposed by Geroch (1972) and Heller (1992), in which spacetime is characterized by its algebra of scalar-valued functions without explicit reference to spacetime points. While similar ideas also feature prominently in noncommutative geometry and its applications to physics (Connes, 1994; Marcolli, 2018), our emphasis here is motivated by the closer proximity of Geroch's Einstein
Spacetime algebraicism based on physical fields
Having arrived at the conclusion that spacetime algebraicism based on scalar-valued functions is problematic, we now discuss how one can develop versions of algebraicism in which all physical fields feature among the primitive structures. This constitutes both a conceptual and a technical development, with the latter leveraging definitions and constructions which are known in the mathematical literature.
Thus consider a physical field, say the electromagnetic field, or more generally a
A general manifold-theoretic formulation of physical fields
In the case of a scalar field, it was clear that its algebraic description would have to be based on the scalar-valued functions. For other types of fields, it is far less clear what the most appropriate mathematical structure is in response to question (0’). For example for the electromagnetic field, there are many different kinds of manifold-theoretic structures which have been entertained to describe it:
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In the non-relativistic context, the electric and magnetic field are described
Algebraic structure from natural operations
We now turn to question (1’), which algebraic structures can be defined on a physical field formulated manifold-theoretically? In light of the previous section, the question becomes this: which algebraic structure does a field in the sense of Definition 4.1 carry? More precisely: what is the richest algebraic structure that can be defined on the field configurations in a way which respects the symmetries? The following technical developments will address this question, considering algebraic
Algebraic structure relevant for a physical field
Finally, we now turn to discussing question (2′) on how to formulate the physics of a field in purely algebraic terms. By considering general instances of these algebraic structures, we can then lift the physics away entirely from the manifold context. We also consider how to prune the algebraic structure further in a way which still allows for a formulation of the physics, and outline a few instances of these structures in which the physics still makes sense despite there not being any
Credit author statement
Lu Chen: Investigation, Writing - Original Draft, Writing - Review & Editing.
Tobias Fritz: Investigation, Writing - Original Draft, Writing - Review & Editing.
Acknowledgements
We thank the anonymous referees at Studies in History and Philosophy of Science for their helpful suggestions on improving the paper.
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