An algebraic approach to physical fields

https://doi.org/10.1016/j.shpsa.2021.08.011Get rights and content

Abstract

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.

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:

  • 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.

References (38)

  • H.R. Brown

    Einstein's misgivings about his 1905 formulation of special relativity

    European Journal of Physics

    (2005)
  • L. Chen

    Smooth infinitesimals in the metaphysical foundation of spacetime theories

    (2020)
  • C. Cherubini et al.

    Second order scalar invariants of the Riemann tensor: Applications to black hole spacetimes

    International Journal of Modern Physics D

    (2002)
  • A. Connes

    Noncommutative geometry

    (1994)
  • J. Demaret et al.

    Local and global properties of the world

  • S. Doplicher et al.

    The quantum structure of spacetime at the Planck scale and quantum fields

    Communications in Mathematical Physics

    (1995)
  • J. Earman

    World enough and space-time. A bradford book

  • J. Earman et al.

    What price spacetime substantivalism? The hole story

    The British Journal for the Philosophy of Science

    (1987)
  • R. Geroch

    Einstein algebras

    Communications in Mathematical Physics

    (1972)
  • Cited by (12)

    View all citing articles on Scopus
    View full text