An algebraic approach to physical fields.

Area of Science:

• Theoretical Physics
• Mathematical Physics
• Algebraic Geometry

Background:

• The algebraic approach to spacetime posits that physical fields exist without an underlying manifold.
• Current implementations often prioritize an undetermined scalar field, which is a limitation.
• This approach is rooted in the concept of thoroughgoing dynamicism.

Purpose of the Study:

• To propose an alternative algebraic framework for spacetime where all physical fields are primitive.
• To address the unjustified primacy of scalar fields in existing algebraic approaches.
• To provide a method for constructing concrete implementations of this new framework.

Main Methods:

• Utilizing the theory of natural operations in differential geometry.
• Classifying diffeomorphism-invariant constructions.
• Developing algebraic structures where physical fields are fundamental entities.

Main Results:

• Demonstrated that existing algebraic approaches lead to the undue prominence of scalar fields.
• Proposed a new framework where all physical fields are treated as primitive.
• Showcased concrete applications of the proposed method to various physical fields, including electrodynamics and Weyl spinors.

Conclusions:

• The proposed algebraic approach offers a more balanced and fundamental treatment of physical fields in dynamicist spacetime theories.
• The theory of natural operations provides a powerful tool for implementing this new framework.
• This work opens new avenues for exploring spacetime and field theories without relying on a background manifold.