Do All Automorphisms in Physics Imply Symmetries in Equations?

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SUMMARY

All physical symmetries are linked to automorphisms in axioms, establishing a foundational relationship in physics. The discussion clarifies that not all automorphisms derived from physical equations necessarily imply symmetries. It emphasizes that many physical phenomena are described by differential equations, where an isomorphism of the solution set represents an automorphism. This isomorphism typically manifests as a coordinate transformation, preserving the form of the governing differential equation and defining a symmetry.

PREREQUISITES
  • Understanding of differential equations in physics
  • Familiarity with the concept of automorphisms
  • Knowledge of isomorphisms in mathematical contexts
  • Basic grasp of coordinate transformations
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  • Explore the role of automorphisms in advanced physics theories
  • Study the implications of symmetries in differential equations
  • Investigate coordinate transformations in various physical systems
  • Learn about the mathematical foundations of isomorphisms and their applications
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Physicists, mathematicians, and students interested in the relationship between symmetries and automorphisms in physical equations.

Garrulo
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All the physical simmetries implicate an automorphism in axioms. But ¿all the automorphisms from a physical equation do implicate a simmetry?
 
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I am not sure I understand, but I think I may know what you have in mind. Many phenomena in physics are governed by differential equations, and we can consider an isomorphism of the solution set of a given equation into itself (an automorphism). Usually such an isomorphism expresses a coordinate transformation of some sort. This transformation maps solutions into each other, and the form of the governing differential equation does not change. We define such a map as a symmetry of the differential equation.
 

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