8 - Field transformations  pp. 169-206

Field transformations

By Mark Burgess

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The previous chapters take a pragmatic, almost engineering, approach to the solution of field theories. The recipes of chapter 5 are invaluable in generating solutions to field equations in many systems, but the reason for their effectiveness remains hidden. This chapter embarks upon a train of thought, which lies at the heart of the theory of dynamical systems, which explain the fundamental reasons why field theories look the way they do, how physical quantities are related to the fields in the action, and how one can construct theories which give correct answers regardless of the perspective of the observer. Before addressing these issues directly, it is necessary to understand some core notions about symmetry on a more abstract level.

Group theory

To pursue a deeper understanding of dynamics, one needs to know the language of transformations: group theory. Group theory is about families of transformations with special symmetry. The need to parametrize symmetry groups leads to the idea of algebras, so it will also be necessary to study these.

Transformations are central to the study of dynamical systems because all changes of variable, coordinates or measuring scales can be thought of as transformations. The way one parametrizes fields and spacetime is a matter of convenience, but one should always be able to transform any results into a new perspective whenever it might be convenient. Even the dynamical development of a system can be thought of as a series of transformations which alter the system's state progressively over time.