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  What phenomena in physics cannot be expressed in terms of differential forms?

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It seems all phenomena in physics can be expressed in terms of differential forms. There are textbooks dedicated to formulating the main theories of physics in terms of differential forms. The more general formulation in terms of Tensors however I think is more often used by physicists. Differential forms seem to carry much advantage in terms of unification, simplicity in higher dimensional theories and generalization. Considering that a differential form is just an antisymmetric covariant tensor on a differential manifold (i.e a specialized form of tensor), what phenomena in physics cannot be re-formulated in the notation of differential forms or can we always reformulate our laws in this notation? If we can reformulate in terms of differential forms, why would we choose not to?

asked Jun 29, 2020 in Theoretical Physics by Kay Bei (10 points) [ no revision ]

1 Answer

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Regarding "If we can reformulate in terms of differential forms, why would we choose not to?" let me ask "Suppose we could reformulate, why should we?". Because some books/physicists firmly advocate the differential form approach? In my opinion certainly not. Physicists are not automatons but individuals with personal tastes and preferences, which to hold they are perfectly entitled.

Considering the various formulations available it should always be kept in mind that the physics behind the formulations is the same. Depending on the specific problem at hand, one or the other of the formulations may be more advantageous.

In Hamilton dynamics, differential forms allow a concise formulation of canoncial transformations or constraints, but can also be largely avoided by using Poisson brackets. In case of specific calculations, i.e. where you have to deal with numbers, you will end up working with some kind of components (e.g. coefficients for a basis of 1-forms or coordinates) anyway.

In electrodynamics, you can for example write down the theory in terms of differential forms, or in relativistic index notation, or in three-dimensional vector notation. The physics is the same. If you are considering a problem in electrostatics, which formulation would you prefer?

There is only one physics, but various formulations. The more formulations you are familiar with, the larger the set of tools you can choose from when addressing a problem.

Not all tensors in physics are antisymmetric. There is the metric tensor, which can be written with 1-forms \(g=g_{ab}dx^a\otimes dx^b\), but which is not a 2-form, or the energy-momentum tensor. Then there are also concepts like spinors.

So there is one physics, but not The One And Only Way of expressing it.

answered Jun 29, 2020 by Flamma (80 points) [ no revision ]

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