Differential Forms or Tensors for Theoretical Physics Today

  • #26
lavinia
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I just do not understand how theoretical physicists can prefer using one mathematical tool over other one. Problem dictates mathematical tool.
I think now a days you are completely correct. While I am not a student of the history of Physics or Mathematics I have the impression that earlier in the 20'th century Mathematics and Physics used different formalisms in some areas although they were talking about the same mathematical structures. A famous but perhaps apocryphal story is that CN Yang was talking to James Simons about his research and Simons said 'Oh. Your talking about a connection'. I would guess that he was talking about a connection on a principal Lie group bundle. Here is a quote from CN Yang

"The beauty and profundity of the geometry of fibre bundles were to a large extent brought forth by the (early) work of Chern. I must admit, however, that the appreciation of this beauty came to physicists only in recent years."
— CN Yang, 1979

Simons was a student of Chern's. Chern was a Differential Geometer and his early work on the geometry of fiber bundles I think was largely done in the 1930's and 40's. Chern has a paper on the mutual recognition by mathematicians and physicists that they were both talking about connections on principal Lie group bundles.

Personal opinion: I think one of the morals is that there is a unity of mathematics and physics. IMO the idea that mathematics is just a tool of physics is passe at best. It reminds me of my sister's ballet teacher who viewed music solely as accompaniment to dance.
 
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Likes dextercioby, romsofia and martinbn
  • #27
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I would also quote

the gauge degree of freedom in EM are related with a local phase degree of freedom in QM ... and the mathematical concept of a connection 1-form and curvature 2-form are related with the physical counter parts of vector potential and field strength tensor, respectively.
From "Fiber bundles, Yang and the geometry of spacetime." (by Federico Pasinato)

or more simply Wikipedia "Applications in physics" paragraph under "differential form":

The EM form is a special case of the curvature form on the U(1) principal bundle on which both EM and general gauge theories may be described... equations can be written very compactly in differential form notation... Also Yang–Mills theory, in which the Lie group is not abelian, is represented in a gauge by a Lie algebra-valued one-form A.
As William O. Straub noticed in "Differential Forms for Physics Students"
Differential forms point to a profound connection between general relativity, electromagnetism and quantum physics. This connection, which is difficult to see without the formalism, is provided by the Cartan structure equations, which all physics students should at least be aware of.
Similarly nLab about Connections in physics describes EM field as connection on U(1), Yang-Mills field more generally on U(n) and
The field of gravity is encoded in a connection on the orthogonal group-principal bundle to which the tangent bundle is associated.
We can conclude with M. Gasperini:
Thanks to the language of differential forms, we can rewrite all equations in a more compact form, where the tensor indices of the curved space–time are “hidden” inside the variables, with great formal simplifications and benefits (especially in the context of the variational computations).
 
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  • #28
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Modern mathematicians and physicists are fluent in index notation and coordinate free notation as @martinbn suggested. You need to learn both.

BTW: Differential forms are skew symmetric tensors. Other tensors may not be differential forms for instance the metric tensor which is symmetric rather than skew symmetric.

IMO index notation is less geometrically clear.
Yes differential forms are defined as antisymmetric tensors. I write here the definition 5.4.1 (page 52) in Michio Nakahara's book. " A differential form of order r or an r-form is a totally anti-symmetric tensor of type (0, r ).". Now if we quote also Gravitation by Wheeler, on page 83 they say "Any tensor can be symmetrized or antisymmetrized by constructing an appropriate linear combination of itself and it's transposes" (they give this as exercise 3.12). Now this is where I get confused, if we put these two facts together, since any tensor can be antisymmetrized, does this also mean we can rewrite any tensor as a differential form? And if this is certainly always possible to do, why would we not want to turn a tensor into differential form notation?
 

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