Differential Forms or Tensors for Theoretical Physics Today

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Thread starter kay bei
Start date Jun 8, 2020
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Differential Differential forms Differential geometry Forms Geometric Physics Tensor Tensors Textbook Theoretical Theoretical physics
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In summary, there are different approaches to theoretical physics, such as using tensors with indices or differential forms. Some textbooks, like Kip Thorne's "Modern Classical Physics" use tensors, while others, like Frankel's "Geometry of Physics" and Chris Isham's "Modern differential geometry for physicists" use differential forms. Mathematicians have mostly adopted the use of differential forms, but physicists may have a preference for one or the other depending on the field. It is possible to rewrite a book like Kip Thorne's using differential forms, but it may be shorter. The notation for these approaches is different and it can impact ease of understanding. In general relativity, the differential form notation is often preferred for its computational power, but the

Jun 12, 2022

giulio_hep

it was believed that the flat Euclidean geometry is the geometry of physical space (regarded by Immanuel Kant as being necessarily true as an « a priori synthetic » proposition) until Einstein’s great discovery that space-time, though locally flat, is in fact curve

From "Non Local Aspects of Quantum Phases" by J. ANANDAN, also noticing:

the electromagnetic field strength of a magnetic monopole belongs to a Chern class that is an element of the second de Rham cohomology group.

More generally I'd say that a differential structure with a tangent bundle is almost always assumed in physics (both classical and quantum, and e.g. including here even Penrose spinor bundles) and I can hardly imagine a generalization to other bundles than those ones.

<h2>1. What are differential forms and tensors?</h2><p>Differential forms and tensors are mathematical objects used in theoretical physics to describe physical quantities such as vectors, tensors, and fields. They are used to represent geometric and physical quantities in a coordinate-independent manner.</p><h2>2. How are differential forms and tensors used in theoretical physics?</h2><p>Differential forms and tensors are used to describe physical quantities in a way that is independent of the coordinate system used. This allows for a more elegant and concise representation of physical laws and equations, making it easier to solve complex problems in theoretical physics.</p><h2>3. What is the significance of differential forms and tensors in modern theoretical physics?</h2><p>Differential forms and tensors have become an essential tool in modern theoretical physics, particularly in fields such as general relativity, electromagnetism, and quantum field theory. They provide a powerful framework for understanding and solving complex physical problems.</p><h2>4. How do differential forms and tensors differ from other mathematical objects?</h2><p>Differential forms and tensors are different from other mathematical objects, such as vectors and matrices, because they are invariant under coordinate transformations. This means that their components change in a specific way under a change of coordinates, making them more suitable for describing physical quantities in a coordinate-independent manner.</p><h2>5. Are there any limitations to using differential forms and tensors in theoretical physics?</h2><p>While differential forms and tensors are incredibly useful in theoretical physics, they do have some limitations. For example, they may not be the most efficient tool for solving certain types of problems, and they may not be applicable in all areas of physics. Additionally, a solid understanding of advanced mathematical concepts is required to effectively use differential forms and tensors in theoretical physics.</p>

1. What are differential forms and tensors?

Differential forms and tensors are mathematical objects used in theoretical physics to describe physical quantities such as vectors, tensors, and fields. They are used to represent geometric and physical quantities in a coordinate-independent manner.

2. How are differential forms and tensors used in theoretical physics?

Differential forms and tensors are used to describe physical quantities in a way that is independent of the coordinate system used. This allows for a more elegant and concise representation of physical laws and equations, making it easier to solve complex problems in theoretical physics.

3. What is the significance of differential forms and tensors in modern theoretical physics?

Differential forms and tensors have become an essential tool in modern theoretical physics, particularly in fields such as general relativity, electromagnetism, and quantum field theory. They provide a powerful framework for understanding and solving complex physical problems.

4. How do differential forms and tensors differ from other mathematical objects?

Differential forms and tensors are different from other mathematical objects, such as vectors and matrices, because they are invariant under coordinate transformations. This means that their components change in a specific way under a change of coordinates, making them more suitable for describing physical quantities in a coordinate-independent manner.

5. Are there any limitations to using differential forms and tensors in theoretical physics?

While differential forms and tensors are incredibly useful in theoretical physics, they do have some limitations. For example, they may not be the most efficient tool for solving certain types of problems, and they may not be applicable in all areas of physics. Additionally, a solid understanding of advanced mathematical concepts is required to effectively use differential forms and tensors in theoretical physics.