I have a doubt about the physical and geometrical interpretation of differential forms. I've been studying differential forms on Spivak's Calculus on Manifolds, but my real intent is to use those concepts in physics.

I was very amazed when I found out that a force can be described as a $1$-form that given a vector gives me the work to move a particle along the vector, cf. this Phys.SE post. I really believe that there's much more use for those concepts in Physics. For instance, writing Maxwell's equations in a more general form.

The problem with those books like Calculus on Manifolds is that all the ones I've found don't care too much about the physical and geometrical interpretation of those concepts. For instance, although they comment about how this fits and geometry and so on, it's not the focus to justify why forms relates to density and how can this be used to model things in physics.

What I want then is to ask if any of you can recommend be books that explains how those concepts fits in physics, how can they be used to give precise descriptions of physical phenomena.

  • $\begingroup$ Related: physics.stackexchange.com/q/111/2451 and physics.stackexchange.com/q/1601/2451 $\endgroup$ – Qmechanic Mar 3 '13 at 17:51
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    $\begingroup$ @Qmechanic I think this question is different and a bit more specific than the links you gave, because he explicitelly and specifically asks about the physical interpretation of differential forms. It would be nice if the question could get an answer which explains things by nice examples directly or link to a shorter than a book tutorial, apart from just recommanding a book. So I liked the "reference-request" tag (which is often better received) better than the "book" tag the question now has ... $\endgroup$ – Dilaton Mar 3 '13 at 18:08
  • $\begingroup$ And if I got this right, the question asks not just about differential forms in classical mechanics, but more generally about what they are useful for generally in different subfields in physics, so I disagree that it is a duplicate. Of course I am interested in a answer too ... :-) $\endgroup$ – Dilaton Mar 3 '13 at 18:13
  • $\begingroup$ One cool application of differential forms in generalizing Maxwell theory to higher dimmensions leads to Branes as a generalization of charged particles $\endgroup$ – Dilaton Mar 3 '13 at 18:22
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    $\begingroup$ There's an OK book by John Baez and friends, called 'Gauge Fields, Knots and Gravity.' It talks about Yang-Mills theory, topics in Chern-Simons theory, gravity etc., in a very geometric way. $\endgroup$ – Vibert Mar 4 '13 at 9:06

I would really recommend the book by Frankel, The Geometry of Physics. He deals with all the fundamental concepts of topology and differential geometry, but gives clear and detailed applications to classical mechanics, electromagnetism, GR and QM. He is not too formal, but develops really a lot of useful tools using differential forms.

Another book, which is a bit more basic and is so to say a light-version of the classical Arnold's Mechanics textbook is Geometric Mechanics, by Richard Talman. One can develop a geometrical and a physical intuition of differential forms. Here the applications are mostly reduced to classical mechanics.

There are of course other good texts, but these are really good starting points.

I also agree with @Muphrid, that Geometric Algebra should be actually preferred as the language for modern physics, instead of differential forms. It is much clearer and familiar. Check the book by Lasenby and Doran and also the dissertation of Anthony Lewis at cosmologist.info, who has a chapter dealing only with the translation between differential forms and geometric algebra.

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    $\begingroup$ In addition, Hestenes has a good section in Clifford Algebra to Geometric Calculus on the relationship between forms and geometric calculus. In particular, he shows how GC handles forms that aren't scalar valued in a more elegant manner. $\endgroup$ – Muphrid Mar 24 '13 at 15:21
  • $\begingroup$ Thanks for the reference to the Lewis thesis. I believe Frankel is not the book to start with. I would recommend the book by Flanders: Differential forms with applications to the Physical Sciences. Well written, clear and cheap. A lot of examples too. It's sometimes difficult to understand everything, but it's pretty easy to make the transition from tensor notations to differential forms ones. $\endgroup$ – FraSchelle Mar 24 '13 at 15:50

As it is, differential forms don't tell you the whole story--strictly speaking, differential forms only deals with covectors and wedge products of covectors and then uses the hammer of the Hodge star to be able to clumsily do inner products. To me, it is too far removed from the vector calculus you may already know.

Instead, I strongly urge you to look into geometric algebra. All of the results of differential forms apply to geometric algebra as well--the former is strictly contained in the latter--but the notation is much more familiar and the emphasis is on geometric interpretation instead of abstract symbol pushing. David Hestenes has several books on the subject. Probably the authoritative piece as far as using geometric algebra to solve physical problems is Geometric Algebra for Physicists by Doran and Lasenby. You can also read some things quickly in this website, written by Gull, Doran, and Lasenby.

I'll give a quick overview. Geometric algebra has a wedge product like differential forms does, but it also lets you directly use a dot product as well. In fact, it combines the two in a useful operation called the geometric product, defined as follows. For two vectors $a, b$, the geometric product $ab$ is

$$ab = a \cdot b + a \wedge b$$

The geometric product is associative (even though the dot product is not!). This makes it very useful. It is also invertible in Euclidean space, as a consequence of that associativity. This makes possible the formula

$$a = abb^{-1} = (a \cdot b) b^{-1} + (a \wedge b) \cdot b^{-1}$$

Geometrically, this decomposes $a$ into $a_{\parallel, b}$ and $a_{\perp, b}$. We emphasize that $a \wedge b$ denotes an oriented plane, and further wedge products yield oriented volumes and more.

Some applications immediate to physics are as follows:

  1. Angular momentum as a bivector. This is one of the first times you "need" a cross product, and using the wedge product instead yields a cleaner interpretation. The angular momentum bivector is exactly the plane in which two objects move in relation to one another. This also generalizes to beyond 3d, so it makes sense to talk about angular momentum bivectors in relativity also.
  2. Unification of integral theorems (the fundamental theorem of calculus). Geometric calculus (like differential forms) makes possible the unification of the divergence theorem, Stokes' theorem, and so on as one basic concept: that the integral of a function over a boundary is equal to the integral of the derivative over the region bounded by that boundary. I think this is a significant quality of life issue; having to remember only one concept is much easier, in my opinion, than remembering many separate integral theorems.
  3. Relativity without indices or classic tensor calculus. Geometric algebra's combination of the dot and wedge products makes possible all the usual operations one usually needs tensor calculus and index notation for. Relativity can be presented using a modest extension of the methods used in 3d electromagnetism. The geometric product makes it possible to boil down Maxwell's equation in vacuum to one equation (instead of two for differential forms): $\nabla F = J$. This emphasizes the interpretation of the EM field $F$ as a bivector field, a field of oriented planes throughout spacetime.
  4. Geometric interpretation of quantum mechanics. A lot of the mathematics in quantum is presented as mystical or special to QM, but most of it is actually inherent to the geometric structure of space and time. Geometric algebra allows one to treat the Pauli and Dirac algebras as the algebras of basis vectors in 3d and 3+1d space. This makes the interpretation of spin and spin operators inherently geometrical.
  5. Construction of spinors. Spinors are things we often deal with in quantum, perhaps with only the understanding that they must be rotated through $4\pi$ instead of $2\pi$ to return to their original form. Geometric algebra shows that spinors underlie all rotations--even those of plain old 3d space. In fact, the spinors of 3d space are quaternions, and the spinors of 2d space are complex numbers. GA gives a framework for constructing spinors and manipulating them like other objects.

Differential forms can do some of these things as well, some of them not (it absolutely can't boil down Maxwell's equations to one expression). Either formalism is a great improvement over traditional methods, however.

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    $\begingroup$ @Vibert Not sure what you mean. Only GA has the $\nabla F = J$ expression that spans across both columns of the table. Yes, forms can have one equation when in terms of the four-potential, and in the Lorenz gauge, but $\nabla F = J$ is always valid, regardless of gauge. $\endgroup$ – Muphrid Mar 4 '13 at 15:05
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    $\begingroup$ On the contrary, equations of this form (which use $\nabla$ in a geometric product, instead of as a divergence or curl) are invertible in the sense that they have Green's functions. In 3d, we often deal with only the Green's function for $\nabla^2$, but GA allows us to construct a vector Green's function for $\nabla$ directly and use it as a practical tool. $\endgroup$ – Muphrid Mar 4 '13 at 22:31
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    $\begingroup$ @Muphrid Thanks for this answer. This Geometric Algebra sounds really like quaternions for me. Is it easy to understand differential geometry from them ? How do they "fit" or "fix" the use of Lie group structure ? One of the main interest of differential forms is that they implement easily the Lie derivative. How to do that algebraically ? By the way, thanks again. $\endgroup$ – FraSchelle Mar 24 '13 at 16:05
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    $\begingroup$ @Oaoa There are two basic approaches to differential geometry. One involves embedding in a flat space and using projections onto the manifold; this is pretty straightforward. The other involves the use of only flat manifolds with gauge fields on top of them; this has been used to model general relativity, and has the benefit of being entirely intrinsic. Hestenes has a chapter on Lie groups in his book Clifford Algebra to Geometric Calculus. He conjectures that all Lie groups are generated by bivectors, hence making GA very useful for understanding Lie algebras, but the claim isn't yet proven $\endgroup$ – Muphrid Mar 24 '13 at 16:21
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    $\begingroup$ @StephenBlake It sounds like you think the dot product is physical only because it's invariant under the actions of $\mathrm O(n)$, but a general principle of gauge invariance under all differentiable remappings of positions supersedes this symmetry (see Lasenby, Doran, and Gull). Bivectors are generated from vectors and transform the same way through "outermorphism": under a linear operator $\underline T$, the bivector $a \wedge b$ is mapped to $\underline T(a \wedge b) \equiv \underline T(a) \wedge \underline T(b)$. $\endgroup$ – Muphrid Mar 24 '13 at 19:21

This may not be exactly what you are looking for, but I am going to recommend two specific texts.

Misner, Thorne, and Wheeler, Gravitation, Chapters 4, 9, and end of 14

Solidly in the realm of physics but they have a lot of tidbits of interpretation in there.

Choquet-Bruhat and DeWitt-Morette, Analysis, Manifolds, and Physics, Chapter IV.C

I mean, this text is incredible. We should all read all of it, all the time. More mathy but that's better in this case. Also more applicable then just E/M + Gravity like MTW. A bit old, but it's nice notation for breaking you in.


As a perhaps "softer" book suggestion, I am finding "The Road to Realty" by Penrose to be quite nice in getting an overview of many mathematical interpretations of physics. Also, there are exercises in this book so although you feel like you're buying some sort of coffee table book, there is plenty to work through if you are willing.


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