Which quantities are naturally forms, and which are (multi)vectors?

[chogg]

Which quantities are "naturally" forms, and which are (multi)vectors?

I'm undertaking a self-study of geometric algebra and differential forms. It is very enlightening, but I find I'm getting a bit confused by which kind of beast is most "natural" for a particular physical quantity.

Where I'm at so far: I feel like I intuitively "get" the picture of contraction (vector with 1-form, bivector with 2-form, etc.), and how these quantities are naturally paired, without any need for a metric. References [1], [2], and [3] (among others) have been very helpful in attaining this intuition. Similarly, I "get" that there's a 1-to-1 mapping between vectors and 1-forms, bivectors and 2-forms, etc., but only if there's a metric.

What I really don't get is: for a given quantity, how do you tell whether it's "naturally" a (multi)vector or a form? Let me give some examples.

First, some easy ones. The displacement vector is obviously a plain old "pointy" vector, not a 1-form. Similarly, the gradient is a 1-form, not a vector, as [3] makes clear. It also seems to me that the reciprocal lattice vectors of crystallography are clearly 1-forms and not vectors; I find it very enlightening to visualize them as the lattice planes themselves.

What about momentum? MTW [2, sec. 2.5] discuss it as both a 1-form and a vector, saying they are equivalent by the dot product (which assumes a metric). I like the association with the wavefronts of the de Broglie wave (which favors the 1-form intepretation), but since it's the derivative of displacement, and displacement is a vector, isn't momentum more "naturally" a vector too?

Angular momentum is another source of confusion for me. The paper in [4] discusses it as a 2-form, based on converting x and p to their equivalent one-forms. But Lasenby and Doran [5, ch. 3] discuss it as a bivector. I think it makes more sense to me as a bivector! Then again, good ol' Wikipedia lists it as a 2-form.[6]

And what about the electromagnetic field? Bivector, or two-form? The venerable MTW [2, ch. 4] discuss it in terms of the latter. John Denker treats it as a bivector [7,8,9,10], but also mentions it as a 2-form [11]. I find it easy to think of F as a bivector, but its formulation as a 2-form seems more natural to me in many ways (i.e. increasing field strength corresponds to increasing density of field lines)

Is there a principled way to take a given physical quantity, and ascertain whether it's best to think of it as a form or a (multi)vector? Any pointers would be appreciated.

Thanks,
Chip

(References:)
[1] Weinreich, Gabriel. "Geometrical Vectors". University Of Chicago Press, 1998
[2] Misner, C., Thorne, K., and Wheeler, J. "Gravitation". W. H. Freeman, 1973
[3] http://www.av8n.com/physics/thermo-forms.htm#fig-bump-hump
[4] http://panda.unm.edu/Courses/Finley/P495/TermPapers/relangmom.pdf
[5] Doran, C. and A. Lasenby. "Geometric algebra for physicists". Cambridge University Press, 2003
[6] http://en.wikipedia.org/wiki/Angular_momentum
[7] http://www.av8n.com/physics/pierre-puzzle.htm
[8] http://www.av8n.com/physics/magnet-relativity.htm
[9] http://www.av8n.com/physics/maxwell-ga.htm
[10] http://www.av8n.com/physics/straight-wire.htm
[11] http://www.av8n.com/physics/partial-derivative.htm#sec-vis

 

[homology]

Whoa Ho! That's sort of my modus operandi as well. Weinreich gives you the answer to some degree, its how they transform. Let me add one more reference: Theodore Frankel's book: The Geometry of Physics. He's all about what's natural.

Its tricky I think. I was just writing (and deleting) how momentum can be defined as fiber derivative of the Lagrangian (fancy words for what you'd see in any mechanics book) and those transform as components of a covector. But then you can also argue that the momenta are components of the Hamiltonian vector field on phase space.

Both have some metrical like properties I haven't fully sorted out yet. Depending on the Lagrangian the kinetic energy can define a metric and thus an inner product on tangent spaces and so you can just associate velocities to momenta via the metric. On phase space there's a very special 2-form which while not a metric, does create relations between contravariant and covariant critters via Hamilton's equations.

That being said momentum certainly seems naturally a 1-form since velocities are contra and the contraction of the two gives energy.

Not really an answer, but a reply :)

 

[lavinia]

I'm undertaking a self-study of geometric algebra and differential forms. It is very enlightening, but I find I'm getting a bit confused by which kind of beast is most "natural" for a particular physical quantity.

Where I'm at so far: I feel like I intuitively "get" the picture of contraction (vector with 1-form, bivector with 2-form, etc.), and how these quantities are naturally paired, without any need for a metric. References [1], [2], and [3] (among others) have been very helpful in attaining this intuition. Similarly, I "get" that there's a 1-to-1 mapping between vectors and 1-forms, bivectors and 2-forms, etc., but only if there's a metric.

What I really don't get is: for a given quantity, how do you tell whether it's "naturally" a (multi)vector or a form? Let me give some examples.

First, some easy ones. The displacement vector is obviously a plain old "pointy" vector, not a 1-form. Similarly, the gradient is a 1-form, not a vector, as [3] makes clear. It also seems to me that the reciprocal lattice vectors of crystallography are clearly 1-forms and not vectors; I find it very enlightening to visualize them as the lattice planes themselves.

What about momentum? MTW [2, sec. 2.5] discuss it as both a 1-form and a vector, saying they are equivalent by the dot product (which assumes a metric). I like the association with the wavefronts of the de Broglie wave (which favors the 1-form intepretation), but since it's the derivative of displacement, and displacement is a vector, isn't momentum more "naturally" a vector too?

Angular momentum is another source of confusion for me. The paper in [4] discusses it as a 2-form, based on converting x and p to their equivalent one-forms. But Lasenby and Doran [5, ch. 3] discuss it as a bivector. I think it makes more sense to me as a bivector! Then again, good ol' Wikipedia lists it as a 2-form.[6]

And what about the electromagnetic field? Bivector, or two-form? The venerable MTW [2, ch. 4] discuss it in terms of the latter. John Denker treats it as a bivector [7,8,9,10], but also mentions it as a 2-form [11]. I find it easy to think of F as a bivector, but its formulation as a 2-form seems more natural to me in many ways (i.e. increasing field strength corresponds to increasing density of field lines)

Is there a principled way to take a given physical quantity, and ascertain whether it's best to think of it as a form or a (multi)vector? Any pointers would be appreciated.

Thanks,
Chip

(References:)
[1] Weinreich, Gabriel. "Geometrical Vectors". University Of Chicago Press, 1998
[2] Misner, C., Thorne, K., and Wheeler, J. "Gravitation". W. H. Freeman, 1973
[3] http://www.av8n.com/physics/thermo-forms.htm#fig-bump-hump
[4] http://panda.unm.edu/Courses/Finley/P495/TermPapers/relangmom.pdf
[5] Doran, C. and A. Lasenby. "Geometric algebra for physicists". Cambridge University Press, 2003
[6] http://en.wikipedia.org/wiki/Angular_momentum
[7] http://www.av8n.com/physics/pierre-puzzle.htm
[8] http://www.av8n.com/physics/magnet-relativity.htm
[9] http://www.av8n.com/physics/maxwell-ga.htm
[10] http://www.av8n.com/physics/straight-wire.htm
[11] http://www.av8n.com/physics/partial-derivative.htm#sec-vis

In general forms and vectors can be transformed into each other with a metric.

The classic case is when the differential of a function is turned into a gradient vector.

df acts on vectors but in the presence of an inner product there is a vector called grad f which has the property that df(v) = <grad f,v> where <,> is the inner product.

In Euclidean space, especially in Physics courses, the distinction is often overlooked and this I guess is why you view the gradient of a function as a 1 form. really it is a vector and the one form is <grad f, > or simply df.

If one takes a force field - say the gravitational field - then it is usually described as a field of vectors. But really one knows the field from the work it does. The work is expressed as the integral of an inner product with the force field and that is a 1-form.

 

[chogg]

Thanks for the replies. A very interesting discussion of velocities and momenta. It made me realize I had been assuming that velocity and momentum are "naturally" the same kind of quantity, having p=mv in my mind or something. Framing it in terms of the Lagrangian helps me see the difference. I think I follow when you say the energy can define a metric (in the abstract phase space), but I would need further study before I become really comfortable with it.