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chogg
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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
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
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