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Does geometric algebra provide an alternative to pseudovectors for representing things like angular velocity? The fact that you have to flip the sign of pseudo

by twic 5y ago

Does geometric algebra provide an alternative to pseudovectors for representing things like angular velocity? The fact that you have to flip the sign of pseudovectors sometimes feels like a hint that they aren't the right representation, somehow.

ogogmad 5y ago
I think angular velocities are precisely the bivectors (which I take to be the grade-2 elements of an exterior algebra or Geometric Algebra). The change of basis is indeed different than for ordinary vectors. The exponential map (the obvious generalisation of e^x) then takes bivectors to the "rotors", which represent rotation operators. In terms of Lie theory, the rotors are a Lie group and the bivectors are the corresponding Lie algebra.
ajkjk 5y ago
Yes (to repeat the other reply here more confidently). Pseudovectors are (n-1)-graded vectors in exterior algebra, ie bivectors in 3d and trivectors in 4d. No sign flip is required required if you write them this way. Algebraically, P(x^y) = P(x)^P(y) (basically by definition), where P is a parity transformation.
jacobolus 5y ago
Yes, a Gibbs/Heaviside “pseudovector” (a.k.a. “axial vector”) is actually a bivector – a magnitude with a planar orientation, rather than linear orientation. Quantities representing a rotation are naturally oriented with a plane of rotation; there is only a unique perpendicular linear “axis” of rotation in 3d space (e.g. in 4D, the subspace fixed under a simple plane-oriented rotation is a perpendicular plane). When you properly treat bivectors as bivectors instead of cramming both bivectors and vectors down into one type of object, all of the trickiness about sign flips under reflection etc. go away. Almost every time you see a cross product in a formula, you should really have a wedge product, and the result should be a bivector. In GA, vector identities all generalize cleanly to any dimension (the trick of representing bivectors as vectors only works in 3D). See http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
- twic 5y ago
  > When you properly treat bivectors as bivectors instead of cramming both bivectors and vectors down into one type of object, all of the trickiness about sign flips under reflection etc. go away. Okay, so next question: if you formulate the laws of physics in terms of geometric algebra, does CP violation still exist?

by twic 5y ago

ogogmad 5y ago

I think angular velocities are precisely the bivectors (which I take to be the grade-2 elements of an exterior algebra or Geometric Algebra). The change of basis is indeed different than for ordinary vectors. The exponential map (the obvious generalisation of e^x) then takes bivectors to the "rotors", which represent rotation operators. In terms of Lie theory, the rotors are a Lie group and the bivectors are the corresponding Lie algebra.

ajkjk 5y ago

Yes (to repeat the other reply here more confidently). Pseudovectors are (n-1)-graded vectors in exterior algebra, ie bivectors in 3d and trivectors in 4d. No sign flip is required required if you write them this way. Algebraically, P(x^y) = P(x)^P(y) (basically by definition), where P is a parity transformation.

jacobolus 5y ago

Yes, a Gibbs/Heaviside “pseudovector” (a.k.a. “axial vector”) is actually a bivector – a magnitude with a planar orientation, rather than linear orientation. Quantities representing a rotation are naturally oriented with a plane of rotation; there is only a unique perpendicular linear “axis” of rotation in 3d space (e.g. in 4D, the subspace fixed under a simple plane-oriented rotation is a perpendicular plane). When you properly treat bivectors as bivectors instead of cramming both bivectors and vectors down into one type of object, all of the trickiness about sign flips under reflection etc. go away. Almost every time you see a cross product in a formula, you should really have a wedge product, and the result should be a bivector. In GA, vector identities all generalize cleanly to any dimension (the trick of representing bivectors as vectors only works in 3D). See http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

twic 5y ago
> When you properly treat bivectors as bivectors instead of cramming both bivectors and vectors down into one type of object, all of the trickiness about sign flips under reflection etc. go away. Okay, so next question: if you formulate the laws of physics in terms of geometric algebra, does CP violation still exist?