I would say that they can't create a cross product. If they can create a cross product, which seems to be the case from the comments below and answer, then is that cross product consider local or non-local? But it seems like to me that such a cross product is just a mathematical artifact and can't be a real physical vector x x y = z, that we could measure. Or can it be measured? If so, how?

  • $\begingroup$ Can you provide an example? What do you mean by "physically separated"? $\endgroup$ – lmr Apr 9 '19 at 6:35
  • $\begingroup$ Say that they are two feet apart from each other. $\endgroup$ – FrediFizzx Apr 9 '19 at 6:36
  • $\begingroup$ What is a physical vector in this context? Could you give an example of a cross-product with physical vectors? $\endgroup$ – Steeven Apr 9 '19 at 6:39
  • $\begingroup$ Two angular momentum vectors. $\endgroup$ – FrediFizzx Apr 9 '19 at 6:41
  • $\begingroup$ Well, I wouldn't think of vectors, but physical quantities. Angular momenta will not be able to "interact" if the objects connected to those quantities are separated from each other. $\endgroup$ – lmr Apr 9 '19 at 6:45

Generally speaking, a cross product of two vectors that are "attached" to two spatially separated points would create a non-covariant object. This is because under a general curvilinear transform the components of each of the two vectors transforms with the Jacobian matrix of the transformation evaluated at the respective point. The fact that the cross product is not covariant means that it cannot appear, for instance, as proportional to a force acting on a physical particle. So it is very unlikely that a cross-product of two vectors will be useful for anything.

However, there are certain special cases where spatially separated vectors can "interact". In particular, one often adds vectors such as momentum vectors to define "total vectors". In undergraduate courses this is rarely commented upon, but the procedure of creating such total vectors is well defined only in Cartesian coordinates, and the "total vectors" certainly do not behave as vectors with respect to general curvilinear coordinates.

Then again, certain physical problems have a clear separation of scales. In that case we can have, for instance, a body whose physical extent is much smaller than any variability length of an outside problem. We then add the momenta of the components of the small body and replace it mathematically with a massive point particle with the total momentum of the body. We then state covariance of this total momentum with respect to general curvilinear transformations. This will actually be very accurate as long as the "curving scale" of the coordinates stays much larger than the size of the body (which it should, since any coordinate system adapted to the "outside problem" will be like that). This is actually the starting point of any undergraduate classical mechanics course!

  • $\begingroup$ Is that non-covariant object something that is physically real or is it just a math artifact. $\endgroup$ – FrediFizzx Apr 9 '19 at 9:28
  • $\begingroup$ @FredDiether Unless otherwise specified, it cannot appear in a typical physical law. So it is likely not physical. Physicists do not say that something is physical based on mathematics, they say it based on whether it works to describe or predict something real. $\endgroup$ – Void Apr 9 '19 at 15:28

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