Edward Green says...

There are a few things that are not immediately clear, to say the
least. Take a 4 + 1 dimensional world vs. 3 + 1; for now, no "rolled
up" dimensions.

First question: is there an analogue/extension of "angular momentum"
in such a world?

Sure, angular momentum makes sense in any number of dimensions.
It's not a *vector* unless there are exactly 3 spatial dimensions,
but the analogous tensor makes sense no matter what the dimensionality.
The angular momentum tensor L^jk = x^j p^k - x^k p^j makes sense
whenever there is a metric tensor, and it is conserved for central
forces in any number of dimensions. In the particular case of
4D space, there are 6 independent nonzero components of this tensor

L^23, L^31, L^12, L^41, L^42, L^43

The first three are the same as the 3-space angular momentum
components L_x = L^23, L_y = L^31 and L_z = L^12. The last
three are new conserved quantities which, from the 3-D point
of view seem to form a pseudo vector Q with components
Q_x = L^41, Q_y = L^42, Q_z = L^43.

It seems to me axial vector only exist in 3
dimensions. Now, that would not mean the posited extension can not
exist, but it might not be very recognizable.

It's recognizable as 3D angular momentum plus another
vector.

Second question: supposing we have answered the first question, what
happens to this extension of angular momentum when we do roll up the
fifth dimension?

The main change that results from rolling up a dimension is that
it tends to make low-energy behavior independent of the curled
up dimension. So physical quantities (scalar, vector and tensor
fields) can depend on x^1, x^2 and x^3, but don't typically depend
on x^4.

In 3 spatial dimensions each component of angular momentum is
conserved separatedly, but they are in some sense fungible: by
applying an arbitrary torque to a body we can create angular momentum
about axes which previously showed none (creating an opposite
increment in the system supplying the torque). This suggests that the
extra component of angular momentum (assuming this "component"
language makes sense, since the total object may not be represented by
a 4-vector) should be coupled to the other 3?

Certainly rotations in and out of the extra dimension are
possible. However, if the extent of the curled-up dimension
is very small, it's hard to get any significant torque.

--
Daryl McCullough
Ithaca, NY