In the Quill series, I will discuss works ranging from quantum gravity to mathematics (especially algebraic geometry). I do not have any specific number of posts to write, so they will come as I see them in this month of February. This has been inspired by This Week's Finds by John Baez.
The Issue with Dirac Strings
There was a paper last year by Gonuguntla and Singleton (https://arxiv.org/abs/2310.06005) that argued that there was an overlooked field momentum in the case
of Dirac string, which makes the model inconsistent with the center of
energy theorem if one accepts that they are truly real.
We start with a simple monopole placed at the origin of ${\mathbb R}^3$
so that the magnetic and electric charges are at rest. The field
momentum of the electric field by this monopole has two components:
Coulomb's term and Dirac's string term. There is a non-zero mechanical
field momentum contribution from the interaction of magnetic charge and
electric field due to the inclusion of Dirac's string, which does not
vanish at all. See https://arxiv.org/abs/2310.06005 for the discussion on this term. It was suggested in same that
there are two takeaways from this non-trivial mechanical field momentum:
1) the first is to say that the center of energy theorem is wrong, which
implies that this term is an error, and 2) the second is to believe in
the center of energy theorem and accept this term as a real contribution
which implies that Dirac's string is real and must be physical even
though how infinitesimally thin we believe it to be. However, then it
becomes a system in which the electric charges generate a monopole-like
magnetic field with a solenoidal magnetic flux.
A comment on that paper appeared (https://arxiv.org/abs/2401.02423v1), which points out that the vector potential taken in the paper of
Gonuguntla and Singleton, is taken over all the space is not possible
because these potentials, which are 1-forms can not be defined globally
but only can be defined in certain overlaps using gauge transformations
and the quantization is defined because of the locally constant cocycle
condition appearing in those overlaps. For basics, see ( T. T. Wu and C. N. Yang, "Concept of Nonintegrable Phase Factors and Global Formulation of Gauge Fields," Phys. Rev. D 12, 3845 (1975) doi:10.1103/PhysRevD.12.3845)
Hi, in mobile view it often displays sentences over one another. This seems to vary between devices, but I think mathjax on mobile view is also an issue here (?)
Nice post, do write lengthier ones though.