Gravity and Strings

254 The Reissner–Nordström black hole
so we find that we can do consistent quantum mechanics ignoring the Dirac string if the
magnetic charge is related to the electric charge by the Dirac quantization condition 26
[323],
qp = n2πc. (8.170)
It is worth remarking that this formula is invariant (up to a global sign) under electric–
magnetic-duality transformations q → p, p →−q.
Using the normalization of Eq. (8.58), the definitions of electric and magnetic charge
that satisfy the Dirac quantization condition in the above form (without any extra factors)
are
q ≡
1
16πG (4)
N
S 2 ∞
⋆
F,
p ≡−
S2 F, pq = 2πn. (8.171)
∞
In a non-simply connected spacetime there will be closed paths that are not contractible
to a point (that is, it will have a non-trivial π1). The wave function will not in general
be single-valued around those closed paths but will pick up a phase, the Aharonov–Bohm
phase [20, 21], which can be detected by interference experiments. The Dirac quantization
condition can be considered as the condition of cancelation of a would-be Aharonov–
Bohm phase around the Dirac string, which physically is unacceptable. The concept of the
Aharonov–Bohm phase is, however, much more general and deals with the non-triviality of
the topology of the gauge-field itself when it is seen as a section of a fiber bundle. To study
the Aharonov–Bohm phase, thus, we first reformulate the Dirac monopole in this language.
8.7.3 The Wu–Yang monopole
Wu and Yang [964] were the first to reformulate the Dirac monopole in the modern language.
The basic idea is to generalize the basic concepts of tensors in manifolds to gauge
fields: 27 a manifold is a topological space that in general is not isomorphic to Rn . Thus it
needs to be covered by patches that are isomorphic to parts of Rn . Each patch provides a
local coordinate system. Neighboring patches must overlap and the two different coordinates
of points in the overlaps are related by diffeomorphisms. Now one can define tensor
fields on a manifold. A given well-defined tensor field will have different components in the
overlaps, corresponding to the different coordinate systems that are defined there, but they
will be related by the tensor-transformation laws corresponding to the diffeomorphisms that
relate the different coordinate systems.
26 There are other ways of finding this condition, such as studying the quantization of the angular momentum
of the electromagnetic field created by the electric and magnetic particles. See, for instance, [459].
27 For aless-pedestrian explanation, there are many reviews and textbooks that the interested reader can consult:
for instance [240, 347, 630, 715, 717].