Download DIRAC`S DREAM: THE MYSTERY OF THE MAGNETIC MONOPOLE

DIRAC’S DREAM: THE MYSTERY OF THE MAGNETIC
MONOPOLE
JOSEPH CHAN
Abstract. In Maxwell’s equations, the symmetry between the electric and magnetic fields is suggestive of the existence of sources of magnetic charge: magnetic
monopoles. In 1931, Dirac showed that the existence of magnetic monopoles are
not only consistent with quantum mechanics but provide us to date, with the only
“reason” why electric charge is quantised. I will lay before you Dirac’s dream of
returning symmetry to Maxwell’s equations and how it haunts us in grand unified
theories.
Before you ask, no we have not yet found fundamental magnetic monopoles. Magnetic monopoles are from their first conception, very curious things. Think of the
bar magnet which we are so familiar with. It has two poles - we say it is a dipole
- and the naïve way of acquiring a monopole is to break the bar magnet into two.
Behold, you have just acquired two bar magnets. So is this a wild goose chase? Even
in 1894, Pierre Curie said, “magnetic monopoles could conceivably exist, despite not
having been seen so far.”
Perhaps it matters not for it’s anyone’s guess which hypothetical particle will
be found next. A fulfilling strategy is to study hypothetical particles which are
mathematically fascinating. Indeed, Dirac said,
“A good deal of my research work in physics has consisted in not
setting out to solve some particular problems, but simply examining
mathematical quantities of a kind that physicists use and trying to
get them together in an interesting way regardless of any application
that the work may have. It is simply a search for pretty mathematics.
It may turn out later that the work does have an application. Then
one has had good luck.”
I am going to tell you what a magnetic monopole is three times, in increasing order
of sophistication.
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DIRAC’S DREAM: THE MYSTERY OF THE MAGNETIC MONOPOLE
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1. Maxwell’s Equations Speak to us
Anyone who has done a course in electromagnetism might recall that electric
charges in some region of space experience electric and magnetic forces represented
by vectors E = (Ex , Ey , Ez ) and B = (Bx , By , Bz ) in R3 . In fact, we talk about
electric and magnetic fields
E : R3 × R → R3
B : R3 × R → R3
which tell us the electric force E(t, x, y, z) and magnetic force B(t, x, y, z) vectors
that a charge at any point (x, y, z) in R3 at a time t would feel.
Maxwell’s equations in a vacuum are four partial differential equations which the
electric and magnetic fields obey
∂
E+∇×B =0
∂t
∂
B+∇×E =0
∇·B =0
∂t
when no electric charges are present. Since we are using geometrised or “natural”
units, the speed of light c and Coulomb’s constant 1/4πε0 are both 1.
The interpretation is this: the two equations on the left are saying that for any
closed surface such as a sphere, the flux entering is the same as the flux exiting
because in a vacuum; that there are no sources. The equations on the right say
that any change of the fields with time is exactly equal to how much the field lines
circulate.
Observe that Maxwell’s equations are symmetrical with respect to the exchange
of E and B. This symmetry is very real - in a relativistic frame moving at some
speed v comparable to the speed of light,
∇·E =0
E 7→ γ (E + v × B)
B 7→ γ (B − v × E)
the electric and magnetic fields mix.
When an electric source ρe , ie. an electron is introduced, the equations lose their
symmetry
∇ · E = ρe
... unless magnetic sources ρm are introduced too
∇ · B = ρm .
DIRAC’S DREAM: THE MYSTERY OF THE MAGNETIC MONOPOLE
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What would this look like? The electric field of an electron E = rq2 r̂ is a “hedgehog”
with charge q and with strength falling off with the squared of the distance. We would
then expect a magnetic monopole to produce a “hedgehog” magnetic field B = rg2 r̂.
A bar magnet looks like two such monopoles put together. However, bar magnets
are secretly the magnetic fields generated by the spin angular momentum of electrons and so can never have their poles separated. They are not being produced by
fundamental magnetic monopoles!
2. The Dirac Monopole
Now that you have some understanding of what we are pursuing, let’s look at
Dirac’s dream of magnetic monopoles.
Actually, the electromagnetic field is an example of a gauge field. You may remember that the equation
∇·B =0
allows us to write the magnetic field in terms of the vector potential A : R3 ×R → R3
B = ∇ × A.
We also write the electric field in terms of A and the scalar potential ϕ : R3 × R → R
∂
A.
∂t
Remember that the derivative of a constant is zero so we can always add “as many
constants as we like” without changing the derivative of something. Likewise, we can
make the gauge transformation
E = −∇ϕ −
∂
λ
∂t
A → A + ∇λ
ϕ→ϕ−
where λ : R3 × R → R is a scalar function of our choosing, without changing the
electric and magnetic fields.
If magnetic monopoles existed, ∇ · B 6= 0 so we can no longer write B = ∇ × A
globally. But all is not lost! We can still attempt to write B in terms of A in the
region outside of the region occupied by the electric charges and then see what goes
wrong.
Since B is the curl of A, to get a point charge B = rg2 r̂, we need to let A have
no r component and the θ component should be a function of ϕ and vice versa. For
example, in Cartesian coordinates,
DIRAC’S DREAM: THE MYSTERY OF THE MAGNETIC MONOPOLE
Ai0 = εij3 r̂j
4
g
z+r
or in spherical coordinates,
A0 = g (cos φ − 1) dθ.
This field resembles the circulation xdy − ydx. The coordinate θ becomes singular
at φ = 0, φ = π. Since cos φ − 1 vanishes at φ = 0, A is okay there. When φ = π,
that is, on the negative z-axis, the field A is singular.
In fact, the general solution to our problem is
Ap = g (cos φ − 1) dθ − p [g (cos φ − 1) dθ − gθ sin φ dφ]
for 0 ≤ p < 1.
The difference between any two of these solutions is some multiple of
g (cos φ − 1) dθ + gθ sin φdφ = d [gθ(cos φ-1)]
which can also be written as ∇ [gθ(cos φ − 1)] so we can go between any two solutions
by a gauge transformation. The gauge transformations can be used to deform the
line singularity to any curve running from the origin to infinity but can never get rid
of the line singularity. Thus, the line singularity is a permanent feature of the Dirac
monopole and we call it the Dirac string.
This seems like a problem since a singularity would be unphysical. But in classical
electromagnetism, there is no way to detect the potential A, only the magnetic field
B which is not singular. So we could come to the conclusion that the potential A is
not physical, just a mathematical tool and call it a day.
But in the quantum mechanical world, there is the Aharonov-Bohm experiment.
If an electron traveled along a closed loop C around a Dirac string, its wave-function
ψ would be multiplied by a phase factor
ˆ
exp iq
A0 · dl = exp (2πiqg(cos φ − 1)) = exp(4πiqg)
C
where the last equality comes from contracting the loop C around the string so that
φ = π. The only way for nature to hide its shame now is if the expression qg were
a half integer so that 4πiqg = 2πn and the phase factor is just 1. This implies that
both the electric and magnetic charges, q and g have to be integer multiples of some
minimum electric charges qmin and gmin respectively.
DIRAC’S DREAM: THE MYSTERY OF THE MAGNETIC MONOPOLE
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This is called the Dirac quantization condition and it explains why charge is quantised in our universe. Furthermore, this magnetic monopole charge g is a topological
invariant. Assuming that the electric charge q is constant, the magnetic charge g
counts the number of times the wavefunction ψ winds around a circle whenever the
electron completes a full circle around the Dirac string. Since there is no way of
deforming this to a different number of windings without “cutting”: it is a constant!
This hints that magnetic monopoles are topological in nature.
3. The ’t Hooft Monopole
Now we get to the kind of monopole that you actually see in quantum field theory
research, the ’t Hooft monopole. Yes, that is the right spelling. The scalar function
λ of the gauge transformation for electromagnetism
A → A + dλ
gives us an element of the real numbers R at every point in the region U at a time
t. Actually, it’s a coincidence that the real numbers R is the Lie algebra of the Lie
group U (1) which is another name for the circle.
One step up from U (1), we have the group SU (2) which we can think of as the
3-sphere S 3 . To imagine this, think of the 2-plane and add infinity to get the 2sphere S 2 . The same procedure applied to three-dimensional space R3 gives us the
3-sphere. The Lie algebra su(2) of SU (2) is a vector space whose elements are of
the form λ = λ1 τ1 + λ2 τ2 + λ3 τ3 where the λ1 , λ2 , λ3 are real numbers and the basis
{τ1 , τ2 , τ3 } are the Pauli matrices multiplied by i, −i, i respectively
"
#
"
#
"
#
0 i
0 −1
i 0
τ1 =
, τ2 =
, τ3 =
.
i 0
1 0
0 −i
Where U (1) gauge theory is electromagnetism, SU (2) gauge theory is, with some
modification, the theory of the weak force. Naïvely, the basis {τ1 , τ2 , τ3 } of the Lie
algebra su(2), correspond to the three gauge bosons Z, W + and W − which play the
same role in the weak force as the photon does for electromagnetism. This is not
strictly true - there is a subtlety involving the mixing of the two forces - but that is
another story so you can just accept this for now.
In the ’t Hooft monopole, in addition to the gauge field A we have a scalar field
Φ : R3 → su(2) which we call a Higgs field. The Higgs field has a vacuum expectation
value which breaks the SU(2) symmetry into U(1). Without giving you a course on
DIRAC’S DREAM: THE MYSTERY OF THE MAGNETIC MONOPOLE
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quantum field theory, I can’t explain this in full. Phyically, you may think of this to
mean that we don’t have a weak force charge, only an electric charge.
The ’t Hooft monopole looks like the Dirac monopole, with a hedgehog magnetic
field B and a line singularity in the field A. However, there is a crucial difference:
the ’t Hooft monopole does not have a singularity at the origin where the monopole
is. This is good news since we would like our objects to not be singular.
The field Φ on a very big 2-sphere which we think of as spatial infinity is a function
from S 2 to SU(2)/U (1) which is equivalent to the 2-sphere S 2 . This is implied by
the symmetry breaking mechanism. In other words, just roll with it. Thus
2
2
2 : S → S .
Φ|S∞
We call the “number of layers” or degree of the image of this function the magnetic
charge and it is a topological invariant π2 (SU (2)/U (1)). We can relate this to the
previous topological invariant by looking at the equator. Here, we see the “layers”
as the winding number which we saw in the case of the Dirac monopole. This
correspondence between the degree of spheres in SU (2)/U (1) with the winding of
loops in U (1) is actually a theorem in topology. Thus, the charge of magnetic
monopoles really are topological invariants and we say that magnetic monopoles are
topological solitons.
4. Conclusion
To conclude, magnetic monopoles are particles with Hedgehog magnetic fields and
with a line singularity called a Dirac string in its potential field. Most importantly,
they have magnetic charges which are topological invariants, things that mathematicians have been studying for a long time with no particular physical application in
mind. This then is truly Dirac’s dream, for he said, “it seems to be one of the fundamental features of nature that fundamental physical laws are described in terms
of a mathematical theory of great beauty and power.”
References
[1] Paul Dirac, Quantised Singularities in the Electromagnetic Field, Proc. Roy. Soc. A 133, 60,
London (1931).
[2] Weinberg, Erick, Classical Solutions in Quantum Field Theory, Cambridge University Press,
New York, USA (2012).