Características y propiedades

FA7→?FA,

?FA7→ −FA,

k7→ −j, j7→k,

where we have used the fact that (?)2 =−1 on 4-dimensional Minkowski space.

By substituting these transformations into the equations of motions, we see that equa- tions (2.2.6) and (2.2.7) transform into one another, thus reproducing aU(1)-gauge theory with the same equations of motion. This defines exactly an electric-magnetic duality be- tween two distinct U(1)-gauge theories with sources.

2.3

The Dirac Monopole

Having now introduced the form aU(1)-gauge theory with electric and magnetic sources would take, we will restrict the theory to the case in which there are no electric sources, and only static magnetic monopoles. Thus we want to construct a monopole in a U(1)-gauge theory with magnetic field given by

~ B = gˆr

r2,

where g denotes the magnetic charge. Here we observe that the magnetic field for a monopole is of the same form as the electric field for an electron, with the electric charge

q has being replaced with the magnetic charge g. This is required to ensure the electric- magnetic duality discussed earlier holds.

Definition 2.3.1 (U(1) Dirac monopole). Let M denote a 4-manifold.

A U(1) monopole arising in a U(1)-gauge theory is given by a gauge potential A ∈ Ω1(M, ad(P))whose gauge field FA has the form

F0j = 0,

Fij =ijk

g0xk

r3

(2.3.1)

and satisfies the equation

dFA=k, (2.3.2)

where k∈Ω3(M, ad(P)) denotes a magnetic source.

As proven in Section 2.2, in order for a magnetic monopole to exist, it is required that the base spaceM be non-trivial. Given we will only be considering static monopoles with no electric field components (that is allk-forms will be independent of the variablet, and

any ‘dt’ term will be trivial), we can pullback the principalG-bundle on which this gauge theory is defined via the embedding

π:R3 →R1,3

(x, y, z)7→(0, x, y, z). (2.3.3)

As a result, we will be considering a principal G-bundle over the 3-manifold consisting of only the three spatial components which we will take to have a Euclidean metric.

Remark 2.3.2. Given this reduction of the base space to 3-dimensions, we can integrate both sides of equation (2.3.2) and equivalently define the Dirac monopole as all gauge potentials A∈Ω1(M, ad(P))satisfying the equation

1 4π

Z

M

FA=g0, (2.3.4)

where g0∈iRdenotes the magnetic charge.

This will be the more frequently used definition.

Given we are interested in topologically non-trivial base spaces, we will consider

M =R3− {0} 'S2, where here'denotes homotopy equivalence andS2 the unit sphere.

Given our interest with monopoles lies in the topological invariants of this base space, we can thus take M =S2_.

Now we want to introduce a magnetic monopole. As such, applying Definition 2.3.1, we need to determine a gauge potential A∈Ω1(M, ad(P)) such that

FA=ijk 1 2Bkdxi∧dxj = i g r3(x3dx1∧dx2+x2dx3∧dx1+x1dx2∧dx3) = i g r3~x·d ~S, (2.3.5) where r=px2

1+x22+x23 and g6= 0 denotes the magnetic charge of the monopole. Here we have writteng0=ig whereg∈R.

Given the Lie algebra ofU(1) is u(1)∼=iR, in order for the gauge potential to locally

take values in g, we require A be pure imaginary. Thus, we require that ig ∈ iR. Now

given we are constructing a magnetic monopole in aU(1)-gauge theory, it is not the gauge field that we are interested in, but its gauge potential. Thus in order for this to be a well defined gauge theory, we need to determine someA∈Ω1(S2, ad(P)) such that FA=dA. Proposition 2.3.3. Let FA be as defined in equation (2.3.5). Then there does not exist

a gauge potential A ∈ Ω1(S2, ad(P)) taking values globally in u(1) over S2, such that

2.3. THE DIRAC MONOPOLE 23

Proof. Assume such a global gauge potential does exist. Let S₊2 and S₋2 denote the upper and lower hemispheres ofM =S2respectively, such that both spaces have boundary given by the equatorC of S2. Orienting C such that the positive direction is clockwise, whilst for the surfacesS2,S₊2 andS₋2 the positive direction is radially outwards from the origin, from Stokes’ theorem we know

1 4π Z S dA= 1 4π Z S2 + dA+ Z S2 − dA ! = 1 4π I C A+ I −C A = 1 4π I C A− I C A = 0. (2.3.6)

Now using the definition ofFA, we know that 1 4π Z S dA= 1 4π Z S FA = 1 4π Z S i g r2xˆ·d ~S = i g 4πR2 Z S dS = i g 4πR2(4πR 2₎ =i g, (2.3.7)

where we have used the fact that the radius of the sphere,R= 1, is constant.

Thus given (2.3.6) and (2.3.7) are equal we get 0 = i g. But g 6= 0, thus we have a contradiction.

Remark 2.3.4. The requirement for Stokes’ theorem is generally that the local covering of M be open. By extending both hemispheres to be open spaces containing the equator, we can get open coverings and the proof is the exact same.

Having just proven that there does not exist a gauge potential A ∈ Ω1(S2, ad(P)) efined globally in u(1) and satisfying F = dA, we now claim that there does however exist two local gauge potentials,A+ andA−, defined on some open neighbourhoodsU+=

S2\{(0,0,−1)} ⊂S2 andU−=S2\{(0,0,1)} ⊂S2. This can be seen by defining

A+=

i g

r(z+r)(xdy−ydx) =i g(1−cos(ϕ))dθ, (2.3.8)

A−=

i g

Then by explicit calculation, it is clear dA+ =F |U+ and dA− =F |U−, as required.

Furthermore we observe that on the intersection of their domains, U−∩U+, the two local

1-forms are related by the gauge transformation g(x) =e2giθ. That is,

A+ =A−+e−2i gθde2i gθ

=A−+ 2i g dθ. (2.3.10)

Thus there exists two local 1-forms A+ ∈ Ω1(U+,u(1)) and A− ∈ Ω1(U−,u(1)) which

defines a gauge potential A∈Ω1_(S2_{, ad(P)).}

Now in order for this connection to be well-defined, the gauge transformation must be continuous for all values ofθ. Thus, we require that

e2i g(0) =e2i g(2π)

=⇒ 1 =e4i gπ

=⇒ 4gπ= 2πk, wherek∈_Z (2.3.11) Thus, we get the quantization condition of the magnetic charge, such that for anyk∈Z

g= k

2. (2.3.12)

Furthermore, we note that the integer karises due to the mapping

f :S1 →U(1)∼=S1 θ7→e2i gθ,

where we have taken the domain to be parametrized by θ ∈ [0,2π). Thus, the integer k

corresponds to the equivalence class of f inπ1(S1), and so we getk∈π1(S1)∼=Z.

Remark 2.3.5. From Theorem 1.2.8, we know that the isomorphism classes of U(1)

principal bundles are in bijective correspondence with the elements ofπ1(U(1))∼=Z. Given

the magnetic monopoles are defined completely by the elements of π1(U(1)), we see that

the magnetic monopoles are completely defined by the U(1) principal bundles of the gauge theory. Therefore we see that the Hopf bundle introduced in the preliminaries defines (up to isomorphism) the k= 1 Dirac monopole.

2.3.1 Dirac quantization condition

From the previous section we know that magnetic charge is quantized. This quantization condition is generally introduced in a slightly different form, which is known as the Dirac quantization condition which relates the electric and magnetic charges to each other.

In order to obtain this quantization condition, we assume that there exists an additional field in our theory that is coupled to the monopole with charge e. As a result of this

2.4. NON-ABELIAN MONOPOLES 25