QuantumPhysics.dvi

of course be non-zero. If we had a magnetic pointlike chargegat the originx= 0, then its

magnetic field would be given by

∇·~ B= 4πgδ(3)(x) B=g x

|x|^3

(7.52)

where we usually refer togas themagnetic charge. Of course, classical Maxwell’s equations

do not allow forg 6 = 0, and we conclude that magnetic monopoles cannot exists as solutions

to the classical Maxwell equations.

Dirac proposed that, nonetheless, magnetic monopoles can exist quantum mechanically.

To see this, imagine a magnetic field pointing outward radially, as above, combined with

magnetic flux brought in from∞through an infinitesimally thin solenoid, orDirac string. If

the incoming flux matches the outgoing radial flux, then flux is conserved and the magnetic

field configuration obeys∇·~ B= 0 everywhere.

Dirac’s remarkable observation is that if the magnetic flux 4πgof the solenoid is actually

an integer multiple of the fundamental magnetic flux quantum Φ(0)B = 2π ̄h/e, then the Dirac

string will be unobservable by any charged particle whose charge is an integer multiple ofe.

This gives the famous Dirac quantization condition on the magnetic chargeg,

4 πg=nΦ(0)B ⇔ g=n

̄h

2 e

(7.53)

It is not too hard to write down the required gauge potential assuming that the Dirac string

is either along the positive z-axis (corresponding to potentialA−which is regular in the

lower hemisphere) or the negativez-axis (corresponding to potentialA+which is regular in

the lower hemisphere),

A±=−g

cosθ∓ 1

rsinθ

nφ (7.54)

in the usual spherical coordinatesr,θ,φ, andnφis the unit vector tangent to the direction

ofφvariation. The difference between these two vector potentials is given by

A+−A−=

2 g

rsinθ

nφ (7.55)

Recalling the expression for the gradient in spherical coordinates,

∇~Λ =∂Λ

∂r

nr+

1

r

∂Λ

∂θ

nθ+

1

rsinθ

∂Λ

∂φ

nφ (7.56)

we see that the difference is actually the gradient of the angleφ,

A+−A−=∇~Λ Λ = 2gφ (7.57)