Charge Quantization&

Magnetic MonopolesArpan Saha

Sophomore UndergraduateDepartment of Physics

IIT Bombay

Student Technical Activities BodyThe 2nd Annual Seminar Weekend

March 19, 2011

It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress.

– Paul Adrien Maurice Dirac

Paul Dirac(1902–1984)

In his 1931 paper, ‘Quantised singularities in the electromagnetic field’, Dirac provided an elegant argument for how the quantization of electric charge can be accounted for by the presence of magnetic monopoles. We shall be seeing a variant of this today.

An Outline of our Argument

Our argument shall take the following course:• Manifestly covariant electrodynamics• Gauge transformations• Bringing in the wavefunctions• Gauge transformations revisited• Dirac phase • Circle or line?• Capturing the monopole

A List of Conventions

We adopt the following conventions:• Units chosen so that c = ħ = 1• Permittivity and permeability are subsumed into charge

density and current density.• Vector and tensor components identified by putting an index

as superscript or subscript• Latin indices run from 1 to 3 (three spatial dimensions).• Greek indices run from 0 to 3 ( time + 3 spatial dimensions).• Einstein Summation Convention: When an index occurs twice

(once as subscript and once as superscript) in a single term, it is summed over.

• 0 := /x0 = /t, 1 := /x1, 2 := /x2 and 3 := /x3

Preliminaries

The permutation symbol or Levi-Civita tensor is defined by

0123 = 1

and that the tensor switches sign whenever two indices are exchanged.In particular, it is zero whenever an index is repeated (no summation convention).

Preliminaries

The Minkowski metric is given by the matrix

–1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

Manifestly Covariant Electrodynamics

We are going to slightly rephrase Maxwell’s Equations (in absence of any monopole) in terms of the 4-potential and the field strength tensor. The resulting equations are more elegant and

simpler to work with.

Maxwell’s Equations

E = (1) E = – B/t (2) B = 0 (3) B = E/t + j (4)

E is the electric field, B is the magnetic field, is the volume charge density and j is the surface current density.

Magnetic Vector Potential

Since we have B = 0

We letB = A (5)

where A is an ordinary 3-vector. A is called the magnetic vector potential.

Scalar Potential

We substitute (5) into (2) to obtain E = – ( A)/t

i.e. (E + A/t) = 0So we let

E + A/t = – (6) is called the scalar potential.

4-Potential and 4-Current

The 3-vector A along with – as the zeroth (i.e. time) component forms a 4-vector, whose components we denote by A. This is called the 4-potential.

The 3-vector j along with as the zeroth (i.e. time) component forms a 4-vector, whose components we denote by j. This is called the 4-current.

Field Strength Tensor

This is defined byF = A – A

The full matrix is thus

0 –E1 –E2 –E3

E1 0 B3 –B2

E2 –B3 0 B1

E3 B2 –B1 0

Dual Field Strength Tensor

This is defined by(F) = –(1/2)F

where is the permutation symbol and is the Minkowski metric. The full matrix is thus

0 –B1 –B2 –B3

B1 0 E3 –E2

B2 –E3 0 E1

B3 E2 –E1 0

Electrodynamics Rephrased

Maxwell’s Equations become F = j

(F) = 0

Meanwhile, the Lorentz force is given bydp/d = qFu

where p is the 4-momentum, is the proper time, q is the charge of the particle and u is the 4-velocity.

Gauge Transformations

Only the field strength tensor directly enters Maxwell’s Equation. Hence, there is more than one possible

choice of the 4-potential that describes a given physical situation. Transformations taking us from one

such 4-potential to another are called gauge transformations.

Gauge Transformations

The transformed 4-potential must yield the same field strength tensor as the original 4-potential.

Consider a smooth scalar function of time and space (x).

Claim: The transformation A A’ where

A’ = A +

is a gauge transformation.

Gauge Transformations

Proof: The transformed field strength tensor F’ is given by

F’ = A’ – A’

= (A + ) – (A + )

= A – A

= F

The transformation A A + is, therefore, a gauge transformation.

Bringing in the Wavefunctions

The 4-potential shall be incorporated into dynamical equations describing the time-evolution of the

wavefunction of a particle. This shall then describe the dynamics of the particle under

influence of an electromagnetic field.

Quantizing Electrodynamics

We start with the nonrelativistic Schrödinger Equation for a particle of mass m in free space.

(–2/2m – i/t) = 0The equation for a particle of spin zero and

charge q subjected to vector potential A and scalar potential is

(–( – iqA)2/2m – i(/t + iq)) = 0

Quantizing Electrodynamics

When spin is nonzero (say ½), or the nonrelativistic approximation breaks down, we have to turn to more complicated equations such as the Pauli Equation and the Dirac Equation.

But in all of them, space and time derivatives occur in the combination ( – iqA) and (/t + iq).

That is, as per our convention, the derivatives occur in the combination ( – iqA).

Gauge Transformations Revisited

The fact that gauge transformations leave a physical situation unchanged must carry over to quantum mechanics. However, it turns out that the solution

to the dynamical equation in question changes with a gauge transformation. But the change

constitutes only multiplication by a phase factor.

Transformation of Wavefunctions

The general dynamical equation may be stated asf( – iqA) = 0

For some function f. We make the following claim.Claim: Given that under identical boundary conditions

f( – iqA) = 0

f( – iqA’)’ = 0

and that A’ = A +

then ’ = eiq

Transformation of Wavefunctions

Proof: To see this, all we need to consider is the effect of the operator ( – iqA’) on eiq.

( – iqA’)eiq = (eiq) – iqA’eiq

= iq()eiq + eiq

– iq(A + )eiq

= eiq( – iqA)

Hence, f( – iqA’)eiq = eiqf( – iqA)

= 0

Transformation of Wavefunctions

This really shouldn’t be surprising, as it is the modulus squared of the wavefunction that has physical significance.

Since ||2 =|’|2, gauge transformations do not change the physical state of a given system.

Dirac Phase

How do we define the notion of phase parallelism of two wavefunctions at a point in spacetime x?

How do we define the same for two wavefunctions at two different points in

spacetime x and x + dx so that it is invariant over gauge transformations?

Wavefunctions at a Point

Two wavefunctions (1) and (2) are said to be phase parallel (i.e. in phase) at x if

(1)(x) = (2)(x)

Does the same idea work for wavefunctions at two points?

No. Why?

Wavefunctions at Two Points

If (1)(x) = (2)(x + dx), then after a gauge transformation A A + , we have

exp(iq(x))(1)(x)

exp(iq(x + dx))(2)(x + dx)

The problem is easily rectified by stipulating that (1) at x and (2) at x + dx are phase parallel iff

exp(iqAdx)(1)(x) = (2)(x + dx)

Wavefunctions at Two Points

As a result, when a gauge transformation A A + is performed,

exp(iqdx) gets multiplied on both sides.

This can be seen as a rule for parallel transporting a wavefunction.

Around Loops

For finitely separated points P and Q joined by a path in spacetime, we replace iqAdx by an integral (iqAdx).

This, in general, is path-dependent, so that integrating around a loopC will give a nonzero phase q where

= C Adx

The phase q is called the Dirac phase.

Around Loops

But all this is applicable to only the case wherein magnetic monopoles are absent. If they were indeed present, we couldn’t have defined any 4-potential.

Fortunately, Stokes’ Theorem offers a way out.

Around Loops

By Stokes’ theoremC Adx = S (A – A)dxdx

= S Fdxdx

Where S is a surface bounded by the loopC.

We redefine as S Fdxdx.

Circle or Line?

The parameter may take on any real value but as it enters the dynamical equations as e–iq, 0 and 2/q

may be identified. Hence, the parameter space (i.e. the gauge group) is a circle. What happens

when there are multiple charges?

Quantized Charges

Consider some charges q(n).

Case (i): They are quantized i.e. there is a maximum quantum of charge q of which all charges are multiples.

Let q(n) = a(n)q where a(n) are integers.

We see that = 2/q may be identified with zero as exp(iq(n)(2/q)) = exp(ia(n)(2)) = 1 n

So, the gauge group is a circle.

Non-quantized Charges

Case (ii): They are not quantized i.e. there is no maximum quantum of charge q of which all charges are multiples.

Hence there is no charge q such thatexp(iq(n)(2/q)) = 1 n

So, gauge group is the real line.We thus see that whether charges are quantized or

not is given by whether the gauge group is a circle or a line.

Capturing the Monopole

Here we finally bring together all the strands and establish a connection between charge

quantization and the existence of magnetic monopoles.

Capturing the Monopole

Consider a closed surface S. We can think of it as being swept out by a family of loopsCt passing through a point P and parametrized by a real number t.

The loops run from a point loop at P to point loop at P.

P

SCt

Capturing the Monopole

We compute for every loop. It depends continuously on t, so it traces

a continuous curve in the gauge group as t varies.

Since = 0 for a point loop, the curve starts and ends at 0 i.e. it is closed.

Capturing the Monopole

What happens if you shrink the surface S to a point? The closed curve in the gauge group should collapse to a point as well.

Is this always possible?

Gauge group: Line Gauge group: Circle

Capturing the Monopole

If the gauge group is a line, this is always possible.

But, if the gauge group is a circle, this may not be always possible.

In the latter case, something blocks us from collapsing the surface S to a point.

That something is a magnetic monopole.How do we see this?

Capturing the Monopole

For any loopCt, there is some ambiguity in choosing the surface bounded by the loop.

In figure, for surfaces S (1) and S (2), we have

(1) = S (1)Fdxdx

(2) = S (2)Fdxdx

S (1)

S (2)

The Monopole Captured

The net outward flux is given by (1) – (2).

If the closed curve in the gauge group can be collapsed to a point, the above becomes zero.

If it makes a full circle around the circle gauge group, because we identified 0 and 2/q, we have

(1) – (2) = (2/q)N

Where N is an integer.

Dirac Quantization Condition

Hence, we conclude there is a magnetic monopole inside whose magnetic charge, being proportional to the outward flux, is quantized in units inversely proportional to q.

This is the celebrated Dirac quantization condition.

The existence of magnetic monopoles thus provides a natural and elegant explanation for why the gauge group for electromagnetic interaction must be a circle and so why electric charge must be quantized.

A better explanation has so far eluded us. As has experimental evidence of a magnetic monopole.

The search for them meanwhile continues.

References· Aitchison, Hey, Gauge Theories in Particle Physics 3rd

Edn., Institute of Physics· Baiz, Muniain, Gauge Fields, Knots and Gravity,

World Scientific (1994)· Chan, Tsou, Some Elementary Gauge Theory

Concepts, World Scientific (1993)· Dirac, ‘Quantised Singularities in the Electromagnetic

Field’, Proceedings of the Royal Society of London: Series A, Vol. 133, No. 821, Sept. 1, 1931

· Kaku, Quantum Field Theory: A Modern Introduction, Oxford University Press (1993)

Thank you!