Department of Physics & Astronomy

Division of Theoretical Physics

Electromagnetic Duality inSO(3) Yang-Mills Theory

Bachelor Thesis

Author : Jim LundinSupervisor : Lorenzo RuggeriSubject Reader : Guido Festuccia2018-06-21

Abstract

We introduce the historical context and motivation for the search for magnetic monopolesor monopole-like objects.

Beginning the theoretical part we investigate the properties of groups as they relateto symmetries in physical theories. Using this as a basis we investigate the requirementsfor global and local gauge invariance for a scalar �eld, the latter giving the non-trivialconnection to a gauge �eld. From this we present the Georgi-Glashow model and developits particle spectrum using the connected Higgs �eld and its associated Higgs mechanism.

We then present the electromagnetic duality by extending the Maxwell's equations toinclude magnetic sources. Using the assumption of magnetic sources we present the Diracquantization condition, motivating the quantization of electric charge.

Returning to our model we present the 't Hooft-Polyakov ansatz and investigate itsde�ning properties as a �nite energy soliton in our Higgs �eld. We show the magneticproperties and motivate its validity as a monopole like object.

Continuing we de�ne BPS-states on the lower bound for the mass of a monopole likeobject with magnetic and electric charge. Giving a BPS monopole as a solution in the veinof 't Hooft and Polyakov.

Returning to the electromagnetic duality we propose the Montonen-Olive conjectureby exchanging massive vector bosons in our model with the BPS monopoles we devel-oped. We shortly comment on evident problems and present supersymmetry as a possiblesolution. Finally we present the Witten E�ect by allowing a CP violating term in our La-grangian. From this we extend the Montonen-Olive conjecture to include invariance underthe SL(2,Z) group.

Sammanfattning

Vi introducerar det historiska sammanhanget och motiverar eftersökningen av magnetiskamonopoler eller monopol liknande objekt.

I början av den teoretiska delen undersöker vi grupper och deras egenskaper i relationtill deras roll som symmetrier i fysiska teorier. Med detta som bas undersöker vi kraven förglobal och lokal gauge invarians, vilket i det andra fallet ger oss den icke triviala kopplingentill ett gauge fält. Utifrån detta presenterar vi Georgi-Glashow modellen och utvecklar desspartikel spektrum ifrån det kopplade Higgsfältet och dess Higgsmekanism.

Fortsättningsvis så presenterar vi den elektromagnetiska dualiteten igenom en expan-sion till Maxwells ekvationer så att dessa innehåller magnetiska laddningar. Under antagan-det av en magnetisk laddning så presenterar vi Diracs kvantiseringskrav, vilket motiverarkvantiseringen av elektrisk laddning.

Vi återvänder till våran modell och presenterar 't Hooft-Polyakov ansatsen, vars egen-skaper vi undersöker i dess mån av en ändlig energi soliton i vårt Higgsfält. Vi visar demmagnetiska egenskaperna och motiverar ansatsens som ett monopol liknande objekt.

Fortsättningsvis så de�nierar vi BPS tillstånd på den lägre gränsen för massan av ettmonopol liknande objekt med magnetisk och elektrisk laddning. Detta ger oss en BPSmonopol som en lösning följande 't Hooft och Polyakov.

Vi återvänder till den elektromagnetiska dualiteten och föreslår Montonen och Olivesförmodan där vi utbyter de massiva vektor bosonerna i vår modell med BPS monopolernavi utvecklat. Vi kommenterar kort på dem uppenbara problemen med vår förmodan ochföreslår supersymmetri som en möjlig lösning.

Slutligen så presenterar vi Witten e�ekten igenom att tillåta en CP brytande term ivår Lagrangian. Utifrån detta expanderar vi Montonen-Olives förmodan till att inkluderainvarians under SL(2,Z) gruppen.

Contents

1 Introduction & Context 1

2 Group Theory 2

2.1 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Classi�cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Groups As Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Gauge Theory 4

3.1 U(1) Global Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Gauge Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 U(1) Local Gauge Connection . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 SU(n) Local Gauge Connection . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Georgi-Glashow Model 8

4.1 Representations and SO(3)/SU(2) Gauge Theory . . . . . . . . . . . . . . . 84.2 Higgs Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2.1 Higgs Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2.2 Vacuum Con�guration . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2.4 Particle-Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Electromagnetic Duality 11

5.1 Maxwell's Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 Dirac Quantization Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.2.1 Dirac String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6 The 't Hooft-Polyakov Monopole 13

6.1 Finite-Energy Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 The 't Hooft-Polyakov Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6.2.1 Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2.4 Asymptotic Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6.3 Topological Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7 Bogomol'nyi-Prasad-Sommer�eld Monopoles 16

7.1 Bogomol'nyi Bound Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 167.2 Saturating the Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8 Montonen-Olive Conjecture 18

9 The Witten E�ect 19

9.1 Dyons and Gauge Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.2 Charge Conjugation-Parity Violation . . . . . . . . . . . . . . . . . . . . . . 20

10 SL(2,Z) Duality 21

11 Conclusion & Discussion 24

References 25

1 Introduction & Context

Dating back to the formulation of classical Electromagnetism by Maxwell our theory ofelectricity and magnetism has implicitly excluded the possibility of magnetic monopolesdue to no empirical evidence suggesting their existence. This theory none the less provideintuitive possibilities for a symmetrical treatment of our electric and magnetic forces invacuo. This symmetry was then generalized to include sources. This development however,necessitated the use of magnetic sources, contradicting the empirical consensus. Furtherresearch due to Dirac [1], showing quantization of the electric charge to be implicit, shouldone �nd a magnetic monopole. This was named the Dirac monopole and provided amotivation for the existence of magnetic sources by virtue of explaining the still as oftoday unexplained quantization of electric charge.

With this motivation in mind the symmetry of electric and magnetic forces forms thebasis for an interesting �eld of research dubbed Electromagnetic duality by virtue of cre-ating a theory, dual, to our original. Substantial results towards this symmetry appearedas, due to 't Hooft [2] and Polyakov [3], topologically non trivial solutions in non-abeliangauge theories provided a possible candidate for the required magnetic source, turning theattention towards theories admitting these topological solutions. This speci�c realisationis dubbed the 't Hooft-Polyakov monopole which also solves many of the problems emer-gent for the Dirac monopole and simultaneously connects our symmetry to the concept ofYang-Mills theories which reins as the current strongest framework of our physical reality.Furthermore we discover that the dual theories have inverse couplings to the gauge �eldsand as such provide a possible connection from a strongly coupled theory to its weaklycoupled dual theory, granting a framework for perturbative expansion in the dual theory,the results of which can be translated back to our original theory. Granting us in total apossible solutions to a strongly coupled theory, a very tantalizing idea. As of yet no realworld physical theory admits the necessary symmetries nor the topological solutions.

With the possible mass of the magnetic monopole being a free parameter in the Diracformulation and therefore lacking in regards to the possible physical realisations of this.This is however one of the problems solved by the 't Hooft-Polyakov monopole as it isrelated to solitons in the vacuo of the theory, and thereby lets us determine the mass of apossible magnetic monopole by a di�erential equation. Prominent research on this is dueto Bogomol'nyi [4], Prasad and Sommer�eld [5], the �rst of whom found an analytical lowerbound to the possible mass of our monopoles, the Bogomol'nyi bound, and the solutionswhich saturates this bound as particle states, BPS-monopoles.

Now based on this we lead to a greater generalisation in terms of duality conjectures.The example for which we will work through in this thesis will lead us to an example ofa duality conjecture which encapsulates the emergent concepts as presented here. For ourpresent case we motivate a conjecture due to Montonen and Olive[6], the Montonen-Olive

conjecture which postulates the aforementioned dual theory formulation wherein the dualgauge theory exchanges the massive vector bosons of our theory with the BPS-monopolesin addition to the regular electromagnetic duality transform.

In this thesis we will be investigating the foundation to group theory and its relevancefor generating gauge symmetries. From this we lay down the groundwork to a gauge theorywhich admits the necessary topological solutions by virtue of a relatively simple example inthe SO(3) Yang-Mills gauge theory. Following this we present the electromagnetic dualityand derive the Dirac monopole and formulate the Dirac quantization condition. Fromthis the 't Hooft-Polyakov monopole is derived and the appropriate considerations areperformed to saturate the Bogomol'nyi bound. Finally this leads to the postulation of theMontonen-Olive conjecture, which we then �nally expand by introducing the Witten e�ect

and demonstrating the emergent SL(2,Z) invariance. Our approach to these subjects willfollow in the steps of Peskin and Schroeder [7], as well as Figueroa [8].

1

2 Group Theory

2.1 Lie Algebras

We will here forego the more mathematical and technical de�nitions of groups and discussthem in a context relative to the theories we will be investigating in the following sections[7].

In this case we ultimately want an operator which obeys unitarity and generates oursymmetry and as such we limit ourselves to groups which follows these same requirements.For a continuous symmetry we look at in�nitesimal shifts from unity, which allows thegeneration of arbitrary shifts trough successive applications. We de�ne an element in ourgroup as the in�nitesimal generator of some symmetry which we would like our theory toexhibit.

g(α) = 1 + iαata

With αa as our in�nitesimal parameter with an index for a complete set of in�nitesimalgenerators in our symmetry group. Given that the parameter is in�nitesimal we can easilysee that this follows the group composition rule - i.e. that two elements applied successivelyresults in another element.

g(α1)g(α2) = (1 + iαa1ta) (1 + iαb2tb) = 1 + i(αa1 + αa2)ta +O(α1α2) ≈ g(α1 + α2)

The above approximation becomes exact in the limit of α1, α2 → 0. We can also letthis de�nition be applied to a �nite parameter β and generate the result. To do as suchwe relate the �nite parameter to the in�nitesimal as a limit by β

n = α.

limn→∞

n∏i=1

g(αi) = g

limn→∞

n∑i=1

αi

= g(β) = limn→∞

(1 + i

βatan

)n= eiβ

ata

Above we used the group composition as we presented earlier for an in�nitesimal pa-rameter, we see that the result is valid for a �nite parameter as well and it comes in theform of a complex valued exponential factor. However, in doing this we have somewhatdisregarded the conditions we require on the coe�cients of our parameter.

For the set of in�nitesimal generators of our symmetry group we require them to behermitian and span the full space of possible in�nitesimal transformations, for our case ofa continuous group this is called a �Lie Group�. For our group composition relation to bevalid we also require the generators to obey the commutation relation as:

[ta, tb] = if cab tc

The vector space spanned by these generators is called a �Lie Algebra�.

2.2 Classifications

If there exists a generator in our set which commutes with all of the remaining generatorswe can from knowledge of the form of our �nite transformation element expand this.

g(β) = eiβ1t1eiβ

ata

Where the generator t1 commutes with all the remaining generators as indexed by a.This generator which commutes and can be isolated is the U(1) symmetry group - we canalways de�ne our generators such that the commuting generator is proportional to identity- and it encompasses the familiar complex phase transformations.

gU(1)(β) = eiβ

2

We dub theories generated exclusively by such a commutative group as �abelian� andtheories which generated by any non-commuting groups as �non-abelian�. The groups ofthese non-abelian theories are classi�ed as �semi-simple� if it contains no U(1) elements and�simple� if it cannot be decomposed into two or more sets of mutually commuting groups.Using knowledge of the irreducible representations for our possible groups we can produceall possible groups as direct products of our irreducible representations. One possiblefamily of groups which satisfy our requirements are the special unitary transformations inn dimensions - SU(n) - where n is the number of �internal� dimensions of our symmetrygroup. This is one choice of possible representations, we can also choose all groups which areisomorphic to the SU(n) groups (for example U(1) ∼= SO(2) as an accidental isomorphismand SO(3) as the double cover of SU(2)) and acquire the the same theories, something wewill discuss more speci�cally for the SU(2) ∼= SO(3) case in a later section.

2.3 Groups As Symmetries

Having shown how our de�nition of groups creates an action which generates some typeof structure which is represented by the generators of the group of which we would like toact. So when we let our group act on an object we �rstly require the object to exist in thesame vector space as the n×n matrix representations of our generators. We also implicitlyrequire our objects to be complex-valued, but this can always be reduced to a real-valuedcase, with some targeted e�ort if it is not trivial in context.

But to generate a symmetry we need to identify the inverse relations between represen-tations of our group. A lone object transforming under a group action will quite obviouslynot exhibit any symmetry for a group element di�erent from identity.

ψ −−→g(α)

(1 + iαata)ψ 6= ψ

But none of the physically relevant quantities we want to investigate come as singleobjects in some representation. Rather, since the quantities that are relevant for thedescription of a physical model - in this case the energy of the system - are all representedby scalars. So for a relevant quantity we need inner products of vectors in our vector space.The usual inner product we prescribe for our vector space is 〈ψ|χ〉 = ψ†χ =

∑ni=1 ψ

∗i χi

where the † notation represents the adjoint of our vector. From this we suspect that theadjoint vector does not transform with the same group representation as we have used upto this point. But the remedy is quite intuitive. We let the adjoint vectors transform bygroups in the adjoint representation.

g†(α) = (1 + iαata)† = 1− iαat†a = 1− iαata

And in the last step we have used our knowledge of the de�nition of our generators ofour Lie Algebra which we know to be hermitian, or rather, t†a = ta. We let this act on ouradjoint vector.

ψ† −−−→g†(α)

ψ†(1− αata)

So if we now look at the transformation of an inner product:

〈ψ|χ〉 −−−−−−−−→g(αψ), g(αχ)

ψ†(1− iαaψta)(1 + iαbχtb)χ = ψ†χ+ iψ†(αbχtb − αaψta)χ+O(αψαχ)

So we now see the clear implications for a possible symmetry as generated by our groupaction. If we were to set this inner product as the norm of a vector 〈ψ|ψ〉 = ψ†ψ = |ψ|2which implies αψ = αχ and we explicitly see that the cross term falls out and leave theidentity term. This is the hallmark of a symmetry of a system and it is under this frameworkwhich we will study some impacts and implications of these symmetries as a grounds forfurther - �grander� - symmetries between physical theories.

3

3 Gauge Theory

3.1 U(1) Global Phase

Taking inspiration from the previous section where we presented a group which transformsas a phase factor in the U(1) example, we can investigate theories which are invariantunder such transforms. To do this we observe the behaviour of a general complex scalarLagrangian under a global phase change β. We also implicitly employ the mostly minusmetric diag

(ηµν)

= (+,−,−,−).

φ→ eiβφ φ† → φ†e−iβ

L = ∂µφ†∂µφ+ V (φ†φ)→ ∂µφ

†e−iβeiβ∂µφ+ V (φ†e−iβeiβφ) = L

So as can be seen for a global phase shift our regular Lagrangian is indeed invariant ifwe limit all of the terms which we allow in our potential to the forms of (φ†φ)2 and lowerexponents, which is indeed a known prerequisite for a number of reasons pertaining to -amongst other things - renormalization. Other terms would explicitly break this symmetry,which represents a rotation in the real and imaginary components of our complex �eld. Asort of �internal� rotary invariance. This is however a quite physically trivial example, asthis does not impact the dynamics of our theory and we have yet to motivate why this isa relevant development.

3.2 Gauge Field

Following from earlier we now inspect the the properties of gauge �elds. A gauge �eld ispresented as a four-vector which has un-physical degrees of freedom as given by possiblegauge transformations of the �eld.

Aµ(x)→ Aµ(x) + ∂µχ(x)

Where χ(x) is more commonly found as ∂µχ(x) = (∂f∂t ,~∂f) for a smooth - twice di�er-

entiable - function f and Aµ is in relation to classical electromagnetism the four-vector of

the electric potential (φ) and the vector potential ( ~A) as Aµ = (φ, ~A).To conform with our Lagrangian formulation we look for gauge invariant Lorentz-scalars

for our gauge �eld. An intuitive ansatz for a gauge invariant term is the well know �eldtensor.

Fµν = ∂µAν − ∂νAµ → ∂µ (Aν + ∂νχ)− ∂ν(Aµ + ∂µχ

)= ∂µAν − ∂νAµ

Since the derivative clearly commutes the result is straitght forward. For the Lorentz-scalar we contract the �eld tensor and observe the emergent terms.

FµνFµν =

(∂µAν − ∂νAµ

)(∂µAν − ∂νAµ) =

= ∂µAν∂µAν + ∂νAµ∂

νAµ − ∂µAν∂νAµ − ∂νAµ∂µAν =

= 2(∂µAν∂

µAν − ∂µAν∂νAµ)

We have simpli�ed by the nature of exchanging the space-time indices. This clearlytakes the form of a kinetic term, and as such we have an essential part of the gauge �eldcomponent of our theory. The remaining term takes care of the gauge invariance of ourtheory and will be determined by gauge �xing when one makes calculations.

4

3.3 U(1) Local Gauge Connection

To lay the foundation to our theory of interest we �rst consider a scalar �eld upon which wewant to introduce a local phase invariance and quickly realise that the standard Lagrangianformulation for a scalar �eld does not allow for this. Our kinetic term (∂µφ

†∂µφ) does notaccount for the new structure we have imposed and some correction needs to be introduced.We now let our scalar �eld transform under a local phase transformation and observe theresulting complication.

φ→ eiχφ φ† → φ†e−iχ

∂µφ†∂µφ→ ∂µ

(φ†e−iχ

)∂µ(eiχφ

)= (∂µ − i∂µχ)φ† (∂µ + i∂µχ)φ

So here we pick up new terms that do not vanish as desired. We can however observethat the unwanted terms seems to be of the type which appears under a gauge transfor-mation of a gauge �eld Aµ → Aµ − 1

g∂µχ. So if we now include this gauge �eld in a newde�nition for our �gauge derivative� Dµ = ∂µ + igAµ (called covariant derivative) we canacquire a theory that is truly invariant under local gauge transformations. We see thisthus:

Dµφ→(∂µ + igAµ − i∂µχ

) (eiχφ

)= eiχDµφ(

Dµφ)†

(Dµφ)→(eiχDµφ

)† (eiχDµφ

)=(Dµφ

)†e−iχeiχ (Dµφ) =

(Dµφ

)†(Dµφ)

Above we have taken a shortcut when calculating the second term and acquired it bycorollary to the term we explicitly calculated. The result from this is a gauge connectionbetween a scalar �eld φ and a gauge �eld Aµ. From this we now want to motivate a �eldtensor and four-potential in correlation to the ones found earlier in this section. To do thiswe investigate the structure of our gauge connection as:[Dµ, Dν

]=[∂µ, ∂ν

]+ ig

([∂µ, Aν

]−[∂ν , Aµ

])− g2

[Aµ, Aν

]= ig

(∂µAν − ∂νAµ

)= igFµν

Above we have solved by virtue of having the commutative property of our gauge �eldand derivatives in general. From this we see that we have recovered a possible �eld tensor,directly from the geometrical structure of our gauge group which we left implicit as complexvalued numbers but is in reality 1 × 1 complex valued matrices, a nuance we will expandupon soon.

We now want to investigate the �eld tensor as given in electromagnetism for comparison.To do this we need to construct a �eld four-potential. By direct relation to the potentialsas we have presented them in the context of electromagnetism earlier. We have the scalarpotential φ and the vector potential ~A. These are straightforwardly combined to produceour four-potential Aµ = (φ, ~A). From this we can - by comparison to above - produce our�eld tensor quite simply as Fµν = ∂µAν − ∂νAµ which is the gauge invariant kinetic termfor our gauge �eld in correspondence with the rest of our theory. This is exactly that ofwhich we calculated for the gauge �eld previously and it matches the calculation where weinspected the curvature of our new geometry. By correlation to previous calculations thisgives us the free Lagrangian for our theory as:

L = −1

4FµνF

µν + (Dµφ)† (Dµφ)

This case will be referred to as a U(1) gauge theory for future reference and it embodiesthe structure of our well known electromagnetism, which is something we would like torecover later in our calculations for parallels to the earlier sections of this thesis. In itspresent form it is an incomplete version of Scalar-Electrodynamics, a well known prototypemodel for regular Electrodynamics, wherein the gauge �eld is coupled to a scalar - (spin-0).particle instead of a fermion - (spin-1

2) - particle.

5

3.4 SU(n) Local Gauge Connection

We now want to expand to a more complicated structure than the straight forward notionfor complex numbers we used above in the case of the U(1) gauge. The connection betweenthe U(1) group and the group of complex phase rotations - SO(2) - is obvious and willserve us when we make a parallel between representations further ahead. The most intuitiveexpansion is to that of the SU(n) groups, for each of which we can �nd a suitable set ofbase matrices ta. We require the base matrices to ful�l the earlier proposed commutationrelation for a Lie Algebra:

[ta, tb] = if cab tc

Where our structure constant fabc will emerge to be antisymmetric (we manipulate theindices so as to conform to our summation convention). We now want to form a gaugeconnection in an similar way as we have done above. Firstly we introduce a local phasetransformation, now with a noticeable connection to out gauge group and a new �n-plet�representation for our scalar �eld φ.

φb → eiαataφb φ†b → φ†be

−iαata φb =

φ1...φn

Here also we make a short comment reminding that the group indices are summed over

in a similar manner to the space-time indices. Our (semi)-arbitrary space time function αais represented in terms of its components with respect to our basis. This n-plet represen-tation of our scalar �eld has matching dimension to that of our gauge group and as suchits representation also matches that of our gauge group, acquiring a group index b.

Now knowing the local transformation we make a slight leap in order to acquire ourcovariant derivative by direct parallel to what we did earlier and connecting our scalar �eldto the gauge �eld W a

µ :

Dµ = ∂µ − igW aµ ta

From this we once again look to �nd a �eld tensor, so we investigate the commutatorof two covariant derivatives:[

Dµ, Dν

]=[∂µ, ∂ν

]+ ig

([∂ν ,W

aµ ta]− [∂µ,W

aν ta]

)− g2

[W bµtb,W

cν tc

]=

=− ig(∂µW

aν − ∂νW a

ν + gfabcWbµW

cν

)ta = −igGaµνta

Although we have taken some liberty with group indices the above calculations are onlya retread of the previous calculation for U(1) and one additional term which was vanishingin the previous example, one which we easily calculate from the commutator for our groupelements.

Ultimately we have now acquired a general �eld strength for our gauge theory

Gaµν = ∂µWaν − ∂νW a

ν + gfabcWbµW

cν

A �nal note on our connection is the covariant derivative acting upon our scalar �eldand a reformulation via the structure constant.

Dµφa = ∂µφ

a − igW bµtbφ

a = ∂µφa + gfabcW

bµφ

c

Which we will use for our central theory.

6

3.5 Symmetry Breaking

Now that we have laid the foundation for our gauge symmetries it is time to observe theinteresting behaviour that appears for a certain form of potential. Inspecting the La-grangians of our theories we know that the allowed potential terms are invariant under ourgauge transformations. We have shown this explicitly for the U(1) case and for consistencywe will now bring the general SU(n) case up to parity.

φ→ eiαata φ† → φ†e−iα

ata φ†φ→ φ†e−iαataeiα

ataφ = φ†φ = |φ|2

An, albeit, trivial result when considering the requirements we have proposed in thesection on group theory. When we discuss the topic of symmetry breaking it is crucial topoint out the symmetries which are in truth broken. The mere statement of this sectionis very much ambiguous as we ask the question of where we can �nd any such brokensymmetries in a theory which consists solely of terms which all obey the gauge symmetrieswhich we have investigated. Or rather, our potential takes the form (the relevant symmetrygroup is left ambiguous as the following discussion is systematically identical for all groups):

V (φ†φ) = a2 + b|φ|2+c2(|φ|2)2

All terms in this potential do indeed obey the gauge invariance we require and anyterms not following the pattern of the above polynomial will indeed explicitly break thesymmetry. However realising some of the parameters in our polynomial are physicallyarbitrary as any constant shift to our �eld and potential would leave the equations ofmotion invariant and we unable to measure any such shift. We rewrite as:

V (|φ|2) = (a+ c|φ|2)2 2ac = b

As there is no explicit broken symmetry we now turn our attention to other physicallyrelevant ideas. Something intrinsically tied to our potential is the de�nition of our vacuum,or rather, our background. For the above presented potential, the background should berepresented by the minimum possible energy con�guration. For c as a positive real numberwe know that the minimum possible value will be when the �eld value is zero φ†φ = 0 andwe will get the result of a2

c2and as such our vacuum coincides with our �eld approaching

zero. If we instead take the constant c bo be real and negative we see that the potentialenergy remains positive but we now have a new criteria for our �eld at the vacuum (c = −1).

min|φ|2∈R≥0

V (|φ|2) = (a− |φ|2)2 ⇒ |φ|2= a2

So what we have here is a shift in the �eld value corresponding with our vacuumcon�guration. So it seems that the proper remedy would be to rede�ne our �eld as toconform with this new shift since, as we brie�y mentioned, a shift to our �eld should leavethe equations of motion invariant. We insert a new de�nition as φ = a+ ρ.

V (ρ†ρ) = (a2 − |a+ ρ|2) = (|ρ|2)2

However we must not fail to inspect the remaining terms in the Lagrangian as we haveyet one more term that involves the scalar �eld.

(Dµφ)† (Dµφ) = (∂µ − igAµ)(a+ ρ†)(∂µ + igAµ)(a+ ρ) =

= (Dµρ)†(Dµρ) + g2a2AµAµ + igaAµ((Dµρ)† −Dµρ)

And now we have acquired terms that explicitly break our symmetries through a mereshift and rede�nition of our �eld. A mere rede�nition should not break any of our postu-lated symmetries and as such we determine that the symmetry is broken for our chosenvacuum and not the theory as a whole, or rather, our choice of vacuum is not invariantunder the full symmetry group as we have an apparent restriction placed upon it by ourtheory. This will be a useful phenomena in analysis of our theory.

7

4 Georgi-Glashow Model

4.1 Representations and SO(3)/SU(2) Gauge Theory

We now turn our attention to the Lagrangian formulation of SO(3) Yang-Mills theory [9],which we have chosen as a suitable example for results to be produced later.

We have yet to discuss the chosen representation for our theory. In this thesis we haveso far spoken of our speci�c choice as SO(3), this will result in a formulation which allowsintuition of the three-vector structure of space to enlighten certain parts of our model. Tocontinue under this formulation we comment on the nuance here. The SO(3) formulationexempli�es the connection to spatial rotations and the concept of familiar vector identities.

The second representation is SU(2) which in contrast exempli�es the non-commutative- i.e. non-abelian - structure of the gauge group and a possible extrapolation from thecommutative - abelian - example of U(1). The most relevant result here is the modi�ed�eld tensor, now giving us something we will di�erentiate by calling it ��eld strength� (inreality these two are considered one and the same), which is dependent upon the structureof our gauge group, given in a general form for the general four-potential W a

µ where thegroup index a is representative of our representation (discussed in the previous section).

Gaµν = ∂µWaν − ∂νW a

µ + gfabcWbµW

cν

(The structure constant in our case of SU(2) actually is the fully antisymmetric ten-sor εabc) We have introduced the coupling constant g in a non-trivial way. This gives aLagrangian with gauge connection, as shown above:

L = −1

4GaµνG

µνa + (Dµφb)

†(Dµφb)

What remains to be introduced in order to complete our formulation is the introductionof the matrix representation of our group as a form of basis which we will interpret now.

In the case of SU(2) the basis matrices are the 2 × 2 traceless (Tr[ta] = 0), hermitian

(t†a = ta) matrices with unit determinant -1 (det(ta) = −1). A possible choice for sucha basis is the three Pauli matrices σi

2 . We now make a note here, seeing as the basematrices for SU(2) number in three, it is here we make the connection to the group ofthree dimensional rotations. Seeing as we have a basis of three matrices, we can insteadchoose to interpret this as a set of basis vectors in a three dimensional space. As such we canpresent the above gauge �elds as components in a vector

# »

W µ = (W 1µ ,W

2µ ,W

3µ) = W a

µ ta.This could be generalised to vectors in higher dimension for gauge groups with more thanthree base matrices.

We now want to rephrase our �eld strength and covariant derivative in this new vectorrepresentation so we contract our group indices as done above.

Gaµνta =#»

Gµν = ∂µWaν ta − ∂νW a

µ ta + gtafabcW

bµW

cν =∂µ

# »

W ν − ∂ν# »

W µ + g# »

W µ ×# »

W ν

Dµφata = Dµ

#»

φ = ∂µφata − gtafabcW b

µφc =∂µ

#»

φ + g# »

W µ ×#»

φ

And here we have realised the cross product identity by connection to the above state-ment declaring the structure constant for our example to be the Levi-Civita symbol, givingus εijkeiajbk = #»a × #»

b , clearly giving us the result above for our example, and some gen-eralisation for higher dimensional examples.

We also would like to reformulate our scalar �eld to reach unity with our currentrepresentation. It follows straightforwardly that

#»

φ = (φ1, φ2, φ3) = φbtb.From this we now once more reformulate our gauge-scalar connected Lagrangian, now

fully in the SO(3) representation:

L = −1

4

#»

Gµν ·#»

Gµν

+ (Dµ#»

φ)† · (Dµ #»

φ)

8

4.2 Higgs Coupling

4.2.1 Higgs Potential

Previously we built a framework for a theory which is locally invariant under transfor-mations of its associated gauge symmetry group. This necessitated the introduction of agauge �eld which was the focus of the previous section.

We now turn our attention to the scalar �eld which we have connected to the gauge �eldin a non-trivial way. This clearly di�ers from our traditional notion of a free gauge �eldand we now need to determine what our scalar �eld represents. It is here we introducea Higgs-potential which enables us to integrate massive vector bosons into our theory,something that is necessary for a theory which intends to describes reality. Although -somewhat incidentally - this is also the same modi�cation which enables the centrepieceof this thesis.

The Higgs-potential as we introduce it is as follows:

V (φ) = λ(|φ|2 − a2)2

Where we have introduced the new parameters λ, a2 and#»

φ† · #»

φ = |φ|2 which we taketo be non-negative. The implication of these will be explained as they become relevant.

From this we now have our complete Lagrangian for SO(3) Yang-Mills with Higgscoupling.

L = −1

4

#»

Gµν ·#»

Gµν

+ (Dµ#»

φ)† · (Dµ #»

φ)− λ(|φ|2 − a2)2

4.2.2 Vacuum Configuration

The Lagrangian presented above allows us to recover a Hamiltonian formulation for whichwe can more easily motivate some of our impending conclusions.

To do this we work by parallel to the early sections wherein we worked with the electricand magnetic �elds instead of our �eld strength. As such we de�ne the new identities forour Hamiltonian as following:

#»

Ei

= − #»

G0i − 1

2εijk

#»

Gjk

=#»

Bi #»

Π = (D0#»

φ)†#»

Π†

= D0#»

φ

From this we can now present a Hamiltonian for our system:

H =1

2(

#»

Ei ·#»

Ei +#»

Bi ·#»

Bi) +#»

Π · #»

Π†

+ (Di#»

φ)† · (Di#»

φ) + λ(|φ|2 − a2)2

From this we now de�ne a vacuum con�guration to be a solution for which our energydensity - or rather our Hamiltonian - vanishes. For our current formulation this impliesH = 0 for which we require all individual terms to vanish.

#»

Ei =#»

Bi =#»

Π = Di#»

φ = λ(|φ|2 − a2)2 = 0

Although we can see that this criteria can be written more simply if we return to ourLagrangian formulation as:

#»

Gµν = 0 Dµ#»

φ = 0 V (|φ|2) = 0

The latter two equations are what we use to de�ne our Higgs-vacuum. We also notethat for the last equation we limit our Higgs-�eld as |φ|2 = a2. This incidentally breaksour SO(3) symmetry for our vacuum con�guration - we are forced to choose an arbitraryvector for our vacuum - and we are left with a new symmetry for our Higgs-vacuum nowrepresented by SO(2) - as the remaining symmetry is the set of all rotations around theaxis of our chosen vector - which is isomorphic to U(1). And thus we have performed aspontaneous symmetry breaking of our vacuum, the solutions of which we will now studyin greater detail.

9

4.2.3 Equations of Motion

From our Lagrangian we can now produce equations of motion for our �elds which will benecessary for our possible solutions. We will recover these by principles of variation and amodi�ed version of the Euler-Lagrange equations, now using our covariant derivatives:

Dµ

(∂L

∂(Dµ#»

φ)†

)=

∂L

∂#»

φ† ⇒ DµD

µ #»

φ = −2λ(|φ|2 − a2)#»

φ

Dµ

(∂L

∂(Dµ# »

W ν)

)=

∂L∂

# »

W ν

⇒ Dν#»

Gµν

= g#»

φ ×Dµ #»

φ

4.2.4 Particle-Spectrum

To conclude our theory we want to investigate the results of rede�ning our scalar �eldto better follow our vacuum de�nition. We also make the shift to a real scalar �eld soas to match the model we have been following so far, namely the Georgi-Glashow model.

Ultimately this gives us#»

φ =#»

φ†

= #»a +#»

ψ with the requirement #»a · #»a = a2 and weexpress our Lagrangian to match this (making suitable replacements as ψ = 1

a#»a · #»

ψ, and

Aµ = 1a

#»a · # »

Wµ).

L = −1

4

#»

Gµν ·#»

Gµν

+1

2Dµ

#»

φ ·Dµ #»

φ − λ

4

(φ2 − a2

)2=

= · · ·+ 1

2

(Dµ

#»

ψ + g# »

W µ × #»a)·(Dµ #»

ψ + g# »

Wµ × #»a

)− λ

4

(2 #»a · #»

ψ)2

=

= · · ·+ g2

2

(# »

W µ × #»a)·(

# »

Wµ × #»a

)− λa2

(#»a

a· #»

ψ

)2

=

= · · ·+ g2

2

((# »

W µ ·# »

Wµ)a2 −

(#»a · # »

W µ

)(#»a · # »

Wµ))− λa2ψ2 =

= · · ·+ g2a2

2

(A2 + 2W+W− −A2

)− λa2ψ2 = · · ·+ g2a2W+W− − λa2ψ2

From this we can read of the masses of our particles quite comfortably as:

MW± = ga MH = a√

2λ MA = 0

Above we introduced the two orthogonal components to our gauge �eld Aµ as W+µ =

1√2(A+

µ +iA−µ ) andW−µ = 1√2(A+

µ −iA−µ ). For our speci�c model we are limiting the case to

a triplet basis in SO(3) and as such we only have three components to our gauge �eld, thisresult is however easily extrapolated to higher dimensionality, although varying for di�erenttypes of breaking schemes, an example of which we have calculated explicitly above. Inmore colourful language, we call that which happens here - the exchange of three real scalar�elds and three massless gauge �elds for one scalar �eld, one, still, massless gauge �eldand two new massive gauge �elds - the Higgs �eating� two scalar �elds and granting theirdegrees of freedom to the now massive gauge �elds.

Our choice of W±µ above is to facilitate a more satisfactory choice of charge for ourparticles. We can determine their charge by how they couple to the gauge �eld Aµ in thecovariant derivative, rather we see this explicitly for an arbitrary choice of the direction of~a in the z direction:

∇µA±ν = ∂µA±ν + gAµ ×A±ν =

{A+ν =

1√2

(W−ν +W+

ν

), A−ν =

i√2

(W−ν −W+

ν

)}⇒

⇒ ∇µW±ν = ∂µW±ν ± igAµW±ν ∇µψa = ∂µψa + gAµ × ψa = ∂µψa

We can see that the W±µ gauge bosons acquire the photon - Aµ - associated charge ±g,and as our scalar �eld ψ is parallel to the Aµ �eld, it is uncharged with respect to it.

10

5 Electromagnetic Duality

5.1 Maxwell's Equations

In this section we lay the foundation for the electromagnetic duality and as such we must�rst present the context in which it explicitly appears. To do this we begin by investigatingMaxwell's equations in vacuo with Planck units.

#»

∂ · #»

E = 0

#»

∂ × #»

E = −∂#»

B

∂t

#»

∂ · #»

B = 0

#»

∂ × #»

B =∂

#»

E

∂t

As we expect from Maxwell's equations, these are invariant under the Lorentz trans-formations, but in this form we can also conclude these equations to be symmetric underthe exchange:

(#»

E,#»

B)→ (#»

B,− #»

E)

And it is this that we dub as the Electromagnetic duality. We now want to expand thisto the Maxwell's equations including electric sources. We also insert imagined magneticsources to make the required introduced terms explicit:

#»

∂ · #»

E = 4πρe

#»

∂ × #»

E = −4π#»

Jm −∂

#»

B

∂t

#»

∂ · #»

B = 4πρm

#»

∂ × #»

B = 4π#»

J e +∂

#»

E

∂t

With the electric and magnetic sources - as well as their corresponding currents - addedwe can see that the electromagnetic duality is conserved if we also transform the sourcesin a similar fashion to the electric and magnetic �elds as done above:

(ρe, ρm)→ (ρm,−ρe) (#»

J e,#»

Jm)→ (#»

Jm,−#»

J e)

Having observed the duality explicitly in the traditional formulation, we now advanceto the covariant formulation for explicit Lorentz invariance and a more compact and moreclosely related connection to the theoretical framework to come.

We do this by de�ning a �eld tensor Fµν and its dual tensor ?Fµν as:

F 0i = −F i0 = −Ei − 1

2εijkF

jk = Bi?Fµν =

1

2εµνλρFλρ

We also rewrite the sources and currents as four-vector currents:

j0 = ρe ji = J ie k0 = ρm ki = J im

From this we can recover the Maxwell equations with sources but in a convenientcovariant form as:

∂νFµν = 4πjµ ∂ν

?Fµν = 4πkµ

We now want to present the electromagnetic duality in this covariant context. Bystraight correlation to the above presented transformations we �nd that the �eld tensorand the four-vector currents transform as:

Fµν → ?Fµν

jµ → kµ

?Fµν → −Fµν

kµ → −jµ

11

5.2 Dirac Quantization Condition

5.2.1 Dirac String

We now attempt to postulate a magnetic monopole as a point source of magnetic �ux byparallel to an electric monopole. As such we determine the charge of the monopole by its�ux through a closed sphere.

#»

B( #»r ) =g

4πr3#»r g =

∫S2

#»

B · d #»

S

However, when we try and solve for a vector potential as given by this point source weencounter a nuance, separating our seemingly simple parallel to the electric case. We �ndour vector potential for a curl free magnetic �eld (

#»

∂ × #»

B = 0 ∀ x ∈ Rn \ 0) - which isthe case for our point source - as

#»

B =#»

∂ × #»

A. From this we consider a spherical solutionto this equation in the vein of Dirac:

#»

A+( #»r ) =g

4πr

1− cos θ

sin θeφ

#»

A−( #»r ) = − g

4πr

1 + cos θ

sin θeφ

These two solutions satis�es the above stated criteria for our vector potential. However,they are singular along a one dimensional line. In our case

#»

A+ is singular along the negativez-axis, and similarly

#»

A− is singular along the positive z-axis. These singular lines are calledDirac strings. These are unavoidable when looking for solutions to Maxwell's equationsfor a magnetic monopole. Their physical motivation is that of a in�nitesimally thin andin�nitely long solenoid.

From the same motivation of our imagined �eld being curl free we can also considerour vector potential to be the gradient of a scalar potential. We call this function χ andde�ne as mentioned i.e.

#»

A+ −#»

A− =#»

∂χ. This appears strikingly similar to a possiblegauge transformation.

5.2.2 Quantization

We now move to the quantization of a charged particle with mass m and dynamics asdescribed by the gauge invariant Schrödinger equation:

− 1

2m

(#»

∂ + ie#»

A)2ψ = i

∂ψ

∂t

Here e represents the coupling constant and - in our units - the elementary electriccharge. We now let our wave function ψ and our gauge �eld transform under a gaugetransformation.

ψ → e−iqχψ#»

A→ #»

A+#»

∂χ

In this framework it is clear that we can select our local gauge transformation arbitrarilyas we de�ne it locally and let it be patched up for the remainder of the domain, in a sensewe can just make local selections such that our transformation is satisfactory for the entiredomain. We make the choice of θ = π

2 and recover our function as χ = g2πφ. However, for

this to be valid we require the wave function to be single valued, which implores our localphase term to contain all possible periodical solutions with the same value.

This gives us the condition:

e−iqg2πφ = e−inφ ⇒ qg = 2πn for some n ∈ Z

And as such we have shown that in the presence of a magnetic charge our regular gaugeinvariant theory implies that both electric and magnetic charge is quantized for a selectionof n.

12

6 The 't Hooft-Polyakov Monopole

6.1 Finite-Energy Solutions

In our search for monopoles we now turn to �nite energy non-dissipative solutions of ourequations of motion. We take the �nite energy requirement to be necessary as we wouldlike to �nd physically realisable solutions. We impose this as requiring the energy integral-∫d3xH - to be �nite and hence exist. This is possible if our �elds approach a vacuum

con�guration asymptotically. Above we have discussed the coupled Higgs �eld, and if weimpose this criteria we require the Higgs �eld to approach the Higgs vacuum at spatialin�nity. The Higgs potential V as we have presented it is clearly a function taking values ofthe �eld at points in a three dimensional space to real values i.e. V : R3 → R. Re�ectingon our rede�nition of the Higgs �eld in the previous section we know this potential to havea set of solutions for V ( #»x) = 0, we de�ne the subset of our vector space which solvesthis equation as M0 ⊂ R3, which we know from before to be the sphere of radius a. Ifwe connect this to our imposed requirement we can see that our �eld must approach thesubset we de�ned above in the limit of spatial in�nity.

#»

φ∞ (r) ≡ limr→∞

#»

φ( #»r ) ∈M0

We now use the fact that the space of functions from a sphere to a sphere is discon-nected, i.e. if we have a function in one connected component of these functions we cannottransform our function to another function in another connected component, these arecomponents are indexed by something called the �degree� of the map. By this we have thedegree of a constant mapping as zero and the identity mapping as degree one.

In terms relating to our work at hand we de�ne the topological number of our �niteenergy con�guration to be the degree of the map of our �eld at spatial in�nity

#»

φ∞. Thecon�guration we have inspected above has as such the topological number of zero, which isevident if we make an arbitrary choice of direction as

#»

φ = ae3. These topological numberare in addition also invariant under all continuous deformations and is therefore invariantunder time evolution and also gauge transformation as these are connected. By this, any�nite energy �eld con�guration at any point in time that has a topological number di�erentfrom zero will remain non-trivial in time evolution and under gauge transformations andas such it will never dissipate. In some sense we have a stable object which we can considerstatic (if it is time-independent and the time component of our gauge �eld

# »

W µ vanishes).

6.2 The 't Hooft-Polyakov Ansatz

6.2.1 Ansatz

We now limit ourselves to spherically symmetric solutions and investigate whether if anysuch stable solutions exist. For a static solution all kinetic terms in our Hamiltonian dropout and as such our Lagrangian agrees with our energy up to a di�ering sign. From thiswe can conclude that the solutions to our equations of motion all extremise the energy.

And now we propose the 't Hooft-Polyakov ansatz for a monopole (with our couplingconstant now as g = −e which brings us fully in line with the Georgi-Glashow model):

#»

φ( #»r ) =#»r

er2H(aer) W i

a = −ε ia j

rj

er2

(1−K(aer)

)W 0a = 0

For some almost arbitrary functions H and K. The reason we say that these are almostarbitrary is because we have to impose that the energy be �nite and as such we acquire aset of boundary conditions for our functions.

We will now investigate these boundary conditions and further characteristics of ournewly proposed monopoles

13

6.2.2 Boundary Conditions

Plugging our ansatz into the energy density above (remembering to make proper adjust-ments for our now real scalar �eld) we work to �nd boundary conditions on our functionsH and K by connecting it to our �nite energy restriction. For simplicity we also de�nethe dimensionless radius ξ = aer.

E =

∫R3

dVH =

∫R3

d3x

(1

2

(#»

Ei ·#»

Ei +#»

Bi ·#»

Bi +#»

Π · #»

Π +Di#»

φ ·Di#»

φ)

+λ

4

(φ2 − a2

)2)

=

=4πa

e

∫ ∞0

dξ

ξ2

[ξ2

(dK

dξ

)2

+1

2

(ξdH

dξ−H

)2

+1

2

(K2 − 1

)2+K2H2 +

λ

4e2

(H2 − ξ2

)2]

From this we inspect to observe what each term requires so as to not diverge in thelimits of our integration. Ultimately the result we get is as follows:{

K → 0,H

ξ→ 1

}for ξ →∞

{K − 1 ≤ O(ξ), H ≤ O(ξ)

}for ξ → 0

We assume above that these limiting behaviours are su�ciently rapid as to match whatwe postulated earlier for our integral.

If we now return to the initial limiting behaviour of our Higgs �eld we can inspect itwith these new conditions in mind.

#»

φ∞ (r) = limr→∞

#»r

er2H(aer) = ar

Which suitably �ts our previously presented notion of a Higgs vacuum. This is inci-dentally also homotopic to the identity mapping and as we discussed previously this meansour ansatz has the topological number of one. So if we can indeed �nd a solution which�ts these requirements - along with the following section - we have a non-trivial solutionwhich will not dissipate with time evolution as discussed previously.

6.2.3 Equations of Motion

We look to �nd equations of motion for our solutions by simply plugging in our ansatz inthe equations of motion that we previously derived.

DµDµ ~φ = −λ

(φ2 − a2

)~φ ⇒ ξ2d

2K

dξ2= KH2 +K(K2 − 1)

Dν#»

Gµν

= −e~φ×Dµ ~φ ⇒ ξ2d2H

dξ2= 2K2H +

λ

e2H(H2 − ξ2)

These equations are di�cult to solve and there are as of yet no solutions for the gen-eral form, however if we now investigate these equations in the limits for which we havedetermined our boundary conditions we acquire new equations which we can easily solve.

ξ →∞

{d2K

dξ2= K,

d2h

dξ2= 2

λ

e2h

}Where for the second equation we have made a convenient replacement of H = ξ + h.

The solutions for these equations are quite simply of the form of second order di�erentialequations:

K ∝ e−ξ = e−MW r h ∝ e−√2λeξ = e−MHr

We have exchanged the constants to return to more physically relevant quantities.From this we can see that our solutions are rapidly vanishing away from the origin and ifwe consider this as a oscillation in the Higgs �eld we see that this is describing an objectwith the size of 1

MWor 1

MHdepending on which is the greater radius (in our natural units

inverse mass has the unit of length), which we can begin to suspect as a possible monopole.

14

6.2.4 Asymptotic Form

We now want to investigate the electromagnetic properties of our ansatz. From what weconcluded earlier we have found the parallel component of our gauge �eld after symmetrybreaking to behave as our typical electromagnetic �eld, given as Aµ = 1

a

#»

φ · # »

W µ, and by

parallel we also have our electromagnetic �eld tensor as Fµν = 1a

#»

φ · #»

Gµν . So if we plugin our ansatz and using the knowledge that our ansatz is static (W 0

a = 0 ⇒ G0i = 0) weknow that the electric �eld is vanishing, as such we observe the remaining components inthe spatial limit so we may use our boundary conditions once more:

Bi = −1

2εijkF

jk = −1

2εijk

1

a

#»

φ · #»

Gjk

= −1

2εijkε

jka raer3

= − 1

er3ri

It seems we have acquired a magnetic �eld in the form of a point charge in the spatiallimit. We can now inspect the �ux this monopole generates by integrating over the shellof the sphere in the spatial limit (g is now our magnetic charge).

g = limr→∞

∮S2

#»

B · d #»

S = − limr→∞

∫ 2π

0dϕ

∫ 1

−1d cos θ

1

er3~r · rr2 =

4π

e

We now compare this to the values acquired with the Dirac quantization conditioneg = 2πn for which the minimum magnetic charge is g = 2π

e and as such we see that ourmonopole has twice the minimum magnetic charge in relation to the Dirac monopole.

This can somewhat intuitively be explained by the structure of our gauge connection.In the above analysis we have taken the adjoint representation of SU(2) which is SO(3),but if we were to consider this by intuition of spin in quantum mechanics this is also thestructure of a spin-1

2 system - something we call isospin - and the eigenvalues of this systemare 1

2 and −12 .

From all this we conclude that we have found an object which seems to ful�l the criteriawhich we demand for a magnetic monopole, however in contrast to the Dirac monopoleand its singular nature this new object appears to be everywhere smooth as demanded byour boundary conditions.

6.3 Topological Charge

Seeing as we have produced these monopoles from non-trivial solutions due to the topo-logical structure of our �eld it is reasonable to suspect that the magnetic charge we haveattributed also follows from this structure.

Knowing that in the spatial limit of any �nite energy solution we approach our vacuumcon�guration we can insert a �nite number of monopoles. This construct is reminiscent ofa gas, but so as to avoid any interactions we demand our monopoles to be well spaced. We�nd the gauge �eld in the limit to be:

Dµ#»

φ = ∂µ#»

φ − e # »

W µ ×#»

φ = 0 φ2 = a2 ⇒ # »

W µ =1

a2e

#»

φ × ∂µ#»

φ

But here we still clearly are missing the part that is parallel to the Higgs �eld, howeverwe already concluded to be 1

a

#»

φAµ. From this we recover our �eld strength once more.

Fµν =

#»

φ

a· #»

Gµν =

#»

φ

a·(∂µ

# »

W ν − ∂ν# »

W µ − e# »

W µ ×# »

W ν

)=

#»

φ

a3e·(∂µ

#»

φ × ∂ν#»

φ)

+ ∂µAν − ∂νAµ

We now take an enclosed volume - closed surface ∂V with a number of monopoles andinspect the �ux passing through in the same manner as before.

g∂V =

∮∂V

#»

B · d #»

S = − 1

2a3e

∮∂Vεijk

#»

φ ·(∂j

#»

φ × ∂k#»

φ)dSi

And here we see that the integral is only determined by the tangential behaviour of∂i

#»

φ on the surface ∂V , which is understood to be determined by our topological number.

15

7 Bogomol'nyi-Prasad-Sommerfield Monopoles

7.1 Bogomol'nyi Bound Estimation

Seeing as our newly produced object is purely topological in nature and a result of thestructure of our Yang-Mills-Higgs system we do not need to introduce a source with a freeparameter mass term. Instead we may attempt to �nd the mass of our monopole as anintrinsic value determined by the breaking scale of our system.

If we take the rest system of our monopoles we can integrate the energy of our spaceto �nd the rest mass of our object.

M =

∫R3

d3x

(1

2

(#»

Ei ·#»

Ei +#»

Bi ·#»

Bi +#»

Π · #»

Π +Di#»

φ ·Di#»

φ)

+ V (φ)

)≥

≥ 1

2

∫R3

d3x(

#»

Ei ·#»

Ei +#»

Bi ·#»

Bi +Di#»

φ ·Di#»

φ)

Inserting a angular parameter θ and completing the squares in our expression with theterms

#»

Ei ·Di#»

φ sin θ and#»

Bi ·Di#»

φ cos θ.

M ≥∫R3

(1

2

(| #»Ei −Di

~φ sin θ|2 + | #»Bi −Di#»

φ cos θ|2))

+

+

∫R3

(sin θDi

#»

φ · #»

Ei + cos θDi#»

φ · #»

Bi

)≥

≥∫R3

(sin θDi

#»

φ · #»

Ei + cos θDi#»

φ · #»

Bi

)If we now integrate these terms individually we may approach a possible numerical

lower-bound for our monopole mass. Firstly we use the equations of motion in the spatiallimit to �nd the result of the �rst term.∫

R3

(Di

#»

φ · #»

Ei

)=

∫R3

(∂i

(#»

φ · #»

Ei

))=

∫Σ∞

#»

φ · #»

EidSi = a

∫Σ∞

#»

E · d #»

S = aq

We have used Stoke's theorem to shift our volume integral into a surface integral whichtakes the form of one of our familiar Maxwell equations. If we now also attempt the sameprocedure for the second term we require a similar motivation as the equations of motion.In this case we make use of the Bianchi identity Dµ

? #»

Gµν

= 0 and �nd our result by exactparallel to above.∫

R3

(Di

#»

φ · #»

Bi

)=

∫R3

(∂i

(#»

φ · #»

Bi

))=

∫Σ∞

#»

φ · #»

BidSi = a

∫Σ∞

#»

B · d #»

S = ag

This gives the above presented lower bound for the mass of a monopole like object withelectric and magnetic charge as:

M ≥ a (q sin θ + g cos θ)

Finding the maxima for this equation we get maxθ∈S1

M ⇒ tan θ = qg . If we return this

into the equation we get:

M ≥ a√g2 + q2

This is the Bogoml'nyi bound for the mass of a charged monopole-like solution to ourHiggs �eld.

16

7.2 Saturating the Bound

Having derived the lower bound we now want to con�rm if there exist solutions such thatthis bound is saturated. In the previous section we calculated the lower bound by excludingintegrals of non-negative terms so for solutions placed on this bound all of the excludedterms should vanish such that their spatial integrals all equate to zero. This is a strongerrequirement than the previous asymptotically vanishing nature of our solutions, as suchwe need these terms to be vanishing in our entire space.

If we follow in the same manner as for the 't Hooft-Polyakov monopole and considerstatic solutions with no electric charge that obey the new requirements we have postulated.As before we get:

#»

Ei = 0 D0#»

φ = 0

In addition to this we can conclude from our postulate that the potential V (φ) shouldvanish in our space. For this to be possible we require λ = 0, but this clearly removes thepossibility to obtain such non-trivial solutions as we have discussed in this text. To solvethis we prescribe a limiting behaviour on the parameter as λ → 0. This lets us keep thebehaviour we observed earlier and still conform to the new setting, this is named as thePrasad-Sommer�eld limit.

Additionally we also get another equation from our postulate:

#»

Bi = ±Di#»

φ

This is dubbed as the Bogomol'nyi equation. Using these features we look for solutionsusing the 't Hooft-Polyakov ansatz. Inserting the Ansatz into the Bogomol'nyi equation(both variants) we get the following:

ξdK

dξ= −KH ξ

dH

dξ= H + 1−K2

And considering the asymptotic behaviour as we required before we can �nd suitablesolutions to this set of equations.

K(ξ) =ξ

sinh ξH(ξ) = ξ coth ξ − 1

These solutions have a clearly distinct appearance to those from the section regardingthe 't Hooft-Polyakov monopole. If we inspect the solution for the monopole as:

h(ξ) = H(ξ)− ξ = 1 +O(e−ξ)

This is not of the same form as a classical Dirac monopole and the similar behaviourof the 't Hooft-Polyakov monopole. This indicates a long range action which does notmatch that of our previous intuition, it is however well justi�ed since we are considering amassless Higgs �eld in taking the limit on λ and a long range action is to be expected.

17

8 Montonen-Olive Conjecture

Now drawing from the conclusions and investigations we have made we follow in the stepsof Montonen and Olive and present their conjecture on the existence of duality transfor-mations for theories which exhibit BPS-monopoles as solutions.

We now take the variant of the Georgi-Glashow model that we have investigated andpresent the qualitative properties we have determined.

Electric MagneticParticle Mass Charge Charge Spin

Photon 0 0 0 ±1Higgs 0 0 0 0

W± boson a|q| ±q 0 1M± monopole a|g| 0 ±g 0

This is the particle spectrum of our bosonic theory with the limit λ → 0. Sincewe have constructed our monopoles to obey the Bogomol'nyi bound it is quite evidentthat our theory admits these solutions and if we recall the simple form of the bound(M ≥ a

√q2 + g2) it is also evident that our W bosons obey this bound. If we now

remember the electromagnetic duality as (q, g) → (g,−q) with W± � M±. We see thatthe particle spectrum is invariant under this transformation, with one caveat we will discussshortly, and we have as such acquired a new theory where the BPS - magnetic - monopolesact as gauge particles and the massive vector bosons act as electric monopoles.

Electric MagneticParticle Mass Charge Charge Spin

Photon 0 0 0 ±1Higgs 0 0 0 0

M± �gauge� a|q| 0 ∓q 0W± �monopole� a|g| ±g 0 1

This new theory represents a dual magnetic representation, the existence of whichis what the Montonen-Olive conjecture discusses. As motivation for the equivalence ofthese theories are the tree level diagrams for the inter-particle forces in the quantum �eldtheories generated by the same type of classical �eld theories discussed here, calculated tobe equivalent by Manton [10]. It is also here that we observe a tantalizing result. TheDirac quantization condition implies a connection between the charge types which statesthem to be inversely proportional. The realisation here is that our duality replaces the�primary� electric charge - a perturbative quantity - with the �secondary� magnetic charge- which is then necessarily non-perturbative as the inverse. We have transformed a weaklycoupled theory into a strongly coupled theory, and as suggested by Manton, calculationsin both theories should be equivalent, so conversely we could transform a strongly coupledtheory into its dual weakly coupled theory and make predictive perturbative calculations.We now address the apparent �aws with this conjecture.

Firstly we have yet to discuss the renormalization properties of these theories. Ra-diative corrections to the �eld interactions could in reality break our calculation for theBogomol'nyi bound making it possibly divergent or otherwise incompatible with our sub-sequent conclusions. Secondly we have not addressed the glaring discrepancy in the spinof our new �gauge� particle as we would expect a gauge-like spin of 1 but we have notobserved anything suggesting we neglect the explicit rotational invariance we have useduntil now. These two problems can be resolved by further expansion and introduction ofsupersymmetry [11]. However, this still leaves the conjecture untestable due to lackingunderstanding of calculations in strongly coupled theories, but possible expansions to thistheory may result in testable quantities as we will discuss in the next section.

18

9 The Witten Effect

9.1 Dyons and Gauge Current

We will assume that the angular momenta of a charged particle in an electromagnetic �eldis separately quantized and using this we will quickly discuss particles with both electricand magnetic charge, called, dyons.

| #»J em|=1

2n n ∈ Z

And for a charged particle - with electric charge q - in the electromagnetic �eld createdby a magnetic monopole - with magnetic charge g - the conserved quantity is the newelectromagnetic angular momenta in addition to the regular momenta as [12]:

d

dt

#»

J tot =gq

4π

d

dt

#»r

r⇒ #»

J tot =#»

L − gq

4π

#»r

r

#»

J em = − gq4π

#»r

r

We see that in the presence of a magnetic monopole we acquire a non-zero time deriva-tive and to compensate this with an invariant quantity we simply include the new term.Following this we now prescribe two dyons with the charges:

d = (q, g) d′ = (q′, g′)

And this gives us for the total electromagnetic angular momenta as:

#»

J em(d) = −g′q

4π

#»r

r

#»

J em(d′) =gq′

4π

#»r

r⇒ #»

J em(d) +#»

J em(d′) =(gq′ − g′q

) #»r

4πr

And invoking our quantization we �nd our result as:

| #»J em(d) +#»

J em(d′)|=(gq′ − g′q

) 1

4π=

1

2n ⇒ gq′ − g′q = 2πn

If we now consider the existence of an elementary electric charge e - an electron -considered as a dyon with charge (e, 0). By the Dirac quantization condition this impliesthe minimum magnetic charge g = 2π

e . If we now observe the di�erence in electric chargefor two dyons with the same minimum magnetic charge we get:

g(q − q′) =2π

e(q − q′) = 2πn ⇒ q − q′ = ne

And once more using the implication of our elementary charge e we conclude that dyonswith minimum magnetic charge will all admit unit multiples of the elementary charge asq = ne.

We will now approach this topic from the properties of our gauge theory and comparethe results.

Firstly we introduce a generator for a gauge transformation denoted N in the isovectorspace around the direction of our �eld

#»

φ with #»v as an arbitrary isovector designating ourtransformation.

δ #»v =1

a

#»

φ × #»v δ# »

W µ = − 1

eaDµ

#»

φ

If we now exponentiate the generator such that it can act as a transformation, muchin the same sense as with our groups, we get the transformation e2πiN . Taking the spatiallimit in our vacuum we �nd the transformation as:

δ #»v =1

a

#»

φ × #»v = φ× #»v δ# »

W µ = − 1

eaDµ

#»

φ ={Dµ

#»

φ = 0}

= 0

19

It is here clear that the transformation maps to identity in the spatial limit for both thegauge and the Higgs �eld i.e. e2πiN = 1. Seeing as we have a continuous symmetry by thetransformation we can conclude that the generator N is conserved. To calculate this welook at the variation of the Lagrangian of our theory with respect to this transformation.

N =

∫R3

(∂L

∂(∂0# »

W µ)· δ # »

W µ +∂L

∂(∂0#»

φ)· δ #»

φ

)=

∫R3

∂L∂(∂0

# »

W µ)· δ # »

W µ

Above we excluded the φ term as it clearly vanishes according to our de�nition. Nowusing the de�nition of our transformation from earlier we continue to modify the expressioninto something more suiting for our goals.

N = − 1

ae

∫R3

∂L∂(∂0

# »

W i)·Di

#»

φ

Evaluating the �rst term as the conjugate momenta of our gauge �eld we remember

the de�nitions of our �eld tensor in terms of the electric �eld − #»

G0i

=#»

Ei

= − #»

Ei. Fromthis we can further rewrite the expression.

N =1

ae

∫R3

#»

Ei ·Di#»

φ =q

e

In the last step we used the result we derived for the Bogomol'nyi bound previously. Weknow that the possible values of the generator N is integer as we mapped the exponentiatedgenerator to identity in our isospace. Rather e2πiN = 1 gives solutions as N ∈ Z.

We see that this garners the same result as derived form the investigation using dyonsand the associated minimum charges, implying that the dyons we suggested are indeed afeature of the structure of our gauge theory and not just a forcibly introduced arti�cialconstruct.

9.2 Charge Conjugation-Parity Violation

This far we have only considered charge conjugation-parity (CP) non-violating terms in ourtheory. However, if we take the well known fact that physical models admit CP violatingterms we are free to consider an additional term constructed from the constituents we haveproduced in this text [13].

Lθ =1

2

e2θ

32π2εαβµν

#»

Gαβ · #»

Gµν

= − e2θ

32π2? #»

Gµν ·#»

Gµν

Where we have re-introduced the Hodge dual tensor as which we introduced in theprevious section on electromagnetic duality. However, if we inspect this term more closelywe will see that it is a total derivative and as such does not impact the dynamics of ourtheory.

? #»

Gµν · #»

Gµν =32π2

e2∂µKµ

{Kα =

e2

16π2εαβµν

(W aβ ∂µ(Wν)a +

e

3εabcW

aβW

bµW

cν

)}

Granted that we integrate this term in our action we can see that it is an multiple tothe parameter θ. If we then consider the possible vacua for a theory with such a term itis clear that it is parametrised by θ. The integral value of the total derivative will dependon the theory at hand and if it admits some number of instanton solutions, which is calledthe instanton number.

20

When including this term in the Lagrangian of our theory we need to adjust the gen-erator N such that the overall symmetry it generates is still respected.

N → N − 1

ae

∫R3

∂Lθ∂(∂0

# »

W i)·Di

#»

φ

If we now calculate this term as we did previously we get the di�erence as:

∆N = − 1

ae

∫R3

∂Lθ∂(∂0

# »

W i)·Di

#»

φ = − eθ

16π2a

∫R3

ε0iαβ#»

Gαβ ·Di#»

φ =

= − eθ

16π2a

∫R3

εijk#»

Gjk ·Di#»

φ =eθ

8π2a

∫R3

#»

Bi ·Di#»

φ =eθ

8π2g

Where this calculation is done in complete parallel to the previous one, making use ofthe integration for the Bogomol'nyi bound and the de�nition of our �eld tensor.

Now presenting our improved equation for N we get:

N =q

e+

eθ

8π2g

And if we return to our 't Hooft-Polyakov monopole with the relation eg = 4π. Usingthis we can �nd the charge of the monopole depending on θ as:

q = ne+eθ

2πn ∈ Z θ ∈ S1

This is a result due to Witten and as such it bears his name as the Witten E�ect.

10 SL(2,Z) Duality

Our complete theory now depends on four undetermined parameters a, λ, θ and e. We canhowever combine θ and e to form a complex parameter τ . For this we �rstly bring thecoupling term out of our �eld tensor (

# »

Wµ → e# »

W µ) and restate our current theory.

L+ Lθ = − 1

4e2

#»

Gµν ·#»

Gµν

+θ

32π2

#»

Gµν · ?#»

Gµν

+1

2Dµ

#»

φ ·Dµ #»

φ − V (φ)

And here we have extracted the e dependence and can now replace it.

τ ≡ θ

2π+ i

4π

e2

We now want to restate the Lagrangian to use the parameter τ so for this we nowintroduce a new identity.

#»Gµν =#»

Gµν + i?#»

Gµν

And we contract this new tensor to �nd a relation to our Lagrangian terms.

#»Gµν ·#»Gµν

= 2#»

Gµν ·#»

Gµν

+ 2i#»

Gµν · ?#»

Gµν

From this we can write the terms as:

− 1

32πIm(τ

#»Gµν ·#»Gµν

)Seeing as the θ term we have introduced is an angular variable we now see a new

emergent symmetry of our theory as θ → θ+2π and as such the transformation of τ → τ+1will leave our Lagrangian invariant. In the case of our previously stated electromagnetic

21

duality we take the case for θ = 0 and get the transformation e → g = −4πe . If we plug

this into the τ term we recover our duality transformation as τ → − 1τ .

This can now be added to the Montonen-Olive conjecture to make an even strongerstatement about theories with possible CP violating terms. With now two possible discretesymmetry transformations.

T : τ → τ + 1 S : τ → −1

τ

It is here that we observe the extrapolation to the SL(2,Z) group of 2×2 unit matriceswith integer entries [14]. (

a bc d

)· τ =

aτ + b

cτ + d

Our previously presented symmetry is clearly just a subgroup of SL(2,Z) with theT transformation represented by a = b = d = 1, c = 0 and the S transformation byb = −1, c = 1, a = d = 0.

And in truth it can be shown that S and T generate the entirety of the group SL(2,Z).So it appears that our current theory is invariant under the whole of SL(2,Z). But toproduce an improved conjecture based upon our previous work we still require this new-found symmetry to respect the Bogomol'nyi bound for our BPS-states in order to preservethe results stated in the Montonen-Olive conjecture.

For a BPS-state with charges (q, g) its mass is given as M = a√q2 + g2 with magnetic

charge determined by the Dirac quantization condition, g = 4πe nm, and with the electric

charge as presented by the Witten e�ect, q = ene + eθ2πnm. We now use this to restate the

mass formula in this context with the number vector #»n = (ne, nm)T .

M2 = 4πa2 #»nT ·A(τ) · #»n

Where we have introduced the matrix A(τ):

A(τ) =1

Im(τ)

(1 Re(τ)

Re(τ) |τ |2

)

The calculation to con�rm this mass formula is straightforward.

M2 = a2(q2 + g2) =

{q = ene

eθ

2πnm g =

4π

enm

}=

= a2e2

((ne +

θ

2πnm

)2

+16π2

e4n2m

)=

= a2e2

n2e +

θ

πnenm +

(θ2

4π2+

16π2

e4

)n2m

=

= 4πa2 #»nTe2

4π

(1 θ

2πθ

2πθ2

4π2 + 16π2

e4

)#»n = 4πa2 #»nT ·A(τ) · #»n

So now all that remains is for this formula to be invariant under a transform of a generalelement G of our group SL(2,Z).

G =

(a bc d

)∈ SL(2, Z)

22

So our transformed term would appear as:

A(G · τ) = Im

(aτ + b

cτ + d

)−1(

1 Re(aτ+bcτ+d)

Re(aτ+bcτ+d) |aτ+b

cτ+d |2

)=

=1

Im(τ)(ad− bc)

(c2|τ |2+d2 + 2cdRe(τ) ac|τ |2+bd+ Re(τ)(ad+ bc)

ac|τ |2+bd+ Re(τ)(ad+ bc) a2|τ |2+b2 + 2abRe(τ)

)=

=1

Im(τ)

(d −c−b a

)(1 Re(τ)

Re(τ) |τ |2

)(d −b−c a

)= (G−1)T ·A(τ) ·G−1

Here we see that for our mass formula to be invariant we require the number vector ofour charges transforms to compensate for this result.

#»n → G · #»n

Making the �nal transformation fully invariant:

#»nT ·A(τ) · #»n −→ #»nT ·GT · (G−1)T ·A(τ) ·G−1 ·G · #»n = #»nT ·A(τ) · #»n

Now we can �nally improve the Montonen-Olive conjecture to include full invarianceunder the SL(2, Z) group. This implies that any theories related by a possible SL(2, Z)transformation are physically equivalent granted that we exchange the magnetic and elec-tric charges as stated just above.

This result has impact on the spectrum of possible BPS-states this predicts. Ouroriginal duality showed that massive vector bosons implied corresponding BPS-monopoles,and now - assuming that every theory determined by τ contains massive vector bosons - wehave found an in�nite spectrum of dyonic states corresponding to the possible transformsof our group.

In order to investigate the stability of these states and their internal consistency weassume that there exists a possible minimum state for all τ with number #»n1 = (1, 0)T .The states generated by an element in the group from this number.

M · #»n1 =

(a bc d

)·

(10

)=

(ac

)Since an element in our group is required to have unit determinant we have a restriction

relating the otherwise arbitrary integers and ad − bc = 1 implies that we have to selectd and b such that this requirement is ful�lled. This means that if we suppose a commonfactor n as d = nd′ and b = nb′ we can rewrite our unitarity as n(a′d − bc′) = 1, but forthis to hold it is clear that n = 1, or rather, the only common factor for these terms is thetrivial case and as such these factors are necessarily coprime.

Returning to physical aspects of our theory we de�ne the clearly non-negative masseigenstate of possible number vectors as | #»n|M≡ M2

#»n . For a combined state #»q = #»n + # »mand the non-negative nature of our mass eigenstates we can clearly see a triangle inequality.

| #»q |M= | #»n + # »m|M≤ | #»n|M+| # »m|MAnd if our state #»q is the one we investigated previously as #»q = (a, c)T with a and c

coprime, and we recall the number dependence of our mass formula we can see that thisinequality is necessarily strict as we can extract no common factor in our expression.

M2#»q = | #»n|M+| # »m|M 6= n2(| #»n′|M+| # »m′|M ) ∀ n 6= 1

It appears as if the state generated by the group action on our state is still a unique stateand not some bound state consisting of smaller dyonic states and as such our presumedstate is stable. Our model predicts an in�nite number of stable dyonic states which shouldbe observable if this invariance is realised in a real physical model.

23

11 Discussion & Conclusion

In a �nal comment to address the somewhat overlooked mention of discrepancies in re-gards to the nature of the Montonen-Olive conjecture. When mentioning the issues ofspin-statistics and possibly non-renormalizable features of our de�nition of BPS-states, asmentioned, an extension to include a more advanced expansion of the allowed symmetriesin our theories - named supersymmetry - is possible. Wherein we take consideration for allpossible symmetries of quantum �eld theories. Expanding our original symmetries to notonly include the Poincaré group and an internal symmetry group, but also a new space-time symmetry relating boson states to corresponding fermion states which serves as apragmatic description of supersymmetry.

It should be quite straightforward to see that this symmetry relating the di�erent spin-statistics should - heuristically - solve the present issues of the statistics of our monopolesand it is indeed a feature of - amongst others - N = 4 supersymmetric theories, howeveras it is outside the scope of this project, by quite some margin, we hope this explanationto be satisfactory.

When considering the issue relating to the radiative corrections we may employ aneven more heuristic motivation. Seeing as our theory consists of bosonic states and corre-sponding fermionic states there exists cancellations between these loop corrections whichoften take care of these issues and is a feature exhibited by N = 2 supersymmetric theories.

In this paper we have presented and discussed the fundamental aspects of groups and theidea of groups as transformations letting us develop the concept of gauge theories andmore speci�cally non-abelian gauge theory. We also investigated the emergent propertiesof �symmetry breaking� where we shifted our vacuum and broke its symmetry. We thencombine this to present a form of the Georgi-Glashow model as a prototype theory for ourobjective. The form of our model is reminiscent of Scalar-Electrodynamics which servesour intuitive parallels in a sense.

After this we presented the emergent duality in Maxwell's electromagnetism and dis-cussed the necessary requirements to ful�l this. We presented an example in the Diracmonopole and found that the existence of magnetic monopoles implies the Dirac quanti-zation condition, a motivation for the quantization of electric charge.

Now with the focus on the possibilities in non-abelian gauge theories, and having devel-oped the motivation for the investigation of this concept we presented a solution that adaptsthe necessary requirements as a magnetic monopole in the 't Hooft-Polyakov monopole.Examining this solution we improved upon it and found that if we want to determine ourmassive states we see that it is only bound from below according to the Bogomol'nyi bound.We modi�ed our solutions in the 't Hooft-Polyakov ansatz to adhere to the requirementson the bound and de�ned BPS-states for our further consideration.

With a massive monopole in hand we de�ned the Montonen-Olive conjecture wherewe suggested the invariance of our physical theories under the electromagnetic duality byreplacing the massive bosons in our theory with the massive monopoles and taking carefulconsideration to all the components presented by the duality.

Having brie�y mentioned the apparent �aws in the presented conjecture we moved toexpand the concept by allowing CP violating terms. We observed the adjustment of ourcharge spectrum with respect to the introduced term and its associated angular variable.

This also necessitated further expansion of the symmetry of our theory to includethe SL(2,Z) group, and we concluded that this also left our conjecture invariant, howeverwith an expanded particle spectrum, now generated by the continuous spectrum of possibleduality transformations.

24

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[3] A. M. Polyakov, Particle Spectrum in the Quantum Field Theory, JETP Letters. 20(6) (1974), 194�195.

[4] E. B. Bogomol'nyi, The Stability of Classical Solutions, Soviet J. Nuc. Phys. 24 (1976),449-454.

[5] M. K. Prasad and C. M. Sommer�eld, Exact Classical Solution for the 't Hooft Monopole

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[12] H. Poincaré, Remarques Sur Une Expérience de M. Birkeland, Comptes Rendus Acad.des Sci. 123 (1896), 530-533.

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grouptheory/SL(2,Z).pdf

25