Electromagnetism and Magnetic Monopoles

Physics & Astronomy

Institute for theoretical physics

Electromagnetism and magnetic monopoles

Author Supervisor F. Carere Dr. T. W. Grimm

June 13, 2018

Image: Daniel Dominguez/ CERN

Abstract Starting with the highly symmetric form of electromagnetism in tensor notation, the consideration of magnetic monopoles comes very natural. Following then the paper of J. M. Figueroa-OFarrill [1] we encounter the Dirac monopole and the ’t Hooft-Polyakov monopole. The former, simpler – at the cost of a singular Dirac string – monopole already leading to the very important Dirac quantization condition, implying the quantization electric charge if magnetic monopoles exist. In particular the latter monopole, which is everywhere smooth and which has a purely topological charge, is found as a finite-energy, static solution of the dynamical equations in an SO(3) gauge invariant Yang-Mills-Higgs system using a spherically symmetric ansatz of the fields. This monopole is equivalent to the Dirac monopole from far away but locally behaves differently because of massive fields, leading to a slightly different quantization condition of the charges. Then the mass of general finite-energy solutions of the Yang-Mills-Higgs system is considered. In particular a lower bound for the mass is found and in the previous ansatz a solution saturating the bound is shown to exist: The (very heavy) BPS monopole. Meanwhile particles with both electric and magnetic charge (dyons) are considered, leading to a relation between the quantization of the magnetic and electric charges of both dyons and, when assuming CP invariance, to an explicit quantization of electric charge. Finally, the Z2 duality of Maxwell’s equations is extended. When PSL(2, Z) duality is assumed for electric and magnetic monopoles satisfying the Bogomol’nyi mass bound a dyonic spectrum is found in the orbit of the electron. Contents

Preface 1

I Electromagnetism and relativity 2

1 Electromagnetism and electromagnetic duality 2 1.1 Electromagnetism in vector calculus notation ...... 2 1.2 Electromagnetism in tensor notation ...... 4

2 A magnetic monopole 6

3 Relativity 7 3.1 The structure of ﬂat spacetime: Minkowski space ...... 7 3.2 Lorentz transformations ...... 8 3.3 Proper time ...... 9

4 Electromagnetism and relativity 11

5 Lagrangian mechanics and actions 13 5.1 Classical mechanics in integral form ...... 13 5.2 The relativistic Lagrangian ...... 14 5.3 The classical electromagnetic Lagrangian ...... 14 5.4 The relativistic electromagnetic Lagrangian ...... 15 5.5 The Hamiltonian ...... 16

II Symmetries, ﬁelds, gauge theories and the Higgs mechanism 17

1 Symmetry 18 1.1 Classiﬁcation of symmetries ...... 18 1.2 Symmetry breaking ...... 20 1.2.1 Explicit symmetry breaking ...... 21 1.2.2 Spontaneous symmetry breaking ...... 22

2 Some classical ﬁeld theory 23 2.1 General forms of the Lagrangian density ...... 23 2.2 Noethers theorem ...... 24 2.3 The Hamiltonian density ...... 25

3 Gauge theory 26 3.1 Classical abelian gauge theory ...... 26 3.2 Gauge invariance of the relativistic electromagnetic Lagrangian ...... 28 3.3 Gauge theory for the Dirac Lagrangian ...... 29 3.4 The general strategy for U(1) ...... 29 3.5 Yang-Mills theory ...... 30 3.6 Again Yang-Mills ...... 32 3.7 The case of SO(3) ...... 33 3.8 Yang-Mills in general ...... 34 4 The Higgs mechanism 36 4.1 The Higgs ﬁeld and its potential ...... 36 4.2 Massless gauge bosons from a broken global symmetry ...... 37 4.3 The Higgs mechanism ...... 38

III Magnetic monopoles 39

1 The Dirac monopole 40 1.1 The search for a global vector potential ...... 40 1.2 The Dirac quantization condition ...... 41

2 Dirac monopole: Dyons and the Zwanziger-Schwinger quantization condition 44

3 The bosonic part of the Georgi-Glashow model 48

4 Static, ﬁnite-energy solutions in the ’t Hooft-Polyakov ansatz 52 4.1 The solution of H and K in the ’t Hooft-Polyakov ansatz ...... 53 4.2 The topological origin of the magnetic charge ...... 56

5 The diﬀerence between the Dirac and the ’t Hooft-Polyakov monopole 61

6 Mass and the BPS monopoles 62 6.1 The Bogomol’nyi bound ...... 62 6.2 The BPS monopole ...... 63 6.2.1 The BPS monopole in the ’t Hooft-Polyakov ansatz ...... 64

7 Electromagnetic duality revisited 67 7.1 The Montonen-Olive conjecture ...... 67 7.2 Quantization of dyons ...... 68 7.3 The Witten eﬀect ...... 69 7.4 PSL(2, Z) Duality ...... 70 7.5 The modular group ...... 70 7.6 Orbifolds and the dyonic spectrum ...... 72

IV Discussion 75

A Tensors 77 A.1 Basistransformations: Co- and contravariant objects ...... 77 A.1.1 Co- and contravariant indices, raising and lowering indices ...... 77 A.2 Transformation properties of tensors ...... 77

B The Hamiltonian density in the ’t Hooft-Polyakov ansatz 78

C The fundamental group of maps between circles 80 C.1 The structure of loops ...... 81 C.2 Homotopy equivalent spaces ...... 82 C.3 The fundamental group of maps between circles ...... 82

D Lie groups and Lie algebras 85 D.1 A note on SO(3)...... 85 Preface: Discussing magnetic monopoles

Around 600 B.C. the Greek philosopher Thales of Miletus described the peculiar phenomena of attractive forces between two substances: Fur and amber [2]. He was likely the first to make a written description of electrical forces. Indeed it is due to the ancient Greeks (and Gilbert) that the term ηλκτρoν (meaning amber) is now used for one of the most fundamental objects in electrostatics: the electron. Furthermore, the Greeks were aware that there existed stones which seemed to attract iron when held close. The magnetic force was found, named after the Turkish city of Magnesia (now Manisa), where the stones were found. These two magical forces turned out to behave very similar. Electric poles were introduced in the theory of electrostatics. Due to a large amount of symmetry in the theories magnetic poles were introduced, in accordance with electrostatics. The first mentioning of magnetic monopoles could well have been in a letter of Pierre of Maricourt in 1269 [3]. The magnetic monopole, however, was – and to this day still is – unobserved. After the discovery of Oersted in 1820 that current deflects a compass needle, electromagnetism was born [4]. This, later on, led to Ampèresmodel of the magnet, in which the magnetic fields of a magnet come from ‘little Amperian currents’ [5]. This model did no good to the theory of magnetic monopoles, since in this model magnetic fields were not produced by a stationary magnetic monopole but moving electric charges. The intrinsic connection between electricity and magnetism was clearly there and Ampère’smodel was the first step to fathom this duality. It was Maxwell’s equations, however, which were able to combine the theory of electrostatics and magnetostatics in a very elegant fashion. The four Maxwell equations, together with the equation for the Lorentz force, are able to fully describe (static and dynamic) classical electromagnetism. One of the four laws of Maxwell, however, is ∇~ · B~ = 0, sometimes called the ‘no magnetic monopole law’. This short history of classical electromagnetism might indicate that it is completely nonsensical to search for and describe magnetic monopoles. Indeed to this day they have not been seen in experiments. However one important argument in favour of the existence of magnetic monopoles is certainly the amount of symmetry in Maxwell’s equations. Including the fact that over the last century gauge theories have become extremely important and very successful in the description of fundamental particles and forces (for instance in the Standard Model), together with the fact that these gauge theories rely on symmetry being an intrinsic property of nature itself, the search for monopoles might seem a bit less odd. Furthermore, in cosmology [6] and in quantum mechanics [7] reasons have come up why it might be that these monopoles have gone unseen and also are not made and observed in particle accelerators. Nevertheless, a reason why these monopoles have not yet been observed could just be the fact that they do not exist. Indeed in 1931 Dirac brought the theory of magnetic monopoles to life [7]. A magnetic monopole similar to the electric monopole was considered, which was therefore coined the Dirac monopole. In 1974 ’t Hooft [8] and Polyakov [9] both found inevitable monopole solutions in a Yang-Mills theory of a spontaneously broken non-abelian classical Lagrangian system. From far away, these monopoles behave in a similar manner as the Dirac monopole but close by they contain more information, including the mass of such a monopole. Another reason was found that the monopoles are hard to find: they are very heavy. In this paper properties of these types of (classical) magnetic monopoles will be described. In part I. it will start with the symmetrical formulation of Maxwell’s equations in electrodynamics, first in vector calculus notation and then going to tensor notation to make it easy on readers with little experience in the latter kind. In the light of the highly symmetric formulation, magnetic monopoles will then be considered in part III, starting with the Dirac monopole and continuing on to the ’t Hooft-Polyakov monopole, though a glimpse of gauge theories and symmetry breaking is given first in part II, which is needed for the discussion of the latter monopole.

1 Part I Electromagnetism and relativity

Electromagnetism is a (special) relativistically invariant theory. Moreover, special relativity and electromagnetism have been two theories, which have seen development that was really intertwining [10]. For example, on completion of Maxwell’s equations in classical electromagnetism, it quickly became clear that light is an electromagnetic wave. Furthermore, from special relativity it became clear that electric fields and magnetic fields are intrinsically connected via Lorentz transformations. Going from one reference frame to another, the interpretation of the electrostatic system might change to an electromagnetic system and vice versa [5]. Thus only electricity and magnetism together make for a complete theory. In this part, electromagnetism will be considered and its relativistic invariance is made explicit. Furthermore a heuristic introduction of Minkowski space is given which, together with the explanation of tensors in the appendix, makes clear that the relativistic invariance is a fact coming from the structure of flat spacetime. This becomes completely oblivious when one formulates electromagnetism in the tensor formulation, where one sees that the electromagnetic fields are defined as properly transforming under special relativistic transformations, because they are tensors. Fi- nally, the classical Lagrangian formalism of electromagnetism is stated, which comes in handy when discussing magnetic monopoles.

1 Electromagnetism and electromagnetic duality

There are various, equivalent, ways to state the laws governing electromagnetism. Two types of formulations will be considered: One in the notation of vector calculus, the other using index notation, to get the reader familiar with the possibly new index notation.

1.1 Electromagnetism in vector calculus notation

In the beginning the SI unit system will be used, such that the speed of light c and the Planck constant ~ remain explicit. If there are no sources present (in vacuum) then Maxwell’s equations are given by [5]

∇~ · E~ = 0 ∇~ × E~ = −∂tB~ (1)

∇~ · B~ = 0 ∇~ × B~ = ∂tE,~ (2) where of course E~ and B~ denote the electric and magnetic ﬁelds, respectively and ∂ = ∂ . Together t ∂t with the Lorentz force law F~ = q(E~ + ~v × B~ ) they summarize the entire content of classical electrodynamics for a particle with charge (electric) q and speed ~v in a system in vacuum without other charges (with ﬁelds E~ and B~ ). This nice set of equations is completely symmetrical under the transformation (E,~ B~ ) 7→ (B,~ −E~ ). When electric charges are present in the system the equations change to

ρe ∇~ · E~ = ∇~ × E~ = −∂tB~ (3) 0

∇~ · B~ = 0 ∇~ × B~ = µ00∂tE~ + µ0J~e, (4) where 0 is the permittivity of free space, µ0 is the permeability of free space, ρe is the electric charge and J~e is the electric current. This set loses its symmetry under the transformation above,

2 however. It is this fact that (arguably) screams for a more symmetric formulation, which comes 1 about by the addition of a magnetic charge ρb and a magnetic current J~b

ρe ∇~ · E~ = − ∇~ × E~ = ∂tB~ + µ0J~b (5) 0

∇~ · B~ = µ0ρb ∇~ × B~ = µ00∂tE~ + µ0J~e. (6)

Together with the new Lorentz force F~ = q(E~ +~v × B~ ) + g(B~ − µ00~v × E~ ), for a particle with speed ~v and electromagnetic charge (q, g), which is called a dyon. The Maxwell’s equations (5) and (6) stated above have two important properties. First of all, it is easy to see that the equations are invariant under electromagnetic duality, deﬁned by the mapping (E,~ B~ ) 7→ (B,~ −E~ ) and (Je, Jb) 7→ (Jb, −Je), (7) T where J = ρe (J~ ) (J~ ) (J~ ) ∈ 4 and J deﬁned in accordance. e 0 e x e y e z R b Furthermore, Maxwell’s equations are invariant under Lorentz transformations (Lorentz invariant), meaning that they are not only useful to describe classical systems but extend to the description of relativistic systems.

The potential formulation in vector calculus notation

There is another way to state the laws of electromagnetism. This formulation uses two objects that describe the electric and magnetic field in a more compact way. 3 3 Specifically, if there is a vector field V~ : R → R such that ∇~ × V~ = 0 then we can define the scalar 3 potential v : R → R such that V = ∇~ v. 3 3 Similarly, if there is a vector field V~ : R → R such that ∇~ · V~ = 0 then we can define the vector 3 3 2 potential W~ : R → R such that V~ = ∇~ × W~ . Nota bene that these potentials are not unique, which will prove useful later and is the basis for electromagnetic gauge theory. Combining the above together with equation (3) and (4) we see that, in the absence of magnetic monopoles it is possible to define B = ∇~ × A~ (8)

~ 3 3 ~ ~ ∂A~ for a vector potential A : R → R . Furthermore if we deﬁne E := E + ∂t , then we can ﬁnd a scalar 3 potential φ : R → R such that E~ = −∇~ · φ, in other words

E~ = −∇~ φ − ∂tA.~ (9)

Note that the Maxwell equations ∇~ × E~ = 0 and ∇~ · B~ = 0 are then satisﬁed. Furthermore the other two Maxwell equations, still in the absence of magnetic monopoles, are satisﬁed if

2 ρe − ∇ φ − ∂t(∇~ · A~) = (10) 0 and ~ ~ ~ ~ ~ 2 ~ ∇ × (∇ × A) = µ0Je − µ00∇∂tφ − µ00∂t A. In other words 2 ~ 2 ~ ~ ~ ~ ~ ~ ∇ A − µ00∂t A − ∇ ∇ · A + µ00∇∂tφ = −µ0Je. (11)

1 Actually, when using a diﬀerent metric system, for example 0 = µ0 = 1, the equations look completely symmetric 2 3 This can be proven using the Helmholtz decomposition of a vector ﬁeld in R see https://en.wikipedia.org/ wiki/Helmholtz_decomposition. It is assumed however, that the space is ‘nice’, meaning for exampl no holes or connectedness.

3 These equations are not very pretty, however, and we can use the non-uniqueness of both potentials to tidy up the equations (10) and (11). It can be shown quite easily [5] that if we have a scalar potential φ and a vector potential A~ then we 4 0 can define, for any scalar function λ : R → R;(~r, t) → λ(~r, t), the scalar potential φ = φ + ∂tλ and the vector potential A~0 = A~ − ∇~ λ such that the pair (φ0, A~0) describes the same system as (φ, A~). This freedom is called gauge freedom3 and the transformation (φ, A~) → (φ0, A~0) is called a gauge transformation, the first of many to come. Using a proper gauge (function λ), namely the so-called Lorenz gauge, one can find an extra equation of the form ∇~ A~ = −µ00∂tV [5]. This is particularly useful if we apply the divergence to it and substitute it in equation (11) to find

2 ~ 2 ~ ~ ∇ A − µ00∂t A = −µ0Je.

Similarly, equation (10) becomes 2 2 −ρ ∇ φ − µ00∂t φ = . 0 2 2 1 2 2 Deﬁning now the d’Alembertian (operator) := µ00(∂t) − ∇ = c2 ∂t − ∇ we see that both equations reduce to ρ ~ ~ V = and A = µ0J. (12) 0 Maxwell’s equations in the Lorenz gauge therefore reduce to these two equations. The Maxwell equations in the above formulation turn out to imply relativistic invariance of the theory (in the Lorenz gauge!).

1.2 Electromagnetism in tensor notation

In tensor notation, relativistic invariance becomes explicit. This is precisely because of the transformation properties of tensors4. To make the symmetry more explicit, the metric system is chosen such that 0 = µ0 = 1, implying c = 1. Specifically, the fields E~ and B~ fields can be joined to form an antisymmetric tensor F of rank 2, which can be written as the matrix   0 −Ex −Ey −Ez Ex 0 −Bz By  F =   (13) Ey Bz 0 −Bx Ez −By Bx 0 called the electromagnetic (EM) tensor. Its dual tensor G is defined by   0 −Bx −By −Bz Bx 0 Ez −Ey G =   . (14) By −Ez 0 Ex  Bz Ey −Ex 0

Or equivalently in index notation

0i 0i i ij k F = −F = −E and F = −ijkB (15) 1 Gµν =?F µν := µναβF , (16) 2 αβ where ? is called the hodge-star operator.

3 Note that the amount of freedom here is in the form of four equations linear in either ∂tλ or ∇~ λ 4For a description of tensors, their transformation properties, and upper and lower indices see appendix A

4 An immediate advantage is that the four Maxwell’s equations (1) and (2) reduce to only two written equations5

µν ∂µF = 0 (17) µν ∂µG = 0, (18) T where ∂µ = ∂t ∂x ∂y ∂z . Indeed evaluating the first equation for ν = 0 and ν = 1, 2 and 3 Gauss’s law and the Ampère-Maxwell law (equation (1)) can be retrieved respectively. Explicitly we find for ν = 0: µ0 i ∂µF = 0 + ∂iE = ∇~ · E~ = 0 Similarly the zero divergence of the B~ field and Faraday’s law (equation (2)) can be found by evaluating the second equation using ν = 0 and ν = 1, 2 and 3 respectively. Again notice that the formulations are invariant under the electromagnetic duality transformation (equation (7)). When electromagnetic charges are present the equations change to:

µν ν ∂µF = (Je) (19) µν ν ∂µG = (Jb) . (20)

The potential formulation in tensor notation

Just like in vector calculus, in tensor calculus the Maxwell’s equations can be formulated with respect to two potentials rather than the two electromagnetic tensors. As was seen above, the equations (17) and (18) are equivalent to the two equations ∇~ · B~ = 0 and ∇~ × E~ − ∂tA~ = 0, which implied the existence of two potentials. In turn, due to Poincaré’s lemma of the cohomology group, we can now state that: µν ˜ 4 4 µν ˜ µ ˜ν ν ˜µ ∂νF = 0 =⇒ ∃A : R → R s.t. F := dA = ∂ A − ∂ A . (21) A˜ is known as a (differential) 1-form, and therefore F is known as a 2-form. 6 Similarly we can find the dual electromagnetic potential A (based again on Poincaré’slemma):

µν 4 4 µν µ ν ν µ ∂νG = 0 =⇒ ∃A : R → R s.t. G = ∂ A − ∂ A (22) This A can again be identified with a differential 1-form and therefore a row vector, moreover A˜ is the dual of A. These formula’s agree with the vector calculus formulation and we find the identification of the differential 1-form with the four-vector A˜ := (φ, A~)T . Indeed, for F 01 we find:

F01 = ∂0A˜1 − ∂1A˜0

= −∂xφ − ∂tA˜x (9) = Ex We can find back Maxwell’s equations in the much simpler form (again in the Lorenz gauge!) ˜ Aµ = ∂µ(Je) and Aµ = ∂µ(Jb) (23) That finishes the formulations of EM. 5In this paper the Einstein summation convention is used and the convention that Greek indices (µ, ν) run over spacetime components 0 to 3 and Latin indices (i, j) run over space components 1 to 3 only 6It is not described what kind of object dA is but in the rest of this paper these objects are not mentioned hence it is not necessary to understand what such are objects, but keep in mind that there exists a completely geometrical formulation of electromagnetism and gauge theories. Just note that in the vector field in which we are working (Flat 4 Minkowski space which can be identified with R ) we can make the identification of a 1-form and a row-vector and a 2-form with a (rank-2 tensor and therefore) a matrix.

5 2 A magnetic monopole

An electric monopole q at the origin produces a ﬁeld E~q(~r) at position ~r = (x, y, z) 6= 0 and, when the particle moves in a system with ﬁelds E~ and B~ , is subjected to the Lorentz force F~ :

q ~r E~ (~r) = and F~ = q(E~ + ~v × B~ ) q 4π r3 As was noted before, due to the presence of high symmetry in Maxwell’s equations, it is very natural to introduce the magnetic monopole g at the origin, which produces a magnetic field B~ g(~r) at position ~r = (x, y, z) 6= 0 and, when the particle moves in a system with fields E~ and B~ , is subjected to the Lorentz force F~ . The definition of the magnetic charge then follows from the symmetry transformations (equation (7)) applied to the electric charge:

g ~r B~ (~r) = and F~ = g(B~ − ~v × E~ ) g 4π r3 As stated before, the existence of magnetic charges g is not considered but just assumed, and is possibly false. The value of the introduction of such a magnetic monopole lies in the theoretical implications, which will be explored later on in this paper.

6 3 Relativity

3.1 The structure of ﬂat spacetime: Minkowski space

To describe the invariance of Maxwell’s equations under Lorentz transformations it pays off to look at the mathematical description of special relativity. The space in which special relativity is described is a four-dimensional smooth manifold N 7, which incorporates three spatial dimensions and a time dimension. Together with an indefinite, non-degenerate, symmetric bilinear B form this manifold is called the Minkowski space M = (N,B). It is the manifold N together with an indefinite form B that makes the space M into a pseudo-Riemannian manifold. However, in special relativity this description, as a manifold, is unnecessary, because the space N has an extra structure, that of a vector space, which greatly reduces the difficulty of the description.

A semiformal deﬁnition

4 Thus, in the context of special relativity, the set N becomes a space: the vector space R . The vector space structure makes things a lot simpler. The resulting form B is called the Minkowski (generalized) metric, and therefore indicates how far points/vectors lie apart in the space. To look at this form we will speciﬁcally take a look at the form B in the special case that the four-dimensional vectors are the spacetime coordinates in other words ~v ∈ M such that ~v = (t, x, y, z) (where the speed of light is still c = 1).

The spacetime interval:

The Minkowski metric gives, as any (proper) metric does, a sense of distance between two points in the associated metric space. In this case the spacetime interval between two vectors ~v = (t, x, y, z)T , ~w = (t0, x0, y0, z0)T in the Minkowski space M is deﬁned to be:

(∆s)2 := (t − t0)2 − (x − x0)2 − (y − y0)2 − (z − z0)2.

A nicer notation to do this is to introduce the bilinear form as a matrix (or actually, a metric tensor) η := diag[1, −1, −1, −1]. The spacetime interval (∆s)2 between ~v and ~w then becomes:

2 T T µν (∆s) = ~v · η · ~w = ~w · η · ~v = vµη wν (24) where the symmetry of the bilinear form η is made explicit. In general relativity things become much harder. The space N loses a lot of its structure, it becomes a manifold again. Also the matrix η is non-diagonal and spacetime dependent because spacetime is curved.

Thus, the matrix η is equivalent to the bilinear form B, which will be called B : M × M → R : (~v, ~w) 7→ B(~v, ~w) = ~vT · η · ~w. Note that this distance can be both negative (called a space-like distance) and positive (time-like distance). The fact that this distance can be negative makes B an indeﬁnite form. B is non-degenerate means that B(~v,~v0) = 0 ∀~w ∈ M =⇒ ~v = 0. The last two properties, together with bilinearity and symmetry make B ≡ η a generalized metric.

7A (smooth) manifold is a set, in which around every point x there is a neighbourhood of points such that a set of numbers/coordinates can be assigned to every point in the neighbourhood of x and the numbers vary smoothly when ‘walking around the neighbourhood’. For example the point x itself can also be written as a set of coordinates x = (x1, ..., xn), xi ∈ R or C. Moreover, any point on the manifold lies in such a neighbourhood and when one jumps from one neighbourhood to another, it can be done smoothly (in the sense that when one walks around in the neighbourhood drawing a line behind oneself, jumping from every point in the line draws another line in the other neighbourhood. Moreover this happens smoothly). Formally a manifold is a set of points together with a structure indicating what smoothness means, an intuitive example being the smoothness of the line in the story above. Physically this means that one can differentiate, integrate and do other useful stuff on these manifolds, which in all generality of the manifold makes proving things easier after mastering the definitions

7 This completes our definition of the Minkowski space M: A 4-dimensional vector space with a(n) (indefinite, non-degenerate, symmetric) bilinear form B which is a generalized metric and can be identified as a metric tensor. Together with the form B again we immediately find a space which is identified as a 4-dimensional pseudo-Riemannian manifold (or actually in the realm of special relativity it is a pseudo-Euclidean space since it is flat).

3.2 Lorentz transformations

The group of orthogonal transformations O(3) acting on a three-dimensional (real) linear space leaves distances between vectors invariant. These kind of actions, that leave distance invariant are defined as isometries. An example of elements in O(3) acting on such a space are linear translations, rotations and reflections. Just like that the set of isometries of spacetime form a group called the Poincarégroup. Again it includes (four-dimensional) linear translations, rotations and boosts, which are transformations connecting two inertial frames i.e. reference frames moving with constant velocity with respect to one another. The subset of the Poincaréconsisting of rotations boosts also form a group and called the Lorentz group. The transformation of a Lorentz boost in the x-direction takes the form x¯0 = γ(x0 − βx1)x ¯1 = γ(x1 − βx0)x ¯2 = x2 x¯3 = x3, (25) if we denote by xµ andx ¯µ, the four-vectors in the two inertial frames, between which the Lorentz v boost is made and where β = c = v (the frames moving with constant velocity ~v with respect to each other). Because of the linear structure of the space, it is easiest to write the boost in matrix notation. For a boost xΛ in x-direction this becomes

 γ −γβ 0 0 −γβ γ 0 0 xΛ :=   . (26)  0 0 1 0 0 0 0 1

Similarly for a transformation in any spatial direction described by the (normalized) spatial vector 8 rˆ we can define rΛ by rotating such that ~r 7→ ~x, then performing the boost xΛ, then rotating backwards [11]. Also in this setting one arrives at the definition of a four-vector. A four-vector µ ∼ 4 is a vector a = (t, x, y, z) ∈ V = R of which the components transform under any Lorentz transformations Λ as ν a¯µ = Λµνa µ 4 Fora ¯ ∈ R as in equation. The definition of a four-vector thus arises from the way it transforms under the isometries in the Lorentz group. In a more general setting this leads one to the important definition of a tensor, which in linear spaces transforms just like that under any linear transformation. This means that elements of Minkowski space M which are 1-tensors can be seen as four-vectors and furthermore that the transformation properties of a four-vector can be slightly extended to ν µ −1 µν a¯µ = Λµνa or equivalentlya ¯ = (Λ ) aν, (27) 4 4 µ for any linear transformation λ : R → R . We have already have seen such a vector: ~v = x := (t, x, y, z)T which is called the four-position vector in Cartesian coordinates. From the fact that isometries in spacetime preserve distance and equation (27) we can conclude that four-vectors have constant spacetime interval when acted upon by elements in the Poincarégroup. One can now generalize from the four-position xµ to other four-vectors aµ by setting xµ 7→ aµ in the definition of the metric, equation (25). Some other examples of four-vectors are (making c explicit and without proof):

8The convention used from this point on is that a four-vector is generally denoted in the form v = vµ = (time component, ~v) such that an arrow sign means the spatial part of a four-vector

8 E The energy-momentum four-vector p := ( c , px, py, pz), where E is the energy and ~p = 2 (px, py, pz) is the linear momentum. This gives the conserved Minkowski interval E = i 2 2 4 pip c + m c . Which is the well known (relativistic) energy-momentum relation (m being the rest mass). Note that the conserved Minkowski interval (squared) was given the value m2c2, which comes about when considering the restframe.

The (electric) current density four-vector (or four-current) Je defined as in equation (7). This ∂ρ ~ ~ gives the conserved Minkowski interval ∂t = ∇·J. Which is just equation for the conservation of charge (the continuity equation). In this case defined for a system where no charge is created or destroyed. ∂ ∂ ∂ ∂ 1 2 2 The four-gradient ∂µ := ( ∂t , ∂x , ∂y , ∂z ). This gives the conserved ‘norm’ − c2 ∂t + ∇ , which is precisely the d’Alembertian operator as defined in equation (12). Hence we can apply, in any inertial reference frame, without further worrying, the d’Alembertian operator on a four-vector and get an object that transforms nicely under actions of the Poincarégroup. Finally, the electromagnetic four-potential Aµ := (φ/c, A~), like in equation (12) (note that the tilde is omitted, it was defined as A˜µ). This conserved interval is not well-determined because of the gauge freedom.

Note that the equation for the Lorenz gauge ∇~ · A~ = −∂tφ is precisely defined such that the value µ ∂µA = 0. In this gauge the last three four-vectors, together with equation (12) can be used to prove relativistic invariance of electromagnetism. Hence fixing of the gauge and looking at the electromagnetic theory in the Lorenz gauge leads to relativistic invariance. Note that the reason for this is that the electromagnetic four-potential, the four-gradient and the four-current are four- vectors, which is not proven. Finally, relativistic invariance holds in other gauges if another gauge 4 transformation is made with a scalar function λ : R → R, as defined in the previous section, that satisfies the property λ = 0. Hence the way one proves that EM is relativistically invariant easily is, in hindsight, to take a very general approach of spacetime and relativity (and omitting the proofs that the stated objects are indeed four-vectors). Then one sees that relativistic invariance of EM is equivalent to the invariance of equation (12) which means that EM is relativistically invariant (in the Lorenz gauge!) if one can prove that the four-gradient, the four-potential and the four-current transform like 1-tensors. Let’s go back to tensor notation. For the EM tensor the transformation property becomes

¯µν µ µ αβ F = ΛαΛβF , (28) see also appendix A. And in general the relativistic invariance of scalars takes an even simpler form in tensor notation: one just needs for every upper index an equal lower index such that the Λ matrices cancel.

3.3 Proper time

It is well known that time and length measurements have different results in different inertial frames. Therefore the relativistically invariant proper time ∆τ and proper length ` have been introduced. Consider two inertial frames S and S0 moving with velocity ~v = vxˆ relative to the first. If we measure the time ∆t and ∆t0 (in frame S and S0 respectively) between two events, that is, points in spacetime, such that the distance is zero in the first frame S: ∆x = 0 then we find:

∆t0 = γ(v)∆t (when ∆x = 0).

This means that the time measured between ‘two things happening’ is always minimized if we measure it in the inertial reference frame such that the spatial distance between the two events is

9 zero. This reference frame is called the (instantaneous) rest frame. Therefore the saying ‘time is relative’ gains its value. However, we have seen that it is nice to work with relativistic invariant objects. In other words properties that stay the same when Lorentz transformations are applied. Hence the proper time (interval) ∆τ is precisely deﬁned as the time interval as measured in the instantaneous rest frame: ∆τ := ∆t. Using the spacetime interval it can be shown that, when one looks inﬁnitesimally close, Z Z dt ∆τ = dτ = =⇒ dτ = γ(v)dt. γ(v(t)))

Similarly the proper length ` can be deﬁned.

10 4 Electromagnetism and relativity

Maxwell’s laws (equations (1) - (4)) give a consistent description of classical electromagnetic systems. However the interpretation of the cause of these phenomena may differ. For example, a Lorentz boost in the x-direction with speed v transforms the electric field E = (Ex,Ey,Ez) and magnetic 9 field B = (Bx,By,Bz) by the following rules

E¯x = Ex, E¯y = γ(Ey − vBz), E¯z = γ(Ez + vBy), v v B¯ = B , B¯ = γ(B − E ), B¯ = γ(B − E ). x x y y c2 z z z c2 y To check invariance of the Lorentz boost we can try to find the four equations in (1) and (2) by evaluating the divergence and the curl of the E¯ and the B¯ field. To make this explicit: We want to show that ∇E = 0 ⇐⇒ ∇¯ E¯ = 0. Where ∇¯ indicates the divergence in the inertial frame of reference. Using equation (25) we find that (∂x¯, ∂y¯, ∂z¯) ≡ ∇¯ = γ(∂x − β∂t), ∂y, ∂z

Resulting in the equation for Gauss’s law in the second reference frame

∇¯ · E¯ = ∂x¯Ex + γ∂y¯(Ey − vBz) + γ∂z¯(Ez + vBy) = γ ∇ · E − xjk∂jBk − ∂tEx (1) = −γ 1jk∂jBk + ∂tEx (1) = 0

Where abc is the Levi-Civita symbol. Similar computations result in the invariance of the other Maxwell equations under Lorentz boosts. This argument alone is of course not enough to generalize relativistic invariance to the whole of EM. Some extra remarks make this a little more general First of all, we now have proven that ∇E = 0 =⇒ ∇¯ E¯ = 0. But the other implication follows from taking an inverted Lorentz boost Secondly, we have proven the equivalence for a Lorentz boost in only the x-direction. However, as was stated before any Lorentz boost can be generated when Lorentz boost in the x-direction is conjugated with a spatial rotation. Thirdly, the equivalence holds not only for Maxwell’s equations in vacuum in the absence of charges. When charges are added the equations stay Lorentz invariant Fourthly, the equivalence has only been proven for a Lorentz boost and not for the more general transformations general the Poincarégroup. Invariance under these transformation will be assumed to hold It was already noted that relativistic invariance is easier proven in the potential formulation, and specifically in tensor notation. Like in the last chapter, again the potential formulation in vector calculus form (equation (12)) is considered. It became apparent already in the last chapter that ¯ µ the d’Alembertian operator satisfies the property = = ∂µ∂ because of the property that the norm of any four-vector is invariant under Lorentz transformations. Furthermore we stated that µ the electromagnetic four-potential A := (φ/c, A~) and the electric four-current Je = (ρ, Jx,Jy,Jz) are both four-vectors and hence transform as in equation (27), such that

¯ 2 ¯µ ¯ 2 µ A = J ⇐⇒ A = J . (29) This means that Maxwell’s equations (in potential form) are relativistically invariant.

9Denoted by bold symbols in this section due to notational convenience

11 Looking now at equation (28) together with the fact that the four-current transforms like a four- vector, equations (17) and (18) can be proven relativistically invariant as well. Both in tensor and vector notation, the trick was to use the transformation properties of four- vectors, which were derived from the more general approach of spacetime and the intrinsic notion of what it means to be relativistically invariant. Note that this proves that EM (described by Maxwell’s equations and the Lorentz force law) is relativistically invariant if we can prove that the Lorentz force Law is also relativistically invariant. This fact will be considered later on.

12 5 Lagrangian mechanics and actions

In Newtonian mechanics it is standard to set up the differential equations (of motion) for a system n with (a set of) particles described by coordinates (q1, ..., qn) =: ~q ∈ R , where n ∈ N, resulting in the equations of motion which, if solvable, can be used to find the trajectory of an object. This is done by using Newtons second law. There is a second way to find the equations of motion of a system containing particles: the Lagrangian formulation. Looking at relatively simple systems the Lagrangian L is defined as L = K − U, where the kinetic energy K minus the potential energy U were used. Similarly to Newtonian mechanics, Lagrangian mechanics provides the equations of motion, which are now deduced from the so-called Lagrange equations (equation (30) below). Based on the principle of least action (Hamilton’s principle) and the definition of that action, it can be shown that the two formulations are equivalent. In summary

d~q ∂L ∂ ∂L L(~q, ~q˙ := ) = T (~q˙)−U(~q) with Euler-Lagrange equations = ∀i ∈ {1, ..., n}. (30) dt ∂qi ∂t ∂q˙i Note that for such a system the coordinates together with their time derivatives in a speciﬁc moment in time are enough to determine the kinematics of the system.

5.1 Classical mechanics in integral form

Note that there are, in electromagnetism, two ways of stating Maxwell’s equations. One in differential form, the other in integral form. Only the former is stated in this paper and it is referred to as Maxwell’s equations in differential form, for it gives a set of differential equations to describe an electromagnetic system. An equivalent set of equations can be stated, in which they take the form of integrals. An analogous case arises in (classical) mechanics. In mechanics the problem arising is almost always finding the motion of a set of objects. Newtonian mechanics ‘in differential form’ is done by using the second law to set up a second order differential equation and find the equations of motion. The other approach gives a set of equations starting not with the second law but, in consistency with electrodynamics, with an integral, called the action S. Using the calculus of variations the integral can be reduced to a set of (differential) equations of motion. An action S can be seen, intuitively, as an object describing the amount of ‘stress’ (in units proportional to energy, since we integrate over the Lagrangian) that a particle experiences when it takes a certain path. The only real information of the path is its starting and final point, which is contained in the boundaries of the action. Mathematically it is a functional, which means that it takes a space of functions as a domain and maps to the (real) numbers. The form of the action in classical mechanics then becomes S[~q(t)] (∈ R), where Z S(~q(t)) = L(~q(t), ~q˙(t), t) dt. t Like before, ~q(t) is the path of the particle, parametrized by the time and the function L is called the Lagrangian of the system. As was stated before, for simple systems the Lagrangian is just the kinetic energy K minus the potential energy U: L = K − U. Therefore a free particle of mass m, ˙ 1 ˙2 which can be described by coordinates (~q, ~q) has the Lagrangian L = 2 m~q . To get from this integral to a set of equations of motion one uses Hamilton’s principle, which states that the action needs to be minimized (or actually extermized). This is physically justified since one then minimizes a value proportional to the energy ([S] = [E][t]). It can then be shown, using calculus of variations, that under certain conditions we then find the Euler-Lagrange equations, see equation

13 (30). Moreover, these equations are consistent with Newtons formulation of classical mechanics. This is easily seen when one considers the Lagrangian of a particle described by coordinates ~q(t) in 1 2 a potential U(q) with kinetic energy K = 2 mq˙ .

5.2 The relativistic Lagrangian

Classical mechanics in both differential form and integral form can be extended to be relativistically invariant, here the latter form will be considered only. Originally, to define the classical Lagrangian, invariance under Galilean transformations was used. To find a Lagrangian consistent with special relativity, a similar argument can be used. Now it must be invariant under Lorentz transformations. Heuristically, the argument can be made quite intuitive. The most important change is that one uses the idea that a free particle follows the shortest trajectory in space, when considered classically. Goin to the special relativistically invariant formulation, the kinetic energy term of a free particle therefore changes, since the particle now takes the shortest path in spacetime. We have seen in a previous section that the spacetime interval (∆s)2 := (t−t0)2−(x−x0)2−(y−y0)2−(z−z0)2 is invariant under any Lorentz (Poincaré)transformation. Denoting now (a − a0) = ∆a for a ∈ {x, y, z, t} and looking infinitesimally (∆s → ds) we find that (making c explicit):

ds2 = −dx2 − dy2 − dz2 + c2dt2 dx2 + dy2 + dz2 = − + 1 c2dt2 v2 = −( − 1)dt2 c2 = γ−2dt, for a particle with speed v and inﬁnitesimal interval ds. The idea now is that the action is determined by the shortest path/spacetime interval, such that Z Z S ∝ ds2 = γ−2dt s(t) t

However this integral is not very well-deﬁned, since ds2 = 2sds, and s is not something that is (well-)deﬁned. The relativistic Lagrangian, up to a constant then becomes r Z Z v2 S ∝ ds = 1 − 2 dt. s(t) t c

q v2 The Lagrangian then becomes L ∝ 1 − c2 . The constant can then be found by a Taylor expansion v with respect to c around zero (approximating in the classical limit v → 0): r v2 L = −mc2 1 − . (31) c2 The Euler-Lagrange equations also change, this will be discussed later. Note ﬁnally that this is the action of a free particle. Such that the ‘stress’ it experiences now depends on the (length of the) 1 2 path in spacetime and therefore the Lagrangian changes from L = 2 mv to equation (31).

5.3 The classical electromagnetic Lagrangian

For a particle with mass m and electric charge q, described by Cartesian coordinates (~r, ~r˙) in a classical electromagnetic system it can be shown that the Lagrangian takes the form: 1 L = m~r˙2 − q(φ − ~r˙ · A~). (32) 2

14 Where A~ is, as before, the magnetic vector potential and φ is the electric potential. Note ﬁnally that the dot indicates a derivation with respect to coordinate time (and not proper time). Using the Euler-Lagrange equations we ﬁnd precisely the Lorentz force law. For example, the x-coordinate gives:

∂L ∂φ ∂A~ = −q( − ~r˙ ) ∂x ∂x ∂x ∂A ∂A ∂A ∂A = mx¨ + q(x ˙ x +y ˙ x +z ˙ x + x ) ∂x ∂y ∂z ∂t ∂ ∂L = , ∂t ∂x˙ since A~ = A~(x, y, z, t). This gives precisely the x-coordinate of the Lorentz-Force law F = q(E + ~r˙ × B~ ) if we use equation (8) and (9).

5.4 The relativistic electromagnetic Lagrangian

From the discussion above it follows that, for the same electromagnetic system (again making c explicit) s ~r˙2 L = −mc2 1 − − q(φ − ~r˙ · A~), (33) c2 Since the kinetic energy changes as in section 5.2 and the potential terms stay the same. Here m is the rest mass of the particle. The Euler-Lagrange equations then give precisely the relativistic version of the Lorentz force law: d~p ∂ (γ(~r˙) m~r˙) = = q(E~ + ~r˙ × B~ ). (34) t dt Now we will use the tensor notation and also consider on magnetic charge g, which are point sources of a B~ field analogous to electrical charge. Without the magnetic charge g we find the relativistic Lagrangian 1 L(τ) = muµu + quµA , (35) 2 µ µ µ µ where u = ∂τ x is the four-velocity. Comparing with the Lagrangian found in vector calculus we 2 1 v − 2 find a factor of γ = (1 − c2 ) difference. However, the Lagrangian remains physically equivalent when multiplied by a factor γ−1. The Euler Lagrange equations become:

∂L ∂ ∂L d ∂L = = . ∂xµ ∂t ∂x˙ µ dτ ∂uµ

ν ∂Aν d Such that we find the equation qu ∂xµ = dτ (muµ + qAµ), or after rearranging terms and using the chain rule: ∂A ∂A d quν ν − µ = (mu ) ∂xµ ∂xν dτ µ And now using the definition of the electromagnetic tensor, we find finally that

d dp quνF = (mu ) = µ , (36) µν dτ µ dτ where the energy-momentum vector pµ := muµ was used. Comparing with the relativistic version of Newtons law, we find that the first term can be identified with a force. Indeed the spatial coordinates µ = 1, 2, 3 just return the relativistic Lorentz force law, equation (34).

15 Taking into account also the magnetic charge, by symmetry arguments (using chapter 2 and equation (35)) we ﬁnd: 1 g µ L(τ) = muµu + quµA + A . 2 µ µ u µ One can check that this gives the modiﬁed Lorentz force d quνF + guν ?F = (mu ), µν µν dτ µ consistent with equation (34).

5.5 The Hamiltonian

Besides the Newtonian and the Lagrangian formulation, there is one more: The Hamiltonian formulation. This formulation has more in common with the Lagrangian formulation than with the Newtonian and it is based on the function H, called the Hamiltonian. Starting with a set of coordinates (~q, ~q˙), one can arrive at the Hamiltonian H(~q, ~p,t) via Lagrangian L(~q, ~q,˙ t) through the equation:

X ˙ H(~q, ~p,t) = piq˙i − L(~q, ~q, t) i

Where p is deﬁned as the conjugate momenta of q : p := ∂L . i i i ∂q˙i The equations of motion become ∂H ∂H q˙i = andp ˙i = − . ∂pi ∂qi The most considerable advantage of the Hamiltonian in this case is the fact that it extends, like the Lagrangian formulation, to quantum mechanics very easily. In fact, one can show from this deﬁnition that the Hamiltonian with respect to the electromagnetic Lagrangian (32) becomes:

(~p − qA~)2 H = + qφ. (37) 2m

This comes from minimal coupling: when a particle with linear momentum ~p is in a system and a magnetic ﬁeld B~ with potential A~ is added to the system the correct Hamiltonian can be found by changing ~p 7→ ~p − qA~.

16 Part II Symmetries, ﬁelds, gauge theories and the Higgs mechanism

An important tool in the discussion of monopoles will be gauge theories and symmetry lies at the heart of these theories. Therefore these topics will be discussed first and a look at the monopoles will be taken in part III. The ’t Hooft-Polyakov monopole, for example, arises as a solution of the dynamical equations in a gauge theory. To describe gauge theories a classification of symmetries will be needed first. The most important distinction that will be made is between physical symmetries and gauge symmetries. A gauge symmetry is, in principle, a symmetry of the theory. One then has a description of nature that can be given in multiple ways. Thus, a gauge symmetry shows a redundancy in the theory. One has ‘too much’ information, which can be extracted and used to ones advantage. An example of this has already been seen: electromagnetism was shown to be invariant under (gauge) transformations on the electromagnetic potential, defined by Aµ 7→ Aµ + ∂µλ. This was used to our advantage by choosing the Lorentz gauge, such that EM became relativistically invariant. A second important distinction is between local and global symmetries, this will be explained later on. Sometimes a theory comes naturally with a global (gauge) symmetry. When one demands this symmetry to be local the dynamical equations change. A new theory then arises, which is precisely a gauge theory. Gauge theories are extremely important in physics. General relativity, which describes gravity, and the Standard Model, which describe the fundamental forces, are both based on gauge theories. Furthermore, for the ’t Hooft-Polyakov monopole one leaves the particle domain and considers fields, so a short introduction of the crossover from a ‘discrete’ Lagrangian for particles and a ‘continuous’ one for fields will be given. Noethers theorem for conserved currents, which has to do with symmetries and conservation laws will be stated there without proof. Finally, the Higgs mechanism will also be explained very superficially. To understand this mechanism, the (spontaneous) breaking of symmetry will be discussed first. The Higgs mechanism is also important for the Standard model, as it is responsible for the mass of some important gauge bosons, like the weak W and Z bosons. For the ’t Hooft-Polyakov monopole one can then identify the massless electromagnetic fields due to the acquiring of mass of other fields, such that a monopole solution can be identified.

17 1 Symmetry

As in the physiognomy of humans, symmetry in theories is often considered aesthetically appealing. Already in classical times symmetries were considered in the describe nature’s elements. It was Plato who wrote in his dialogues as the persona Timaeus of Locri [12]. He described the elements as geometrical shapes with a high degree of symmetry: regular polyhedra. In Kepler’s Mysterium Cosmographicum Kepler drew the elements of nature and their shapes as seen in ﬁgure 1.

Figure 1: The elements as in Kepler’s Mysterium Cosmographicum. Image thanks to https: //en.wikipedia.org/wiki/Platonic_solid

As the ages went by, a strong mathematical theory was developed that could describe symmetries. In modern mathematics, they are described by groups and in this chapter this is precisely the definition of a symmetry that is used such that every symmetry comes automatically with a (symmetry) group G and vice versa. Furthermore, every symmetry comes with a symmetry transformation, which is defined to be an action of the group on a specific system. A system is invariant with respect to the (transformations of a certain) symmetry if a transformation preserves the physical properties of the system. For example, Lie groups describe (continuous) transformations. The transformation of the Lie group G = SO(3) consists of rotations in three dimensional space, which can be seen (in the fundamental representation) as matrix multiplication with a matrix g ∈ SO(3) applied to vectors 3 in physical space R . As an action, SO(3) causes any three-dimensional space to rotate. This group will be important in the description of the ’t Hooft-Polyakov monopole. Another example we encountered of a symmetry group is the Lorentz group consisting of the Lorentz transformations, which is a subgroup of the Poincaré(symmetry) group consisting of the isometries of Minkowski spacetime. In particular, the system being invariant means – in the Lagrangian formalism – that a symmetry transformation leaves the action invariant. The equations of motion derived from this action are then the same, however the description of the path of the particle/history of the fields can be changed. For example, when one rotates a spherically symmetric system, the physics does not change, but the actual path of a particle will be rotated. It is symmetry (EM duality) that started this paper and it is symmetry that will be the topic of this chapter. As was stated before, we will see that it provides some tools, necessary to describe an extremely important theory in theoretical physics: gauge theory. Furthermore we will see that the breaking of symmetry in such a theory has led physicists to a mechanism, again extremely important in physics, called the Higgs mechanism: A mechanism that gives mass to particles in the standard model such as the weak force gauge bosons W ± and Z.

1.1 Classiﬁcation of symmetries

There are many ways to classify symmetries and the use of them [13]. The first distinction that can be made is the way one uses symmetry: as an argument or as a principle. In specific physical systems where one identifies a certain symmetry and uses this to derive specific consequences, then it is used as an argument. This is done for example in Noether’s theorem, see subsection 2.2, part

18 II. Symmetry can also be used, as in the case of inertial frames in special relativity, as a principle. Then it is fundamental for the theory. Another important example of a symmetry principle is that nature is assumed to be fundamentally invariant under certain types of local gauge transformations, which gives rise to gauge theories. These are to be defined later on. Furthermore we have already seen that a magnetic monopole can be expected to exist if EM duality symmetry is used as an argument. Besides the use of symmetry, one can classify symmetries (together with the system they are acting on) according to certain properties, for example: Continuous vs discrete Global vs local Physical vs gauge Abelian vs non-abelian Exact vs approximate The first distinction is quite intuitive. A discrete (continuous) symmetry is linked to a discrete (continuous) group. An example of a continuous group is the previously encountered group G = 3 SO(3), given physically by rotations in physical space R . An example of a discrete symmetry is given by the parity transformation in physical space, usually denoted by P . It is represented physically by a point reflection with respect to the origin. A symmetry is global when elements in the group g ∈ G acting on the system are independent of the spacetime coordinates. When elements of the group are dependent of spacetime g = g(xµ) for µ 3 x ∈ R , the symmetry of the group is called local. For example, if a system invariant under transformations of the non-abelian group G = SO(3) then it is global if every point in space is rotated by the same amount (g ∈ SO(3) constant over spacetime). It is local when g can vary (continuously) over spacetime. Then the matrix elements are functions of spacetime and as a result every point in spacetime can be rotated by a different amount. Note finally that local (symmetry) invariance implies global invariance. The third distinction is between physical and gauge symmetries. A physical symmetry is one where the group acts on an object which can be measured/seen in the system. For example the regular polyhedra in empty space can be physically rotated in certain ways without changing the configuration. A gauge symmetry is a symmetry of the theory. Any gauge symmetry implies an intrinsic redundancy in the theory, more freedom than is needed. The non-uniqueness of the electromagnetic potential as seen in section 1, part I, is an example of a gauge symmetry. The group 4 linked to the symmetry is the group of differentiable scalar functions {λ : R → R| λ differentiable}, the operation being summation. Recall from part I that the transformation is given by Aµ 7→ Aµ + ∂µλ. Note that a gauge transformation can be seen as a transformation to an entirely new description of the same physical system, in the sense that a gauge invariant theory defines a set of equivalent theories, linked together by gauge transformations. A(n) (non-)abelian symmetry is one such that the corresponding symmetry group is (non-)abelian. In an abelian group the elements commute. The final distinction can be stated compactly: An exact symmetry is always valid; An approximate symmetry is only valid sometimes, for a certain set of values of some parameters (e.g. above a certain temperature). In the latter case the symmetry can be broken, which means that the amount of symmetry of the system reduces and the system is now invariant only under transformations of a subgroup of the original group H ⊂ G (possibly trivial). A better look at the meaning of local symmetries reveals an interesting feature [14]. Consider the

19 path of a particle evaluated by some action. Since a local gauge transformation can change the path of the particle locally, the endpoints of the path can be left unchanged. However then we have a symmetry transformation where the physical state (described by the endpoints of the path) is left unchanged, but the theory changes. So two theories, which should imply the same physical states, imply different physical states. In other words the theory is not well-defined. To have a well-defined theory (a deterministic theory), this must mean that the symmetry must be a gauge symmetry. So, any local symmetry in a deterministic theory must be a gauge theory. Furthermore, any gauge symmetry reveals a non-physical object in the theory. When one wants to retrieve the physics for a system. One must often fix a gauge (as done in the case of the electromagnetic potential). Otherwise one will be working with a non-unique description of a system such that the configuration between two systems (or a system at different spacetime coordinates) will be ill-defined. However, when really fixing the gauge, the new/chosen/representative theory with fixed gauge loses it’s invariance with respect to the local gauge symmetry (gauge invariance). It is precisely local gauge transformations that are important in physics. And the demand for invariance under varying gauge transformations at different points in spacetime requires the introduction of an object that carries information about the variation of the transformation/group element for every point in spacetime. This results in the introduction of fields, called the gauge fields. This will become clearer when discussing gauge theories.

1.2 Symmetry breaking

It was already stated that a property of a symmetry of some system is that it can be broken. Math- ematically, this means that the symmetry group is projected onto one of its subgroups. Intuitively, this means that the degree of symmetry is reduced. This usually happens when a system that has an exact symmetry is changed such that it becomes approximate and for certain values of the parameters (when they cross a so-called critical value) symmetry can be broken. Small fluctuations then decide the fate of the system (for physical symmetries). Symmetry breaking can be distinguished into two categories: Explicit symmetry breaking and spontaneous symmetry breaking (SSB). A set of examples will be used to clarify this. 1: Consider a bendable rod between your fingers, which you hold vertically, see figure 2. When applying no force the rod is locally, rotationally symmetric around its own axis. The symmetry group is the subgroup of rotations around the symmetry axis SO(2) ⊂ SO(3).

Figure 2: Example of symmetry breaking, both explicit and spontaneous SSB. Image thanks to [14]

20 2: One can also consider a big three-dimensional ferromagnet spread out over (a big part of) empty space. This ferromagnet will be modelled by a three-dimensional lattice with spins at the lattice points. A simple Hamiltonian (of the Ising model) then consists of the sum of the inner product of P neighbouring spin orientation of the elements in the crystal H = γ neighbours ~s(~ri) · ~s(~rj), where 3 3 s(x, y, z) ∈ R , γ < 0, ~ri ∈ Z are the spins s, the coupling constant between spins γ and positions ~ri of lattice point i on the lattice respectively. At high temperature the spins are oriented randomly, no long-range physical symmetry is present. However, there is a local (gauge) symmetry present due to the random orientation: rotations. The symmetry group is the Lie group of rotations SO(3). 3: One ﬁnal example is a ball in a Mexican hat, as shown in ﬁgure 3. When the ball rests on the unstable equilibrium at the top there is global rotational symmetry. Again the symmetry group is SO(2).

Figure 3: The Mexican hat geometry. A scalar particle/ball in the middle is invariant under physical, global, rotational symmetry transformations around the z-axis. When the ball rolls down into the trough and comes to rest the rotational symmetry is broken. The potential shown here is given by V (x, y) = ((x2 + y2) − 25)2.

In all three examples symmetry can be broken, both explicitly and spontaneously.

1.2.1 Explicit symmetry breaking

A symmetry is broken explicitly if the dynamical equations (in this paper the Lagrangian density) describing the system are not invariant under the symmetry transformation. In other words there

21 is an asymmetric term in the dynamical equations. Therefore the physical system and specifically the ground state of the system is not symmetrically invariant. For example, the weak theory turns out not to be invariant under parity transformations, thus a parity variant term was introduced in the Lagrangian. In the three examples above this can take different forms. 1: For the rod we can apply a force in a direction different to the axis of rotational symmetry. As shown in figure 2 2: For the ferromagnet we can add to the system a magnetic field such that the spins align. The symmetry is then broken due to an explicit term added to the Hamiltonian. 3: For the case of a ball in the Mexican hat geometry, again one can add a force beside gravity to the system such that the symmetry is broken explicitly.

1.2.2 Spontaneous symmetry breaking

This type of SSB in systems where the dynamical equations are invariant under a symmetry transformation but the ground state of the system is not. The ‘spontaneity’ of the breaking is due to the absence of an explicit asymmetrical term in the theory. Let’s look at the examples again. 1: When you apply too much force to the rod, it bends. The extra force is again rotationally invariant such that the theory keeps it symmetry in a way. The physical system loses its symmetry however, since the rod may bend in any horizontal direction with equal probability. The new ’ground state’ of the rod can be in the direction of any axis and ground states are related by the action of the old (unbroken) symmetry group. 2: When the temperature crosses below a critical value, the Curie temperature, the spins of the ferromagnet align. The orientation of the spins in the ground state is arbitrary, it can be along any axis with equal probability. The different ground states are related by the action of the old symmetry group. 3: For the ball on the Mexican hat, small fluctuations may cause the ball to roll in the trough. Symmetry is broken and any position in the through is equally probable. Different ground states are related by the old symmetry group. Note that two properties are reoccurring when a symmetry is spontaneously broken: The ground states are degenerate and all ground states are equally valid Symmetry is broken but becomes ’hidden’: The old symmetry group gives a well-defined and surjective action on the set of the above degenerate ground states The SSB of continuous local (gauge) symmetries turn out to be useful. In the Higgs mechanism, which provides an explanation for the mass of certain gauge bosons in the Standard model, is based on this type of symmetry breaking. It is a Mexican hat shaped potential that is the backbone of the Higgs mechanism, which will be explained later on.

22 2 Some classical ﬁeld theory

Before arriving at the Higgs mechanism, gauge theories need to be explained. Before arriving at gauge theories, the dynamics of continuous fields need to be explained. Therefore, the classical Lagrangian density describing fields is considered. When describing the Lagrangian formulation the path of a particle was considered. Many systems do not only have localized objects (particle) but also objects that are spread out over spacetime, called fields. The concept of a field comes very natural given that the electric field E~ and magnetic field B~ have already been introduced. The goal remains to extremize the action S, though two things change. First, the extremized action gives us the equations of motion. In the case of particles, these equations of motion describe the path. In the case of fields, the described dynamics is called the history. Second, the important µ object is not just the Lagrangian but the Lagrangian density L(φ, ∂µφ, x ), a function of the fields, their derivatives and the coordinates. The action becomes the Lagrangian density integrated over spacetime coordinates dt d~x. Z Z S = L dt = L d~xdt

This is precisely because the ﬁelds are spread out over space as well, whereas the position of a particle (described by the Lagrangian) was just spread out over time. Finally, note that the Lagrangian density is often just referred to as the Lagrangian. The Euler-Lagrange equations written with respect to the Lagrangian density are:

∂L ∂L ∂µ = . (38) ∂(∂µφ) ∂φ

2.1 General forms of the Lagrangian density

Here three forms of general Lagrangian densities are given: for a scalar field, a vector field and a spinor field [15].

4 For a scalar ﬁeld φ : R → R the general Lagrangian density takes becomes the Klein-Gordon Lagrangian (making c explicit):

µ 1 mc 2 2 L = ∂µφ∂ φ − ( ) φ , (39) 2 ~ consisting of a kinetic term and a mass term respectively.

µ 4 4 For a vector ﬁeld A : R → R we get the Proca Lagrangian:

1 µν 1 mc 2 ν L = − F Fµν − ( ) A Aν, (40) 16π 8π ~ where F µν = ∂µAν − ∂νAµ. The equation of motion becomes: µν mc 2 ν ∂µF + ( ) A = 0 (41) ~

These equations bear almost the same structure as Maxwell’s equations in vacuum. Setting the mass to zero m = 0 and identifying Aµ as the electromagnetic potential with absence of sources we find that it is a massless vector field. Indeed photons, the quantizations of the field, are in general

23 1 T massless. For a spinor ﬁeld (with spin- 2 particles) ψ = ψ1 ψ2 ψ3 ψ4 , for wavefunctions ψµ, the Lagrangian becomes the Dirac Lagrangian: µ 2 ¯ L = i~cψγ ∂µψ − (mc )ψψ, (42) where ψ¯ is the adjoint of ψ and γµ are the Dirac matrices. In general the Lagrangian density of a scalar ﬁeld φ takes the form

∞ 2 2 X i L = (∂µφ) + C + αφ + βφ + γiφ , (43) i=3 for C, α, β, γi ∈ R. Here the first term represents the kinetic energy of the field. The second term is a constant of no importance for the theory. The third term has no direct interpretation. The fourth, quadratic term is the mass term, where β is a function of the mass (β > 0 =⇒ m > 0). The higher order terms are interaction terms of the field (with itself). Later it will be shown that the Lagrangian of a massless scalar field φ together with the Mexican hat potential are one of the starting ingredients for the Higgs mechanism.

The Lagrangian density for an electromagnetic system

Up until now we have only considered the regular Lagrangian (both relativistic and non-relativistic) L(t) or L(τ). The Lagrangian density for an electromagnetic system will be given here. 1 µ µ The starting point lies at the Lagrangian L(τ) = 2 mu uµ + qu Aµ. This Lagrangian consists of the (first) kinetic term and the interaction term. Of course when we go from particles to fields, the charges get spread out and become currents µ µ µ qu 7→ J . The second term then becomes J Aµ. The kinetic term changes a bit less obvious, like in the change from the classical Lagrangian to the relativistic Lagrangian of a free particle. In field theory, the kinetic term is often called the field term and turns out to be − 1 F µνF (making µ explicit). The reason to consider this term is 4µ0 µν 0 because it is is one of the only terms that is both gauge invariant (seen later) and relativistically invariant. Considering the above discussion together with Maxwell’s equations and equation (41), the ‘Maxwell’ Lagrangian density for an electromagnetic system is then given by:

1 µν µ L = Lﬁeld + Linteraction = − F Fµν − AµJ (44) 4µ0

2.2 Noethers theorem

It was noted that any symmetry comes with a group. Now, without proof, it is stated that any continuous transformation of the fields φj (not necessarily scalar fields) and (spacetime) coordinates xi that leaves the action S = R L dx invariant corresponds to conserved function of the fields and k their derivatives Jn [16].

In particular when one has an inﬁnitesimal (continuous) transformation of the ﬁelds φj and the (spacetime) coordinates xi, which then lies in the Lie algebra g of some Lie group G, see appendix D, i 0i i i 0 0 x 7→ x = x + δx and φj(x) 7→ φj(x ) = φj(x) + δφj(x). We then get the following conserved currents

k ∂L m k Jn = − Φj,n − ∂mφI Xn − LXn (45) ∂(∂kφj)

24 m Where the matrices (notice infinitesimal transformations are linear transformations) Xn and Φj,n are defined such that i i δx = Xnδωn and δφj = Φj,nδωn, for δωn the parameters of the representation of the Lie algebra so the index goes up to the dimension of the representation. Moreover, one finds the equation Z µ 0 ∂µJ = 0 =⇒ ∂tPn := ∂t( Jn d~x) = 0 (46) V such that one gets a ‘conserved charge’ Pn, in the space V . For the electromagnetic Lagrangian (later shown to be one-dimensional U(1) symmetry so n = 1), one finds the conserved current Je and the electric charge q is precisely the conserved charge. k Finally, when a total derivative ∂kFn is added to the Lagrangian density, such that the equations of motion do not change, one finds that the conserved current changes to

k k k Jn 7→ Jn − Fn (47)

2.3 The Hamiltonian density

In accordance with the Lagrangian formalism, the ﬁeld dynamics can also be written in the Hamil- tonian density formulation. It is written in terms of a set of ﬁelds φi(~x,t) indexed by integers i and ∂L their conjugate momenta pi(~x,t) = . The Hamiltonian density H for ∂φ˙i

H(φi, pi, t) = φ˙ipi − L(φi, φ˙i, t) and the equations of motion become δH δH φ˙i = and pi = − δpi δφ˙i where δ indicates a variation of the ﬁelds.

25 3 Gauge theory

For all purposes a gauge theory can be seen as a theory which has a local continuous (gauge) symmetry. Generally, a gauge theory is built from demanding a globally continuous gauge symmetric theory (which is already present) to also be locally invariant. Gauge theories are extremely important in the standard model and general relativity and the realm of gauge theories will be explored in this chapter. Of course the ﬁrst locally invariant gauge symmetry has already been seen in electromagnetism, where the four-potential can be transformed Aµ 7→ Aµ + ∂µλ for a scalar function 4 λ : R → R.

3.1 Classical abelian gauge theory

The starting point of a gauge theory lies in turning a global gauge symmetry local. An example will be given here.

Suppose we consider a complex scalar field φ = φ1 + iφ2 with the standard Klein-Gordon La- grangian density ∗ µ mc 2 ∗ L0 = ∂µφ ∂ φ − ( ) φ φ, ~ where ∗ indicates a complex conjugate. If we consider an electromagnetic system then the field φ can be associated to a charged particle of mass m. The most simple gauge transformation is connected to the symmetry Lie group U(1), which consists of the complex numbers with modulus 1. The Lagrangian above is invariant under global gauge iα transformations with respect to the group U(1), since φ 7→ e φ, for some angle α ∈ R which is constant in spacetime. Again, the crux of gauge theory is finding a system which is invariant under a global gauge transformation and then demanding it to be locally invariant as well. In this case this means that the angle is a function of the spacetime coordinate φ 7→ eiα(xµ)φ.

Note that the Lagrangian L0 is not invariant under the local gauge transformation since ∂µφ 7→ iα(xµ) iα(xµ) iα(xµ) µ ∂µ(e φ) = e ∂µφ + e φ i∂µα(x ). This means that we get an extra term, namely the transformed Lagrangian becomes

∗ ∗ µ µ mc 2 ∗ L0 = (∂µφ − iφ ∂µα)(∂ φ + iφ∂ α) − ( ) φ φ ~ which is obviously not the same such that the former Lagrangian is not gauge invariant. The extra term is an indication that the transformed field not only changes (infinitesimally) due to the change of the field, but also due to the infinitesimal variation of the transformation. To make a gauge invariant theory one introduces a new set of degrees of freedom in the form of a vector field called the gauge field and denoted as Aµ(xν), which will (be defined to) contain information about the variation of the field φ. In the case of an electromagnetic field φ with mass m, the object turns out to be the electromagnetic four-potential Aµ. The reason for introducing this object immediately makes clear that the gauge field also must transform under the gauge transformation. The first step into making the former Lagrangian gauge invariant is to lose the cross terms of the derivative caused by the transformation. A proper way to do this is to introduce the so-called covariant derivative Dµ = ∂ + i g Aµ, where g is the coupling constant indicating the strength of µ ~c (the interaction of the) gauge field, and then to replace every partial derivative in the Lagrangian with the covariant derivative. For an electromagnetic system the coupling constant equals the electric charge g = q, but in general often the term e = q is used. Note that the covariant derivative ~ (the infinitesimal linear change of an object in the gauge invariant theory) consists of an infinitesimal spacetime variation ∂µ together with an infinitesimal gauge transformation i q Aµ. ~c

26 After transformation, the Lagrangian density takes the form

∗ ∗ g ∗ µ µ g µ mc 2 ∗ L1 = (∂µφ − iφ ∂µα − i A˜µφ )(∂ φ + iφ∂ α + i A˜ φ) − ( ) φ φ ~c ~c ~ where A˜µ stands for the transformed gauge field. We see that the scalar field becomes coupled to the scalar field by the last terms in the brackets.

From the above we see immediately that we can make this Lagrangian density (locally) invariant µ ˜µ µ c~ µ if we make the gauge ﬁeld transform as A 7→ A = A − g ∂ α. This new, invariant Lagrangian density takes the form µ mc 2 ∗ Linv = DµφD φ − ( ) φ φ. (48) ~

Summarizing, it was shown that the demand of a scalar Lagrangian which is invariant under local gauge transformation in U(1) necessitates the introduction of the covariant derivative Dµ and the gauge field Aµ. Furthermore, the gauge field contains information about how the scalar field φ transforms infinitesimally10.

The former Lagrangian density of the scalar ﬁeld has changed in quite a nice way, just substitute the covariant derivative. The theory is not done however, we have a new vector ﬁeld Aµ which has its own Lagrangian density.

We saw that a vector ﬁeld Aµ has comes with a general Proca Lagrangian (equation (40))

1 µν 1 mc 2 ν LB = − FµνF − ( ) AνA . 16π 8π ~

µν µ ν ν µ µν Since F = ∂ A −∂ A is gauge invariant, we find that the kinetic term F Fµν is gauge invariant, However, the mass term of the gauge field 1 ( mc )2AνA is not gauge invariant such that the gauge 8π ~ ν field must be massless, m = 0. This means that, when the gauge field is quantized, massless particles called gauge bosons come about in the system. The final Lagrangian is then the Linv +LB such that the gauge field is massless [15]. Since we added a field to the system, we must also add its Lagrangian density to the invariant Lagrangian density Linv. We then end up with the Lagrangian density

1 L = L + L = L − F µνF µν inv B inv 16π

Finally consider again the gauge invariant Lagrangian (equation (48) above). We see that we have a general Klein-Gordon Lagrangian if not for the covariant derivative. Geometrically this takes on 4 a very nice form. Instead of taking the partial derivative ∂µ in the space R , we add to the flat Minkowski space the vector field Bµ. The space is no longer flat and the mathematical concept of curvature becomes important11. The covariant derivative, which takes the curvature in account, now becomes the infinitesimal change in the field (which is the kinetic energy), hence we must substitute the partial derivative with the covariant one.

10Due to the nature of Lie groups, they have a corresponding infinitesimal version around the identity, called the Lie algebra. The gauge field, defined to be in the Lie algebra, therefore contains (linear!) information about how the group action changes infinitesimally. An object that contains linear infinitesimal information about a function is precisely a first derivative. Therefore the covariant derivative is chosen 11If the space is considered as a two-dimensional space. An analogy can be taken by considering yourself walking on a flat square measuring the temperature. Now small hills and troughs are added to the space, making it curved. The partial derivative is analogous to the change of temperature of the flat square. The covariant derivative is the change when walking over the flat square together with the added change in temperature due to the hills and troughs. Compare the addition of curvature also with the previous footnote, footnote 10

27 3.2 Gauge invariance of the relativistic electromagnetic Lagrangian

Finally, consider again the relativistic Lagrangian (not the density!) for a particle in an electromag- 1 µ µ netic field L(τ) = 2 m0u uµ + qu Aµ, equation (35). By considering the former gauge transformation we find that the Lagrangian changes to L(τ) = 1 µ µ µ 2 mu uµ + qu Aµ − u ∂µα. Considering now the (relativistic) action we see that the final term just equals the difference in phase of between the endpoint and the starting point of the trajectory Z Z µ µ δS = u ∂µα dτ = dx ∂µα = α|boundaries of trajectory (49) which is, classically, physically insignificant since this does not change the path of the particle, just some kind of angle (which turns out to be the phase of the corresponding wavefunction in quantum mechanics, see below). This also means that the relativistic Lagrangian is also gauge invariant under the mentioned gauge transformations. This is a very nice result. It is now proven that EM is relativistically invariant, gauge invariant ándinvariant under EM duality. This is both the case For particles carrying charge in an EM system, as described by the equations at the start of part II together with the Lorentz force or in other words the gauge invariant action, equation (35) For fields carrying charge in an EM system, described by the Lagrangian density Z Z 1 S = L + L dxµ = − F F µν − A J µ − dxµ, e b 4 µν µ e

µ µ This is because the electric four-current Je satisfy ∂µJe = 0 such that the term introduced by the transformation becomes a total derivative (see also equation (47)) Note finally that in the last subsection, it was the U(1) symmetry of a scalar field that motivated the introduction of a gauge field Aµ with the transformation properties Aµ 7→ Aµ + ∂µλ. In the electromagnetic (classical) formalisms we have seen thus far, no such symmetry is present, it was just noted that the electromagnetic four-potential was invariant under a set of gauge transformations. The transformation property is no coincidence however. There is a U(1) symmetry present in electromagnetism but it becomes explicit in the quantum mechanical formulation of electromagnetism. It is there that the coupling constant becomes identified with the charge q and furthermore i q α the gauge transformation of the state vectors becomes |ψi 7→ e ~ for some angle α. In particular the change in phase factor for a particle with charge q moving along a path ` when a magnetic field is presents becomes precisely: Z Z q ~r˙ · A~(~r) dt = q A~(~r) · d~r, (50) t ` which comes in useful when considering the Dirac monopole. Note finally that the state vector |ψi can be identified with the scalar field φ from the previous section, therefore we now have a quantum mechanical U(1) gauge transformation:

q i λ(~r) iδS/~ |ψi 7→ e ~ |ψi = e |ψi and A~ 7→ A − ∇~ λ, where δS is the change of the action due to the gauge transformation, see (49). The change in phase eiδS/~ can be generalized in a nice intuitive way. Note that it was stated that the path from classical mechanics to quantum mechanics is easier via the Lagrangian or the Hamiltonian formalism. Via the Lagrangian formalism, the phase of the wavefunction of a particle going along any path ` can be identiﬁed with the value eiS(`)/~. The actual (classical) path taken by the particle, is the path for which the action S is extremized, for now assume w.l.o.g. it is minimized. In a

28 small neighbourhood (a set of paths) around the classical path of the particle, the action is strictly greater, meaning that the phase of these paths are all ‘pointing in the same direction’ (imagine a circle). The phases of the paths very close to the path of minimized action then interfere positively. In any small neighbourhood around any path other than the classical path the action of those neighbouring paths also diﬀers very little, but in both directions. Because ~ is so small, the phases become completely randomly distributed such that they cancel each other out. Physically (and heuristically) this implies that we only see the classical path of the particle [17].

3.3 Gauge theory for the Dirac Lagrangian

A second example is encountered when one considers the Dirac Lagrangian density

µ 2 ¯ L = i~cψγ ∂µψ − (mc )ψψ.

Again this is invariant under global, gauge transformations with respect to the group U(1), which − iq λ(xµ) can be made local: ψ 7→ e ~c ψ. The recipe is the same. The Lagrangian changes to

µ ¯ µ 2 ¯ L = i~cψγ ∂µψ − qψγ ψ∂µλ − (mc )ψψ. (51)

It is now necessary to add a new element to the system: The vector ﬁeld Bµ, the gauge ﬁeld. The following works out nicely:

µ ¯ µ µ 2 ¯ L = i~cψγ ∂µψ − qψγ ψB − (mc )ψψ. (52)

Such that the interaction between the gauge field and the spinor ψ is determined first. If the gauge field transforms as Bµ 7→ Bµ + ∂µλ then the Lagrangian becomes gauge invariant. 1 µν Again, the kinetic term − 16π F Fµν must be added and the gauge field must be massless for the Lagrangian to remain gauge invariant. Finally we get the Lagrangian density 1 L = i cψγµ∂ ψ − qψγ¯ µψBµ − (mc2)ψψ¯ − F µνF . (53) ~ µ 16π µν

µ Notice again that if we substitute the partial derivative ∂µ with the covariant derivative D = ∂µ +i q Bµ in the original Lagrangian density (and add the kinetic term of the gauge ﬁeld), then we ~c µ − iq λ(xµ) µ end up with the ﬁnal invariant Lagrangian density, since D ψ 7→ e ~c D ψ when transformed.

3.4 The general strategy for U(1)

We have seen two examples of Lagrangian densities, invariant under global U(1) gauge transformations. The strategy was equal both times and it pays off to look at it again. The demand for local invariance arose from the idea that local gauge invariance (of the Lagrangian) is a principle of nature. The consequence was an extra demand, the local invariance of the La- grangian, which could not be met with the variables (degrees of freedom) we had. To overcome this obstacle we introduced new degrees of freedom, the gauge field, and used the demand to work out the properties that the gauge field must have. Then it was found that the gauge field was massless but more importantly the covariant derivative was to replace the partial derivative and a α(xµ) α(xµ) transformation of the gauge field was introduced such that Dµφ 7→ Dµe φ = e Dµφ. This was because the introduction of the gauge field could be seen as the change of spacetime from flat to curved. Equivalently, as was noted before, the gauge field(s) caries information about the variation

29 of the transformation (infinitesimally, linearly) across spacetime, precisely because it lies in the Lie algebra of the corresponding Lie (gauge) group. Therefore it was not a surprise to find the gauge field in the covariant derivative. Summarizing, the crucial step in gauge theories is the introduction of a derivative Dµ which changes ‘in a covariant way’: Dµ 7→ D¯ µ = Dµeiα, such that the Lagrangian becomes locally gauge invariant. This was precisely the covariant derivative and a new set of degrees of freedom (the gauge fields) had to be introduced to be able to construct such an object. The rest of the gauge theory followed from that. In the following sections it will be shown that for groups other than U(1) the strategy is, although a bit more subtle, in principle the same. One final word about the degrees of freedom mentioned in the previous discussion. The general, globally U(1) gauge invariant theory had a ‘redundancy’ in it. There was more information in the physics than needed to describe the physical system. Because the global symmetry was made local, however, every point in space was assigned a group element of U(1), in other words every point in space could be represented by an angel α(xµ). In principle this does not have to bring along any problems. If the Lagrangian density would be completely local, depending only on one point (of the history) in space being considered, then the theory would not change. However, derivatives bring along linear information about a neighbourhood of the point, such that a whole set of (continuously varying) angles now enters the theory. The theory is now overloaded with new parameters and new degrees of freedom, (linear approximation of the group elements called) the gauge fields, are introduced to overcome the overloading. Physically these fields bring in new kinetic terms, which are then added, which change the space in a way that the angles can still be chosen continuously varying in other words the gauge invariance is not to be broken. This translates beautifully in the theories above. A completely local mass term does not violate local gauge invariance, the kinetic (differential) term does however. Note that this argument only holds in general for gauge theories with O(n) or U(n) as the gauge group. The cross-over to Yang- Mills comes very natural now. Yang-Mills theories are gauge theories, like above, but now with the non-abelian groups SU(n).

3.5 Yang-Mills theory

The above two examples of gauge theories were symmetries with respect to an abelian group: U(1). Yang-Mills theory is deﬁned more generally, including a range of non-abelian groups (SU(n)), which makes the theory a bit more tricky. One of the simplest groups after U(1) which is often considered 3 in gauge theories is the non-abelian group SU(2), which includes rotations in R .

Suppose we consider two spinor ﬁelds ψ1 and ψ2 with Lagrangian density:

L = L1 + L2, (54) where Li is the Dirac Lagrangian as seen in equation (42) for i = 1, 2.

The equations are linear in ψi implying they can be written as matrices: ¯ µ 2 ¯ L = i~cψγ ∂µψ − c ψmψ, (55) ψ where ψ = 1 and ψ¯ = ψ†, the hermitian conjugate of ψ. By assumption the mass of the spinor ψ2 ﬁelds is taken to be m1 = m2 = m. 2×2 † We now not only have U(1)×U(1), but global U(2) = {U ∈ R | U U = 1} gauge symmetry. Now any U ∈ U(2) can be decomposed as a product of λ ∈ U(1) and V ∈ SU(2) = {V ∈ U(2) | det V = 1}. This makes it possible to look at gauge invariance with respect to the group SU(2), since the other part was already considered.

30 The strategy for looking at symmetry invariance is to look infinitesimally, whic like before means at the Lie algebra of the lie group SU(2). The Lie algebra of SU(2), can be spanned by the Pauli ~σ·~λ ~ 3 matrices σi for i = 1, 2, 3. It follows that the matrix V can be written as V = e , for λ ∈ R . The global symmetry transformation then becomes ψ 7→ V ψ. The demand that the symmetry is local gives λ = λ(xµ) and ψ 7→ V ψ = e~σ·~λ(xµ)ψ. Like before, the Lagrangian L is not invariant and the covariant derivative D = ∂ + i q ~σ · A~ is µ µ ~c µ introduced, for three gauge fields Aµ, i and a coupling constant q ∈ R which couples the interaction of the gauge fields and the spinor fields. Furthermore the transformations of the gauge fields should imply the transformation property Dµψ 7→ DµV ψ = VDµψ. So the transformation property is given by:

c ~σ · A~ 7→ ~σ · A~0 = V (~σ · A~ )V −1 + i~ (∂ V )V −1 (56) µ µ µ q µ

Furthermore, to evaluate this term we approximate the terms with V in it inﬁnitesimally/linearly: q V = 1 − i ~σ · ~λ ~c q V −1 = 1 + i ~σ · ~λ ~c q ∂µV = −i ~σ · ∂µ~λ ~c such that c ~σ · A~0 = V (~σ · A~ )V −1 + i~ (∂ V )V −1 µ µ q µ q = ~σ · A~µ + i [~σ · A~µ, ~σ · ~λ] + ~σ · ∂µ~λ ~c where terms up to linear in ~λ and A~µ were considered.

To evaluate the commutator bracket [~σ · A~µ, ~σ · ~λ] = σjAµ,j − σjλj the identity (~σ · ~a)(~σ · ~b) = ~ ~ ~ 3 ~a · b + ~σ · (~a × b) = ajbj + iσjjklajbl for ~a, b ∈ R is useful and the transformation property becomes

0 q σjAµ,j = σj(Aµ,j + ∂µλj − 2 jklAµ,kλl). ~c In other words

0 q ~ ~ 0 ~ ~ q ~ ~ Aµ,j 7→ Aµ,j = Aµ,j + ∂µλj + 2 jklλkAµ,l or Aµ 7→ Aµ = Aµ + ∂µλ + 2 (λ × Aµ). (57) ~c ~c

Like before, the Lagrangian density

¯ µ 2 ¯ L = i~cψγ Dµψ − mc ψψ is gauge invariant with respect to the group SU(2), but the gauge fields A~µ bring their own (gauge invariant) Lagrangian density LA~µ into the system: 1 L = − F~ µν · F~ , A~µ 16π µν µν µ where Fj is the field strength for gauge field Aj , which in the case of non-abelian groups not µ ν ν µ necessarily equal to ∂ Aj − ∂ Aj . Again the mass terms are not gauge invariant such that the gauge fields are massless.

31 Note that under the gauge transformation

µν µ ν ν µ 0µν µ 0ν ν 0µ Fj := ∂ Aj − ∂ Aj 7→ Fj := ∂ Aj − ∂ Aj µ ν ν q ν ν µ µ q µ = ∂ (Aj + ∂ λj + 2 jklλkAl ) − ∂ (Aj + ∂ λj + 2 jklλkAl ) ~c ~c µν q µν µ ν ν µ = Fj + 2 jklλkFl + ∂ λj Ak − ∂ λj Ak ~c or in the notation of vector calculus q F~ µν 7→ F~ µν + 2 ~λ × F~ µν + ∂µ~λ × A~ν − ∂ν~λ × A~µ. (58) ~c

µν The ﬁeld strength F~ F~µν should be gauge invariant, however one can check that using the expression (58) does not leave us with a gauge invariant term. Luckily, considering the gauge transformation of the term A~µ × A~ν we ﬁnd that

µ ν 0µ 0ν µ µ q µ ν ν q µ jklAk Al 7→ jklAk Al = jkl(Ak + ∂ λk + 2 kmnλmAn)(Al + ∂ λl + 2 labλaAb ) ~c ~c = ∂µ~λ × A~ν − ∂ν~λ × A~µ

We see then that we get the gauge transformation becomes 2q 2q G~ µν := F~ µν − (A~µ × A~ν) 7→ G~ 0µν := F~ 0µν − (A~0µ × A~0ν) ~c ~c 2q = F~ µν + (~λ × F~ µν). ~c

µν It is easily checked that this results in a gauge invariant term in the Lagrangian G~ µνG~ , [15]. Finally, the gauge invariant Lagrangian density is 1 L = i cψγ¯ µD ψ − mc2ψψ¯ − G~ · G~ µν. (59) ~ µ 16π µν

3.6 Again Yang-Mills

In this section general transformations are considered of a field φ~, a multiplet (vector) of fields φj for j ∈ N. We consider the (linear) representation of elements g in a Lie group G = O(n), the group of orthogonal matrices, acting on the space of fields (canonically identified with a vector space). The representation of g will be denoted with O. As in the previous section, the start is at a Lagrangian density

µ L = ∂ φ~ · ∂µφ~ − V (φ~ · φ~) (60) for some potential V (φ).. The start of the gauge theory is the set of ﬁelds φ~ together with the gauge transformation φ 7→ Oφ. Then the transformation is made local meaning O = O(xµ) such that the Lagrangian density L is no longer gauge invariant:

0 µ µ µ L 7→ L = ∂ φ~ · ∂µφ~ + (∂ O)φ~ · O∂µφ~ + O∂ φ~ · (∂µO)φ~ − V (φ~ · φ~) 6= L

The strategy is the same. Like before, the covariant derivative which acts as Dµφ~ = ∂µφ~ − eW µφ~ µ µ for a gauge ﬁeld W and a coupling constant e ∈ R. W lies in the Lie algebra and can therefore be identiﬁed with a matrix.s

32 Using the demand that the covariant derivative transforms in a covariant way Dµφ~ 7→ (Dµφ~)0 = O(x)Dµφ~ one ﬁnds

~ 0 0 ~0 (Dµφ) = (∂µ − eWµ)φ 0 ~ = (O∂µ + (∂µO) − eWµO)φ demand = O(∂µ − eWµ)φ~ such that 1 W 0 = OW O−1 + (∂ O)O−1. (61) µ µ e µ 0 Note that Wµ is again in the Lie algebra of O(n). Furthermore, if an object B in the Lie algebra of a matrix group G transforms as B = gBg−1 for g in the Lie matrix group, then B is said to transform under the adjoint representation of the matrix group G. Thus W µ does not quite transform under the adjoint representation, but it can easily be shown that it still performs under a group action: The adjoint action together with an extra term. As in the case of SU(2), the field strength F µν = ∂µW ν − ∂νW µ as defined for the abelian case is µν no longer gauge invariant. Considering the fact that an FµνF terms comes into the Lagrangian, a definition for the gauge field strength is wanted to transform in a covariant way, such that the term becomes gauge invariant. The definition of F µν can be properly extended from the abelian case by considering the term proportional to [Dµ,Dν]φ~, which transforms in a covariant.

[Dµ,Dν]φ~ = e(∂µW ν − ∂νW µ − e[W µ,W ν])φ~ where [.,.] is the Lie bracket, which in the case of the matrix groups is the commutator bracket. Note then that the transformation for (the matrix) F µν := ∂µW ν − ∂νW µ − e[W µ,W ν] becomes

F µν 7→ (F µν) = OF µνO−1, because of the transformation properties of Dµ and φ. The ﬁnal Lagrangian becomes 1 1 L = − tr(F µνF ) + Dµφ~ · D φ~ − V (φ~ · φ~), (62) inv 2 µν 2 µ

µν µν −1 µν which is invariant, since the trace tr(F Fµν) 7→ tr(OF FµνO ) = tr(F Fµν).

3.7 The case of SO(3)

In this paper, invariance of the Lagrangian under the symmetry group SO(3), consisting of the orthogonal matrices with determinant 1, will be important for the study of magnetic monopoles. Extra attention will be given here for the deﬁnition of our objects. The start lies again with the scalar ﬁeld φ which transforms under the adjoint representation of SO(3). Note that the action of the adjoint representation of matrix groups consists of conjugation with matrices. φ lies in the lie algebra of SO(3), denoted by so(3). so(3) is a vector space, which can be generated by the matrices:

0 0 0   0 0 1 0 −1 0 Lx = 0 0 −1 Ly =  0 0 0 Lz = 1 0 0 . (63) 0 1 0 −1 0 0 0 0 0

Together with the commutator bracket it forms the Lie algebra so(3).

33 The Lagrangian density for φ is given by

µ L = tr(∂ φ∂µφ) − V (tr(φφ)). (64) which is invariant under the global gauge transformations φ 7→ OφO−1. Invariance under this transformation is equivalent to saying that the system described by the Lagrangian density is invariant under the action of global rotations acting on the field φ, in other words with respect to the action of the adjoint representation of SO(3). The covariant derivative takes the form Dµ(·) = ∂µ11 · +e[W µ, ·] for some gauge fields W µ ∈ so(3). The covariant derivative takes the form of a rank-2 tensor, therefore it can be identified with a µ µ γ µ γ matrix with components (D )αβ = δαβ∂ + gfαβWγ , for the structure constants fαβ defined by the Lie bracket c [va , vb] = fabvc, (65) for a basis {va}1≤a≤s of the Lie algebra and 1 ≤ a, b, c ≤ s, with s the dimension of the Lie algebra. g is the coupling constant and will define the coupling strength between the scalar field and the gauge field. The demand Dµφ 7→ (Dµφ)0 = ODµφ O−1 implies that Dµ 7→ ODµO−1. Like before, one finds

−1 −1 −1 0 −1 O(∂µφ − e[Wµ, φ])O = O∂µφ + (∂µO)φO + Oφ∂µO − e[Wµ, OφO ].

When one makes a first ansatz that the gauge field transforms as the field does (namely under the adjoint representation/in a covariant way) then one finds, after a simple calculation the transformation property 1 W 0 = OW O−1 + ∂ OO−1 (66) µ µ e µ

Like before we deﬁne the matrix F µν ∝ [Dµ,Dν] as the gauge ﬁeld-strength, which is in this case the same as for the other Yang-Mills theories

F µν = ∂µW ν − ∂νW µ − e[W µ,W ν]. (67)

Then F µν transforms in a covariant way: F µν 7→ OF µνO−1. The invariant Lagrangian takes the form 1 1 L = − tr(F µνF ) − trDµφ D φ − V tr(φφ). (68) inv 2 µν 2 µ

3.8 Yang-Mills in general

The general strategy for local gauge invariance of a Lagrangian density of scalar fields φ~ = (φ1, ..., φn) under the symmetry group O(n) for n ∈ N becomes. µ Start with globally symmetric Lagrangian density of scalar with kinetic term ∂ φ~ · ∂µφ~, which transforms as φ~ 7→ Oφ~ for O ∈ O(n) a constant matrix Demand local invariance such that φ 7→ O(xµ)φ~ but then the Lagrangian loses its invariance Introduce covariant derivative Dµ = 11∂µ − evaW µ,a for basis-elements va of the Lie algebra o(n) and gauge fields W~ µ = (W µ,1, ..., W µ,n). In particular we take the basis to be the matrices defined below by equations (170) - (172) for the lie algebra so(3) By demanding the covariant transformation of the covariant derivative Dµ 7→ ODµO−1 one ~ µ µ −1 1 µ −1 finds the transformation property of the gauge fields: W 7→ OW O + e ∂ OO

34 The gauge ﬁeld has its own (gauge invariant!) kinetic term, with Lagrangian 1 1 − tr(F~ µνF~ ) = − tr(vaF µν,avbF b ) 2 µν 2 µν 1 = − tr(vavbF µν,aF b ) 2 µν ab µν,a b = δ F Fµν µν,a a = F · Fµν

~ µν a µν,a a a a for the gauge field-strength F = v F defined as [Dµ,Dν] = ev Fµν, such that Fµν = a a a b c a ∂µWν − ∂νWµ − efbcWµWν , for the structure constants fbc defined by equation (65). Further- more the basis elements are assumed to satisfy tr(vavb) = −2δab, which often holds for the most natural choice of basis-elements in the orthogonal groups. The final, locally gauge invariant Lagrangian density has the form 1 1 L = − F~ µν · F~ + Dµφ~ · D φ~ − V (φ~) inv 4 µν 2 µ 1 = tr(F 2) − tr((Dµφ~)2) − V (φ~), 2

2 a µν,a b b ~ µν a µν,a a 3 when we write F = v F v Fµν and F =e ˆ F , fore ˆ the canonical basis of R . It is thus seen that the SO(3) gauge invariant Lagrangian can be rewritten with a gauge field 3 3 strength in R , this is because the Lie algebra so(3) can be identified with R , leading to a final note, see D.1.

35 4 The Higgs mechanism

The Higgs mechanism is essential for the Standard model. We have seen that U(1) and SU(2) gauge theory both predict massless gauge bosons. In the case of electrodynamics, the predicted boson is the existent massless photon. In the weak theory, the three predicted (massless) bosons are the massive W ± and Z bosons, thus there is a shortcoming in the description of weak theory using gauge theories. An adaptation to the theory is needed to provide mass to these bosons. The first demand for this theory is to be compatible with the gauge theories, since the amount of predicted bosons has been experientially verified. Eventually it was some 20 years between the making of gauge theories and the invention of the Higgs mechanism: The mechanism which predicts the massive gauge bosons. In the 1940s, the first (U(1)) gauge theory, quantum electrodynamics (QED), was finished [14]. QED was experimentally verified with extremely high precision, which motivated physicists to apply gauge theory to other fundamental forces. The weak interaction, however, left people at a loss. In 1933 Fermi gave one of the first descriptions of the weak force, he described it as a contact interaction. In turn, the success of QED motivated the idea of a particle mediated weak interaction described by gauge theory. An important difference between the electric force and the weak force is that the gauge bosons are, in this case of the weak interaction, massive. For the purpose of describing the weak theory, Yang and Mills constructed non-abelian gauge theory in 1954 using the SU(2) group. This gauge theory was flawed however. A mass term for the gauge field would imply the massive gauge bosons but was not possible due to the breaking of gauge invariance12. It was the Higgs mechanism which explained the mass generation of the bosons and moreover was (needed to be) consistent with the already present (renormalizable) gauge theories of the standard model. The Higgs mechanism was based on an analogous situation in condensed matter theory: supercon- ductivity. The latter process relies on the SSB of a continuous gauge symmetry of electromagnetism. In 1961 SSB of a symmetry led to the Goldstone theorem: when a (global gauge) symmetry is spontaneously broken a massless particle is predicted. These so-called Goldstone bosons were not known to exist. A year later, in 1962, SSB was combined with gauge invariant field theories which led to the Higgs mechanism, predicting massive gauge bosons (in a way such that the theory was renormalizable). Thus, it is the SSB of a continuous (local) gauge symmetry which forms the basis of the Higgs mechanism. Since any gauge theory comes with such a symmetry the trick was to break it spontaneously in the weak interaction such that massive gauge bosons could be predicted correctly. The trick lies in the introduction of a scalar field called the Higgs field. This field is subject to a potential in the shape of a Mexican hat, which we saw could be spontaneously broken.

4.1 The Higgs ﬁeld and its potential

The Higgs field plays a fundamental role in the Higgs mechanism. More precisely the Higgs potential is the important part. Consider the complex (Higgs) field φ = φ1 + iφ2 and its complex conjugate ∗ 1 2 2 2 φ , then the Higgs potential is given by V (φ) = 4 λ(|φ| − a ) for some a ∈ R and λ ∈ R≥0, which is shown in figure 3 and a cross section is shown in figure 4. The first thing that stands out is that the vacuum configuration (the configuration of lowest energy) of the field φ is non-zero: the field has modulus a 6= 0. Furthermore, the potential is gauge symmetrical with respect to the transformation φ 7→ uφ for u ∈ U(1), in other words rotations around the symmetry axis.

12That was not the only problem though, since a mass term would also break the so-called renormalizability of the theory

36 600

500

400

300

200

100

-6 -4 -2 2 4 6

∗ 2 Figure 4: Cross section of mexican hat potential V (φ1, φ2) = (φφ − 25) , meaning the vacuum conﬁguration has modulus a = 5 in this case.

An example of SSB of a physical symmetry of the Mexican hat potential was given. It will be shown that the SSB of the local continuous gauge symmetry of the Higgs potential is the crux of the Higgs mechanism13.

4.2 Massless gauge bosons from a broken global symmetry

From equation (43) it is indicated that one looks at the prefactor of the term that is proportional to |φ|2, to determine the mass of the particle associated with the field. However, this is based on perturbation theory around a minimum (Feynmann theory), such that this statement only holds for a field with zero vacuum configuration. In general, to determine the mass term, one needs to expand around the vacuum configuration. In the case of the Higgs field φ the expansion needs to be made around a value of the field with modulus a.

Now the determination of the mass of particles generated by a ﬁeld φ = φ1 +iφ2 in a Higgs potential will be considered. The general form of the Lagrangian density is

µ ∗ L = ∂µφ∂ φ − V (φ) 1 = ∂ φ∂µφ∗ − λ(φφ∗ − a2)2 µ 4 1 1 1 = ∂ φ∂µφ∗ + λa2|φ|2 − λ|φ|4 − λa4. µ 2 4 4

The constant will be disregarded. If prefactor of the quadratic term would be considered as proportional to the mass mass term we would ﬁnd an imaginary mass, a consequence of the fact that the perturbation theory was not used correctly and an indication of the fact that at φ = 0 the conﬁguration is unstable.

13Actually, the crux of the Higgs mechanism is the non-zero vacuum conﬁguration, see [14]

37 To find the mass of the (particles of the) field, a perturbation around the configuration φ = a ∈ R is considered, in other words φ2 = 0. It is convenient to consider the fluctuations in the real and imaginary direction (around the minimum) such that the field η = φ1 − a and χ = φ2 are naturally defined. After the substitution φ = η + a + iχ we find the Lagrangian 1 1 L = ∂ (η + a + iχ)∂µ(η + a + iχ) − λ[(η + a + iχ)(η + a − iχ)]2 + λa2(η + a + iχ)(η + a − iχ) µ 4 2 1 1 = ∂ η∂µη + ∂ χ∂µχ + λa2[(η + a)2 + χ2] − λ[(η + a)2 + χ2]2 µ µ 2 4 1 1 1 1 = [...] − λη2a2 + λa4 − λη4 − λaη3 − η2χ2 − λaηχ2 − λχ4. 4 4 2 4 Or disregarding the constant and considering up to quadratic terms in the field we get

µ 2 2 µ L = ∂µη∂ η − λa η + ∂µχ∂ χ. (69)

Since a perturbation around the minimum was performed,√ one can now consider the quadratic terms as the mass terms. Then a massive field η with mass λ a and a massless field χ is seen. Note also that by substitution, the symmetry is broken: The Lagrangian density with respect to the new fields η and χ is not invariant under the gauge transformation. There is a theorem which generalizes this example: The Goldstone theorem, which states that a massless scalar field appears whenever a continuous global symmetry is broken. Actually the amount of massless fields appearing depends on the dimensions of the original and broken symmetry groups. The Higgs mechanism (when the symmetry is made local) will now ‘eat the massless Goldstone boson and produce massive ones’.

4.3 The Higgs mechanism

It was explained that one makes a global gauge symmetry local, such that the Higgs mechanism occurs. If we demand local U(1) gauge symmetry then the Lagrangian density takes the form: 1 1 L = D φDµφ∗ − λ(φφ∗ − a2)2 − F F µν, µ 4 16 µν with the covariant derivative Dµ and the gauge field Aµ, as in section 3.1 (massless gauge field). Again a perturbation around the point φ = a + η + iχ is considered. The difference in the potential does not change and up to quadratic terms in η and χ is −λa2η2. The kinetic term changes as:

µ ∗ q ∗ µ q µ Dµφ(D φ) = (∂µ + i Aµ)φ (∂ − i A )φ ~c ~c µ µ q µ µ µ q 2 µ 2 2 = ∂µη∂ η + ∂µχ∂ χ + 2 Aµ(χ∂ η − a∂ χ − η∂ χ) + AµA (a + η) + χ ~c ~c such that the total Lagrangian becomes

µ 2 2 µ µ 1 µν qa2 µ qa L = ∂µη∂ η − λa η + ∂ χ∂ χ − FµνF + AµA − 2 ∂µχ + [...], 16π ~c ~c where the other terms [...] are not written out. √ Like before the η field has mass λ η and the χ field is massless. Finally the gauge field Aµ has acquired mass qa , from the term −( qa )2A A . In principle this is how the Higgs mechanism ~c ~c µ µ works, when considering a Yang-Mills-Higgs theory later on this becomes important.

38 Part III Magnetic monopoles

In part I both the Galilean invariant form and the special relativistic invariant form of Maxwell’s equations and the Lorentz force law were considered. Explicitly, stated together with the names of the laws

Vector notation and Jb = 0 Tensor notation

Gauss’ law ∇~ · E~ = ρe µν ν ∂µF = Je

Ampere’s law ∇~ × B~ = ∂tE~ + J~e

No magnetic monopole law ∇~ · B~ = 0 ∗ µν ν ∂µ ( F ) = Jb

Faraday-Maxwell law −∇~ × E~ = ∂tB~ + J~b

˙ ˙ ν ? Lorentz Force law ∂t(γm~r) = q(E~ + ~r × B~ ) ∂τ muµ = u (qFµν + g Fµν)

The most important properties are relativistic invariance, gauge invariance and, together with the extra duality for the electric and magnetic charge, invariance under the electromagnetic duality stated in equation (7),

(E,~ B~ ) 7→ (B,~ −E~ ) and (Je, Jb) 7→ (Jb, −Je) and (q, g) 7→ (g, −q). (70)

The fact that the theory contains this high amount of symmetry motivates one to consider magnetic monopoles. The simplest being of the form of section 2, Part I

g ~r B~ (~r) = 4π r3 Having seen the properties of non-abelian gauge theories and the Higgs mechanis, we can now start the discussion on magnetic monopoles: First the Dirac monopole, which is built from a point magnetic charge with precisely this field. A second monopole is then considered by examining the so-called Yang-Mills-Higgs action, which in turn produces an intrinsically different monopole but looks, from far away, again like a Dirac monopole. In the sense that the second kind of monopole brings extra local information, the second can be seen as an improvement of the first. Among others, a very important result of these considerations will be the quantization of electric charge, which is of course a well known property of nature. Thus the high amount of symmetry in the theory and the fact that symmetry is thought to be an intrinsic property of nature, together with the fact that one of the results of existent monopoles is charge quantization leads one to want to believe that they indeed exist. Sadly, the experimental search has been unsuccessful to this day.

39 1 The Dirac monopole

We saw earlier that the absence of magnetic sources implied the existence of the four-potential Aµ ∗ µν (equations (20) and (22)), since ∂µ F = 0. If magnetic sources are present, a global four potential might not exist. One can try to cover the system by the domains of a set of (locally defined, where magnetic sources are absent) four potentials, but sometimes the topology of the system does not allow every part of the system to be covered; A singular region might exist. This is the case for the Dirac monopole: a magnetic monopole in free space, as defined in section 2: g B~ (~r) = ~r (71) 4πr3 Due to the magnetic charge we precisely get a singular region, where the electromagnetic potential cannot be defined, called the Dirac string since it often is a linear singularity in space.

1.1 The search for a global vector potential

We have seen that a particle-like magnetic monopole with charge g creates a magnetic field B~ (~r) = g rˆ 4π r2 , where ~r = (x, y, z) is the position of the second particle (mass m, electric charge q) in Cartesian coordinates. In this case (as for the electric monopole) it is found that that g is equal to the magnetic flux Φ, Z Φ = B~ · dS~ = g, ∂V for any volume V enclosing the magnetic monopole, with orientated boundary (∂V, S~). In the complement of the origin ∇~ · B~ = 0 so a vector potential A~ as in equation (8) might exist for this region. However a global, well-defined vector potential cannot exist. On the origin, at the Dirac monopole, one finds ∇~ · B~ (~r) = gδ(~r), which can be found by considering the magnetic flux R ~ ~ Φ through any closed surface ∂V enclosing the origin. Indeed one then finds that Φ = ∂V B · dS = R ~ ~ ~ V ∇·B dV = g. However, if a global vector potential A exists then, for a closed surface ∂V = S one R ~ ~ R ~ ~ ~ R ~ ~ finds ∂V B · dS = ∂V (∇×A)·dS = ∂S A·d` = 0, which is in contradiction with the results above. Since the divergence of the magnetic field is well-defined on any non-empty V containing the origin, there must be at least one point on the surface ∂V which cannot be described by a vector potential. Furthermore, this holds for any closed surface surrounding the origin. When one considers the set of 2-spheres, parametrized by the radius, one finds that for every distance r from the origin there is at least one point which cannot be described. In specific (non-global) vector potentials, which will be seen later, one finds that this singular region has the same spherical angles such that it becomes a string. Now such a vector potential will be looked for. A mathematical formulation of the topology of such a monopole can bring up intrinsic properties of the system. Here a vector potential will be found by looking at a specific configuration. Note that the system of the magnetic monopole at the origin is equivalent to an infinitely thin solenoid, stretching from the origin along the negative z-axis, with its positive magnetic pole at the origin, see figure 5. The magnetic field of such a system is given by g B~ = ~r − gθ(−z)δ(x)δ(y)ˆz = B~ − B~ , (72) sol 4πr3 string where in this equation θ is the step function, not to be confused with the azimuthal angle.

In this case the flux indeed vanishes on both sides, ∇~ · B~ sol = 0 and one can look for a global vector potential A~sol. Thus, the magnetic field of the monopole is described by the (hopefully global) vector potential A~sol together with a singular, string-like field.

Now an explicit form of A~sol is found by considering rotational symmetry around the z-axis, such ˆ that one tries A~sol = A(r, θ)φ, for spherical coordinates (r, φ, θ). This can be found using Stokes’

40 Figure 5: The magnetic field of a Dirac monopole as described in the text, which is a sum of the magnetic field of a solenoid and a string-like magnetic field. Image thanks to [18]. law on a constant-θ~ curve ` = ∂Sˆ14, for a spherical cap Sˆ on the unit sphere S1, which has boundary `: Z Z Z θ ~ 2πg 0 0 1 2πr sin θA(r, θ) = A~sol · d` = B~ · dS~ = sin θ dθ dθ = g(1 − cos θ), ` Sˆ 4π 0 2 such that g 1 − cos θ A~ =: A~ = φ,ˆ sol + 4πr sin θ which is singular on the negative z-axis. This means that because the string was introduced, a vector potential is found which is not global. Again, note that this vector potential A~+ corresponds to a magnetic monopole at the origin, together with a half-infinite stretching from the origin along the negative z-axis. Finally, one more notion about the string. Note that the gauge transformation

g g 1 + cos θ A~ 7→ A~ = A~ − ∇~ φ = − φˆ (73) + − + 2π 4πr sin θ transforms the Dirac string from the negative to the positive z-axis, meaning that the string changes under a gauge transformation. This implies that the string is non-physical (see subsection 1.1, Part II). The change in orientation of the singular string is caused by the fact that important term in the gauge transformation, χ, is singular on both the old and the new locations of the Dirac string. Furthermore, any vector potential has at least a one-dimensional singular region so the Dirac-string is really an intrinsic property of the system. No gauge transformation can remove the singularity, it can only change shape.

1.2 The Dirac quantization condition

From Maxwell’s equations we saw that classically, a particle interacts only explicitly via the EM ﬁeld-tensor F µν (equation (36)). Quantum mechanically, the Hamiltonian (equation (37)) describes the system, meaning that the existence of the electromagnetic potential Aµ is necessary for the (quantized) description of the system.

14Note, however, that Stokes’ is not always valid, the assumption is made that A~ is deﬁned on the entire spherical, cap. The solution of A~ will indeed satisfy this demand, but this is only because the mistake of ignoring the conditions for which Stokes’ law holds turns out to be irrelevant.

41 In 1931, Dirac studied the problem of a particle with mass m and (electric) charge q interacting with a magnetic ﬁeld from a magnetic monopole [7]. The non-existence of the potential mentioned above can therefore obstruct the quantum theory of a particle interacting with the monopole. Dirac then found a relation between the electric charge of the particle and the magnetic charge of the monopole: The Dirac quantization condition. Recall the two potentials describing the Dirac monopole:

g 1 − cos θ g 1 + cos θ A~ (~r) = φˆ and A~ (~r) = − φˆ (74) + 4πr sin θ − 4πr sin θ

Where (r, φ, θ) are the spherical coordinates. Notice again that ∇~ × A~+ = B~ everywhere but on the negative z-axis (θ = π) and similarly ∇~ × A~− = B~ everywhere but on the positive z-axis (θ = 0).

Furthermore, we see that outside of the z-axis we have ∇~ × (A~+ − A~−) = 0, which, as we have 3 seen before (Helmholtz theorem) implies the existence of a scalar potential χ : R → R such that ~ ~ ~ 3 A+ − A− = ∇χ. However, the complement of the z-axis R − {(x, y, z)|x = y = 0} is not simply connected, which means that χ might not be defined globally as well. Actually, the χ mentioned ~ ~ g here is just a gauge transformation, as shown in the final comment in the last section, A+ + ∇ 2π . ~ ~ ~ g ~ For any θ 6= 0 or π, in other words over their common domain, indeed A+ − A− = ∇( 2π φ) := ∇χ. However, we see from the above (keeping r and z constant) that over the polar angle φ the function χ cannot be continuous, since it depends on the angle, This can be shown explicitly by considering 2 π n the flux over the unit sphere, or actually Σ := S − {(r, φ, θ)|θ = 2 }, where S , n ∈ N is the n-dimensional unit sphere. In other words the globe without the equator E (which is of measure zero): Z Z g = (∇~ × A~+) · dS~ + (∇~ × A~−) · dS~ Σ+ Σ− Stokes Z Z = A~+ · d~` − A~− · d~` E E Z = ∇~ χ · d~` E = χ(2π) − χ(0).

Where E is the equator, and ~` the induced orientation from the upper and lower hemispheres, Σ+ and Σ− respectively. The minus sign arises precisely from the opposite induce orientation (the right hand rule). Since the flux is non-zero, χ must be discontinuous, in other words it is multivalued. The quantization condition can now be by invoking the quantum mechanical description of an electrically charged particle in the magnetic field of the monopole. If the state of the particle is represented as a wavefunction ψ, then we can write the Schrödingerequation as

∂ψ Hψˆ = i , ~ ∂t where Hˆ is the Hamiltonian operator. Then Hˆ becomes, by the minimal coupling to the magnetic potential (equation (37)):

Πˆ 2 (ˆp − qAˆ)2 2 Hˆ = + V (~r) = = − ~ (∇~ 2 − uA~)2. (75) 2m 2m 2m

Where (working in the position basis) Π := ∇~ − iqA~ and u := q , for q the magnetic charge. ~ Recall that the Schr¨odingerequation is invariant under U(1) gauge transformations

−iuχ −i q χ A~ 7→ A~ + ∇~ χ and ψ 7→ e ψ = e ~ ψ,

42 as was mentioned in subsection 3.2, part II. Considering now the non-relativistic action (32) of a particle, which we take to move across a path `, one finds that the change in action due to a gauge transformation χ becomes (due to equation (50)), Z Z q ~r˙ · ∇~ χ dt = q ∇~ χ · d~` = δS (76) t ` and thus the change in phase of eiδS(`)/~ for the wavefunction describing the particle. Now the phase of a wavefunction is not relevant, but phase differences are. If two particles have the same beginning and endpoints but a different path, then the change in phase cannot depend on a gauge transformation (as described in section 1.1, part II) in other words δS = 2πn~ for an n ∈ Z.

To have a well-determined theory, it must be possible to gauge transform the A+ potential to the ~ ~ g A− potential by the gauge transformation (in their common domain) of ∇χ = ∇ 2π φ. Considering now a particle moving in a closed loop on the equator E one ﬁnds from the discussion above that: Z δS = 2πn~ and δS = q ∇~ χ · d~` = gq, `=E leading to the famous Dirac quantization condition

qg = 2π~n, where n ∈ N.. (77)

Note that the integral over the equator could be taken around any closed curve around the Dirac string. It turns out that the number n depends on the winding number of the map from the curve to the phase change in the wavefunction, see appendix C. Another argument is given by the fact that the phase shift of a wavefunction ψ by a gauge trans- g formation must be physically invariant. The the gauge transformation deﬁned by χ = 2π φ gives a −i q χ −i qg φ phase shift of e ~ = e ~2π . Since this holds at any non-zero point described by the polar angle φ, one must have that qg = 2π~n for some n ∈ Z. Finally, notice that the Dirac quantization condition indeed implies quantization of the electric charge q if a monopole exists. Summarizing we get A global vector potential cannot be found because it contradicts the no magnetic monopole law. A singular Dirac string in the vector potential is intrinsic to the system.

Considering the specific vector potentials A~+ and A~−, one then finds the Dirac quantization condition, if one considers invariance under gauge transformations of the quantum mechanical wavefunction of an electrically charged particle One also finds the Dirac quantization condition when noting that the variation of the action δS for the same potentials along a closed path around the singular strings must remain invariant under a variation caused by the gauge transformation. As a final note, something about the Aharanov-Bohm effect. When a particle in a system which has a non-zero electromagnetic potential goes around in a closed loop `, the particle picks up a phase change ∆φ of q Z ∆φ = A~ · d~`. ~ ` This effect, called the Aharanov-Bohm effect, after some of its founders [19], was experimentally proven. The fact that the Dirac string is unobservable translates into the absence of the Aharanov- Bohm effect for the Dirac monopole.

43 2 Dirac monopole: Dyons and the Zwanziger-Schwinger quantization condition

The Dirac quantization condition can be derived via a different route. Namely looking at the conservation of angular momentum L~ = ~r × m~r˙, again of a particle of mass m and charge q at position ~r in the presence of a magnetic monopole at the origin. Invoking the Lorentz force law we find: dL~ d qg = rˆ. dt dt 4π Where the final identity can be found by invoking the BAC-CAB identity of vector calculus and the fact that ~r˙ and ~r are perpendicular in the system of a point charge in a magnetic field. So the conserved quantity is qg qg J~ := L~ − rˆ = ~r × m~r˙ − rˆ (78) 4π 4π which is known since 1896 due to Poincaré. This is not the most intuitive guess, however, since rotational symmetry of the system usually implies conservation of the angular momentum L~ = ~r × m~r˙. The angular momentum hence is now not identified with L~ but with J~. The quantity J~ turns out to be the total angular momentum of the particle, with a correction from ~ R ~ ~ 3 the total angular momentum of the fields Jem = 3 ~s × (E × B) d s, where one integrates over all R 3 space (except the origin) ~s ∈ R − {0} is the position, E~ (~s −~r) is the electric field due to the charge 3 (which is at position ~r ∈ R ) and B~ (~s) is the magnetic field due to the magnetic monopole (at the origin). Indeed one finds: Z ~ gq 1 3 (Jem) = 2 3 3 ~s × ((~s − ~r) × ~s) d s (4π) R3 s |s − r| g Z ~s = (E~ · ∇~ ) d3s 4π R3 s qg = − rˆ 4π Where the BAC-CAB identity of vector calculus was used and partial integration was performed, together with the fact that ∇~ · E~ = ρe = qδ(~s − ~r). Thus the total angular momentum is just the sum of the angular momentum of the electric charge and the angular momenta of the fields. Since we have complete spherical symmetry, total angular momentum in every direction is conserved. Considering now the time derivative of equation (77) we find that (redefining also ~r as the old ~s) qg ~r × F~ = ∂ rˆ 4π t where one defines the force classically as F~ = m~r¨. This means that the force is known in the ~ ~ ~ direction perpendicular to ~r, which will be defined as F⊥ such that F = F⊥ + Fkrˆ. Taking the cross product gives 2 ~r × (~r × F~ ) = ~r × (~r × F~⊥) = −r F~⊥ combining these two equations we find qg qg 1 1 qg F~ = − ~r × ∂ rˆ = − rˆ × ( ~r˙ − r∂ˆ r) = − rˆ × ~r˙ (79) ⊥ 4πr2 t 4πr r r3 t 4πr2 Meaning that this component of the force, forces the particle to move in the direction opposite to the direction of the particle angular momentum, perpendicular to the velocity with constant ~ magnitude. This means that in the absence of a Fk term the particle moves across a circle ‘cut out of’ the surface of a cylinder with symmetry axis through the origin, see figure 6. Considering both components of the force together with equation (77), the energy and the (total

44 Figure 6: The cone of movement for an electric monopole in the presence of a magnetic monopole. Image thanks to [20]. and) angular momentum in the radial direction one finds that the particle moves on a cone with the top at the magnetic monopole. Again the quantization condition arises when one considers quantum mechanics. In quantum me- 1 chanics the total angular momentum J is quantized in the form J = 2 n~ where n ∈ Z. Using this and equation (77) one finds: 1 qg |J~ | = n = − =⇒ qg = 2π n (80) em 2 ~ 4π ~ which is consistent with the information we found before (where u = q ). ~ The advantage of the latter proof is that we can retrieve a bit more information about a system consisting of two dyons (particles with both electric charge q and magnetic charge g). If we have two dyons (q, g) and (q0, g0) then we find the angular momentum of the total field (by linearity of the fields): Z Z ~ ~ ~ ~ ~ Jem = ~r × (Eq × Bg0 ) dV + ~r × (Eq0 × Bg) dV. R3 R3 We can deduce, by the same reasoning as before, and using electromagnetic duality (equation (70))

45 that 0 0 (qg − q g) = 2π~n for n ∈ Z, (81) meaning that for two dyons we have another quantization condition. This quantization condition for two dyons can be rewritten in a nice way when considering a system of two dyons with equal magnetic charge. If we have two dyons of equal magnetic charge (q = u~, g) 0 0 0 0 and (q = u ~, g) then we ﬁnd from equation (81) that g(u − u ) = 2πn or g(q − q ) = 2π~n. Furthermore, the existence of the electron (e, 0) implies that

2π g = ~n, (82) e for any dyon with magnetic charge g. If the magnetic charge g is positive and minimal in other 2π~ 0 words g = e then we ﬁnd the diﬀerence between electric charge of the two dyons (q, g) and (q , g)

q − q0 = ne, (83) where still n ∈ Z. This result, that two electric monopoles with the same magnetic charge differ by an integer times the charge of the electron is a nice extension of the known charges of the (fundamental) particles. We cannot say anything further about the magnitude of q or q0. Actually, considering another symmetry, charge-parity (CP) transformations, it is possible to say something about the absolute magnitude of q. Since electric charge remains equal under a parity transformation: q 7→ −q. Since the magnetic field is a pseudo-vector one finds g 7→ g. If, finally, it is assumed that the system is invariant under a CP transformation then one can consider a dyon with 2π~ electric charge q and minimal magnetic charge in other words of the form (q, g = e ). Considering now the CP-invariance, and the condition in equation (83) for this particle one finds n qg = πn =⇒ q = e. (84) 2 Note that this is the most strict equation that one can find when considering any pair of dyons. So the charge can take any half integer values From the equation q − q0 = ne we find then that the charges lie integer steps apart, i.e 1 either q = ne or q = ne + e. (85) 2 This equation was derived by Zwanziger [21] and Schwinger [22]. Later this equation will be properly adapted, since it turns out that this system is not CP invariant (see section 7). Finally note that the quantization condition predicts the minimal strength of the field of the magnetic monopoles: g2 4πr2 g2 1 4π 1 1372 ( ) = = ( ~)2 = = ≈ 5 · 104, (86) 4πr2 q2 q2 4 e2 4α2 4 1 for α ≈ 137 the fine-structure constant, such that the magnetic charge/field-strength is much larger then the electric field strength. The results found from the last sections can be summarized as: For a particle with electric charge q in the presence of a magnetic monopole, charge q (and vice versa) one finds the Dirac quantization condition qg = 2π~n for n ∈ Z. Either via the angular momentum or via the consideration of the vector potential with a singular string. Considering dyons, the existence of the electron implies a ladder of values for the magnetic 2π~ charge of any dyon: g = e n

46 For two dyons (q, g), (q0, g) of equal magnetic charge, the electric charges diﬀer by an integer times the electron charge q − q0 = ne. 0 2π~ Similarly for two dyons of equal electric charge q = e one ﬁnds g − g = e n, which was already implied by the existence of the electron If the system is CP invariant, then the absolute magnitude of the electric charge of dyons with 2π~ 1 minimal magnetic charge g = e is given by q = ne or q = ne + 2 e

47 3 The bosonic part of the Georgi-Glashow model

A new kind of monopole will be considered in this section. A monopole which is everywhere smooth and moreover inevitably arises in the system instead of Dirac monopole, which is placed in the system in an ‘ad hoc’ way. The monopole solutions arise in a model with a compact simple or semi- simple gauge group which gets spontaneously broken into the U(1) subgroup [23]. In particular, in any Grand unified theory, unifying the strong, weak, and electromagnetic force, such monopole solutions arise. The latter group being also the gauge group of the Dirac monopole. An example of such a system is the bosonic part of the Georgi-Glashow model. The Georgi-Glashow model was an early proposal to describe the electroweak interactions. The bosonic part of the model is used which consists of an SO(3) Yang-Mills theory coupled to a Higgs field in the adjoint representation. This is described by a Lagrangian of the form 1 1 L = − F~ · F~ µν + D φ~ · Dµφ~ − V (φ), (87) 4 µν 2 µ called the Yang-Mills-Higgs Lagrangian, after the founders of its components. As in section 3, part II, the objects in the gauge theory are identified with objects in the Lie ∼ 3 3 algebra so(3) = (R , ×). Due to the commutativity of R , all further inner products commute. The components that we find in the Lagrangian are: The Higgs field φ~ which transforms as transformation φ~ 7→ O(xµ)φ~ under an O(xµ) ∈ SO(3) gauge transformation, where the spacetime dependence of O is made explicit 1 ~ ~ 2 2 The Higgs potential V (φ) = 4 λ(φ · φ − a ) , which transforms trivially ~ µ ~ µ ~ µ −1 1 µ −1 The gauge fields W , which transform as W 7→ OW O + e ∂ OO . Where the fields W~ µ are first transformed in so(3) and then the identification with 3 is made. Also e = q R ~ has taken the role of u, which is not to be confused with the electric charge of an electron. Note that the transformed field lies again in so(3), since (OW~ µO−1)T = −OW~ µO−1 and (∂µOO−1)T = O∂µO−1 = −∂µOO−1, where the second equality is found if one derives 11= OOT The covariant derivative is, as before, defined as Dµ· = ∂µ11 · −eW~ µ × · The gauge field strength F~ µν = ∂µW~ ν − ∂νW~ µ − eW~ µ × W~ ν, which, like the gauge fields and the Higgs field, is a three-dimensional vector of matrices in so(3) or equivalently a three- dimensional vector. 1 ~ µν ~ Again, the objects are defined as in section 3, Part II. The Lagrangian density L = − 4 F · Fµν + 1 µ ~ ~ ~ 2 D φDµφ − V (φ) is then gauge invariant. This is not the only symmetry, however. The ’t Hooft- Polyakov ansatz arises from the consideration of all the symmetries of the above Lagrangian. Now the dynamics of the fields are derived. The first equation follows quickly from the Jacobi identity of the Lie bracket [A, [B,C]]+[C, [A, B]]+ [B, [C,A]] = 0, for A, B, C in the corresponding Lie algebra, one finds, by filling in the covariant derivatives A = Dµ, B = Dν and C = Dρ, which are in the Lie Algebra due to the fact that [Dµ,Dν] ∝ F~ µν is in the Lie algebra.

[Dµ, [Dν,Dρ]] + [Dρ, [Dµ,Dν]] + [Dν, [Dρ,Dµ]] = 0,

µν which implies, since [Dµ,Dν] ∝ F~ that

∗ DµF~νρ + DρF~µν + DνF~ρµ = 0 ⇐⇒ Dµ F~µν = 0, (88)

∗~ µν 1 µνλρ ~ if we deﬁne the hodge adjoint F = 2 Fλρ. Equation (88) is called the Bianchi identity.

48 Furthermore by varying the action one can ﬁnd the equations:

µν µ µ 2 DνF~ = −eφ~ × D φ~ and DµD φ~ = −λ(φ~ · φ~ − a )φ,~ (89) such that the dynamics of the ﬁeld is determined by equations (88) and (89). The following identity is useful for the derivation of the dynamical equations: ~ ~ (δ~a × b) · ~c = ijk δajbkci = δ~a · (b × ~c). (90)

First we vary the action by introducing a variation δW~ µ of the corresponding ﬁeld. The ﬁrst term becomes

1 Z (90) Z − δ(F~ µν · F~ ) dx = δW~ ν · ∂µF~ + e(W~ µ × F~ ) · δW~ ν dx, 4 µν µν µν Z ν µ = δW~ · D Fµν dx where partial integration was performed and dx = d~xdt. The second term becomes Z Z µ µ µ (δD )φ~ · Dµφ~ dx = (δ(∂ φ~) − eδW~ × φ~) · Dµφ~ dx Z (90) µ = − e φ~ × (Dµφ~) · δW~ dx

By looking at the last two equations and using Hamilton’s principle we get precisely the first of equations (88), since the Higgs potential and the coordinates are left unchanged. Varying now the Higgs field φ~ with variation δφ~ one finds Z Z δV (φ) dx = λ (φ2 − a2)φ~ · (δφ~) dx.

Furthermore one ﬁnds the term 1 Z ~ Z δ(Dµφ · D φ~) dx = (∂µδφ~ − eW~ µ × δφ~) · D φ~ dx 2 µ µ Z µ = δφ~ · D Dµφ~ dx.

Again, adding the above two final equations and noting both that the kinetic term of the gauge µν field ∝ F Fµν is unchanged and that the two equations form the variation of the action δS = R R R 2 2 µ δ( L dx) = δ(L) dx + Lδ(dx) = δφ~(λ(φ − a )φ~ + D Dµφ~) dx, which is zero by Hamilton’s principle, one finds exactly the second equation of equations (88). Now the system will be rewritten in the Hamiltonian formalism. First consider the conjugate momenta of W~ ν and φ~ respectively ∂L ∂L ~pν := = −F~0ν and Π~ := = D0φ.~ (91) ν ∂(∂0W~ ) ∂(∂0φ~) ~ ~ ~ 0i ~ 1 ~ In accordance with the electromagnetic tensor the quantities Ei := −F0i =F , and Bi := 2 ijkFjk are defined (equations (15) and (16)), which is later seen to be similar to an electric and magnetic field. Then the Hamiltonian density becomes

∂L ν j H = · ∂0W~ + F~0j · ∂0W~ + Π~ · ∂0φ~ − L ν ∂(∂0W~ ) j = E~j · ∂0W~ + Π~ · ∂0φ~ − L.

49 Note that the Lagrangian takes the form 1 1 1 1 L = E~ · E~ − B~ · B~ + Π~ · Π~ − D φ~ · D φ~ − V (φ~), 2 i i 2 i i 2 2 i i due to the metric signature (+, −, −, −) and the fact that

ij F~ · F~ij = ijkijlB~ k · B~ l = 2B~ k · B~ k.

The Hamiltonian density is then found to be equal to 1 1 1 1 H = E~ · E~ + Π~ · Π~ + B~ · B~ + D φ~ · D φ~ + V (φ~) + eΠ~ · (W~ × φ~) + E~ · (∂ W~ + eW~ × W~ ). 2 i i 2 2 i i 2 i i 0 i i 0 0 i

Now the gauge will be ﬁxed such that W~ 0 = 0. The Hamiltonian density reduces to: 1 1 1 1 H = E~ · E~ + Π~ · Π~ + B~ · B~ + D φ~ · D φ~ + V (φ~). 2 i i 2 2 i i 2 i i

A vacuum configuration is defined to be a configuration such that the Hamiltonian density vanishes. In the other words µ F~µν = 0 D φ~ = 0 V (φ~) = 0. (92) Note that for the Higgs potential to vanish one must demand φ = |φ~| = a. Moreover, if the kinetic ~ µ 1 µ term of the Higgs field vanishes simultaneously one finds the that ∂µφ = a∂µnˆ(x ) = e W in all µ 3 of spacetime, for some normalized vectorn ˆ(x ) ∈ R , dependent on the spacetime coordinate. The configuration W~ µ = 0 and φ~ = azˆ in all of spacetime is an example of a vacuum configuration. The Higgs vacuum is defined as the configurations precisely such that the energy of the Higgs field is zero. In particular the second two equations in (92) should be satisfied. We have seen that the configurations of the Higgs field in the Bosonic part of the Georgi-Glashow model are gauge invariant w.r.t. the group SO(3). Note that we are working in the Lie algebra 3 ∼ 3 (R , ×) = so(3), so one can visualize the action of SO(3) on the Lie algebra as rotations in R . 3 However, this R is not physical space but the so-called isospin space in which φ~ lives. The rotation group SO(3)phys acting on physical space, however is again an action that is a stabilizer of the Lagrangian. Now one already has the symmetry group SO(3) × SO(3)phys. The configurations of the Higgs field in the vacuum spontaneously breaks the symmetry since one can only rotate the field φ~ = anˆ around then ˆ axis. The former symmetry group SO(3) is therefore spontaneously broken into the group SO(2) of rotations around then ˆ axis, defined to be the stabilizer of the Higgs field. Note that this is precisely the spontaneous breaking of a local gauge symmetry as seen in chapter 4, part II. Therefore one can now expect to identify massive gauge bosons coming from the gauge fields W~ µ. Indeed this is the case. Considering again a small perturbation ψ~ of φ~ = anˆ, around the vacuum equilibrium, φ~ = anˆ + ψ~ =: ~a + ψ~, one can find a perturbed Lagrangian density. W.l.o.g it may be assumed (using rotational invariance of the physical space) that at a specific point ~a = 0 0 aT . The gauge group SO(3) then gets spontaneously broken into the subgroup SO(2), similar to figure 3. Because the dimension of the gauge group is reduced from 3 to 1, two massive bosons and one massless boson are expected. Because of the choice of ~a it is useful to further specify the vector W~ µ and split the covariant derivative W µ 1 e W~ µ = W µ and Dµ = Dµ| + Dµ| where Dµ| = ∂µ11 − Aµ~a × · .  2  3 3⊥ 3 a Aµ Note that the covariant derivative (an operator) was split up into operators on two subsets. One being the subspace of the broken group (here rotations around the z-axis), such that one expects the vector boson Aµ to be massless and the other two components of W~ µ to be massive.

50 Then the perturbed kinetic term of the Higgs ﬁeld becomes

µ µ µ 2 2 µ µ Dµφ~ · D φ~ = Dµψ~ · D ψ~ − 2e (W~ µ × ~a) · D ψ~ + e a (W~ µ · W~ ) − (W~ µ · ~a)(W~ · ~a)   Wµ,2 ~ µ ~ µ ~ 2 ~ ~ µ ~ ~ µ = Dµψ · D ψ − 2ea(−Wµ,1) · ∂ ψ + 2ae (Wµ · W )ϕ − (Wµ · ψ)A 0 2 2 µ µ + e a W1 Wµ,1 + W2 Wµ,2)

Furthermore, the middle two terms can be written as

 W  0  W  µ,2 e µ,2 −2e(W~ × ~a) · ∂µψ~ + 2e2(W~ × ~a) · (W~ µ × ψ~) = 2ea∂ −W + A 0 × −W · ψ~ µ µ µ  µ,1 a µ    µ,1 0 a 0

Also the potential term becomes 1 V (~a + ψ~) = λ(ψ4 − 4aϕψ2 + 4a2ϕ2 − 4aψ2 − 8a2ϕ + 4a2). 4

Taking a look at equation (43) and redeﬁning the mass as

1 MH 2 2 1 MW 2 ± µ,± L = ··· + ( ) φ + ( ) Wµ W + ··· 2 ~ 2 ~ and deﬁning a charge q via the coupling with the gauge ﬁeld in other words the constant in the covariant derivative (explained in the following section)

µ µ µ µ µ e µ µ µ q µ D = D |3 + D |3 where D |3 = ∂ 11 − A ~a × · = D |3 = ∂ 11 − A ~a × ·, (93) ⊥ a ~a µ +,µ one then ﬁnds the following masses and charges for the new ﬁelds (note W1 = W )

Field Mass Charge

Aµ 0 0 √ φ MH = a 2λ~ 0

± Wµ MW = ae~ ±e~

Table 1: The perturbative spectrum after the broken local symmetry

Indeed two massive fields have come up, which in accordance to the weak force bosons are called the ± µ Wµ fields. The massless gauge field A pointing in the direction of ~a can be identified with the electromagnetic potential/the photon. Thus one encounters an electromagnetic system in the solutions of the Yang-Mills-Higgs Lagrangian, which will lead to the description of the ’t Hooft-Polyakov monopole.

51 4 Static, ﬁnite-energy solutions in the ’t Hooft-Polyakov ansatz

Here finite-energy solutions of the system will be considered. For these types of solutions the integral R E = V H d~x must converge when the volume V extends to all of physical space, meaning that the fields must go to a vacuum configuration asymptotically as one moves away from the origin.

A first property of the finite-energy solutions is that the solutions can be defined in a set of equivalence classes considering homotopic properties. 3 ~ µ ~ µ 1 2 2 2 The Higgs potential can be seen as a function V : R → R; φ(x ) 7→ V (φ(x )) = 4 λ(φ − a ) , ~ µ ∼ 3 3 since φ(x ) ∈ so(3) = (R , ×). Of course the (vacuum) sphere M0 ⊂ R of radius a contains all possible values of the Higgs field φ~ in the vacuum configuration (compare with figure 3). One can then consider the function φ∞(ˆr) := lim φ~(~r) ∈ M0 that defines the asymptotic configuration of r→∞ the Higgs field, which must be in M0 by the above discussion. Note finally that the domain of the function φ~∞ will be taken to be ’a sphere at spatial infinity’ S, which turns out to be well-defined[1].

Having defined this function, one can now exploit the mathematical knowledge about the homotopy classes. The space of continuous functions C(S, M0) from one sphere S to another M0 is discon- 15 nected since it is isomorphic (in the sense of groups) to (Z, +) . So the space in which φ∞ lives is disconnected and indexed by an integer n called the degree of the map. The topological number of a finite-energy configuration is defined as the degree of the corresponding map φ∞.

The topological number of a field configuration is invariant under any connected, continuous defor- 2 mation, since C(S, M0) isomorphic to the fundamental group π1(S ) of the unit sphere. In particular we find that the topological number is invariant under time evolution by considering the 2 2 homotopy F : I = [0, 1] × S → S ;(t, x) 7→ F (t, x) = φ∞(t, x), when the Higgs field is also considered time dependent. Note that the time-evolution of the Higgs field is continuous because of the equations of motion. Furthermore the topological number is invariant under the group action performed by the gauge group SO(3), in other words the topological number is a gauge invariant. Just like time- 2 2 2 evolution, the action of SO(3) on π1(S ) can be seen as a homotopy F : I × S → S ; F (t, x) = 16 φ∞(t, x) This can be done since SO(3) is path-connected .

The importance of the topological number of a finite-energy solution is seen by noting that such a solution must stay in its own homotopy class, when continuous, (connected), changes are being made. For example, a configuration with energy density H = Diφ~ ·Diφ~ 6= 0 due to a sufficient gradient of the Higgs field (such that the topological number is non-zero) can never dissipate to a zero-energy configuration, without gradient. So in a sense it will be stable.

Now we will investigate if such a stable solution exists. This is done by restricting to spherically symmetric static solutions. A (finite-energy) configuration is static if it is time-dependent and W~ 0 = 0 throughout spacetime. ∂L ~ ~ ∂L ~ ~ Note then that i = Ei = −F0i = 0 and = D0φ = Π = 0. This implies that H = −L, ∂(∂0W~ ) ∂(∂0φ~) meaning that one can extremize the Hamiltonian density to find the equations of motion.

15Every result mentioned about homotopy can be found, or easily derived by information written in appendix C 16Using Euclidean topology one can perform any rotation by continuously increasing the Euler angles from (0, 0, 0), corresponding to the identity, to the wanted rotation. This increasing of angles is then parametrized by a single t such that it becomes a path.

52 An ansatz that turns out fruitful is the ’t Hooft-Polyakov ansatz, given by ~r φ~(~r) = H(aer) (94) er2 0 Wk = 0 (95) rj 1 W i = (1 − K(aer)) or equivalently W~ i = eî × ~r (1 − K(aer)) (96) k kij er2 er2 For some functions (smooth) H,K : R → R. This ansatz might come out of the blue, but can be found by consideration all the symmetries in the Yang-Mills-Higgs Lagrangian density (87). First of all, as stated before we have SO(3) gauge invariance w.r.t the space in which φ(xµ) lives (called isospin space). Furthermore the Lagrangian is relativistic, thus from part I we know that it is also invariant under the group SO(3)phys of rotations in physical space. One is not left with a lot more than the ’t Hooft-Polyakov ansatz [24]. When substituting this into the expression for the Hamiltonian density one finds Z Z ∞ 4πa dξ 2 dK 2 1 dH 2 1 2 2 2 2 λ 2 2 2 E = H dV = 2 ξ ( ) + (ξ −H) + (K −1) +K H + 2 (H −ξ ) , (97) R3 e 0 ξ dξ 2 dξ 2 4e where ξ = aer. This is derived in appendix B. Now one can look at the boundary conditions for H and K in the two limits ξ → ∞ and ξ → 0. For ξ → ∞ we want the Hamiltonian density to vanish. The dominant terms for this limit are H2 2 2 H2 H2 proportional to ξ( ξ2 − 1) and K ξ2 . For the first term to vanish we demand that ξ2 → 1 ‘sufficiently fast’, such that the second term vanishes if K → 0 ‘sufficiently fast’. The other terms H then vanish as well, because the limits of ξ and K are asymptotic such that the derivatives vanish as well. 1 (K2−1)2 For ξ → 0 the term 2 ξ2 implies that K must approach 1 at least linearly in ξ as ξ → 0 and H the term proportional to ξ2 implies that H must approach zero at least linearly in ξ as ξ → 0. Notice that al the other terms are also kept from diverging with the above boundary conditions. Summarizing we get H K → 0 and → 1 ‘sufficiently fast’ as ξ → 0 (98) ξ K − 1 ≤ O(ξ) and H ≤ O(ξ) for ξ → 0. (99) Note furthermore that plugging these boundary equations into equations (166) - (168) one indeed finds that the terms in the Hamiltonian density vanish. With these boundary conditions we can conclude that the map φ~∞(ˆr) in the ’t Hooft-Polyakov ansatz has topological number 1 and therefore the solution is stable in the sense that it will not dissipate because of a previous discussion. Its topological number is 1 since ~r H φ~∞(ˆr) = lim H = lim arˆ = arˆ r→∞ er2 ξ→∞ ξ which is homotopic to the identity map (by the canonical homotopy F (t, ~r) = arˆ + t(1 − a)ˆr). The reason that the ’t Hooft-Polyakov ansatz is interesting is precisely that its topological number is non-zero. Also the gauge fields W µ there vanish. Later it will be seen that this implies that the monopole-like solution (found in the next section) has a magnetic charge, which is purely topological.

4.1 The solution of H and K in the ’t Hooft-Polyakov ansatz

If a solution for the SO(3) Georgi-Glashow model in the ’t Hooft-Polyakov ansatz exists then it is non-dissipative, because of its topological number. The solution is expressed in the dynamical equations of the ﬁelds H and K, which can be found in two ways:

53 The energy E as in (97) is extremized, since H = −L One substitutes the ansatz (94) - (96) in the equations of motion (89) Here the energy is extremized. Beforehand, it is noted that H = −L since we are looking for time independent solutions. For invariance under H 7→ H + δH note that Z ∞ 4πa 1 2 2 2 λ 2 2 δE = dξ 2 − ξ ∂ξ H + 2HK + 2 (H − ξ )H δH e 0 ξ e such that δE λ = 0 =⇒ ξ2∂2H = 2K2H + H(H2 − ξ2) (100) δH ξ e2 Varying now w.r.t. K one finds Z ∞ 4πa dξ 2 2 2 δE = 2 ξ 2∂ξK∂ξδK + 2K(K − 1)δK + 2H KδK e 0 ξ Z ∞ 4πa dξ 2 2 2 2 = 2 − 2ξ ∂ξ K + 2K(K − 1) + 2H K δK e 0 ξ such that δE = 0 =⇒ ξ2∂2K = H2K + K(K2 − 1). (101) δK ξ The equations (100) and (101) describe the dynamics of the solution for the SO(3) Georgi-Glashow model in the ’t Hooft-Polyakov ansatz: λ ξ2∂2H = 2K2H + H(H2 − ξ2) and ξ2∂2K = H2K + K(K2 − 1). (102) ξ e2 ξ It turns out that there exists a solution to these differential equations[25] but the exact form of the solutions is unknown. Therefore only the asymptotic form of the solutions will be considered in other words (what turns out to be) the monopole from far away. Numerical analysis of the differential equations (102) have been performed of which the results are shown in figure 7. The asymptotic form of these dynamical equations in the limit ξ → ∞ are λ ∂2K = K and ∂2h = 2 h, ξ ξ e2 For h = H − ξ. Note that the boundary condition for h becomes h → 0 as ξ → ∞. In this limit the equations have a solution compatible with the boundary conditions M r M r K ∝ e−ξ = exp − W and h ∝ exp − H (103) ~ ~ ~ where MW and MH are as in table 1. The fields K and h now have a typical size of the order of MW and ~ respectively (of course the bigger the mass, the more localized the object). Comparing with MH equation (98) and (99) we see that the boundary conditions which should be satisfied are indeed satisfied. Thus, The solution in ’t Hooft-Polyakov ansatz is an object of finite size. This effective size is given by the largest of the two effective sizes of the field K and h. Note that the ’t Hooft-Polyakov solution cannot yet be identified with some kind of monopole (no charges/magnetic fields have been identified), but now it can be identified with a type of localized, particle-like object. Equation (103) also implies the behaviour in the far-field limit

rˆ rˆ φ~ = H(ξ) = (e−MH r/~ + ξ) such that in the limit ξ → ∞, φ~ → arˆ er er 1 1 W~ i = eˆi × ~r(1 − K(ξ)) such that in the limit ξ → ∞, W~ i → eˆi × ~r, er er

54 H(ξ) Figure 7: The fields ξ (different in graph) and K(ξ) as solutions of the differential equations (102), for different values ofs the Prasad-Sommerfield limit λ, which is discussed in subsection 6.2. For stronger values of the Higgs potential i.e. bigger values of a or λ one finds a smaller monopole. One also clearly sees the asymptotic behaviour of the fields, as described in equations (98) and (99). Image thanks to [23]. as it should be according to (98) and (99). The identification of the ’t Hooft-Polyakov solution with a monopole starts here. In the asymptotic µ 1 ~ ~ µ µ limit (where, φ∞(ˆr) = arˆ)) the electromagnetic potential is seen to be A = a φ · W (since A was the component of W µ that does not change when rotating around the Higgs field), which is zero in µν 1 ~ ~ µν µν the ansatz, such that the electromagnetic field becomes G = a φ · F . Because G is precisely the gauge field-strength in the direction of the electromagnetic potential Aµ one can expect that the components of Gµν form the electric and magnetic fields, compare to equation (13). Indeed, one can find a connection to the Dirac monopole when one looks at the asymptotic form of the ’t Hooft-Polyakov solution:

ij 1 ~ ij H(1 − K) ik l rl lim G = lim φ · F~ = lim ljk δ r = ijl (104) ξ→∞ ξ→∞ a ξ→∞ ae2r4 er3

~ 1 jk such that one can define the magnetic field B = − 2 ijkG eî, in other words: 1 ~r B~ = . (105) e r3 This means that the field of the ’t Hooft-Polyakov solution from far away, equation (105), looks like the magnetic field of a Dirac monopole. From the Dirac quantization condition we know that 2π the minimum (magnetic) charge of a monopole is equal to g = e . Comparing now with the ’t 1 Hooft-Polyakov monopole(!) from far away (charge e ) one sees that, when looking at the standard

55 form of a magnetic monopole, equation (71), the magnetic charge of the ’t Hooft-Polyakov monopole is twice the minimum magnetic charge. This is because the electromagnetic SO(2) is embedded in such a way that the electric charge is an eigenvalue of the T3 isospin operator [23], which in the adjoint representation has integer isospin (compare with the half-integer values of the angular momentum operator in the section about the Dirac-monopole. The difference is due to the SO(3) isospin invariance, such that the T3 operator only takes on integer, versus SU(2) isospin invariance, such that the T3 operator takes on half-integer values). In particular, formally the operator of the chargeq ˆ is identified with the generator of the SO(2) =∼ U(1) group, such that the coupling constant e becomes the electric charge q. This means that the free parameter in the one-dimensional group SO(2) is identified with (the operator of the) charge (up to a constant), such that the charge can be found by the way a field couples to the electromagnetic potential q Qˆ = ~ φ.~ a This means that the covariant derivative (of the broken symmetry) takes the form 1 1 Dµ = ∂µ − (eW~ µ) · φˆφˆ × · = ∂µ − e Aµφ~ × · = ∂µ − AµQˆ × ·, a ~ such that the charge becomes q = e , like in the Dirac monopole (there e 7→ u). Compare with ~ equation (93). Note that this was also the case in the Dirac monopole, which exhibited U(1) symmetry. Now it is explicitly stated that the electric charge in the Dirac monopole also was the generator of the U(1) group. So the ’t Hooft-Polyakov solution describes an object of finite size, which from far away looks like a Dirac monopole with twice the magnetic charge. Moreover, note that the monopole is everywhere smooth. There is no presence of a singular region like the Dirac string. This is because of the massive fields h and K that become relevant as we approach the core.

4.2 The topological origin of the magnetic charge

In this section it will be seen that the magnetic charge of the ’t Hooft-Polyakov monopole is purely topological. Furthermore, the ’t Hooft-Polyakov monopole is everywhere smooth, again, due to the massive fields becoming important at the origin, which also distinguishes it from the Dirac monopole. The most special result of this subsection will be that the magnetic charge arises purely from behaviour of the Higgs field far away from the core. As seen in equation (103), a large distance away the massive fields K and h of the ’t Hooft-Polyakov monopole vanish. By looking at the definition of φ~, equation (94), one sees that the Higgs field approaches the Higgs vacuum outside of the monopole of typical radius R, for R the effective size of the ’t Hooft-Polyakov monopole described above, such that (since the gauge fields behave like k rj Wµ = er2 kij): 2 2 Dµφ~ = 0 and φ = a (106)

−r with a correction term of order O(e R ). When considering the particles/quantization of the fields this means that R is dictated by the mass of the heavy particles. This is true since the far away behaviour of the monopole is dictated by low order terms. Now we generalize from the ’t Hooft-Polyakov solution and consider any finite-energy solution of the equations of motion of the SO(3) Georgi-Glashow model. Since the massive field solutions always decay exponentially we can assume that, in the far-field limit, the Higgs field of these general solutions always satisfy equations (106). In particular this is satisfied everywhere apart from some localised regions in space, in other words a gas of monopoles is considered, which from far away go into the Higgs vacuum and are solutions of the equations of motion, equation (88) and (89).

56 ~ ~ ~ µ 1 µ ~ ~ µ Now we look at the Higgs vacuum, in which Dµφ = 0 such that φ × W = − e ∂ φ. Then W is determined in the orthogonal complement of φ~. Extracting this information by computing the term φ~ × (φ~ × W~ µ) = φ~(φ~ · W~ µ) − W~ µa2 and comparing we ﬁnd that 1 1 W~ µ = φ~ × ∂µφ~ + φA~ µ, (107) a2e a µ 1 ~ ~ µ where A = a φ · W like before. The ﬁeld strength F~ µν then takes the form

F~ µν = ∂µW~ ν − ∂νW~ µ − eW~ µ × W~ ν 2 1 1 = ∂µφ~ × ∂νφ~ + φ~(∂µAν − ∂νAµ) − φ~φ~ · (∂µφ~ × ∂νφ~), a2e a a4e since φ~ · ∂µφ~ = 0 and 1 1 1 1 −eW~ µ × W~ ν = −e( φ~ × ∂µφ~ + φA~ µ) × ( φ~ × ∂νφ~ + φA~ ν) a2e a a2e a 1 ~~ µ ~ ν ~ 1 ~ ~ µ ν ~ ν µ ~ 1 µ ν ~ ν µ ~ = − 4 φ φ · (∂ φ × ∂ φ − 3 φ (φ · (A ∂ φ − A ∂ φ)) − (A ∂ φ − A ∂ φ)) a e a | {z } a 0 where the identity (A × B) × (C × D) = (A · (B × C))D − (A · (B × D))C was used. One can then say that the field strength F~ µν points in the φ~ direction: 1 F~ µν = φ~ Gµν, (108) a for 1 Gµν = φ~ · (∂µφ~ × ∂νφ~) + ∂µAν − ∂νAµ. (109) a3e Most importantly, this term is gauge invariant [18] such that it can be defined as an object describing a physical property of the system such as the electromagnetic potential. µν µ µ µν From the equation of motion DνF~ = −eφ~ × D φ~ = 0 in the vacuum, one finds that D G = 0. ∗~ µν 1 µνλρ ~ µ ~ From the Bianchi identity Dµ F = 2 Dµ Fλρ = 0 one finds, again since D φ = 0, the equation Dν ∗Gµν = 0. This means that Gµν satisfies Maxwell’s equations in vacuum, equations (19) and (20). Indeed Gµν can now be identified with the electromagnetic potential. This is 1 ~ ~ µν not completely surprising since in the ’t Hooft-Polyakov ansatz, a φ · F was also viewed as the electromagnetic potential. Moreover, when the Higgs field is in a trivial (zero winding number) configuration, we get the regular electromagnetic potential Gµν = ∂µAν −∂νAµ, such that Maxwell’s equations become ∂µ ∗Gµν = 0. This reassures us that the assumption made above was quite reasonable. Furthermore, one sees that any solution that in the far field limit goes to the Higgs vacuum can be seen as a monopole in vacuum, but with what charge? Note that, by the discussion above, it becomes clear that the magnetic charge is defined by the term 1 ~ µ ~ ν ~ a3e φ · (∂ φ × ∂ φ). In particular, the magnetic current ((Jb)ν = kν|) becomes (equation (20)) 1 1 1 k = ∂µ ∗G = ∂µGλρ = ∂µφ~ · (∂λφ~ × ∂ρφ~) (110) ν µν 2 µνλρ 2a3e µνλρ a3e which is independent of the gauge field. Moreover from the antisymmetry in λ and ρ one finds that

ν ∂ kν = 0. (111) Indeed one ﬁnds a current. However, this current is not a Noethers current, it is conserved due to its deﬁnition. Indeed the magnetic charge then becomes (equation (46)) Z Z 1 1 i~ j ~ k ~ g = k0d~r = − 3 3 ijk∂ φ · (∂ φ × ∂ φ) d~r. (112) V 2ea V 2a e

57 Consider a set of magnetic monopoles in the Higgs vacuum and a (closed) surface Σ enclosing them. As in the treatment of the Dirac monopole, the flux of the magnetic fields through Σ defines the ~ 1 i jk magnetic charge. The magnetic field is defined, as usual, as B = 2 ijkeˆ G . The flux/charge gΣ then becomes Z gΣ := B~ · dS~ Σ Z 1 ~ ~ ~ = − 3 ijkφ · (∂jφ × ∂kφ)dSi 2ea Σ

i j k k j since ijkeˆ (∂ A − ∂ A ) = 2∇~ × A~ integrates to zero by Stokes’ theorem. Reassuringly, the definition of the magnetic charge makes sense when looking both at the Maxwell’s equations and the abelian electromagnetic theory. Furthermore, in the last integral only the partial derivatives in the direction perpendicular to dS in other words tangential to Σ contribute to the integral. This can bee seen by using equation (90). Since these partial derivatives are the only measure of local behaviour, this means that the magnetic flux, and therefore the magnetic charge, only depends on the behaviour of the Higgs field φ~ on the surface Σ. Furthermore, it turns out that the magnetic charge only depends on the homotopy class of φ~, seen as φ~ :Σ → M0. Thus the origin of the magnetic charge of the monopoles in the Higgs vacuum is purely topological, in particular depending on the homotopy class of the Higgs field in the Higgs vacuum

The fact that the magnetic charge only depends on the homotopy of the map φ~ :Σ → M0 can be shown by considering infinitesimal (continuous) deformations of the Higgs field in the Higgs vacuum, such that its mapping remains well-defined. This means that variation δφ~ of φ~ is considered, such that Dµδφ~ = 0 and φ~ · δφ~ = 0 (113) Indeed, one sees that Z ~ ~ ~ δgΣ ∝ ijkδ(φ · (∂jφ × ∂kφ)) dSi Σ Z ~ ~ ~ ~ ~ ~ = ijk 3δφ · (∂jφ × ∂kφ) − 2∂j δφ · (∂kφ × φ) dSi, Σ where partial integration was performed (and Σ is a closed volume). Note that the second term looks like dS~ · (∇~ × [...]), such that, due to Stokes’ theorem, the integral over this term vanishes when a closed volume Σ is taken. The first term is set to zero by noticing ~ ~ ~ ~ ~ that, as noted before, in the Higgs vacuum, φ · ∂µφ = 0, which implies that φ × (∂jφ × ∂kφ) = 0. Hence one can use (113) to see that the first term vanishes. This means that δgΣ = 0, implying again that the magnetic flux in the Higgs vacuum is invariant under continuous deformations generated by a variation in φ~ satisfying equations (113). Physically this means that configurations in different homotopy classes have infinite potential bar- riers between them. We have already noted that time-evolution and a continuous gauge transformation leave the homotopy class unchanged. A third homotopy is given by continuous changes of Σ within the Higgs vacuum such that (113) is not violated.

Because the flux is additive we find that the magnetic charge gΣ is also additive. Notice also that 4π the magnetic charge can be rewritten as gΣ = − e NΣ, where NΣ is defined as Z Z 1 ~ ~ ~ 1 NΣ = 3 ijkφ · (∂jφ × ∂kφ) dSi = 3 ijkmpqφm∂jφp∂kφq dSi (114) 8πa Σ 8πa Σ

58 R If the integrand is denoted by Ai, such that the integral becomes proportional to Σ Ai dSi, then ~ ~ ~ ∂iAi = ∂i(ijkφ · (∂jφ × ∂kφ)) ~ ~ ~ = ijk ∂iφ · (∂jφ × ∂jφ)

= 6 ∂1φ~ · (∂2φ~ × ∂3φ~)

Furthermore notice that the Jacobian j of the map φ~ :Σ → M0 is given by

j = ∂1φ~ · (∂2φ~ × ∂3φ~) 1 = ∂ A 6 i i such that the integral NΣ becomes 3 Z NΣ = 3 j dV 4πa V where V is the (finite) volume enclosed by Σ. This volume integral, over the Jacobian of the determinant of a map between spheres, is another definition of the topological number of the map ~ φ meaning that NΣ is an integer (called the Brouwer degree). This leads us again to a quantization condition, namely that gΣe = 4πNΣ. (115) This is not quite the Dirac quantization. The minimum magnetic charge is precisely twice the minimum charge of the Dirac monopole as stated before. Furthermore, the magnetic charge depends (only) on the winding number NΣ of the map φ~ and therefore (only) on the topological number. Thus, when considering the Bosonic part of the SO(3) gauge invariant Georgi-Glashow model, described by the Yang-Mills-Higgs Lagrangian, it is shown in the last few chapters that The energy vanishes when a solution of the equations of motion (88) and (89) is in the Higgs vacuum. In the Higgs vacuum the SO(3) local gauge invariance reduces to a local SO(2) invariance, rotating around the direction in which the Higgs field points After Higgsing, one finds two massive vector boson, W ± of mass M = ae with electric √ µ W ~ charge ±e~, one scalar boson ϕ with mass MH = a 2λ~ and a massless, chargeless vector field which (still in the vacuum) is identified with the electromagnetic potential/the photon (see table 1). In the Higgs vacuum, the Higgs field can be characterized by its topological number. In particular, non-trivial behaviour in the far-field limit implies a non-zero energy which is then stable under connected continuous deformations. The ’t Hooft-Polyakov ansatz has precisely such non-trivial behaviour of the Higgs field, in particular φ~∞ = arˆ. The gauge fields vanish asymptotically in the far-field limit, but first k rj behave like Wµ = er2 kij. The equations of motion, acquired by varying the energy, in the far-field limit showed us that the solution in the ’t Hooft-Polyakov ansatz has a finite typical length because of the massive fields K and h = H − ξ, which furthermore must satisfy equations (98) and (99), both of which were indeed satisfied by the behaviour of the solutions in equation (103). The ’t Hooft-Polyakov solution from far away looked like the Dirac monopole with twice the magnetic charge.

59 When generalizing to solutions with massive fields in the Higgs vacuum, it was found that the gauge field-strength pointed in the φ~ direction (precisely the direction of the electromagnetic potential) and from there one could show that the magnetic flux/charge of such a solution, satisfying equation (106), is homotopically invariant. Because the magnetic flux/charge of the solution only depended on the behaviour of φ~ on the surface Σ measuring the charge, the homotopy invariance implies that the magnetic charge of these solutions depends purely on the topological number of the far-field φ~∞. One then found the (non-abelian variant of the) Dirac quantization condition up to a factor 2, where the charge was connected to the winding number of the map φ~∞

60 5 The diﬀerence between the Dirac and the ’t Hooft-Polyakov monopole

There are two important differences between the Dirac monopole and the ’t Hooft-Polyakov monopole: The origin of the magnetic charge and the size of the monopoles. A transition from the ’t Hooft- Polyakov to the Dirac monopole can be made, precisely by an SO(3) (isospin) gauge transformation of the Higgs field. From this it turns out that the Dirac monopole is similar to the ’t Hooft-Polyakov monopole but with a trivial (vacuum) configuration of the Higgs field. It was seen that for finite-energy solutions of the bosonic part of the Georgi-Glashow model the electromagnetic potential becomes of the form (changing G to F ) 1 F µν = ∂µAν − ∂νAµ + φ~ · (∂µφ~ × ∂νφ~). (116) a3e For the Dirac monopole this becomes F µν = ∂µAν − ∂νAµ,. (117) Note that from equation (117) it follows that for the Dirac monopole the charge comes from the gauge fields (together with the string singularity). Similarly, for the ’t Hooft-Polyakov monopole the charge was seen to be completely determined by the second term in equation (116), containing the Higgs field. Indeed, the charge was determined completely by the winding number of the Higgs field in the Higgs vacuum17. Thus, the information about the magnetic charge of the monopoles is contained in different terms. Is there a way to relate these monopoles? To find this relation between the monopoles one can note that the system of a static Dirac monopole can be described by the Lagrangian density of the form (87), with a trivial Higgs field in the Higgs vacuum, for example φ~ = azˆ. Indeed the kinetic term vanishes and the Higgs potential just becomes a physically insignificant constant. In the Hooft-Polyakov ansatz on the other hand, the Higgs field (in the vacuum) points radially outward. The relation between the monopoles is given by a singular SO(3) gauge transformation of the Higgs field. Because of the singularity, it turns out that it does not leave the winding number invariant. Indeed the winding number of the Higgs field must be changed from 0 for the Dirac monopole to 1 for the ’t Hooft-Polyakov monopole. Now we will look at the specific potentials in equation (74), related by the gauge transformation in equation (73). The singularities of the element in SO(3) then has a singular term on both the north and the south pole, since there the polar angle φ is ill-defined. This translates directly into the positions of the Dirac strings. Thus, the Dirac string has its origin coming from a singular gauge transformation, meaning that the Dirac string is a defect coming from a faulty description of the system, not from nature itself (as for example a topological defect would be). Indeed, when the vacuum sphere M0 is considered with the radially outward pointing Higgs field, one cannot ‘comb the ball’ with continuously varying gauge transformations such that the transformed configuration is equal to the trivial configuration, compare with the Hairy Ball theorem, shown in figure 8. Therefore, in principle the introduction of the Higgs field provided enough information to spread out the Dirac singularity across the space to produce a smooth monopole and the gauge transformation transported the information of the charge from the gauge fields (and the singularity) to a specific, current-like term (see equation (110)) containing solely the Higgs field. So it is seen that the magnetic charge of both monopoles arises, although in both cases having to do with the topology, in a different way. The difference in size has already been noted. The Dirac monopole is a singularity at the origin, the ’t Hooft-Polyakov monopole is everywhere smooth and has a core with typical size of the Compton wavelength of either the H or the K fields, as noted after equation (103).

17For the Dirac monopole the charge was determined by the winding number of the wavefunction of the electric particle, and therefore the Chern class of the gauge ﬁelds [23], such that the magnetic charge also has a topological root

61 Figure 8: A visualization of the problem with combing the ball. Indeed in our case, like in the picture, one has a gauge transformation parametrize by the polar angle φ, which produces a singularity and one zero of the ﬁeld. Image thanks to https://en.wikipedia.org/wiki/Hairy_ball_theorem.

6 Mass and the BPS monopoles

Now the mass of the monopoles will be considered. Just like the point-like sources of electric ﬁelds, the mass for Dirac monopoles is a free parameter. It cannot be calculated by considering the electromagnetic properties. As we have seen, all the sources for mass of the ’t Hooft-Polyakov monopole are in the Yang-Mills-Higgs Lagrangian density, such that it should be calculable. A lower bound will be given for its mass (the Bogomol’nyi bound) and a solution which saturates the bound, called the BPS monopole, will be found.

6.1 The Bogomol’nyi bound

Because of relativistic invariance of the Hamiltonian density, one can find the equation for the mass M (note c = 1) Z 1 1 1 1 M = ( E~i · E~i + B~ i · B~ i + Π~ · Π~ + Diφ~ · Diφ~ + V (φ~)) dV (118) R2 2 2 2 2 1 Z ≥ E~i · E~i + B~ i · B~ i + Diφ~ · Diφ~ dV (119) 2 R3 0 since the Higgs potential V (φ~) ≥ 0 by definition and Π~ · Π~ = D0φ~ · D φ~ ≥ 0 due to the metric signature. Now the mass is rewritten with respect to an angular parameter θ, indicating the strength of the interactions E~i ·Diφ~ and B~ i ·Diφ~, between the momenta of the gauge fields and the (spatial) gradient of the Higgs field. In other words, we rewrite the bound of M as a function of E~i · Diφ~ sin θ and B~ i · Diφ~ cos θ: 1 Z 2 2 ~ ~ ~ ~ ~ ~ ~ ~ M Ei − Diφ sin θ + Bi − Diφ cos θ + Ei · Diφ sin θ + Bi · Diφ cos θ dV 2 R3 Z ≥ E~i · Diφ~ sin θ + B~ i · Diφ~ cos θ dV R3

62 for the standard Euclidean norm k·k.

Since B~ i · (W~ i × φ~) + (W~ i × B~ i) · φ~ = 0 we ﬁnd that the second term becomes

Z Bianchi Z Z B~ i · Diφ~ cos θ dV = cos θ ∂i(B~ i · φ~) dV = a cos θ B~ · dS~ = ag cos θ (120) 3 3 R R Σ∞

1 ~ i ~ 18 where the Bianchi identity 2 Di 0ijkFjk = D Bi = 0 and Stokes’ theorem was used . In a completely similar way, but instead using the ﬁrst of the two equations of motions (89) (such that φ~ · DiE~i = 0) instead of the Bianchi identity, ﬁnds Z Z E~i · Diφ~ sin θ dV = a sin θ E~ · dS~ =: aq sin θ (121) R3 R3 which is similar to the magnetic charge, such that q is seen as the electric charge19. Combining these two we get the bound

M ≥ ag cos θ + aq sin θ (122) which can be maximized to get the Bogomol’nyi bound for the mass of monopole solution of the Yang-Mills-Higgs model p M ≥ a g2 + q2. (123)

± When looking at the mass of the weak mediator bosons W we ﬁnd that MW = ae~ = aq ≈ 90 GeV (remember e = q~, see equation (105)). The mass of the ’t Hooft-Polyakov monopole MHP then becomes 4πa 4π 1 M ≥ a|g| = = M ~ = M ≈ 12TeV HP e W q2 W α where the Dirac quantization condition for the ’t Hooft-Polyakov monopole, equation (115) was 2 used and the ﬁne-structure constant α = q ≈ 1 was used. Note that 12 TeV is very high, 4π~ 137 the LHC could collisions with energy up to 14 TeV in 2015 [26], so the change of creating an ’t Hooft-Polyakov monopole is very small.

6.2 The BPS monopole

Now the question arises if there are static ﬁnite-energy solution of the Yang-Mills-Higgs system which saturates the bound. There are and they are called BPS monopoles. It will be shown that such solutions must satisfy a relatively simple equation, the Bogomol’nyi equation. To ﬁnd theses solution we just have to set the non-negative terms, which have disappeared due to 3 the inequalities, to zero for every point in R . Notice that these states are the states such that at least V (φ~) = 0 and D0φ~ = 0 everywhere, since the terms are integrated over the entire space. Moreover, like before, a system with solely magnetic monopoles will be considered, meaning that ~ π no electric sources are present in other words q = 0 and Ei = 0 and in particular θ = ± 2 . The mass bound, equation (122), becoming an equality, then gives the Bogomol’nyi equation

B~ i = cos θDiφ~ = ±Diφ~ (124)

It turns out that this equation is very strong. Together with the Bianchi identity it determines the dynamics of the monopole system being considered.

18 i jk R Also remember B~ = ijkeˆ G such that B~ · dS~ ∝ gΣ=Σ , compare with (104) Σ∞ ∞ 19Note that E~ is derived from the gauge-field strength tensor in the Aµ direction, such that it can be identified with the electric field, similar for the magnetic field

63 The first property for the system that becomes apparent is that we must consider it as a limiting ~ λ 2 2 2 case. In particular, for the Higgs potential V (φ) = 4 (φ − a ) to vanish everywhere one must have that λ = 0. This follows from the Bogomol’nyi equation. If λ 6= 0 then we get that φ2 = a2. In other words φ~ lies on the sphere of radius a. Because of this one finds that φ~ · Diφ~ = φ~ · ∂iφ~ = 0, where in the second equality the fact was used that the tangent planes of the sphere are perpendicular to the sphere. i Using the Bogomol’nyi equation one finds that in the Higgs vacuum B~ ∝ eˆ φ~ · B~ i = 0 meaning that the magnetic field vanishes. Thus, due to the type of system being considered (with in general a non-zero magnetic field), it may be assumed that λ = 0. For the purpose of finding an analytical answer to the equations of motion it is useful to look at the limit λ ↓ 0, because the system is well-behaved in this limit due to continuity in this limit w.r.t the Yang-Mills-Higgs Lagrangian density. In particular, in this limit it follows that in the Higgs vacuum on retains the condition φ = a2, which is meaningless at λ = 0. This limit is called the Prasad-Sommerfield limit, called after the founders, [27]. It was already noted that the Bianchi identity together with Bogomol’nyi equation (of which the positive solution is considered) determine the equations of motion (89), since

−eφ~ × D1φ~ = −eφ~ × B~ 1 = eφ~ × (∂2W~ 3 − ∂3W~ 2 − eW~ 2 × W~ 3)

And

D2F~ 12 = −D2B~ 3 = D2D3φ~ = ∂2∂3φ~ − e∂2(W~ 3 × φ) − eW~ 2 × ∂3φ~ + e2W~ 2 × (W~ 3 × φ~), where the Bogomol’nyi was used twice. By considering (cyclic) permutations one now ﬁnds the ﬁrst equation of motion

12 3 13 2 3 3 2 2 3 3 2 1 1 D2F~ + D F~ = D B~ − D B~ = D D φ~ − D D φ~ = −eφ~ × B~ = −eφ~ × D φ.~

The second equation of motion for λ → 0 follows directly from the Bianchi identity

i i D Diφ~ = D B~ i = 0.

6.2.1 The BPS monopole in the ’t Hooft-Polyakov ansatz

The non-linear coupled differential equations (102) in ’t Hooft-Polyakov ansatz do not have a known exact solution. In the Prasad-Sommerfield limit (or, as seen before, in the far-field limit) an analytical solution is known. The Bogomol’nyi equation provides two simple solutions for the H and K fields in the ’t Hooft- Polyakov ansatz ~r φ~(~r) = H(aer) er2 0 Wk = 0 rj 1 W i = (1 − K(aer)) or equivalently W~ i = eî × ~r (1 − K(aer)). k kij er2 er2 for i = 1, one finds like in section 4

2 (1 − K2) x − y∂ K − z∂ K 1 r2 2 3 B~ = 2 xy 1 2  (1 − K ) r2 + ∂2Kx  er 2 xz (1 − K ) r2 + ∂3Kx

64 and  x2 HK˜ 2 2  H − 2H r2 + ∂1Hx − r2 (y + z ) 1 xy HK˜ D1φ~ =  −2H + ∂ Hy + xy  er2  r2 1 r2  xz HK˜ −2H r2 + ∂1Hz + r2 xz. One can then see that

i i 2 Diφ~ = (B~ i) =⇒ (1 − K ) − 2r∂rK = −H + r∂rH + 2HK (125)

The other six just imply ~r×∇~ K = ~r×∇~ H = 0, which is due to the definitions of the fields. Looking for solutions which also satisfy the previously found second order differential equations (102), one finds 2 ξ∂ξK = −KH and ξ∂ξH = H + 1 − K . (126)

Prasad and Sommerfield found in 1975 that [27] ξ H(ξ) = ξ coth ξ − 1 and K(ξ) = (127) sinh ξ are solutions of the differential equations (126), which can be checked by substitution. Now one must check that they are proper solutions of the Yang-Mills-Higgs Lagrangian, meaning that they must satisfy the boundary conditions (98) and (99) H K → 0 and → 1 ‘sufficiently fast’ as ξ → 0 ξ K − 1 ≤ O(ξ) and H ≤ O(ξ) for ξ → 0.

Indeed in the limit ξ → ∞ we ﬁnd

1 + e−2ξ H(ξ) = ξ − 1 → ξ 1 − e−2ξ ξe−ξ K(ξ) = 2 → 0 1 + e−2ξ and by performing a Taylor expansion we ﬁnd

2 ξ ξ↓0 H(ξ) ≈ + O(ξ3) → 0 3 2 ξ ξ↓0 K(ξ) − 1 ≈ − + O(ξ4) → 0 6 Then the corresponding fields become rˆ rˆ rˆ φ~ = ξ coth ξ − → raˆ − er er er 1 ξ 1 W~ i = eî × rˆ(1 − ) → eî × r.ˆ er sinh ξ er Again the fields are localized in space, forming a particle-like monopole, of course now with mass a|g| for g the magnetic charge. Furthermore, fields automatically satisfy the equations of motion because they were derived from the Bogomol’nyi equation, and the magnetic field was seen to be indeed divergenceless. . However, the asymptotic behaviour of the Higgs field of the BPS monopoles is different from the one seen in the ’t Hooft-Polyakov monopole with λ 6= 0. In particular, when looking again at the Higgs field one finds that (taking the limit less rigorously)

H(ξ) − ξ =: h ≈ 1 + O(e−ξ). (128)

65 This decay is much slower then the decay of the regular soliton solutions in equation (103) in other words it becomes long range in a sense. This is not in contradiction with the boundary conditions (98) however, since the Higgs field can be seen as massless for λ ↓ 0 such that the interaction can be long range. The radial behaviour of these fields are shown in figure 7 (for λ ↓ 0).

The mass contribution in the BPS monopoles comes from the B~ i and the Diφ~ terms, see equation (118). Therefore, the Bogomol’nyi equation (124) implies that the mass contribution is equal. This means that the mass density in the ‘radial tail’ of the monopole is twice that of the Dirac or the ’t Hooft-Polyakov monopole [18], in the assumption that the Higgs field is non-zero (λ > 0). It turns out that gravitational interactions are observably stronger, which can help in experimentally verifying the BPS monopoles. Moreover, it can be shown that the mass density at the origin is finite. Note that the mass density is proportional to the term (for a derivation see appendix B) 1 D φ~ · D φ~ = ((r∂ H − H)2 + 2H2K2) = const + O(ξ2), i i e2r4 r where a Taylor expansions around ξ = 0 was performed, for H and K as in (127), for a constant const. Indeed then the mass density turns is finite for the BPS monopoles. This can also be derived in a less brute force way by using the second boundary equation DiDiφ~ = 0 such that then 1 D φ~ · D φ~ = ∂ φ~ · D φ~ + eφ~ · (W~ × D φ~) + φ~ · D D φ~ = ∂ (φ~ · ∂ φ~) = ∂ ∂ (φ~ · φ~) i i i i i i i i i i 2 i i such that then we find a2 H2(ξ) a2 1 H2(ξ) − ∇~ 2( ) = − ∂ r2∂ ( ) = const + O(ξ2) 2 ξ2 2 r2 r r ξ2 which is precisely the same answer and again implies a finite mass density at the origin. The long range force of the Higgs field turns out to always be attractive and, for static monopoles, equal in magnitude to the inverse square law magnetic force (far away). Therefore the static, finite- energy BPS monopole solutions behave differently from the regular ’t Hooft-Polyakov monopoles: For equally charged monopoles the forces cancel, they remain static. For a monopole, anti-monopole pair the interacting forces double, all shown by Manton [28]. To show this one can consider equation (120) and two BPS monopoles of charges g and g0 = ±g, and note that the sum of the mass energy of the two monopoles is equal to the mass energy of the monopole with both charges (note also that the magnetic charge was an additive property). Thus, the BPS monopole differs from the earlier seen ’t Hooft-Polyakov, and therefore also the Dirac monopole in the long-range behaviour.

66 7 Electromagnetic duality revisited

In this section the electromagnetic symmetry from part I will be reconsidered. There it was shown that the dynamical equations satisfy the duality transformation

(E,~ B~ ) 7→ (B,~ −E~ ) and (Je, Jb) 7→ (Jb, −Je) and (q, g) 7→ (g, −q).

The ﬁnal transformation, the interchanging of the electric and magnetic charge (Z4 duality, which together with CP-invariance reduced to Z2 duality), will be generalized, leading to the Montonen- Olive conjecture. Afterwards a CP-violating term is introduced and the Witten eﬀect will suggest an improved (P )SL(2, Z) duality.

7.1 The Montonen-Olive conjecture

In the Prasad-Sommerﬁeld λ ↓ 0 the bosonic part of the SO(3) Georgi-Glashow looks like:

Particle Mass Electric charge Magnetic charge Spin

Photon (massless gauge boson) 0 0 0 ±1

Higgs 0 0 0 0

W± vector gauge boson aq = ae~ ±q = ±e~ 0 0

M± BPS-monopole ag 0 ±g 0

One can see the following two properties

The particles satisfy the Bogomol’nyi bound, as they should

The system is invariant under Z2 duality, if one also exchanges the deﬁnition of the massive gauge boson W± and the BPS-monopole M±

The second property is implied by ﬁrst, together with the fact that the Bogomol’nyi equation (124) is invariant under electromagnetic duality.

The assumption of the Dirac monopole came in this paper from the Z2 duality of the Maxwell equations. Similarly, Montonen and Olive [29] conjectured that the spectrum description of this gauge theory (in our ‘electric world’) has a dual description (probably the standard theory of magnetoelectrism invented in a dual ‘magnetic world’). As stated above, the gauge particles are needed to be the BPS monopoles and the gauge vector particles are needed to be the new (in that world most likely unobserved) ‘electric monopoles’. Thus, if duality is assumed, the existence of the vector gauge bosons implies the existence of the BPS monopoles. As seen before, the magnetic charge/coupling g was about 104 times stronger than its electric version q in the original theory, thus the coupling constants in the dual theory are precisely ordered the opposite way. For example, in the dual Yang-Mills-Higgs Lagrangian density, the broken group U(1) has a much stronger conserved charge g and the gauge ﬁelds couple to the other ﬁelds in a much stronger coupling constant g . Thus, the nice properties of the particle spectrum above lead ~ Montonen and Olive to conjecture that there exists such a dual theory.

67 7.2 Quantization of dyons

Here the dyonic spectrum mentioned in section 2 will be reconsidered. In particular, one ﬁnds a quantization of the electric charge, which was there stated to be either: 1 q = ne or q = ne + e. 2

When considering infinitesimal gauge transformations of SO(3) around the Higgs field φ~, i.e. the generator of the SO(2) group with small angular parameter , one gets the infinitesimal transformation properties for any isovector ~v (meaning living in the same space as the Higgs field):

δ~v = φˆ × ~v, precisely because the cross product is acquired in the linear approximation: ∞ X n O(xµ) ~v = (φˆ(xµ) · L~ )n · ~v ≈ ~v + L~ · ~v 7→ ~v + φˆ × ~v, (129) n! φ n=0 | {z } ~ :=Lφ

µ for O(x ) ∈ SO(3) rotating with small angle around φˆ, compare with the start of section 3. Also L~ = Li is a vector of the generators of the Lie algebra of so(3). Note that the arrow 7→ goes from 3 so(3) to (R , ×). Similarly one gets for the gauge ﬁeld transformations (ﬁrst seen as in so(3), so as a matrix) 1 O(xµ) W~ µ(O(xµ) )−1 ≈ W~ µ + [φ,ˆ W~ µ] 7→ W~ µ − W~ µ × φ~ and (∂µO )O−1 7→ 11∂µφ,ˆ (130) e e

µ µ Lφ(x ) µ î µ Since O(x ) = e and we are taking up to linear terms in and Lφ(x ) = φ (x )Li ∈ so(2) ⊂ ∼ 3 ˆ so(3) = R is the generator of the Lie algebra of rotations around φ, for Li the generators of so(3) as defined in (170) - (172). From equations (129) and (130) we then find δ~v = φˆ × ~v and δW~ = D φˆ (131) µ e µ

In the Higgs vacuum, equation (92) we get that, Dµφ~ = 0 such that any (not necessarily infinitesi- µ mal) rotation around φˆ of angle θ(xµ) can be written as eθLφ(x ). Because of the rotational axis, in θL the Higgs vacuum one then finds that e φ = 11,such that θLφ = 2πn for n ∈ N. If the rotations in the Higgs vacuum are all of angle 2π then we find N := Lφ ∈ Z. Note that the gauge rotations transform the Higgs and the gauge fields (in isospace), but leave the action invariant, therefore one can use Noethers theorem and its associated current to find another expression for N, see equation (45), it can be identified with the charge of the Noether’s current, see (46) Z ∂L ∂L Z ∂L N = − · δW~ + · δφ~ dV = − · δW~ ) dV, (132) ~ µ ~ ~ µ R3 ∂(∂0Wµ) ∂(∂0φ) R3 ∂(∂0Wµ) since rotations around φ~ gives δφ~ = 0. Looking back at equation (91) one finds that Z Z 1 1 q N = F~0ν · Dµφ~) dV = − E~i · Diφ~) dV = , (133) e R3 e R3 e where (121) was used. 2π~ 2π Now, in subsection 2 it was seen that a dyon of (minimal magnetic) charge (q, g = e = q ) has a charge spectrum of either 1 q = ne or q = ne + n for n ∈ 2 N

68 Here we can conclude indeed that, since N ∈ Z

q = Ne forN ∈ Z (134)

Finally, we can compare this statement with the Dirac monopole, and speciﬁcally the use of the U(1) group. It was stated before that the charge is the generator of U(1) group, which is precisely what is shown above (for the Yang-Mills-Higgs Lagrangian only, but one can generalize to the Dirac monopole from far away).

7.3 The Witten eﬀect

One can always introduce another term to the action if it is a total derivative, compare with equation (49). This was done by Witten [30], who used a then recently found angle called the vacuum angle 20 θ in his extra Lagrangian term in the action Lθ

e2θ e2θ L = F~ · F~ = F~ · ∗F~ . (135) θ 64π2 αβµν αβ µν 32π2 µν µν Even though this term is a total derivative and therefore does not change the action, it does change the physics of the system. Furthermore, its integral is an integer multiple of θ called the instanton number. Therefore θ can indeed be seen as an angle, which turns out to parametrize diﬀerent vacuum conﬁgurations. As stated before, the conserved charge now changes to

1 Z ∂L L 7→ L + L =⇒ N 7→ N + θ · D φˆ dV. (136) θ ~ i e R3 ∂(∂0Wi)

This term can be computed similarly to equation (133):

1 Z ∂L Z eθ eθ θ · D φˆ dV = − B · D φˆ dV = g, ~ i 2 i i 2 e R3 ∂(∂0Wi) R3 16π 8π where g is the magnetic charge. This implies that

q eθ N = − g. e 8π2 Obviously, as simple check when θ = 0 gives the old quantization condition for dyons.

For the ’t Hooft-Polyakov monopole (NΣ = 1) of this Lagrangian density the quantization condition read qg = 4π~, equation (115), such that eθ q = Ne + . (137) 2π Diﬀerent from the previous quantization condition for dyons. Indeed, in diﬀerent theories, parametrized by θ one gets a shifted quantization of the electric charge. Finally the condition in equation (137) can be generalized a bit more, by bringing in the quantization condition in equation (115) eθ q = Ne + m , (138) 2π where both m and N are integers.

20This term is violating CP invariance, but not C invariance, such that it is consistent with the duality conjecture

69 7.4 PSL(2, Z) Duality

π Here the Z2 duality will be extended from rotations in the (q, g) plane of angle 2 to another kind of group (P )SL(2, Z) called the modular group. This is done by introducing a complex parameter τ θ 4π τ := + i 2π e2

Note that the Lagrangian density is now of the form 1 θ 1 L + L = − F~ · F~ µν + F~ ·∗ F~ µν + D φ~ · Dµφ~ − V (φ), (139) θ 4e2 µν 32π2 µν 2 µ where the transformation W~ µ 7→ eW~ µ was made to make the e dependence more explicit, in favour of the τ notation.

Now the Lagrangian density is rewritten in terms of τ, such that the SL(2, Z) group can come into play. This is done by introducing the term

∗ F~µν = F~µν + i F~µν (140) such that 1 1 θ − Im(τF~ · F~ µν) = − F~ · F~ µν − F~ · ∗F~ µν 32π µν 4e2 µν 32π2 µν The Lagrangian density then becomes 1 1 L + L = − Im(τF~ · F~ µν) + D φ~ · Dµφ~ − V (φ). (141) θ 32π µν 2 µ

Since θ was an angular variable, it can be assumed to lie in the interval [0, 2π). This means that the system is invariant under the rotations θ 7→ θ + 2π, or equivalently, τ 7→ τ + 1. Furthermore, now the conjecture that the Yang-Mills-Higgs system is invariant under Z2 duality 4π will be extended. for θ = 0 one finds that the duality transformation e 7→ g = − e , which 1 translates to τ 7→ τ . Henceforth it will be assumed that for any θ the theory is invariant under the transformations: 1 T : τ 7→ τ + 1 and τ 7→ . (142) τ But on the upper part of the complex plane these transformations are precisely the actions of the 2 modular group PSL(2Z). Note furthermore that e > 0 such that τ naturally lives on the upper half of the plane. Note finally that using this assumption, the Yang-Mills-Higgs system is completely defined by the parameters τ, θ, a and λ.

7.5 The modular group

The modular group PSL(2Z) can be created from the projective space of the matrix group SL(2, Z). Taking now τ ∈ C × C>0 and consider ﬁrst the group of real 2 × 2 matrices with unit determinant SL(2, R). This group acts on the upper half-complex plane by a b aτ + b g(τ) = τ = (143) c d cτ + d Note also that if τ = x + yi, where of course y > 0, then ax + b + ayi (ax + b + ayi)(cx + d − cyi) [...] + yi = = , cx + d + cyi (cx + d)2 + c2y2 (cx + d)2 + c2y2

70 where the terms left out are all real and ad − bc = 1 was used. This means that

Im(g(τ)) > 0. (144)

In other words all orbits of the action are subsets of the upper half plane. Now when the matrices ±11are considered it is seen that the action is not acting in an injective way (faithful). Furthermore, considering any two matrices ±M ∈ SL(2, R) one finds that these matrices act completely similar. Of course then the projection SL(2, R)/±11then defines another group. When taking only integer entries we find the modular group PSL(2, Z) := SL(2, Z)/±11. Note that the transformations in equation (142) can be precisely defined with this group. The actions defined there can be easily related to the matrices 0 1 1 1 S = ± and T = ± (145) −1 0 0 1

Moreover, using this it is very easy to show that

S2 = 11 and (ST )3 = ±11. (146)

These matrices lie in SL(2, Z), however. Such that the above identities are not completely concrete, only up to a sign. To identify the actions (142) with elements in the modular group we need to choose a sign. Furthermore we get the identities

Sˆ2 = 11 and (SˆTˆ)6 = 11, (147) where the hat indicates the mapping into the projective group (the modular group). Finally, by considering the transformation T dST cST bST a (hat emitted) one ﬁnds that these two elements Sˆ and Tˆ generate the whole group.

Now how does PSL(2, Z) ﬁt into the rest of the picture (of the Yang-Mills-Higgs system). It was already noted that the equations of motion are completely left unchanged, therefore the results of section 7 and 8 remain invariant. The BPS monopoles might change, however. When looking at the Bogomol’nyi bound, we see it can be written in the form

M 2 = a2(g2 + q2) = 4πa2 ~nT · A(τ) · ~n, (148) where 1 1 Reτ A(τ) = (149) Imτ Reτ |τ|2 and ~n = (n, m)T , where n and m are defined as 4π g = m From equation (115) (150) e eθ q = ne + m From the Witten effect, equation (137). (151) 2π Indeed one then finds θ 2 2 T n + m 2π 2 2 2 2 θ 2 e θ 2 2 2 2 ~n · A(τ) · ~n = θ θ2 4π2 = a n e + 2nme + m 2 + g = a (q + g ) (152) n 2π + m( 4π2 + e4 ) 2π 4π

a b Now let B ∈ SL(2, ) = , then Z c d

Im(τ) ImB(τ) = |cτ + d|2 ac|τ|2 + (ad + bc)x + bd ReB(τ) = |cτ + d|2

71 Figure 9: The visualization of the modular group acting on the upper-half of the complex plane. The lines get mapped onto the circles by the action S, the lines and the circles get mapped onto another line or circles respectively by the action T . Image thanks to [1] from which it follows that A(B · τ) = (B−1)T · A(τ) · B−1 (153) Which means that the mass bound equation stays the same, provided that we simultaneously perform the transformation ~n 7→ B · ~n. Thus, we see that if physics is PSL(2Z) invariant, then we can transform any theory by rotating the vacuum angle θ over 2π implying that the electric charge 1 eθ q is raised by me, or transform τ 7→ τ , meaning that q 7→ ±(me − n 2π ).

7.6 Orbifolds and the dyonic spectrum

First of all, the modular group is very well known such that one can use group-theory, topology and geometry to examine the action of the modular group on the upper-half of the complex plane. Second, the dyonic spectrum will be considered when PSL(2, Z) invariance is assumed. The action of the modular group on the upper-half of the complex plane can be visualized as in figure 9. It turns out that the fundamental domains D, where 1 D = {τ ∈ |Imτ > 0, |Re(τ)| ≤ , |τ| ≥ 1}, (154) C 2 has the property that the orbit spans the entire plane. In other words, all the information about any invariant theory is contained in the fundamental domain. Moreover the action is faithful on that interior of the fundamental domain, meaning that one cannot find a smaller subset of the upper-half of the complex plane than the interior of the fundamental domain such that the information about every invariant theory is contained within that domain. There exist generalizations of manifolds called orbifolds, which can be used to describe this system very well. In this case one can ‘glue’ the sides of the fundamental domain, such that a cylinder is made. More gluings can be made, such that one glues a specific topological space, which can then be described nicely using orbifold geometry.

72 Now consider the dyonics spectrum. When Z2 invariance is assumed, it was found that the existence of the gauge bosons implies the existence of the BPS monopoles. When PSL(2, Z) invariance is assumed and furthermore it is assumed that for every value of τ a massive vector bosons exists, then there are an infinite number of dyonic states possible. 2 Note furthermore that the set of vectors ~n ∈ Z can be made into a normed vector space. When one 2 2 2 T defines the norm k~nk − M~n = 4πa ~n · A(τ) · ~n, for the mass matrix A(τ) as defined in equation (149) and the mass of the dyon Mn as in equation (148) To see that its a norm, note that the mass-matrix is positive-definite, because Im(τ) > 0. Therefore Mn > 0 can be (well-)defined by taking the positive root. Absolute homogeneity follows immediately because of the square root. This means that the Bogomol’nyi saturated bound determines the length of the vector. Taking that into one can show that the triangle inequality k~n + ~mk ≤ k~nk + k~mk holds, since q q ~nT · A(τ) · ~n ~mT · A(τ) · ~m ≥ ~nT · A(τ) · ~m (155)

eθ and if we define the dyons described by ~n = n1 m1 as having charges (q1 = n1e + m1 2π , g1 = 4π e m1) and similarly for ~m = n1m2 , then we find that indeed it satisfies equation (155) and therefore the triangle inequality since

2 2 2 2 2 (g1 + q1)(g2 + q2) ≥ (g1g2 + q1q2) (156) since 2 2 2 2 2 g2q1 + g1q2 ≥ 2g1g2q1q2 ⇐⇒ (g2q1 − g1q2) ≥ 0. (157) 2 So we now have a discrete normed vector space Z with an action on it. If for every τ the electron exists, i.e. a dyon with charge q = e, described by the vector ~n = 0 1T then (by assumption) the modular group implies the existence of a set of dyons. In particular, when one looks at the orbit of τ, then for every g ∈ PSL(2, Z), deﬁned as in equation (143), there exists another dyon state. As seen above, when looking at the direct orbit, after one transformation21, the dyon state can be described by the transformation matrix g as:

a ~n 7→ g · (~n) = (158) c

Note, however, that g has unit determinant ad − bc = 1, which of course means that a and c are coprime (assuming both non-zero).

Say we have such a state ~w = a cT . The norm on the vector space of dyons tells us about the stability of such a state. In particular, it is claimed that it is stable. Indeed, assume that there exist two states such that q is the combined state of which i.e. there exist two non-zero vectors ~v and ~w such that ~w = ~n + ~m. Also we demand that both a 6= 0 6= c. Assume w.l.o.g. that ~n = r s. The triangle inequality tells us that k~wk ≤ k~nk + k~mk. The inequality becomes an equality when ~m and ~n and therefore also ~q are linearly dependent22 This at least implies that all entries of the vectors are non-zero. Also this means that ∃λ ∈ Q such that λ r s = a c , for a properly chosen λ. This means in particular that sa = rc. Now it was seen that a and c are coprime, which then must mean that s = na, since sa and rc have common divisor itself, which should be unambiguously split in its prime factorization. Then sa ∝ ac and since sa ∈ Z, we get that s = na for some n ∈ Z. However then s ≥ a which is a contradiction, since ~m had non-zero entries.

21 Actually, because S and T generate PSL(2, Z), one ﬁnds that the whole orbit of the electron can be identiﬁed with a new particle with coprime entries 22Note that we not only have a normed vector space, but an inner product space. The theorem of Cauchy-Schwarz can now be used to conclude that an ‘triangle equality’ implies the linear dependence of its composites.

73 Figure 10: The dyonic spectrum a versus c, implied by the orbit of the electron in out theory of electromagnetism. Dots denote existing states, crosses the absence of one. Note in particular that 2 every line with rational slope, only the ﬁrst point on the grid Z is indicated with a dot, since the entries must be coprime. (For |c| = 1 we apparently ﬁnd that every state is allowed). Image thanks to [1].

One of the assumptions is wrong, which then means that ~w 6= ~m + ~n, meaning then that Indeed we see that ~w is a stable state. So every state, which is of the form (149) is a stable state, meaning that the existence of the electron implies the existence of many other stable states (meaning likely fundamental particles!), again only if we assume that for every τ such a state actually exists. In particular, the spectrum of dyons is shown in 10.

74 Part IV Discussion

Maxwell’s equations together with the Lorentz force law can predict the dynamics of a classical (relativistic) electromagnetic system. It was shown that the Maxwell’s equations contained a high amount of symmetry, as seen in table at the start of part III. In the light of that symmetrical formulation, the Dirac monopole was introduced, a complete copy of the electric particle, transformed under the duality transformation, equation 70. The Dirac monopole brought us to a very important proposition, the Dirac quantization condition. This condition explains the observed quantization of electric charge and allows for an estimate of the strength of the radial field of the magnetic monopole, which was shown to be of order 104 stronger. Thus a higher amount of energy is needed to create such a (particle with that kind of) field, but it should also be more easily measurable because of this. The Dirac monopole had its (topological) flaws however. A global vector potential could not be introduced without the addition of an unobservable singular string-like region: The Dirac string. This string ‘carried away the magnetic flux’, such that the ‘no magnetic monopole law’ still was satisfied. Quantum mechanical consideration of the single-valuedness of the wavefunction of an electric charge in the presence of such a monopole brought us to the Dirac quantization condition, equation (77). This was the first occurrence of a gauge theory, with U(1) as the gauge group. The quantization condition could be extended to dyons when considering the quantization of angular momentum. The consideration of two dyons with equal non-zero magnetic charge together with the assumption of a CP-invariant system gave as a result the quantization of electric charge. This e quantization ladder started at either 0 or 2 and going in integer steps modulo electron charge (both in positive and negative direction). The existence of the electron, together with the assumption that 2π~ the electron charge is the minimal possible charge, then implied a minimal magnetic charge g = e and moreover the magnetic charge itself is also quantized in integer multiples of the minimal charge, the latter being already implied by the Dirac quantization condition. Then an isospin (SO(3) gauge-)invariant Yang-Mills-Higgs Lagrangian density was introduced, equation (87), consisting of a vector gauge field and a scalar Higgs field, together with the Higgs potential which allowed for spontaneous symmetry breaking. Indeed in the Higgs vacuum the symmetry was broken and two massive vector (gauge) bosons arised, which could in principle be identified with the weak gauge bosons. More importantly, the reduced symmetry group SO(2) =∼ U(1) was seen to be the stabilizer of the Higgs field, and the massless component of the gauge field pointing in the direction of two-dimensional rotation could be identified with the electromagnetic potential. When looking at finite-energy solutions of the dynamical equations (88) and (89) of the Yang-Mills- Higgs system it was seen that in the far-field limit the system should be in the Higgs vacuum. From consideration of the homotopy classes it was seen that the important behaviour of the Higgs field could be summarized in one number: the winding number. Continuous deformations such as time- evolution or a gauge transformation did not change that number. Furthermore, a non-zero winding number implied a non-zero energy state, hence it could be seen as a stable solution in the sense of not losing its energy. When considering static, finite-energy solutions and the symmetry of the system (also relativistic invariance), one arrived at the ’t Hooft-Polyakov ansatz, equations (94) - (96). Rewriting the Hamiltonian density in terms of the ansatz, boundary conditions (98) and (99) for the fields K and h were found, corresponding to the Higgs and the gauge fields respectively. In the far-field limit, it could be seen that the Higgs field hat topological winding number 1 and that the solutions to differential equations obtained from the dynamical equations implied a localized object, where the field fell off exponentially fast, far away from the core. Moreover, from far away

75 the solutions to the equations of motion in the ’t Hooft-Polyakov ansatz (now identified with a monopole) looked like the Dirac monopole with double the charge. Then it was assumed that any finite-energy solution had an exponential decay of the massive fields, such that in the far-field limit the Higgs field was in vacuum configuration. From these considerations it was shown that the (gauge-)field-strength tensor, in the Higgs vacuum, pointed in the direction of Higgs field i.e. the direction of the massless gauge field. Furthermore, this field-strength tensor satisfied Maxwell’s equations of motion, such that the massless gauge field in the direction of the Higgs field could be identified with an electromagnetic potential. Moreover, it was shown that the magnetic charge (defined as the flux in the Higgs vacuum) was dependent only on the topological number of the Higgs field, therefore being both gauge-invariant and time-independent. A second quantization condition was then (equation (115)) derived for the ’t Hooft-Polyakov solution of the dynamical equations, which then could be identified with a magnetic monopole, linked to the winding number of the solution. Afterwards a relation between the Dirac monopole and the ’t Hooft-Polyakov monopole was found: a singular SO(3) gauge transformation of the Higgs field. Because of the singularity the winding number of the Higgs field could be changed and therefore the origin of the charge was then seen to be different for both monopoles. Also the singularity indicated position of the Dirac string, again confirming that the string is a non-physical object. Then the mass of general solitons in the Yang-Mills-Higgs Lagrangian was considered. The Bogo- mol’nyi mass bound was found, putting a lower limit on the mass. Static solutions that saturated the bound (BPS monopoles), were shown to exist and in the Prasad-Sommerfield limit, the Bogo- mol’nyi equation together with the Bianchi identity were equivalent to the equations of motion. In the ’t Hooft-Polyakov ansatz, it was shown that this BPS-monopole behaved a little differently in the far-field limit than the other two monopoles.

Finally, electromagnetic duality was extended and it was seen that simple Z2 duality already implied the existence of the BPS monopoles. Then a CP-violating term was introduced, which contained a vacuum angle connecting different kind of theories. A new kind of quantization condition was derived for the different kind of theories, parametrized by the angular value of the vacuum angle θ. Duality invariance was extended to PSL(2Z) invariance, under the action of the modular group. This group acted on a complex number τ in the upper-half plane, in which the information of all the dual theories was contained. The important quantities were no longer the electric and magnetic 2 charge, but two integers quantizing them: ~n = (n, m) ∈ Z one coming from the quantization condition (115), the other from the Witten effect, (137). An inner-product vector space (Z, < ., . >) was then made, the inner product being equation (148). The action of PSL(2, Z) on the vector space could be deduced by the invariance of the Bogomol’nyi mass bound. The triangle inequality finally could imply a range of possibly existing dyons in the orbit of the electron under the action of the modular group, as shown in figure 10.

76 A Tensors

We have seen that the electromagnetic ﬁeld tensor and it’s dual play an important role in the compact description of electromagnetic duality. The mathematical theory of tensors is known for it’s importance in the description of general relativity. Tensors can be described over many types of spaces, but for all intents and purposes we will stick to tensor products of vector spaces. After all, the tensors are used in ﬂat Minkowski space. In this paper the rigorous mathematics is not needed, instead I will aim for an intuitive understanding of the object.

A.1 Basistransformations: Co- and contravariant objects

Covariant (contravariant) objects in a vector ﬁeld are, simply said, objects that transform in the same (opposite) manner as a basis. To get an intuitive feeling of this let’s consider an example.

Take the real numbers R, which is of course a 1 dimensional vector space. Now suppose that one measures a position ~x ∈ R and that it is measured in meters. We can then say that the vector can be decomposed in the manner ~x = x · ~η where ~η is the basisvector. The vector ~x is an object in the vector space R and is well-defined, irrespective of the basis. Suppose that the position ~x is 1 meter, then we might just as well say that it is 1000 millimetres. We see that in the first case x = 1 and ~η = 1 meter, which can be considered as the arrow pointing from 0 to 1. In the second casex ˜ = 1000 and ~η˜ = 0.001 meter. Noting that the process of going from one scale to another (meters to millimetres) is just a basis transformation and we see that x transforms tox ˜ precisely in an inverse way to the basisvector ~η. This is precisely how co- and contravariant objects are defined. Any covariant (contravariant) object transforms in the same (opposite) manner when a basistransformation is performed.

A.1.1 Co- and contravariant indices, raising and lowering indices

Since spacetime in special relativity is ﬂat in other words a vector space, we have co- and contravariant objects. Covariant objects will be denoted by superscripts and contravariant objects by subscripts. Furthermore, in a vector space with a metric they can be linked to each other via the metric tensor, which deﬁnes distances and was seen to be (equation (24))

µν η = ηµν = diag(1, −1, −1, −1) (159)

µ A covariant object Fµ and its contravariant dual F are then linked by the equation

µ µα F = η Fα (160)

A.2 Transformation properties of tensors

By far the most important properties of tensors is the way they transform. The properties of a tensor are defined by the amount of upper and lower indices. When V ∗ is the dual-space of the vector space V of dimension dim V =: n ∈ N then a tensor on V is defined as: Definition A.1. Tensor A rank (p, q), p, q ∈ N0 tensor is a multilinear map

∗ ∗ T : V × ... × V V × ... × V → R, (161) | {z } | {z } ptimes qtimes written also component-wise as T i1,...,ip (162) j1,...,jq

77 n×n such that it transforms under a basis transformation A ∈ R as

0 0 0 0 i1,...,ip −1 i1 −1 ip i1,...,ip j1 jp T 0 0 = (A ) ...(A ) T (A) 0 ...(A) 0 (163) j1,...,jq i1 ip j1,...,jq j1 jp

Meaning that the contravariant object (upper indices) transform inversely with respect to the basis and the covariant objects transform just like the basis. Finally note that this implies the nice transformation properties of the four-vectors as tensors. Moreover, a rank (1, 0) or (0, 1) tensor is seen as a four-vector (row or column vector respectively) and the rank (2, 0), (1, 1), (0, 2) tensors can be seen as matrices.

B The Hamiltonian density in the ’t Hooft-Polyakov ansatz

The Hamiltonian density in the ’t Hooft-Polyakov ansatz was stated in equation (97) as Z E = H dV R3 Z ∞ 4πa dξ 2 2 1 2 1 2 2 2 2 λ 2 2 2 = 2 ξ (∂ξK) + (ξ∂ξH − H) + (K − 1) + K H + 2 (H − ξ ) . e 0 ξ 2 2 4e

This follows since λ V (φ~) = (H2 − (aer)2)2 4e4r4 and furthermore

i i i 2 i 2 i Diφ~ · D φ~ = ∂iφ~ · ∂ φ~ − 2e(W~ i × φ~) · ∂ φ~ + e (W~ i · W~ )(φ ) − (W~ · φ~)(W~ i · φ~) .

Considering the three terms separately, deﬁning K˜ = (1 − K) and noting that

HK˜ HK˜ (W~ i × φ~) = − (~r × (êi × ~r)) = (~rri − eîr2), (164) e2r4 e2r4 one finds 1 ∂ φ~ · ∂iφ~ = (−3H2 + 2H~r · ∇~ H − r2(∇~ H)2) i e2r4 and

(164) HK˜ −2e(W~ × φ~) · ∂iφ~ = −2 (~rr − eˆ r2) · ∂iφ~ i er4 i i H2K˜ = 4 e2r4 and ﬁnally 2 2 (164) H K˜ e2(W~ × φ~) · (W~ i × φ~) = −2 . i e2r4 Collecting terms one then ﬁnds

1 D φ~ · Diφ~ = − ((r∂ H − H)2 + 2H2K2) i e2r4 r where in the last equality the gradient operator has been expressed in the basis of spherical coordinates with |~r| = r.

78 The ﬁnal term of the Hamiltonian density is proportional to

µν j k k j k j F~µν · F~ = ∂jW~ k · ∂ W~ + ∂kW~ j · ∂ W~ − 2∂jW~ k · ∂ W~ j k j k − 2e ∂jW~ k · (W~ × W~ ) + 2e ∂kW~ j · (W~ × W~ ) 2 j k + e (W~ j × W~ k) · (W~ × W~ ). First note that. 1 rj ∂ W~ = (ˆe × K˜ eˆj − eˆ × 2~rK˜ +e ˆ × ~r∂ K˜ ), (165) j k er2 k k r2 k j

jk 12 13 23 Also F~jkF~ = 2(F~12 ·F~ +F~13 ·F~ +F~23 ·F~ ) such that only the ﬁrst term needs to be calculated:

12 1 2 2 1 2 1 F~12 · F~ = ∂1W~ 2 · ∂ W~ + ∂2W~ 1 · ∂ W~ − 2∂1W~ 2 · ∂ W~ 1 2 1 2 − 2e ∂1W~ 2 · (W~ × W~ ) + 2e ∂2W~ 1 · (W~ × W~ ) 2 1 2 + e (W~ 1 × W~ 2) · (W~ × W~ ).

Like before, the terms are treated separately.  0   z  Furthermore W~ 1 ∝ eˆ1 × ~r = −z , W~ 2 ∝ eˆ2 × ~r =  0 , giving y −x

1 x2 x2 x ∂ W~ ·∂1W~ 2 = K˜ 2−4K˜ 2 −2xK∂˜ K+4K˜ 2 (z2+x2)+4K∂˜ K (z2+x2)+(∂ K)2(z2+x2) 1 2 e2r4 r2 1 r4 1 r2 1 meaning that 1 y2 y2 y ∂ W~ ·∂2W~ 1 = K˜ 2−4K˜ 2 −2yK∂˜ K+4K˜ 2 (z2+y2)+4K∂˜ K (z2+y2)+(∂ K)2(z2+y2) 2 1 e2r4 r2 2 r4 2 r2 2 also we get that 2 y2 + x2 x2y2 −2∂ W~ · ∂2W~ 1 = K˜ 2 − 2K˜ 2 − K˜ (x∂ K + y∂ K) + 4K˜ 2 1 2 e2r4 r2 1 2 r4 (∂ Kxy2 + ∂ Kyx2) + 2K˜ 1 2 + xy∂ K∂ K r2 1 2 and z2K˜ 3 −2e∂ W~ · (W~ 1 × W~ 2) = −2 1 2 e2r6 since zK˜ 2 W~ 1 × W~ 2 = ~r. e2r4 Similarly z2K˜ 3 2e∂ W~ · (W~ 1 × W~ 2) = −2 2 1 e2r6 and ﬁnally z2K˜ 4 e2(W~ × W~ ) · (W~ 1 × W~ 2) = . 1 2 e2r6 Collecting terms we then get 4 (x2 + y2) z2(x2 + y2) + (x2 + y2)2 F~ · F~ 12 = K˜ 2 − 2K˜ 2 + K˜ 2 12 e2r4 r2 r4 1 1 z2(−K˜ 3 + 1 K˜ 4) + z2(∂1K˜ )2 + (∂2K˜ )2 + (x∂1K˜ + y∂2K˜ )2 + 4 . 4 4 r2

79 Meaning that, by performing cyclic permutations (123), we get

jk 12 13 23 F~jkF~ = 2(F~12 · F~ + F~13 · F~ + F~23 · F~ ) 4 1 = ( (K2 − 1)2 + r2(∂ K)2). e2r4 2 r

We then have the three equations λ V (φ~) = (H2 − (aer)2)2 (166) 4e4r4 1 1 1 D φ~ · D φ~ = − ( (r∂ H − H)2 + K2H2) (167) 2 i i e2r4 2 r 1 1 1 F~ · F~ jk = ( (K2 − 1)2 + r2(∂ K)2) (168) 4 jk e2r4 2 r such that the Hamiltonian density becomes 1 1 1 λ H = r2(∂ K)2 + (r∂ H − H)2 + (K2 − 1)2 + K2H2 + (H2 − (aer)2)2 (169) e2r4 r 2 r 2 4e2 and the energy becomes Z E = H dV R3 Z ∞ 4πa dξ 2 2 1 2 1 2 2 2 2 λ 2 2 2 = 2 ξ (∂ξK) + (ξ∂ξH − H) + (K − 1) + K H + 2 (H − ξ ) e 0 ξ 2 2 4e where spherical symmetry is used and ξ = aer. This result is precisely what was stated in equation (97).

C The fundamental group of maps between circles

Topological spaces can be distinguished using homotopy groups. This is done by associating a group to every point in such a space. The basis lies at the continuous deformation of loops in the space, where two loops are defined to be homotopic to each other if one can ‘continuously deform’ into another. This homotopic equivalence of loops will be an equivalence class such that a proper group can be defined. The intuitive definition is quickly generalized and one arrives at the definition of homotopic functions between spaces, which again can be ‘continuously deformed’ into one another. Finally one arrives at the definition of homotopy equivalent spaces, which is slightly less of a strong equivalence then homeomorphic spaces, thus one looks at relatively elementary properties of topological spaces. The group operation as well as the set structure comes about most naturally when considering loops and can then be extended when considering functions between topological spaces. One starts with the set of loops starting at the same point. An operation (not yet the group operation) on the set of loops in a is given by the ‘composition’ of loops (∗ operation). This operation, on the set of loops, almost forms a nice algebraical structure: a group. However, the set of loops must undergo some changes for the operation to become a properly defined group operation. One now quickly finds an equivalence relation defined by the ability to continuously deform a loop into another (in other words formed by the ∗ operation). The new set of equivalence classes together with the induced operation of continuously deforming forms a well-defined group. The elements in the equivalence class are called homotopy classes and the formed group is called the fundamental group. The fundamental group is the easiest of a set of groups called the homotopy groups. An example of a characterizing topological property by which homotopy groups can distinguish spaces is the existence of ‘holes’ in them, made by objects such as the Dirac monopole.

80 C.1 The structure of loops

The most intuitive definition to start with is the one of a path, since continuous deformation can be made quite intuitive in this way. A special case of a path is a loop (same starting and ending point). These loops are, as discussed above, a nice starting point for the discussion of the fundamental group. Definition C.1 (Path, loop). Let X be a topological space. A continuous map α : [0, 1] → X, such that α(0) = xi and α(1) = xf for xi, xf ∈ X is called a path from xi to xf . A loop α in X with base point x0 ∈ X is a path from x to x. From this point on the object X defines the topological space in which the loops lie and I = [0, 1]. Taking all the loops with same base point together one finds a set which can be made interesting.

Remark C.2. The set of loops with base point x0 ∈ X is denoted by Ax0 = {α : [0, 1] → X| α continuous, α(0) = α(1) = x0} The interesting part of the set is that it has a (intuitive) structure which is found in the following operation.

Definition C.3 (∗ operation). If α, β ∈ Ax0 then one defines the object α ∗ β by α ∗ β : [0, 1] → X; t 7→ (α ∗ β)(t), where the latter is defined as

( 1 α(2t) t ≤ 2 (α ∗ β)(t) = 1 β(2t − 1) t ≥ 2

Note that α ∗ β is again a loop in Ax0 . In other words one goes around both loops, α then β therefore having to go twice as fast. When trying to form a group structure the constant loop cx0 deﬁned by cx0 (t) = x0 is the object naturally taking the role of the identity. The object a−1 given by α−1(t) = α(1 − t) is the natural inverse of −1 the loop α. However, one does not ﬁnd the demand α ∗ α = cx0 to form the group, for this to work one looks at homotopy of loops. Here is where the continuous deformation comes into play:

Deﬁnition C.4 (Homotopic loops, homotopy). Let α, β ∈ Ax0 , then the loops are said to be homotopic if ∃ a continuous map F : I × I → X, such that F (·, t): I → X continuously deforms α into β in other words

F (0, t) = α(t) F (1, t) = β(t) ∀t ∈ I

F (s, 0) = F (s, 1) = x0 ∀s ∈ I

One writes α ∼ β if the loops are homotopic. The map F is called a homotopy between α and β. For example Example C.5. Take the paths α(t) = (cos(t), sin(t)) and β(t) = (cos(t), 2 sin(t)), where both lie in 2 R . The map F (s, t) = (cos(t), (1 + s) sin(t)) is a homotopy between α and β. 2 3 Note that this does not work if we take X = R \ (0, 2 ), therefore a ‘hole’ can be detected by the homotopy of loops. Example C.6. We view C(Y, R), the space of continuous functions from a vector space Y over R to the vector space R. Then we take α(t) = ft defined by ft(y) = h(y), for some linear map h : Y → R 1 and β(t) = gt defined by gt(y) = 2|t − 2 |h(y), then the map F (s, t) = (1 − s)ft + sgt is a homotopy between α and β. Note that C(Y, R) itself is canonically identified with a vector space over R. One now tweaks the set of loops by noting that the homotopy relation ∼ is an equivalence relation.

The equivalence classes will be denoted by [α] for some representative loop α ∈ Ax0 . For a loop β ∈ [α] the equivalence class it belongs to is called the homotopy class of β.

81 The set Bx0 := Ax0 / ∼ of homotopy classes of loops (with the same base point) does have a properly deﬁned group operation when considering ∗ deﬁned above.

One first checks that the operation is well-defined (by [α] ∗ [β] := [α ∗ β]) for the new set Bx0 . In other words, one must check that the ∗ operation is independent of the representative being used, which is proven easily. Then one finds that the homotopy class map cx0 forms the identity element and the inverse of a class [α] is given by the homotopy class [α−1], which was defined before.

Finally a group is formed by the dual (Bx0 , ∗), which is called the fundamental group. It is often denoted by π1(X, x0), where the 1 denotes that this is the simplest of homotopy groups.

C.2 Homotopy equivalent spaces

One can extend the homotopy equivalence from loops in a topological space X with base point x0 ∈ X to maps between two topological spaces X,Y

Deﬁnition C.7 (Homotopic maps, homotopy). Let x, Y be topological spaces. Two maps f, g : X → Y are homotopic, or f is homotopic to g if ∃ a continuous map F : X × I → Y such that

F (x, 0) = f(x) F (x, 1) = g(x) ∀x ∈ X.

This relation is again denoted by ∼, in other words f ∼ g means that f is homotopic to g and F is called the homotopy between f and g. With this deﬁnition and the fact that we can relate two topological spaces on a homotopic scale, one can now look at similarities between topological space. Deﬁnition C.8 (Homotopy type). Let X,Y be two topological spaces. X and Y are of the same homotopy type if ∃ two continuous maps f : X → Y and g : Y → X if f ◦ g ∼ idX and g ◦ f ∼ idY where id denotes the identity map. This is denoted by X =∼ Y . The maps f and g above is called the homotopy equivalences. It turns out that homeomorphic spaces are also of the same homotopy type. Furthermore, if two spaces are of the same homotopy type then the fundamental groups with respect to the base point x0 and its image under the homotopy equivalence map f are isomorphic, in other words ∼ π1(X, x0) = π1(Y, f(x0)).

C.3 The fundamental group of maps between circles

The space C(S1) of continuous functions with the unit circle S1 as (co)domain will now be considered via the fundamental group of the circle S1.

1 Theorem C.9. The fundamental group π1(S ) is isomorphic to (Z, +) and this implies that 1 ∼ 1 (C(S ), ·) = (Z, +), where · is the product induced from the S group structure. First we note that we can speak of one well-defined fundamental group of S1, since all fundamental groups with respect to different base point in S1 are isomorphic. This is because S1 is path- connected, meaning that for every two points in the space there exists a path from one to the other. 1 This means that for any two loops α, β ∈ π1(S , x0) a homotopy can be found between them if and −1 −1 only if the mapsα ˜ = h ◦ α ◦ h , β˜ = h ◦ β ◦ h are homotopic, for some x1 ∈ X and a path h −1 from x1 to x0. Just take the homotopy h ◦ Fx1 ◦ h, which then defines a group isomorphism if it is noted that there is a one-to-one correspondence between loops in Ax0 and loops in Ax1 . This means that in S1 it does not matter which base point is chosen for the loops. Here we view S1 together with the structure of a topological group, with induced topology from 2 1 1 1 R . Working with continuous maps in C(S ) = {f : S → S : f continuous} can be made easier

82 (both intuitively by making paths explicit and algebraically by going to R) by noting that we 1 ∼ have a group isomorphism S = [0, 2π) = R/2πZ using the complex exponential, together with the complex logarithm, to be the group isomorphism p : [0, 2π) → S1; x 7→ eix. There is a further structure on R/2πZ, namely a topology, induced by R or equivalently the quotient topology (needed to make sense of continuity). The isomorphism and the fact that we now have a topological space [0, 2π) =∼ [0, 1)23 allows us to identify maps in C˜(S1) = {f ∈ C(S1): f(1) = 1} with paths f¯ in S1 with base point 1. Note that one then defines f¯(1) = f¯(0) to get a well-defined path. This identification is (subjectively) very intuitive. Furthermore a group isomorphism between the group 1 1 π1(S , 1) and C˜(S ) is then found, which will follow from theorem 7.10.

Now the mathematics is simplified by going to R or, in particular, a group isomorphism onto the space C˜(R) = {f˜ : R → R : f continuous, f(0) = 0, ∃n ∈ Z s. t. f(x+2π) = f(x)+2πn}. Note that the second condition ensures that the exponential map preserves the identity. The final condition implies that eix = eiy =⇒ eif˜(x) = eif˜(y). In figure 11 the realisation of the number n, such that f˜(x + 2π) = f˜+ 2πn is shown, which will be called the degree of the map deg(f) = n.

Note that the property eix = eiy =⇒ eif˜(x) = eif˜(y), implies that all the information of the function eif(·) is contained in the domain [0, 2π). Therefore we can modify the domain of the functions in ˜ ˜ C(R) such that we get a well-deﬁned (group) isomorphism between the set C([0, 2π)) = {f|[0,2π) : f ∈ C˜(R)} and C˜(R). Moreover, using this isomorphism one can make another one between the set 1 1 1 C˜(R) and the set C˜(S ), = {f ∈ C(S )| f(1) = 1} of continuous maps with (co)domain S . This is ˜ ˜ achieved by a bijection between the spaces (groups) C([0, 2π)) = {f|[0,2π) : f ∈ C(R)} and by com- 1 if˜(·) position with the complex exponential map, arriving at the mapp ˜ : C˜(R) → C(S ); f˜ 7→ f := e .

Theorem C.10. The complex exponentialp ˜, as above, deﬁnes a bijection (or actually a group isomorphism) between C˜([0, 2π)) and C˜(S1).

Idea of proof. Indeed, the regular complex exponential map p : [0, 2π) → S1 deﬁnes a isomorphism. So we have an induced isomorphism, ensured by the property f(1) = 1, f˜(0) = 0) given by p0 : C˜([0, 2π)) → C˜(S1) of the maps from the two sets (groups). Now one just takes the composition, see ﬁgure 12.

This means that we have a one-to-one correspondence between maps f˜ ∈ C˜(R) such that for some 1 fixed n ∈ Z f˜(x + 2π) = f˜ + 2πn and f˜(0) = 0 and maps f ∈ C˜(S ), such that f(1) = 1, all continuous. We now define the degree of f as deg(f) = deg(f˜) = n and the meaning of n becomes 1 ∼ 1 more intuitive from figure 11. Furthermore it is now easy to show that π1(S , 1) = C˜(S ). There will be two final important lemmas needed to prove theorem 7.9.

1 Lemma C.11. For every n ∈ Z, ∃f ∈ C˜(S ) such that deg(f) = n. ˜ Proof: Take f˜ ∈ C(R) such that f˜(x) = nx, which has degree n.

Lemma C.12. For f, g ∈ C˜(S1), we have that deg(f) = deg(g) ⇐⇒ f is homotopic to g.

Proof. We proof this via the space C˜(R). =⇒ : Assume deg(f) = deg(g). Take the according f,˜ g˜ ∈ C˜(R) with winding number n. The trick is to take the homotopy F˜ : I × R → R;(s, t) 7→ sf˜(t) + (1 − s)˜g(t). Since p is continuous one ﬁnds that f ∼ g by the homotopy F (s, t) = eiF˜(s,t). Finally note that this does not work if def(f) 6= deg(g), ∼ since a problem arises with the continuity of F at the point 0 = 2π, limx↑2π F (x, t) 6= limx↓0 F (x, t), for any s ∈ (0, 1), where the metric is of course (induced) from Euclidean topology.

23together with the remark that, if we ﬁx w.l.o.g. by the discussion above the base point as 1 for all elements in C(S1)

83 Figure 11: The winding number n depicted. In the left is the variable in the domain going in circles. On the right is the codomain going in n circles when the variable in the domain makes 1 full turn. Image thanks to [31]

Figure 12: The composition to make the isomorphism.

84 ⇐= : Assume f ∼ g, then ∃ a homotopy F : S1 × I → S1. The homotopy F˜ between f˜ andg ˜ (found again by the continuous bijection p) we ﬁnd that F˜(x + 2π, t) = F˜(x, t) + 2πn due to the properties f˜(0) = 0 =g ˜(0).

1 This means that we can make a bijection between the group Z and the fundamental group π1(S ). 1 Finally it is easy to show that the product of loops in π1(S ) is equivalent to the sum in Z. Moreover, note that due to the fact that there is only one (distinct) fundamental group and the fact ˜ 1 ∼ 1 1 ∼ that C(S ) = π1(S ), it is proven that C(S ) = Z such that theorem 7.9 is proven. 2 2 ∼ ∼ 2 More importantly, this can easily be extended to the sphere S , meaning that π1(S ) = Z = C(S ).

D Lie groups and Lie algebras

In physics a theory is often linearised. One makes performs a Taylor expansion around a properly chosen point and then considers up to linear terms. In this case one loses information earlier acquired by the set up of the analytical equations of the system. When dealing with functions on a space which come from (the representation) of a Lie group, however, this information is not lost. The group properties ensure that linear information in a very small neighbourhood implicitly is already global, precisely because a group expresses symmetries i.e. local structures becoming global. In matrix groups the map going from the local structure, called the Lie algebra, to the Lie group is the matrix-exponential map. The only important Lie group in this paper are the rotation groups. The one-dimensional Lie group U(1) and the three-dimensional Lie group SO(3) are encountered. The latter one will be considered here.

D.1 A note on SO(3)

The Lie group SO(3) consists of matrices that are orthogonal and have unit-determinant. The most 3 natural way to see them is when they act on R in the fundamental representation. The lie algebra so(3) consists of the skew-symmetric matrices together with the commutator bracket as algebra 3 structure and can be identiﬁed with R together with the cross product for a Lie bracket. Indeed, the map

0 0 0  1 Lx = 0 0 −1 7→ 0 (170) 0 1 0 0  0 0 1 0 Ly =  0 0 0 7→ 1 (171) −1 0 0 0 0 −1 0 0 Lz = 1 0 0 7→ 0 (172) 0 0 0 1 defines a Lie algebra isomorphism. This map is used throughout the paper. µ µν µ µν 3 Then the fields (matrices) φ, W ,F ∈ so(3) are identified with vectors φ,~ W~ and F~ ∈ R .

85 References

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