Is Dynamical Mass Generation Via Spontaneous Symmetry Breaking in Conformal Gravity Possible with a Single Scalar ﬁeld?

BIELEFELD UNIVERSITY

FACULTY OF PHYSICS

MASTERTHESIS

Is dynamical mass generation via spontaneous symmetry breaking in Conformal Gravity possible with a single scalar ﬁeld?

1st Supervisor: Prof. Dr. Dominik SCHWARZ [email protected] Bielefeld University Author: Nils Frederik BERTRAM [email protected] Bielefeld University 2nd Supervisor: Yuko URAKAWA [email protected] Bielefeld University

December 30, 2019

Contents

1 Introduction & Motivation 1

2 General Relativity 3 2.1 General coordinate systems ...... 3 2.2 Geometric objects ...... 6 2.3 The geodesic equation ...... 8 2.4 The Einstein equations ...... 9 2.5 A problem of General Relativity ...... 11

3 Quantum Field Theory 14 3.1 Some difﬁculties of relativistic quantum mechanics ...... 14 3.2 The principle of least action and the Lagrangian density ...... 15 3.3 The quantum mechanical Klein-Gordon ﬁeld ...... 18 3.4 Renormalization and General Relativity ...... 19

4 Conformal Gravity 23 4.1 Conformal transformations ...... 23 4.2 The Bach equations ...... 27

5 Real & Complex scalar fields in Conformal Gravity 29 5.1 Real scalar field ...... 29 5.2 Complex scalar field ...... 30 5.3 The parameter ...... 30

6 The abelian Higgs model in Conformal Gravity 33

7 Conclusion and outlook 39

References 40 Declaration of Authorship 42

Acknowledgements 43

1 Introduction & Motivation

One of the biggest tasks, if not the biggest one, of modern theoretical physics is to find a theory of quantum gravity. This theory would describe phenomena that happen on smallest scales quantum physically, as well as (gravitational) phenomena, present on large scales. Currently we have two different theories for this. General Relativity describes the world as a four dimensional manifold, called spacetime, that is curved by matter and energy. Trajectories of objects, called world lines, are straight lines across this manifold. With this set up, the majority of gravitational effects, and large scale effects in general, can be described. As an example, the equations of General Relativity, the Einstein equations, are used to derive the Friedmann equations, which describe the evolution of the Universe in cosmology. Quantum Field Theory is currently the best accepted theory, describing the building blocks of matter. Every elementary particle in Quantum Field Theory is understood as an excitation of a quantum physical field. Thus, there is only one electron field with multiple excitations, that we see as multiple electrons. This gives an explanation why all electrons are indistinguishable. One of the major successes of Quantum Field Theory is the theory of Quantum Electro Dy- namics, describing the interaction between charged particles via the exchange of photons. The predictions of this theory are confirmed with great precision by various collider experiments. The big problem with these two theories is, that no one yet managed to unify them into one single theory, describing the physics of large, as well as small scales. However, there are multiple candidates, that could provide a theory of quantum gravity, like, e.g., string theory, loop quantum gravity and Conformal Gravity, which is discussed in this thesis.

In its most basic form, Conformal Gravity is nearly as old as Einstein General Relativity. In 1918, three years after Einstein published his theory, Hermann Weyl published a paper about “pure inﬁnitesimal geometry” [1]. In this paper, Weyl claimed, that the Riemannian geometry Einstein used for General Relativity was not a good choice for a theory forbidding action at a distance. He stressed, that in Riemannian geometry the line elements, which measure distances, of two arbitrarily chosen points can be compared. In Weyls geometry this is avoided,

1 and invariance under conformal transformation is obtained as a feature of this geometry. However, this approach was not much pursued.

Seventy years later, Weyls theory was rediscovered by Peter Mannheim and Demosthenes Kanzas [2]. They found, that some of the problems of General Relativity are solved in Con- formal Gravity [3]. For example galaxy rotation curves can be described without the need for dark matter. Additionally Conformal Gravity is a renormalizable theory, unlike General Relativity, which makes it a candidate for quantum gravity.

In this thesis we will explore mass generation in Conformal Gravity. Because of the demanded invariance under the conformal symmetry, standard mass terms cannot be included into the Lagrangian density, and masses have to be generated dynamically via spontaneous symmetry breaking. We will have a look at whether this is possible with a single scalar field or not. In the second chapter we will review the methods and equations of General Relativity, before we take a look at some of the principles used in Quantum Field Theory in chapter three. Chapter four deals with the basics of Conformal Gravity. Thereafter, in the fifth chapter, we investigate the behaviour of matter in Conformal Gravity by going through the calculations with the real scalar field presented in [4]. These calculations are then applied to a complex scalar field, before we take a closer look at the sign used in the Lagrangian so far, like it is also done in [4]. In the sixth chapter the method of spontaneous symmetry breaking is re- viewed, based on the abelian higgs model, before we try to apply it to Conformal Gravity. The conclusion and some outlook is presented in chapter seven. Throughout this thesis we use natural units, i.e., c = ~ = 1.

2 2 General Relativity

When Michelson and Morley performed their famous experiment in 1887 they did not know that it would eventually lead to the best conﬁrmed and arguably most beautiful theory modern physics has right now. By proving that there is no aether light travels through and that light has a constant velocity, they provided one of the basic ideas of Special Relativity, which was found by Einstein in 1905. With rather simple thought experiments one can deduce from the basic idea that light travels at a constant velocity, that time is not measured the same by all observers. One other thing that needs to be considered when deriving Special Relativity is that physics should be invariant under certain transformations. These transformations are translations in space, translations in time, rotations and boosts. Translations in space and time means repeating the experiment at another location or time respectively. Invariance under rotations means that the results of an experiment are the same, if it is observed from a different direction. Invariance under boosts is given, if two observers with a relative uniform motion along a straight line with respect to each other get the same result when doing an experiment. A simple example would be a pendulum in a train observed by two observers, one inside the train and one outside. These transformations form the Poincaré group. The result that two observers with relative motion to each other experience passage of time differently, forces us to change our view. A three dimensional space and a one dimensional absolute time, like used in Newtonian physics, is not sufﬁcient to describe our world. We have to change them to a single four dimensional object called spacetime.

2.1 General coordinate systems

To describe positions in this spacetime, we choose a coordinate system by choosing 4 four dimensional vectors (“four-vectors”) eµ as base vectors. As in all of this thesis, greek indices run from 0 to 3, with 0 denoting the time direction or component. Using Einstein’s summation convention, we can now write an inﬁnitesimal four-vector dx which describes a position in our

3 coordinate system as

µ dx = eµdx , (2.1) where the dxµ are the components of the four-vector.

To go from Special Relativity to General Relativity, we stop restricting ourselves to transformations of the Poincaré group, and enforce the general principle of relativity [5, p.250]:

All systems of reference are equivalent with respect to the formulation of the fundamental laws of physics.

So, let us look how different objects transform under a general change of basis. If we pick 0 another set of base vectors eα, we could express the same inﬁnitesimal vector as above as

0 0α dx = eαdx , (2.2) with the dx0α being the components in the new frame. With (2.1) and (2.2) we ﬁnd a relation between the base vectors of the ﬁrst and the second basis as dxµ e0 = e . (2.3) α µ dx0α

dxµ Note that dx0α are the components of the Jacobian matrix of the coordinate transformation. If we assume, that the first basis belongs to a reference frame, in which the spacetime looks (locally) flat,1 we can define the components of the Jacobian matrix as an object called Vielbein µ eα [6] dxµ eµ := . (2.4) α dx0α To measure the four dimensional “distance” between two events in spacetime the line element

2 µ ν ds := dx dx gµν (2.5)

2 is used. Here gµν is the (pseudo-)metric tensor . If we assume that we work in a frame where the spacetime is (locally) ﬂat, the metric is typically denoted with ηµν. ηµν can either be

1We can always do this since spacetime is defined as a four dimensional manifold, when defined properly. Local flatness is a property of manifolds. 2 Actually gµν denotes not the tensor, but its components. As it is done usually in theoretical physics we will often refer to the components of a tensor as the tensor itself.

4 diag(−, +, +, +) or diag(+, −, −, −). In this thesis we will use the latter convention.

It will later be helpful to look at some simple properties of the metric. One can deﬁne the so called “inverse metric”3 gµν by demanding that

αν ν gµαg = δµ, (2.6)

ν where δµ is the Kronecker delta, which is 1 if µ = ν and 0 else. Using the metric and the inverse metric, we can now express objects with upper indices using objects with lower indices and vice versa:

µ µν ν a = g aν, aµ = gµνa . (2.7)

Since the metric is a tensor, we use the transformation laws for tensors to get a relationship between gµν and ηαβ as

α β gµν = eµηαβeν . (2.8)

Knowing this, we ﬁnd an expression for the Jacobian determinant, which we need to use the integration measure in general coordinates. We start by re-expressing (2.8) in terms of tensors instead of their components:

g = JT ηJ ⇔ g := det (g) = det JT ηJ = − det (J)2 √ ⇒ det (J) = −g. (2.9)

If we want to be able to differentiate tensor fields, we need to define a total derivative, or connection, ∇. Demanding that ∇ behaves like a derivative, i.e., that it obeys, e.g., linearity, additivity and a product rule, one finds that there is still some freedom of choice left. However, the demands from above only allow for ∇ to act on a vector c as (c.f. [7]) ∂cν ∇ cν = + Γν cα, (2.10) µ ∂xµ αµ 3Strictly speaking this is not the inverse metric. Since the metric is mathematically defined as a map with the cartesian product of two vector spaces as the domain and R as target space, its inverse should also have those spaces as domain and target space, but swapped. The object we define here instead maps objects from the cartesian product of two dual vector spaces to real numbers. Since it is often called “inverse” metric we will also use this notion.

5 ∂ ν where ∂xµ =: ∂µ denotes the partial derivative. The Γαµ are a priori arbitrary functions depending on the position in spacetime. These functions are called the connection coefﬁcient functions. After choosing a connection, i.e., choosing a set of connection coefﬁcient func-

µ1µ2...µr tions, we can differentiate any tensor Tν1ν2...νs : ∂ ∇ T µ1µ2...µr = T µ1µ2...µr ρ ν1ν2...νs ∂xρ ν1ν2...νs µ1 αµ2...µr µ2 µ1α...µr µr µ1µ2...α (2.11) + ΓαρTν1ν2...νs + ΓαρTν1ν2...νs + ... + ΓαρTν1ν2...νs

α µ1µ2...µr α µ1µ2...µr α µ1µ2...µr − Γν1ρTαν2...νs − Γν2ρTν1α...νs − ... − ΓνsρTν1ν2...α .

µ Note that Γνρ does not form a tensor, as can be easily checked by investigating its transformation behaviour [8].

One special connection we will choose is the Levi-Civita connection. It is the unique connection (proof see [7, p. 156]) with the additional properties of metric compatibility

∇µgνρ = 0, (2.12) and vanishing torsion

∇µ∇νa = ∇ν∇µa for every scalar object a (2.13) µ µ ⇔ Γνρ = Γρν. The connection coefﬁcient functions of the Levi-Civita connection are called Christoffel symbols. They are fully determined by the metric via 1 Γµ = gµα (∂ g + ∂ g − ∂ g ) . (2.14) νρ 2 ν ρα ρ να α νρ

2.2 Geometric objects

In general, spacetime is not flat but can be curved. We need to be able to describe the curvature of spacetime, not only for the sake of generality, but also to work with the second big idea of General Relativity. Aside from demanding that the general principle of relativity holds, we assume that gravitation is not a force, but an effect of curved spacetime. In order to be able to describe curved spacetime, we need to introduce some geometric objects. The first object that we will define is the Riemannian curvature tensor, often only called curvature tensor or Riemann tensor. It is defined as

µ µ µ α µ α µ Rνρσ := ∂ρΓνσ − ∂σΓνρ + ΓνρΓασ − ΓνσΓαρ. (2.15)

6 Note that sometimes the Riemann tensor is defined with a global minus sign in the definition above. The Riemann tensor has certain symmetry properties. It is antisymmetric in the first and last indices

Rµνρσ = −Rνµρσ = −Rµνσρ , (2.16) and symmetric under exchange of the ﬁrst two with the last two indices

Rµνρσ = Rρσµν. (2.17)

In addition, it obeys the ﬁrst

Rµνρσ + Rµρσν + Rµσνρ = 0, (2.18) and second Bianchi identity

∂λRµνρσ + ∂ρRµνσλ + ∂σRµνλρ = 0. (2.19)

The next object we deﬁne is the Ricci tensor

ρ Rµν := Rµρν. (2.20)

Note that, like the Riemann tensor, it is also sometimes deﬁned with a global minus sign in its deﬁnition. Due to the symmetries of the Riemann tensor the Ricci tensor is symmetric under exchange of its indices

Rµν = Rνµ. (2.21)

The last object we will deﬁne before turning to the equations describing General Relativity is the Ricci scalar

µν R := g Rµν. (2.22)

Like the Ricci and Riemann tensors it can be deﬁned with a global minus sign.

Now we have deﬁned all the quantities needed to derive the equations describing General Relativity.

7 2.3 The geodesic equation

As mentioned above, in General Relativity gravitation is not a force, but the effect of spacetime being curved by masses. This means that in the absence of (non gravitational) forces particles move along straight lines. Since spacetime is curved in general, straight lines are not necessarily looking the way we expect them to look. For example the elliptic motion of earth around the Sun is a motion along a straight line in a four dimensional curved spacetime. To find out how straight lines look like in such a curved spacetime, we can use the fact that straight lines are, like in a flat spacetime, also the shortest lines [9, p.166]. Thus a particle will take the shortest path4 xµ(τ) from a given point to another. Here τ is just a parameter describing the path that the particle takes. To find out what the shortest path is between two given points we have to minimize the length given by (calculation similar as in [8, p.295]) Z l = ds. (2.23)

Let us ﬁrst take a look at the variation of the line element

p µ ν δds = δ gµνdx dx r dxµ dxν = δ g dτ µν dτ dτ dxµ dxν dxν dδxµ δgµν + 2gνµ = dτ dτ dτ dτ dτ. (2.24) q dxµ dxν 2 gµν dτ dτ

1 Renaming the denominator as A , we can use (2.24) to look at the variation of S Z dxµ dxν dxν dδxµ δl = A ∂ g δxα + 2g dτ dτ dτ α µν νµ dτ dτ Z 1 dxα dxν d dxν = 2A ∂ g − g δxµdτ (2.25) 2 dτ dτ µ αν dτ µν dτ =! 0.

If δS should vanish for every variation δxµ we get d2xν dxν dxα 1 dxν dxα 0 = − g + ∂ g − ∂ g µν dτ 2 dτ dτ α µν 2 dτ dτ µ αν d2xν 1 dxν dxα ⇔ 0 = gµβ g + (∂ g + ∂ g − ∂ g ) . (2.26) µν dτ 2 2 dτ dτ α µν ν µα µ αν

4Note that the path should be timelike everywhere, i.e., the inﬁnitesimal distance ds2 travelled by the particle should be positive for every τ.

8 Using now (2.14), we ﬁnally get d2xµ dxν dxα + Γµ = 0, (2.27) dτ 2 να dτ dτ which is the geodesic equation.

2.4 The Einstein equations

Now that we know how particles move through a given spacetime, we need to describe spacetime itself, and how it is influenced by the matter in it. This relationship is given by the Einstein equations. We will derive them like we already derived the geodesic equation: by using the principle of least action. The action we want to minimize consists of two parts, a curvature part SC and a matter part SM . The curvature part of the action is defined as [8, p. 321 & p.427] 1 Z √ S = − −g (R + 2Λ) d4x, (2.28) C 16πG which is known as the Einstein Hilbert action. Varying the action becomes much easier, when we first look at the variation of some other objects. Using the rules for differenciating the inverse of a matrix, one can show that the perturbation of gµν is

µν µα νβ δg = −g g δgαβ. (2.29)

One can also show that

µν δg = gg δgµν, (2.30) which leads to [6] √ 1√ δ −g = −ggµνδg . (2.31) 2 µν µ Taking a look at (2.15) and (2.14), and using (2.11), δRνρσ can be derived as

µ µ µ δRνρσ = ∇ρδΓνσ − ∇σδΓνρ, (2.32) with the variation of the Christoffel symbols being 1 δΓα = − gαµgβν (∂ g + ∂ g − ∂ g ) δg ρσ 2 ρ σβ σ ρβ β ρσ µν 1 + gαβ δµδν ∂ δg + δµδν ∂ δg − δµδν∂ δg . (2.33) 2 σ β ρ µν ρ β σ µν ρ σ β µν

9 We then calculate the variation of the Ricci tensor 1 δR = [∇α∇ δg + ∇α∇ δg − δg − ∇ ∇ gµνδg ] . (2.34) ρσ 2 ρ ασ σ ρα ρσ ρ σ µν Also we get

µν µρ νσ ρσ µρ ν RµνδR = Rµνδ(g g Rρσ) = R δRρσ − 2R Rρδgµν, (2.35) which will be important later, when we derive the Bach equations. Finally, the variation of the Ricci scalar is

µν µ ν µν δR = −R δgµν + ∇ ∇ δµν − g δgµν. (2.36)

The matter interaction will be discussed in Chapter 3, so for now let us simply assume that there is a Lagrangian density describing matter LM , so that Z √ 4 SM = −gLM d x. (2.37)

Taking everything together we now look at the variation of the whole action with respect to the metric gµν. The variation is

δS = δ (SC + SM ) 1 Z √ √ √ = − δ −g (R + 2Λ) + −gδR − 16πGδ −gL d4x 16πG M Z √ 1 √ 1 µν µν µν 16πG δ ( −gLM ) 4 = − −g g R + g Λ − R − √ δgµνd x, 16πG 2 −g δgµν where we have used the fact that total derivatives vanish in the integral when looking at varia- tions. To simplify this expression we deﬁne the energy momentum tensor T µν as5 √ 2 δ ( −gL ) T µν := √ M , (2.38) −g δgµν and the Einstein tensor Gµν as 1 Gµν := Rµν − gµνR. (2.39) 2

Demanding that the variation of the action vanishes for every δgµν, we get

−Gµν + gµνΛ = 8πGT µν. (2.40)

These are the Einstein equations, describing how spacetime behaves in the presence of matter.

5Note, that some authors (e.g. [8]) have a minus sign in this deﬁnition.

10 2.5 A problem of General Relativity

Now, that we have the equations of General Relativity, we will investigate, how well it works. It is important to note that General Relativity works very well on the length scales of the solar system [10]. Additionally it explains observations, that are not explained by classical Newto- nian gravity. These observations are for example the clock effect, the gravitational redshift and the perihelion motion of Mercury. However, there are some problems with General Relativity. On the smallest length scales, there is the problem that General Relativity is not renormalizable, leading to the fact that it cannot be described as a Quantum Field Theory. This problem will be looked at in section 3.4.

Another big issue is the so called Dark Matter problem, which is present on large length scales, starting at galactic scales. For detailed information on Dark Matter see [11]. This problem is easiest to see if we try to describe how a galaxy behaves using Newtonian gravity, which is a good enough approximation for our intent. Galaxies are mainly composed of stars and gas (mainly hydrogen). Other components, like planets, make up such a little part of the mass of the galaxy6, that they can be neglected. Using various methods, one can now measure how fast the masses of a galaxy rotate around the galactic center. This rotational velocity is found to be dependent on the distance from the galactic center. The rotational velocity of the masses as a function of the radial distance is called galaxy rotation curve. We can not only measure these rotation curves, but also use the theory of Newtonian gravity to predict what it will look like.

In ﬁgure 2.1, one clearly can see, that the predicted rotation curve deviates from the measured one in the outer parts of the galaxies. Thus there are two possibilities. Either the theory we used, being General Relativity7, does not work on these length scales, or there is some other matter that is not yet accounted for in the calculations. Since this matter is not measured yet, it should not interact, except gravitationally. This was indeed postulated and is called Dark Matter to emphasize the fact that it does not interact like conventional matter. Taking into account a Dark Matter Halo, one can match the predicted curve much better, as is shown in ﬁgure 2.2.

6In our solar system for example over 99% of the mass is made up by the Sun. 7Actually, in this overview we only used the limit of Newtonian gravity. However, the predictions of General Relativity in the case of galaxy rotation curves are similar, and do not ﬁt the observations either.

11 Figure 2.1: Radial light distribution in the upper panels and galaxy rotation curves in the lower panels for the two galaxies NGC2403 (a) and NGC3198 (b). The dots in the lower panel are measured using the 21 cm line of neutral hydrogen. The solid line represents the rotation curve expected by Newtonian gravity. For the calculation, stars, hydrogen and helium were taken into account. The graphs are taken from [12].

Figure 2.2: The rotation curve for the galaxy NCG3198. The measured points are the same as in ﬁgure 2.1. This time not only the measured matter (“disk”) is accounted for in the calculation of the solid line representing the theoretical expectation, but also the Dark Matter Halo. This graph is taken from [13].

12 Here one clearly can see that Dark Matter solves the problem with the galactic rotation curves. It is noteworthy, that Dark Matter also solves many other problems on large length scales, e.g., it explains how structures formed in the early Universe. One is now tempted to say that there is no problem with Dark Matter, since it appears to be the solution. Nevertheless the question what Dark Matter consists of remains unsolved. There are many theories about what Dark Matter could be, but so far none is supported or singled out by experimental data. On the other hand Dark Matter is not the only possible explanation for the shape of the galactic rotation curves, since there are many theories of modiﬁed gravity, that try to explain them without the need of Dark Matter. The question remaining open is if Dark Matter exists, and if it does, what it consists of, or whether General Relativity has to be modiﬁed. This is the Dark Matter problem.

13 3 Quantum Field Theory

In the early 20th century, the theory of General Relativity was not the only great discovery in theoretical Physics. In that time, Planck, Schrödinger and many others developed the theory of quantum mechanics. The basic ideas are, that energy is transferred in discrete packages called quanta, and that physical systems are described by states. The state, a physical system is in, when it is not observed is an overlap of all the possible states, that can be measured. If we, for example, observed an electron and tried to measure its spin, there are two possible results. The spin of the electron could either be up |↑i or down |↓i. According to quantum mechanics however, the electron does not need to be in one of those states before we measure it. When the spin of the electron has not been measured yet, the probabilities that it is measured in one of the two states are 50% for each state. This means, that the state the electron is in before measurement √ is 1/ 2(|↑i + |↓i), which is a superposition of the states mentioned above. This theory provides explanations for a large amount of effects. Nevertheless it is somewhat difficult to combine it with the idea of relativity. To circumvent those difficulties we stop working with “simple” objects like particles, and instead work with fields in a Quantum Field Theory. The ideas of Quantum Field Theory are explained with more detail in, e.g., [14, 15]. In the following sections we will discuss some of the aspects of it.

3.1 Some difﬁculties of relativistic quantum mechanics

One result of Special Relativity is, that mass and energy are equivalent, leading to the fact, that pairs of particles and antiparticles are produced in some processes. Thus, it becomes quite difﬁcult, to describe even the simple case of a single particle, since we cannot describe this system in a single particle state. If it has enough energy, then the production of particle- antiparticle pairs has to be accounted for. Even if there is not enough energy in the particle to produce a pair of particles, energy conservation can be violated for inﬁnitesimal time intervals,

14 leading to production and immediate destruction of pairs of virtual particles and antiparticles. One could now try working with states of the Fock space, which is the uniﬁcation of all the n-particle spaces. But even doing that, we get the Problem, that particles can, with a small, but yet non vanishing probability, travel spacelike distances in spacetime. The amplitude of a particle propagating from a start point x0 to a point x in a time t can be calculated as[14]

−iHtˆ U(t) = hx| e |x0i , (3.1) where Hˆ is the Hamilton operator measuring the energy of a system. From Special Relativity we know, that E = pp2 + m2. We can plug this into (3.1), if we exchange the momentum p for the momentum operator pˆ. Then we can calculate the amplitude in Fourier space. √ −it pˆ2+m2 U(t) = hx| e |x0i Z ∞ √ 1 −it p2+m2 = 2 p sin(p|x − x0|)e dp. 2π |x − x0| 0

This can be evaluated using the method of stationary phase for x2 t2, which is true if the particle moves faster than the speed of light. The result is [14]

√ U(t) ≈ e−m x2−t2 , (3.2) which is not zero, meaning that the probability of a spacelike motion does not vanish. This problem does not appear if we describe particles as excitations of quantum mechanical ﬁelds, as shown in [14].

3.2 The principle of least action and the Lagrangian density

Before we really turn to Quantum Field Theory we should investigate some of the tools needed for it. If we describe a physical system, we might want to know which path a particle takes between two given points in spacetime. In theory there are infinitely many paths connecting the two points, but we want to find out, which one is physically realized? To find an answer, let us first look at the energy of the test particle in an entirely classical, non relativistic set up (compare [15]). Let the path the particle takes be given as a function of time ~x(t) and the potential as a function of the location of the particle V (~x). We then calculate the mean kinetic and potential

15 energies V¯ and T¯. 1 Z τ V¯ = V (~x(t)) dt (3.3) τ 0 1 Z τ 1 h i T¯ = m ~x˙(t) dt (3.4) τ 0 2 To ﬁnd out how these two quantities vary when we slightly alter the path, we calculate the functional derivatives with respect to ~x(t). δV¯ 1 = ∇~ V (~x(t)) (3.5) δ~x(t) τ δT¯ 1 = − m~x¨(t) (3.6) δ~x(t) τ Using Newtons second law −∇~ V = m~x¨ we get

δT¯ = δV,¯ (3.7) meaning, that the mean kinetic energy and the mean potential energy vary in the same way, when the path of the test particle varies. This motivates us to deﬁne the Lagrangian function as the difference between the kinetic energy and the potential

L := T − V. (3.8)

To connect this to (3.7) again, we look at the time integral of the Lagrangian function, which is deﬁned as the action S. Z S := Ldt (3.9)

With (3.7) we can now see that the variation of the action with respect to the path the test particle takes vanishes

δS = 0. (3.10)

This is known as the principle of least action1. However, for our purposes it might be more fruitful the use the Lagrangian density L instead of the Lagrangian function. The Lagrangian density is deﬁned in such a way, that its spacial integral is the Lagrangian function Z L = Ld3x. (3.11)

1It is noteworthy, that the principle of least action does not state, that the action is minimal. It rather states that the action is stationary, but for historical reasons it is called principle of least action instead of principle of stationary action.

16 Now, it is quite easy to generalize the principle of least action to a curved spacetime, where it looks like2

Z √ δS = δ −gLd4x. (3.12)

Now, to describe any Physical system, we just need its Lagrangian density L. With this we can immediately find the equations of motion for the system. In general, we do not necessarily need to vary the action with respect to the path the particle takes. We only do this, when we are interested in the equations describing the trajectory of the particle, like in section 2.3. However, if we are interested in the equations of motion of, e.g., a field, we vary the action with respect to the field. We already did this in section 2.4 to derive the Einstein equations.

There, we took gµν as a ﬁeld, and found the Einstein equations as its equations of motion. To use this in quantum physical theories like in the next section, it is helpful to also introduce the Hamiltonian function

H := piq˙i − L. (3.13)

where the qi are the coordinates of the particle we want to describe, and pi are the components of the canonical momentum

∂L pi := . (3.14) ∂q˙i

When working with ﬁelds however, it is more helpful to use the Hamiltonian density H instead of the Hamiltonian function. It is deﬁned by

X ˙ H := πiφi − L, (3.15) i where the φi are the ﬁelds, and the πi are the conjugate momenta deﬁned by

∂L π(x) := . (3.16) ˙ ∂φi

The Hamiltonian density and the Hamilton function are connected via Z H = Hd3x. (3.17)

√ 2Note, that in some literature −g is deﬁned as part of L.

17 3.3 The quantum mechanical Klein-Gordon ﬁeld

To be able to also describe quantum fields, we take a closer look at the Klein-Gordon field, as in [14], and try to connect it to a well known quantum mechanical system. We start with the Lagrangian density 1 1 L = ∂ φ(x)∂µφ(x) − m2φ(x)2, (3.18) 2 µ 2 where φ(x) is a real field. With the principle of least action we can find the equations of motion for φ as

µ 2 (∂µ∂ + m )φ(x) = 0. (3.19)

This is the Klein-Gordon equation, therefore, φ(x) is called the Klein-Gordon ﬁeld. Let us write the Klein-Gordon ﬁeld in Fourier space Z d3p φ(~x,t) = ei~p·~xφ(~p,t). (3.20) (2π)3 Then the Klein-Gordon equation is

2 2 2 ∂0 + (|~p| + m ) φ(~p,t) = 0. (3.21)

This is the same equation of motion as for a simple harmonic oscillator. Forgetting for a moment, that φ(x) is a ﬁeld, and treating it as the coordinate describing the oscillator, we can write down the Hamilton operator for the harmonic oscillator 1 1 Hˆ = pˆ2 + ω2φ2. (3.22) 2 2 ~p Here pˆ is the momentum operator, and

p 2 2 ω~p = |~p| + m (3.23) is the frequency of the oscillator. It is well known, that we can reexpress the Hamiltonian for the simple harmonic oscillator and the coordinate and momentum operators in terms of the ladder operators aˆ and aˆ†. The operators are then 1 Hˆ =ω aˆ†aˆ + , (3.24) 2 1 φ = √ aˆ +a ˆ† , (3.25) 2ω rω pˆ = − i aˆ − aˆ† . (3.26) 2

18 The ladder operators aˆ and aˆ† are deﬁned via their action on the n-th eigenstate |ni of the Hamiltonian operator, which is √ aˆ |ni = n |n − 1i , (3.27) √ aˆ† |ni = n + 1 |n + 1i . (3.28)

We can see, that the operator aˆ† adds an energy quantum to the system, while the operator aˆ removes one. It is now important to note, that in (3.22) the frequency of the oscillator ω~p depends on the momentum, while in (3.24) it did not. Thus we interpret the system described by (3.22) as a collection of simple harmonic oscillators, distinguished by the momentum. Interpreting φ(x) as a field again (3.24)-(3.26) then become [14, p.21] ! Z d3p 1 φ(x) = aˆ ei~p·~x +a ˆ†e−i~p·~x , 3 p ~p ~p (3.29) (2π) 2ωp Z d3p rω π(x) = (−i) ~p aˆ ei~p·~x − aˆ†e−i~p·~x , (3.30) (2π)3 2 ~p ~p Z d3p 1 Hˆ = ω aˆ†aˆ + [â , aˆ†] . (3.31) (2π)3 ~p ~p ~p 2 ~p ~p Since the commutator in the second term is 1 for every ~p it gives an infinitely big constant. However, this is not a problem since the energies themselves are not physical and only energy differences are measured. Now, that φ(x) is an operator that we can represent as shown in (3.29) we take a look at what this operator does to the ground state of our system. Z 3 d p 1 −i~p·~x φ(x) |0i = 3 e |~pi (3.32) (2π) 2E~p

Here, E~p = ω~p is the energy of a quantum in the momentum eigenstate

p † |~pi = 2E~paˆ~p |0i . (3.33) The interpretation of equation (3.32) is, that by acting on the vacuum state |0i, the ﬁeld φ creates a particle at a location ~x.3

3.4 Renormalization and General Relativity

After we have discussed the basic ideas of Quantum Field Theory, we will summarize parts of its remaining derivation, before we take a closer look at renormalization.

3Note that we work in the Schrödinger picture where operators are time independent. However, the generalization to the Heisenberg picture is straightforward [14, p.25 ff].

19 The main purpose of Quantum Field Theory is, to calculate cross sections of processes of elementary particles. The elementary particles in the calculations are, as discussed in the previous sections, interpreted as excitations of ﬁelds. Thus, ﬁeld operators like φ and φ† create and annihilate particles. The cross section we want to calculate mainly depends on |M|, with M being the quantum mechanical amplitude of the process [14, p.5]. However, this amplitude can not be calculated explicitly; we have to use perturbative methods.

To brieﬂy explore this we will look at the simple case of an electron and an antielectron annihilating each other to create a muon-antimuon pair, as in [14, chap. 1]. The ﬁrst order of this perturbation series is

+ − + − M1 = µ µ HI e e . (3.34)

In this equation HI is the interaction part of the Hamilton operator. Since this interaction part does not couple electrons and muons directly, but to a photon γ, this order in the perturbation series vanishes. The leading order is now

+ − ν + − M2 = µ µ HI |γi hγ|ν HI e e . (3.35)

In this expression the photon states have vector indices ν, because the photon ﬁeld is a vector ﬁeld. If we now want to calculate M in higher than leading orders, we would have to keep expanding, leading to a growing number of processes contributing to that order. Luckily, Richard Feynman found out, that each of the processes that are part of the perturbation series can be translated to a picture called Feynman diagram, using a set of rules called Feynman rules. Due to the Feynman rules, every part of the picture is a direct representation of a part of the formula needed, to calculate the contribution of the process to M.

Taking (3.35) as an example we will look at how such a Feynman diagram arises. The right- most and leftmost terms of (3.35) are the initial and final state respectively. in the Feynman diagram, they will be expressed as an external line for each particle. The term HI , which couples the fields to each other, will be represented by a vertex, at which different lines meet. ν Finally, the |γi hγ|ν, describing the propagation of a (virtual) photon, will be represented by an internal wiggly line, connecting the two vertices. The resulting picture is shown in figure (3.1.a)

20 Figure 3.1: a: The Feynman diagram describing the process (3.35). b: A Feynman diagram contributing to the fourth order of the perturbation series of the process e+e− → µ+µ−. Without renormalization, the loop causes this contribution to diverge. In both diagrams time increases from left to right. The diagrams are taken from [16].

Using the Feynman rules we could now collect all the pictures of given order n, i.e., in Quantum Electro Dynamics all diagrams with n vertices, translate them to their contributions and add them, to get the n-th order of the perturbation series. Let us now take a look at the diagram of fourth order shown in (3.35.b). According to the Feynman rules we have to integrate over the momentum for the Fermions in the loop, causing the contribution of this diagram to diverge. This is of course nonsense, since this divergence would make the probability of this process to occur inﬁnitely large. Seeing this, we have to repair the theory. This repairing process is called renormalization, and is described in more detail in the second part of [14], and in part seven of [15]. The basic idea is, to add terms, so called counterterms, to the Lagrangian density describing the theory, to cancel the diverging parts in the perturbation series of M. It turns out, that this does not only solves the problem with the divergences, but also makes the parameters, like m, in the Lagrangian density describe physical parameters, such as the measurable mass of a particle, instead of simply being an “arbitrary” number. Nevertheless, this does not work in all theories. For some theories an ever increasing number

21 of counterterms in every order would be needed. Such theories are called non-renormalizable. So how do we recognize such a non-renormalizable theory? The answer is surprisingly simple. If the interaction part in the Lagrangian density has a coupling constant of negative mass dimension4, it is non-renormalizable [15, p.293]. Knowing this, let us next take a look at General Relativity again. The coupling constant of General Relativity is Newtons gravitational m3 constant. In SI unit it has a dimension of kg·s2 which in natural units is a mass unit of −2. This already hints to the fact, that General Relativity is non-renormalizable. A more thorough discussion of the non-renormalizability of General Relativity is given in [17].

4In natural units, when c = ~ = 1 all mechanical quantities have a power of mass as dimension. The exponent of this power is called mass dimension.

22 4 Conformal Gravity

So far, we have investigated the best theories for large scale, and small scale physics respectively. However, we also saw, that there are some problems with those theories. In section 2.5 we saw, that without Dark Matter some largest scale effects cannot be described by General Relativity, though Dark Matter has not been directly measured yet. Afterwards, we found in section 3.4, that Quantum Field Theory and General Relativity can not be merged together to one theory of everything. Currently, there are many approaches to solving these problems investigated. One, very promising, approach is Conformal Gravity. The basic idea of a relativistic theory of gravity that is invariant under a conformal transformation was already published by Weyl in 1918 [1]. However, it was not pursued much. Later, in 1988, it was rediscovered by Mannheim and Kanzas as a theory that could describe galactic rotation curves without the need for Dark Matter [2]. Additionally it is shown, that, like General Relativity, Conformal Gravity predicts the existence of Gravitational waves [4], that were measured in 2014. But Conformal Gravity does not only solve the Dark Matter problem, it is also claimed by Mannheim, that it also solves many other problems, like the smallness of the cosmological constant or the zero point problem [3]. Another advantage of Conformal Gravity is, that its coupling constant is dimensionless, unlike Newtons constant, as we will see later. This leads to Conformal Gravity being renormalizable [15, 3], when expressed as a quantum theory. In this chapter we will ﬁrst have a look at the conformal transformation, before we derive the Bach equations, which are the equations of motion for Conformal Gravity.

4.1 Conformal transformations

Since the actions in Conformal Gravity are demanded to be conformally invariant it is important to understand what this conformal transformation is. In the literature conformal transformations are often defined as transformations that leave the angles between two vectors invariant. However there is another approach to define a conformal transformation which is more practical for discussing Conformal Gravity. We define the conformal transformation as

23 a transformation that transforms the metric tensor gµν in the following way:

0 2 gµν → gµν = Ω(x) gµν, (4.1) where Ω(x) is a positive function, whose x dependence is dropped hereafter. Note that this transformation changes the length scales of spacetime locally, affecting physical lengths in a way, that cannot be compensated by a change of unit. Thus we ﬁnd that the conformal transformation changes the physical structure when applied, showing that it is no “simple” coordinate transformation like usually discussed in General Relativity. Therefore we see, that demanding the action to be invariant under conformal transformations in addition to coordinate transformations indeed gives us further restrictions.

We should check, if the deﬁnition we want to use, coincides with the deﬁnition given in the beginning of this chapter. We do this by checking if (4.1) leaves angles between vectors invariant. The angle α between two vectors a and b can be calculated as ! a · b aµg bν α = arccos = arccos µν , (4.2) √ µ νp µ ν |a||b| a gµνa b gµνb which transforms to ! aµΩ2g bν α0 = arccos µν p µ 2 νp µ 2 ν a Ω gµνa b Ω gµνb Ω2a · b (4.3) = arccos Ω|a|Ω|b| = α.

We see, that the transformation we will use does leave angles between vectors invariant. In the beginning of this section it was stated, that the conformal transformation changes the length scales in spacetime. As mentioned in section 2.1, lengths are measured by the line element deﬁned in (2.5). Using (4.1) we ﬁnd, that lengths change like

ds2 → ds20 = Ω2ds2. (4.4)

Since, in general, Ω can be different at different points in spacetime, we also can have different measures of length at different points. However, if we change the unit of length, we always do this in the same way at every point. Taking another look at (4.4), and remembering that Ω2 is positive everywhere, we see that positive distances stay positive etc., showing that the causal

24 structure is kept when performing a conformal transformation.

Before we explore the transformation behaviour of different geometric objects we should take a look at the deﬁnition of the Levi-Civita connection given in section 2.1, especially at the metric compatibility (2.12)

∇ρgµν = 0. (4.5)

Performing a conformal transformation on (4.5) we get

2 ∇ρgµν → ∇ρΩ gµν = ∇ρgµν + 2Ωgµν∇ρΩ = 2Ωgµν∇ρΩ 6= 0, (4.6) showing that the Levi-Civita connection is not invariant under conformal transformations. At ﬁrst glance, this looks problematic, but remembering, that we can ﬁnd a Levi-Civita connection for every metric, we see that there is no problem, if we change the connection to the new Levi-Civita connection every time we perform a conformal transformation. In this thesis, we assume, that this is done every time.

The following part mainly presents results of [18]. Now that we know from (4.1) how the metric transforms, it is easy to show, that the inverse metric (2.6) transforms like

g0µν = Ω−2gµν, (4.7)

µν 0 0µν by demanding that gµνg = gµνg = 4. The Jacobian determinant, which we found in section 2.1 to be the square root of the determinant of the metric tensor (2.9) transforms like

√ p √ g → g0 = Ω4 g. (4.8)

Since we change the connection when we perform a conformal transformation, as explained above, we should look at how the new Christoffel symbols look like after we perform the transformation. The new Christoffel symbols, expressed in terms of the old ones (2.14) are

1 Γ0µ − Γµ + δµ∂ Ω + δµ∂ Ω − g ∂µΩ . (4.9) νρ νρ Ω ν ρ ρ ν νρ

In the literature (e.g., [18]), this is often presented as the transformation of the Chistoffel symbols. With the above discussion in mind, we nevertheless adopt this notion. Since we now know the transformation behaviour of the Christoffel symbols, we can ﬁnd the transformation

25 behaviour of the Riemann tensor (2.15) 1 R0 =R + g ∇ ∇ − g ∇ ∇ Ω + g ∇ ∇ Ω − g ∇ ∇ µνρσ µνρσ Ω µσ ν ρ µρ ν σ νρ µ σ νσ µ ρ 2 + g (∂ Ω)(∂ Ω) − g (∂ Ω)(∂ Ω) + g (∂ Ω)(∂ Ω) − g (∂ Ω)(∂ Ω) Ω2 µρ ν σ µσ ν ρ νσ µ ρ νρ µ σ 1 + g g − g g (∂ Ω)(∂αΩ), Ω2 µσ νρ µρ νσ α (4.10) the Ricci tensor (2.20) 1 1 R0 = R − 2∇ ∇ Ω + g Ω + 4(∂ Ω)(∂ Ω) − g (∂αΩ)(∂ Ω) , (4.11) µν µν Ω µ ν µν Ω2 µ ν µν α µ with = ∇µ∇ , and the Ricci scalar (2.22) 1 Ω R0 = R − 6 . (4.12) Ω2 Ω To write down the equations of motion for Conformal Gravity, which we will derive in the next section, it will be helpful to introduce another object describing the curvature of spacetime. The Weyl tensor is deﬁned as 1 1 C := R + (R g − R g + R g − R g ) + R (g g − g g ) . µνρσ µνρσ 2 µσ νρ µρ νσ νρ µσ νσ µρ 6 µρ νσ µσ νρ (4.13) Its transformation is

0 2 Cµνρσ → Cµνρσ = Ω Cµνρσ. (4.14) To investigate the transformation behaviour of matter ﬁelds it makes sense to introduce the notion of conformal weight. The conformal weight of a quantity A is the exponent p of the Ω term it gets multiplied by, when a conformal transformation is applied: A → ΩpA ⇔ "A has conformal weight p" (4.15)

From the definition above we can see that, e.g., gµν has conformal weight 2, L has conformal weight −4, while the Ricci scalar has no well defined conformal weight. To find the transformation behaviour of the matter fields we can use the fact that if an object has a conformal weight, it equals the negative mass unit of the object. To show this for the metric we need to choose the coordinates dxµ to be dimensionless. In order to make the line element 2 µ ν ds = dx dx gµν a quantity with mass dimension −2, gµν now has to have mass dimension −2 which coincides with it having a conformal weight of 2. Using the above we see that a scalar field φ has conformal weight −1, a fermion field has 3 conformal weight − 2 and a gauge field has conformal weight 0.

26 4.2 The Bach equations

To derive the equations of motion for Conformal Gravity, as we did for General Relativity we need an action to vary. As in section 2.4 the action should look like Z √ 4 S = SC + SM = gLd x. (4.16)

√ This action should be invariant under conformal transformations. Since g has conformal −4 weight 4, L should transform to Ω L. Let us ﬁrst ignore the matter part SM and concentrate on what the curvature part SC will look like. The most general scalar built from at most quadratic geometric objects is

µνρσ µν 2 L = a · R Rµνρσ + b · R Rµν + c · R + d · R + e. (4.17)

Higher orders in geometric objects can not be included, if one requires that L transforms to Ω−4L, which we do now. Higher orders in any geometric objects would give higher negative orders in Ω, that would not be cancelled by any other term in the action. Invoking the transformation behaviour and using (4.10), (4.11) and (4.12) we get

1 L = a · RµνρσR − 2a · RµνR + a · R2. (4.18) C µνρσ µν 3 This is the unique curvature part of a conformally invariant action. It can be rewritten in terms of the Weyl tensor (4.13) as

µνρσ LC = −αGC Cµνρσ, (4.19) which is the form typically given in the literature.

It is noteworthy, that the parameter αG is dimensionless.

To ﬁnd the underlying equations of motion one has to perturb the action S with respect to the metric tensor gµν. Before we do this however let us introduce another tensor, the Bach tensor W µν can be written as [19] 1 W µν = W µν − W µν, (4.20) 2 3 1 1 W µν = 2∇µ∇νR − 2 Rgµν − 2RRµν + R2gµν, (4.21) 1 2 1 1 W µν = ∇µ∇ Rαν + ∇ν∇ Rµα − Rµν − Rgµν − 2RµαRν + RαβR gµν. (4.22) 2 α α 2 α 2 αβ

27 R √ 4 Introducing now a generic matter part SM = −gLM d x in the action, and using the fact that the Lanczos Lagrangian LL, which is also known under the name Gauss-Bonnet term

µνρσ µν 2 LL = R Rµνρσ − 4R Rµν + R , (4.23) gives only a surface term, we get:

Z √ 1 δS = δ −g −2α RµνR − R2 + L + L d4x (4.24) G µν 3 L M Z √ µν 2 δ( −gLM ) 4 = 2 −4αGW + √ δgµνd x, (4.25) −g δgµν which, deﬁning the energy-momentum tensor as √ 2 δ( −gL ) √ M =: T µν, (4.26) −g δgµν gives the Bach equations

µν µν 4αGW = T . (4.27)

In Conformal Gravity the Bach equations replace the Einstein equations from General Rela- tivity.

Looking at the Bach equation, we see, that the dimensionless parameter αG takes the place of

Newtons gravitational constant. Since αG is dimensionless, this theory is renormalizable, as stated before in section 3.4, and as mentioned in [3].

28 5 Real & Complex scalar ﬁelds in Conformal Gravity

5.1 Real scalar ﬁeld

To get a complete theory with quantum gravity we now need a matter part in our interaction. As a first theory, let us have a look at a real scalar field φ with fermions and antifermions ψ and ψ¯ respectively, as done in [4]. Such an action has to be conformally invariant, thus we introduce it in the following way: Z √ (∂ φ)(∂µφ) φ2R ↔ S = −g µ + − λφ4 + iψ¯Dψ/ − ξφψψ¯ d4x. (5.1) M 2 12 ↔ µ µ Here ADB/ is defined as Aγ DµB − (DµA)γ B, with Dµ the total (fermion-) derivative de- µ fined with the fermion spin connection Γµ [20] as Dµ := ∂µ + Γµ. γ (x) are the vierbein dependant Dirac-Gamma matrices. Note that only the combination of the first two terms is conformally invariant. The parameter in front of those terms can take values of +1 or −1. The consequences of this choice are investigated in section 5.3.

Perturbing this matter action with respect to the scalar and fermion field will give the equations of motion for those fields: φR φ − + 4λφ3 + ξψψ¯ = 0, (5.2) 6 µ µ 2i∂ψ/ + i[γ , Γµ]ψ + iψ∇µγ − ξφψ = 0. (5.3) Using (ψ¯·(5.3)) we can derive the energy momentum tensor defined by (2.38) 2(∇ φ)(∇νφ) gµν(∇ φ)(∇αφ) φ∇µ∇νφ gµνφ φ 1 1 T µν = − µ + α + − + φ2 Rgµν − Rµν 3 6 3 3 6 2 (5.4) µν 4 µν − λg φ + TF ,

29 µν where TF is the energy momentum tensor of the fermionic part

µν ¯ µ ν ν ¯ µ TF = iψγ D ψ − i D ψ γ ψ + (µ ↔ ν). (5.5)

5.2 Complex scalar ﬁeld

When considering a complex ﬁeld ϕ instead of a real one φ the matter part of the Lagrangian has to be changed in order to remain real:

1 1 ↔ 1 L = (∂ ϕ∗)(∂µϕ) + Rϕ∗ϕ − λ(ϕ∗ϕ)2 + iψ¯Dψ/ − ξϕ∗ψψ¯ + ξϕψψ¯ . (5.6) M 2 µ 12 2

Doing the same calculations as above again, one can get the equations of motion in this theory. Since most parts of the new action are the same as, or at least very similar to, the action given in section 5.1, we will only discuss the ϕ equation of motion:

1 ϕ − ϕR + 4λϕ3 + ξψψ.¯ (5.7) 6

This equation looks just like (5.2), except that ϕ is a complex ﬁeld, while φ is not. For a better understanding what this difference means let us express ϕ as a combination of two real ﬁelds φ(x) and α(x):

ϕ(x) = φ(x)eiα(x). (5.8)

Using this, one can get to a complex equation of motion where both the real and imaginary have to vanish: µ 1 3 ¯ φ + φ(∇ α)(∇µα) − Rφ + 4λφ + ξ cos(α)ψψ 6 (5.9) µ ¯ +i 2(∇ α)(∇µα) + φα − ξ sin(α)ψψ = 0

Note that this differs from (5.2) if α(x) 6= 0. The α ﬁeld cannot be gauged away here since the action at hand does not carry a U(1) symmetry.

5.3 The parameter

To investigate the consequences of choosing a sign as the parameter we will look at two aspects of the theory: The stability of the scalar ﬁeld and the limit αG → 0. For our investiga- tions we use the action (5.1) with a real scalar ﬁeld.

30 Stability of the φ-ﬁeld For a look at the stability properties of the scalar ﬁeld φ we only need to look at the "kinetic" part of the Lagrangian density:

R L = (∇ φ)(∇µφ) + φ2 . (5.10) 2 µ 6

Using this we can deﬁne the conjugate momentum

δL 1 πφ := = ∇0φ, (5.11) δ∇0φ 2 with which we can compute the Hamiltonian density:

H := πφ∇0φ − L 1 R (5.12) = ∇ φ∇ φ + (∇φ)2 − φ2 . 2 0 0 6

Here ∇φ is the (spatial) gradient of φ.

We can see that for = −1 the energy density has no lower boundary, i.e., φ is a ghost. Interpreting R/6 as the mass of the φ we can see that for = +1 we get a non-ghost particle with negative mass. We will use this fact later in chapter 6, to try to generate masses via spontaneous symmetry breaking.

General Relativity as a limit of Conformal Gravity For this next part we use the fact that we can choose Ω(x) in such a way, that by applying the conformal transformation φ becomes constant:

φ(x) Ω(x) = . (5.13) φ0

Note, that this is only possible, if φ does not vanish anywhere. Applying a conformal transformation with the Ω given above to the energy momentum tensor given in (5.5) we get

1 1 T 0µν = φ2 Rg0µν − R0µν − λg0µνφ4 + T 0µν. (5.14) 6 0 2 0 F

After rewriting some constants

φ2 1 0 = 6φ2λ = Λ, (5.15) 6 8πG 0

31 where G is Newtons gravitational constant and Λ is the cosmological constant, we can express the energy momentum tensor as (primes are dropped) 1 1 T µν = − Gµν − Λ gµν + T µν, (5.16) 8πG 8πG F

µν 0µν 1 0µν where G = R − 2 Rg is the Einstein tensor. In the limit αG → 0 in the Bach equations (4.27) we can rearrange some terms to get to

−Gµν = −8πGTµν + Λgµν. (5.17)

To get Einsteins equations (2.40) in this limit with a positive Newtons constant G and the cosmological constant Λ, would have to be negative.

Result In the above parts we see that if is negative we have General Relativity as a limit of Con- formal Gravity, but the scalar field becomes a ghost field. On the other hand, if is positive we get a healthy scalar field φ, but negative Newtons and cosmological constants. However it seems to be possible to get a positive Newtons constant and a well behaving scalar field, by interpreting the conformal symmetry as a gauge symmetry, like presented in [21]. Since in this thesis we do not look at the limit to General Relativity, we will set to +1 for the remainder of this work.

32 6 The abelian Higgs model in Conformal Gravity

In order to get a more complete theory of quantum gravity, we should have a look at the generation of masses. To investigate this, we will use the toy model of the abelian Higgs model with the matter Lagrangian [22] 1 L = (∂µ + ieAµ)ϕ∗(∂ − ieA )ϕ − µ2ϕ∗ϕ − λ(ϕ∗ϕ)2 − F F µν, (6.1) M µ µ 4 µν which contains a complex scalar ﬁeld ϕ and a gauge boson Aµ to make the theory invariant under U(1) symmetry. e is the coupling strength of the gauge boson. The kinetic part of the gauge boson contains its ﬁeld strength tensor Fµν := ∂µAν − ∂νAµ. Let us, for a while, simply explore this theory as in [22], and ignore that we want to demand invariance under conformal symmetry.

Assuming, that µ2 is negative we have a potential

V (ϕ) = µ2ϕ∗ϕ + λ(ϕ∗ϕ)2, (6.2)

2 that does not have a global minimum in the origin, but at |ϕ| = √1 ν, with ν2 = − µ . Since 2 λ ϕ is a complex ﬁeld, we can express it with a real valued radial part, and a real valued angular part. The same is true for the minimum, and we can express it as

1 iϑ ϕ0 = √ νe . (6.3) 2 This is a minimum for all ϑ, so like the Lagrangian density, it is invariant under U(1) symmetry. However, the ﬁeld can only be in one minimum, so at some point, when the ﬁeld goes to its minimum, a ϑ is chosen freely among all possible values. This is called spontaneous symmetry breaking. Now we reexpress ϕ in terms of its radial part h(x) and its angular part θ(x) as 1 ϕ → √ ν + h(x) eiθ(x)/ν. (6.4) 2

33 The radial part is chosen in such a way, that ϕ is in its minimum, when h(x) is small. The new fields h(x) and θ(x) are called “Higgs field” and “Goldstone field” respectively. To get rid of the Goldstone field, we interpret the angular part as a U(1) transformation, and substitute 1 A → A + ∂ θ, (6.5) µ µ eν µ accordingly. This leads us to

0 1 µ 2 2 3 1 4 L = (∂µh)(∂ h) − λν h − λνh − λh 2 4 (6.6) 1 1 1 + e2ν2A Aµ + eνA Aµh + e2A Aµh2 − λν4. 2 µ µ 2 µ 4 √ This is now a theory with a Higgs ﬁeld of mass 2λν, and a gauge boson Aµ of mass eν. Note that the last term is constant and can simply be dropped. If we wanted to introduce other massive particles, they would simply have to be coupled to ϕ by a Yukawa coupling. The masses of those particles would then be proportional to the Higgs mass and to the coupling constant of the Yukawa coupling.

Using the method of spontaneous symmetry breaking we can now generate masses for all sorts of particles if µ2 < 0. However, this theory is not yet invariant under conformal transformation, but we can make it conformally invariant by setting 1 µ2 = − R. (6.7) 6 This leads to spontaneous symmetry breaking being possible if R > 0, and a Higgs mass p of mh = R/3, as is shown later. Now there seem to be two problems. First of all, the Ricci scalar is spacetime dependent, leading to the Higgs mass, and thus all the masses, being spacetime dependent. Secondly, this spontaneous symmetry breaking only works for positive R, but we cannot assume that this, in general, is true for all x. However, the ﬁrst problem turns out to be nonexistent. If we use a spacetime dependent parameter µ(x) as in (6.7) then the √ mass of the gauge boson from (6.1) becomes mA = (e/ λ)µ(x). Note that these quantities may be spacetime dependent, but they are also not physical, in the sense that they can not be measured directly. On the other hand the relation between two masses can be measured directly. This relative mass is m e A = √ . (6.8) mh 2λ We note that this relative mass does not depend on µ, and thus on spacetime, in any way, so we can deﬁne µ as we like, as long as µ2 is negative.

34 If the Ricci scalar is positive, we can insert (6.7) into (6.1) to get to the conformally invariant Lagrangian density 1 1 L = (∂µ + ieAµ) ϕ∗ ∂ − ieA∗ ϕ + Rϕ∗ϕ − λ (ϕ∗ϕ)2 − F F µν. (6.9) M µ µ 6 4 µν √ Here, ϕ has a minimum at |ϕ| = √ 1 R. Analogous to (6.4) and (6.5) we transform ϕ and 12λ Aµ as √ √ 1 1 i 12λ √θ ϕ → √ √ R + h e R , (6.10) 2 6λ √ 12λ θ Aµ → Aµ + ∂µ √ . (6.11) e R This leads to √ √ √ √ 1 µ 1 µ 1 µ 1 µ LM = ∂ R∂µ R + √ ∂ R∂µh + √ ∂ h∂µ R + ∂ h∂µh 12λ 2 6λ 2 6λ 2 2 2 √ 2 e µ e µ e µ 2 + A AµR + √ A Aµh R + A Aµh 12λ 6λ 2 √ √ 1 2 1 2 λ 3 λ 4 1 µν + R − Rh − √ Rh − h − F Fµν. (6.12) 144λ 6 6 4 4 √ Identifying ν(x) = √ R we arrive at 6λ 1 1 1 1 L = ∂µν(x)∂ ν(x) + ∂µν(x)∂ h + ∂µh∂ ν(x) + ∂µh∂ h M 2 µ 2 µ 2 µ 2 µ 1 e2 + (eν(x))2AµA + ν(x)e2AµA h + AµA h2 2 µ µ 2 µ 1 1 1 + λν(x)4 − λν(x)2h2 − λν(x)h3 − λh4 − F µνF . (6.13) 4 4 4 µν Here we indeed ﬁnd the Higgs mass to be √ √ R mh = 2λν(x) = √ . (6.14) 3 The problem that remains to be discussed is, that this mechanism of spontaneous symmetry breaking only works, if R > 0 for all xµ. However, this is not true for a general Ricci scalar. For example the Ricci scalar of ﬂat Euclidean spacetime is 0 everywhere. To get the Ricci scalar to be positive everywhere, one could try to transform it, using a conformal transformation. The transformed Ricci scalar R0 is (4.12) 1 Ω R0 = R − 6 . (6.15) Ω2 Ω

35 Since Ω is a positive function, R0 is positive, if and only if the term in brackets is positive. To ﬁnd out what is needed to make this term positive, we assume, that there is a function f(x) that is smaller than R at every point in spacetime. This does not work, if there are points in spacetime, where R has negative singularities. However, close to singularities, physics does not work in standard ways, so it does not make much sense to wonder about spontaneous symmetry breaking close to those points. For this reason, we will omit these points in our discussion. With our function f given, the Ricci scalar becomes positive by a conformal transformation, if the equation 1 Ω f(x) = (6.16) 6 Ω can be solved by a function Ω, that is positive everywhere. To implement the condition, that Ω has to be positive, we will rewrite it as

Ω(x) = eω(x), (6.17) with a real function ω(x). Plugging this into (6.16) yields 1 f = ω + ∇ ω∇µω. (6.18) 6 µ This is a four dimensional nonlinear differential equation of second order. To further discuss it, we will look at it in one dimension only. 1 f = ω00 + ω02 (6.19) 6 We see, that this differential equation does not contain ω itself, but only its derivatives. Sub- stituting ω0 for z(x) we get 1 f = z0 + z2, (6.20) 6 which is a Ricati equation. For this equation a general solution can be found, if a particular solution is known, but in general one can not say, if this equation is solvable or not. This means, that we can also make no statement about the existence of a solution of (6.18). Thus, it can not be proven, nor disproven, that we can ﬁnd a conformal transformation, connecting a general Ricci scalar to a positive one. To solve our problem of the Ricci scalar not being positive, we could restrict ourselves to only using Ricci scalars that are either positive themselves, or connected to such a scalar via a conformal transformation. This approach, however, would not be very scientiﬁc, since the

36 only reason to discard the other Ricci scalars would be to make the theory work, and it is not clear, if we could describe the real world with the remaining choices for Ricci scalars.

To summarize, spontaneous symmetry breaking with a single scalar ﬁeld in Conformal Gravity is possible, or impossible, depending on whether (6.18) has a solution or not. How- ever, even if (6.18) has no solution, this does not mean, that spontaneous symmetry breaking does not work at all in Conformal Gravity. In 2015, Oda presented a way of getting a Higgs mechanism in Conformal Gravity, using two independent scalar ﬁelds [19]. We will now have a look at his theory.

First of all, we should have a look at the matter Lagrangian 1 1 1 1 L = ∂ φ∂µφ − (D H)†DµH + φ2R − HH∗R − F F µν − V (φ, H). (6.21) 2 µ µ 12 6 4 µν In this Lagrangian φ is a real scalar field, and H is a complex scalar field, the Higgs Field. The covariant derivative is defined as

DµH := (∂µ + ieAµ)H, (6.22) with the coupling strength e and a gauge boson field Aµ like in the Lagrangian (6.1) for the abelian Higgs model discussed earlier. As in the model above, the field strength tensor is defined as

Fµν := ∂µAν − ∂νAµ. (6.23)

The Higgs doublet H can be rewritten, using the unitary gauge (see [19]) as 1 H = √ eiαθ(x)(0, h)T . (6.24) 2 Using the U(1) invariance of H, the Nambu-Goldstone boson θ(x) can be gauged away, simi- larly to the way it is done in (6.5). Doing this, we arrive at [19] 1 1 1 1 L = (φ2 − h2)R + (∂ φ)2 − (∂ h)2 + e2A2 h2 − F F µν − V (φ, h). (6.25) 12 2 µ 2 µ µ 4 µν Looking at the kinetic terms of φ an h, we see, that φ is a ghost ﬁeld, while h is a healthy ﬁeld. In order for the theory to be conformally invariant, V (φ, h) has to have conformal weight −4, which is only possible, if it is of fourth order in φ and h. Thus the potential has the form [19]

4 2 2 4 V = a1h + a2h φ + a3h . (6.26)

37 To get rid of the ghost ﬁeld, and to get a Higgs like potential, Oda uses the conformal invariance. A conformal gauge, the Einstein gauge, is chosen, such that [19]

2 2 02 02 2 (φ − h ) → (φ − h ) = 6Mp , (6.27)

2 2 with Mp being the Planck mass. It is noteworthy, that this only works, if (φ − h ) is positive everywhere. In reduced Planck units, which we adopt here, the Planck mass is simply set to 1. Looking again at (6.25), we see, that the ﬁrst term becomes the Lagrangian of the Einstein Hilbert action (2.28). So by only using the Einstein gauge, we already recover General Rela- tivity in the limit αG → 0. To investigate, how the choice of the Einstein gauge changes the potential V we rearrange (6.27) to

φ2 = 6 + h2 (6.28) ⇒φ4 = 36 + 12h2 + h4. (6.29)

Plugging this into (6.26), we get

4 2 (a1 + a2 + a3)h + (6a2 + 12a3)h + 36a3. (6.30)

With the choices for the ai given in [19]

1 λ ν2 2 a = Λ + 1 + , (6.31) 1 36 8 6 1 λ ν4 ν2 a = − Λ − + , (6.32) 2 18 8 18 3 1 λ ν4 a = Λ + , (6.33) 3 36 8 36 we get to λ λ λ V → V 0 = h4 − ν2h2 + ν4 + Λ, (6.34) 8 4 8 which is indeed a Higgs like potential, that even gives a cosmological constant, when Λ is interpreted as part of the Einstein Hilbert action in the ﬁrst term of 6.21.

38 7 Conclusion and outlook

In this thesis we investigated the possibility of spontaneous symmetry breaking in Conformal Gravity. We saw, that Conformal Gravity solves some of the problems of General Relativity, as it is, for example, renormalizable. As was shown in [4], the implementation of real scalar quantum fields is straightforward, as is the generalization to complex ones. We showed, that in regions with positive Ricci scalar spontaneous symmetry breaking works fine. The vacuum expectation value of the scalar field in this case is spacetime dependent, resulting in a spacetime dependent Higgs mass. However, we argumented, that the Higgs mass itself is only a parameter, that cannot be measured by experiment, since in order to measure a mass, we need to choose a mass unit. Since, by Yukawa coupling, all masses depend on the spacetime position in the same way, it would make every natural choice of mass unit, e.g., the mass of an electron or of the Higgs boson, also depend on the spacetime position in this way.

In regions with negative or vanishing Ricci scalar, the Higgs potential becomes a central potential. Due to the vacuum expectation value in such a potential being zero, there would be no spontaneous symmetry breaking happening. The question, that remains open, is if the Ricci scalar can be transformed in a way, that it is positive nearly everywhere. To explore this, it might be fruitful, to further investigate the transformation behaviour of the Ricci scalar under conformal transformation. However, such an investigation might be problematic, since most special cases of Ricci scalars, e.g., the Ricci scalar of the Schwarzschild solution, use the assumption, that smallest scale features can be neglected. If we couple the Ricci scalar directly to a quantum ﬁeld, as is done in Conformal Gravity, this assumption no longer holds, since in a Quantum Field Theory we are explicitly interested in smallest scale features.

As an alternative to Conformal Gravity with one scalar field, we looked at the case of two scalar fields. Here we found, that spontaneous symmetry breaking is not only possible, but it also gives a constant Higgs mass, unlike the case with one scalar field discussed earlier. In this theory however, we have to assume that (φ2 − h2) is positive, to go into the Einstein frame.

39 It would be interesting to investigate the implications of this restriction. In addition to that it should be investigated, how the fermionic sector changes with two scalar ﬁelds in contrast to just one. An interesting feature of this theory would be, that every fermion is coupled to both scalar ﬁelds, and not just to one. In [19] it is claimed that the addition of a fermionic sector does not change anything for the spontaneous symmetry breaking, but there might be changes in the fermionic sector itself by the addition of a second Yukawa coupling.

To Summarize, Conformal Gravity is a very fruitful theory, capable of ﬁxing many problems of General Relativity. Nevertheless, the implementation of spontaneous symmetry breaking is not yet fully clear. There are multiple methods to implement spontaneous symmetry breaking into Conformal Gravity, but for each one, there are still some details or restrictions present, with the implications not fully understood.

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42 Declaration of Authorship

I hereby declare that the thesis submitted is my own unaided work. All direct or indirect sources used are acknowledged as references.

I am aware that the thesis in digital form can be examined for the use of unauthorized aid and in order to determine whether the thesis as a whole or parts incorporated in it may be deemed as plagiarism. For the comparison of my work with existing sources I agree that it shall be entered in a database where it shall also remain after examination, to enable comparison with future theses submitted. Further rights of reproduction and usage, however, are not granted here.

This paper was not previously presented to another examination board and has not been published.

Bielefeld, December 30, 2019

43 Acknowledgements

I want to thank all the people who supported me, when I was writing this thesis. Without them this work would not exist in this form.

First of all I want to thank my supervisor Professor Schwarz for the interesting subject and the great care. The meetings and discussions we had were really interesting.

Secondly, I want to thank Patrick for his help when I got started with this subject.

I also want to thank Hendrik, Friederike and Michaela who proofread my work, so that some awkward errors could be removed.

Finally I want to thank my parents, for always encouraging me, and making it possible for me to study physics.