Electromagnetic Duality in Linearized Conformal Gravity

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Electromagnetic Duality in Linearized

Conformal Gravity

Hap´

e Fuhri Snethlage

10 augustus 2020

Bachelor thesis Physics & Astronomy Supervisor: dr. Sergio H¨ortner

Institute of Physics

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Abstract

We show that the Hamiltonian action principle for linearized conformal gravity can be cast in a manifestly duality symmetric form. To do this first the relevant mathematical tools are developed in the form of Hamiltonian constrained systems. Afterwards we ex-plore electromagnetic duality in multiple theories, with the focus on showing that this duality emerges not only on the level of the equations of motion but also as a symmetry of the more fundamental action. The first theory is that of classical electromagnetism. Where we solve Gauss’ law to obtain a second potential that induces a duality transfor-mation that leaves the Hamiltonian action principle invariant. Then a similar analysis is done for linearized gravity where the corresponding constraints are solved. Again the duality transformation between the two potentials leaves the Hamiltonian action invari-ant. Lastly the techniques learned from these two examples are applied to linearized conformal gravity. Under a small assumption on the form of the potential that solves the constraint on the conjugate momentum, the Hamiltonian action principle is shown to be invariant under this duality.

Title: Electromagnetic Duality in Linearized Conformal Gravity Authors: Hap´e Fuhri Snethlage, hape98@hotmail.com, 11194324 Supervisor: dr. Sergio H¨ortner,

Graders: prof. dr. Jan de Boer, dr. Alejandra Castro End date: August 10, 2020

Institute of Physics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.iop.uva.nl

Institute for Theoretical Physics Amsterdam University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.itfa.uva.nl

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Introduction

Duality is an interesting phenomenon that appears in both physics and mathematics. While there are many similarities between the dualities found in different theories, there are usually just as many differences. This makes it virtually impossible to write down one all encompassing definition of duality. As Sir M. Atiyah put it: “Duality in mathematics is not a theorem, but a principle.” [1].

In its most basic form, duality is characterized by the existence of two formulations or constructions in a theory that are equivalent. In physics this takes on many different forms, including but not limited to: Position and momentum in the harmonic oscillator where the Fourier transform links the two concepts together. Kramers-Wannier dual-ity where there is a relation between the high and low temperature expansions of the partition function of the 2d Ising model, and most important for our purposes, electro-magnetic duality in four dimensional Maxwell theory.

It is interesting to notice the similarity between electric-magnetic duality in Maxwell theory and the Hodge theory of harmonic forms. While in electromagnetism we deal with harmonic forms of degree 2 (in four dimensional Minkowski background), Hodge theory is about harmonic forms of any degree (in Riemannian manifolds) in any dimension. Atiyah even suggests that it is this example that motivated Hodge to develop the theory [1]. In any case the example of electromagnetism is what sparked a wide spread search for structures like it in other physical theories. Although it was first thought that the only way to formulate this duality was, on the level of the equations of motion, not on the level of the more fundamental action principle, this was later disproven for: electrodynamics [2], even later for linear gravity [3] and we shall do so here for linear conformal gravity. The formalities of constrained systems are important to describe this type of duality. Typically, covariant Lagrangian action principle do not display duality symmetries. The goal of our approach is then to define an equivalent action principle that does display a certain duality symmetry. The way to do this is to introduce the Hamiltonian formalism so first we will discuss the generalities of the Hamiltonian formalism. Afterwards the ideas of constraints are added in a way to retain equivalence with the original Lagrangian formalism. These constraints can be used in the theories we are looking at to obtain a duality symmetric action principle. The transition to the Hamiltonian formulation does come at the cost of losing manifest space-time covariance. This is what will be discussed in the first chapter of the thesis

In the second chapter, after the mathematical framework is put into place, the dis-cussion will turn to the first example. To begin any disdis-cussion of duality in a physical theory it is illuminating to first consider the classical theory of electromagnetism. For it is a theory in which these concepts manifest themselves clearly and in a way that does not require the technicalities of a field theory on a curved space, even though

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it is possible to generalize these concepts to a curved background. So we will intro-duce the notion of electromagnetic duality in the source-free Maxwell equations and use the framework developed in the first chapter to show that the symmetry can also be de-scribed on the level of an action principle. Afterwards the discussion will turn a bit more speculative when sources are considered. We will discuss an extension of the source-full Maxwell equations that is duality symmetric. This is done via the introduction of the still undiscovered magnetic monopole. This extensions is interesting because, from all the possible extensions of electromagnetism that have no experimental backing, the one that respects duality invariance offers a mechanism that accounts for the quantization of electric charge.

Once we have been properly warmed up to the ideas involved with duality in a theory, we will continue to the first theory of gravity that shows dual structures, the theory of linearized gravity. This chapter will be kept brief since it serves mainly as an introduction to theories of gravity than anything else.

Lastly, we will come to linear conformal gravity, where we will use all the previously learned ideas and techniques to show that it also contains dual structures in its action functional.1

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1 Constrained Hamiltonian Systems

1.1 Principle of least action

One of the first accounts of the principle of least action was that of Pierre de Fermat, in the seventeenth century. Considering optics, he figured out that the path light travelled will always be the one of least time and derived the law of refraction with this insight. This was, unbeknownst to him, an example of a far more general idea in physics. The first general description of the principle of least action was done by Maupertuis, further devolpments by the likes of Euler, Lagrange and Hamilton [4, 5, 6, 7] have culminated in a statement that still stands at the basis of most contemporary physics. It states that the correct equations of motion for a physical system are those that extremize the action. Of course this is an empty statement until the quantity action is defined. Before that it is necessary to know the quantities to be determined to describe a physical system.

Let us start our discussion by considering a system of N interacting particles. In a system containing N particles one must determine the location of the particles at a certain time t1to begin to describe the physical system, but even then it is not guaranteed

that the system can be described at a time t1+ δt. Particles can, and often will, be

in motion. So if one has a complete set of positions {q1_{, . . . , q}N_{}, it is required that}

their velocities ˙qi := dq_dti are known as well. It turns out that this is enough, meaning the way the velocities change from time t1 to time t1+ δt1 can be determined by the

knowledge of the positions and velocities at t1 alone. The job of a physicist is then to

find the relationship between the positions qi, the velocities ˙qi and the accelerations ¨qi, also called the equations of motion.

Say a system evolves between some time t1 and a later time t2 from positions qi(t1)

to qi(t2). The principle of least action says that the system evolves in such a way that

the integral

S[q] = Z t2

t1

L[qi, ˙qi]dt, (1.1)

is extremized, this integral is called the action [8]. The integrand of the action is called the Lagrangian, and it is this quantity, together with boundary conditions, that will determine the equations of motion. To derive conditions on the variables so that the action functional is extremized, one uses the fact that a function is extreme when it varies second order if the input is altered first order. In the case of the action that means: If there is some path q(t) that extremizes the action and it is altered in a way

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that the endpoints remain fixed, by a small variation δq(t), then the difference

δS[q] = S[q(t) + δq(t)] − S[q(t)] = 0, (1.2)

should be zero. Since the variation of the input is linear, the variation in ˙q is ˙q + δ ˙q. The definition of the action (1.1) then gives the expression for this difference

δS = Z t2

t1

L[qi+ δqi, ˙qi+ δ ˙qi] − L[qi, ˙qi] dt. (1.3)

The integrand here can be recognized as the variation of L, δL, allowing it to be rewritten as δS = Z t2 t1 ∂L ∂qiδq i₊ ∂L ∂ ˙qiδ ˙q i dt, (1.4)

using the multivariable chain rule. The second term can be integrated by parts to get δS = ∂L ∂ ˙qiδq i t2 t1 − Z t2 t1 d dt ∂L ∂ ˙qi − ∂L ∂qi δqidt = 0. (1.5)

The condition that the endpoints remain fixed by the variation imply that the boundary term vanishes. What is left is that the integrand should be zero, however the variation was specifically chosen to be non zero. Thus the other term must be identically zero, arriving at d dt ∂L ∂ ˙qi − ∂L ∂qi = 0, (1.6)

the Euler-Lagrange equations.

This derivation was done under the assumption that the physical quantities are discrete points in space, particles if you will. Often it will be the case that the physical quantity that is to be determined, is continuously distributed through space. Such a quantity will be called a field Φ. The generalization of the E-L equations for fields is important and will be used throughout this thesis. It is however not that different from the particle case, so only the main differences will be outlined here and everywhere when deemed necessary. The big difference up till now is the fact that in field theory the Lagrangian is composed of a Lagrangian density, which is first integrated over space, transforming the action from (1.1) into

S[Φ] = Z t2 t1 Z Ω L[Φ, ∂βΦ] d3x dt. (1.7)

The corresponding E-L equations derived from extremizing the action will now take the form ∂µ ∂L ∂(∂µΦ) − ∂L ∂Φ = 0, (1.8)

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where the greek index takes on all the spacetime indices [9].

An interesting property that occurs in both cases is that the equations of motion are invariant under addition of a total time derivative to the Lagrangian. Consider two Lagrangians differing by a total time derivative

L, L0= L + df (q

i_{, t)}

dt ‘, (1.9)

the actions take the form S = Z t2 t1 Ldt, (1.10) S0 = Z t2 t1 L +d(f (q i_{(t), t))} dt dt. (1.11)

By the fundamental theorem of calculus, the second equation becomes S0 =

Z t2

t1

Ldt + f (qi(t2), t2) − f (qi(t1), t1). (1.12)

The assumption that was made regarding the variation of the endpoints of the time evolution make it so that the last two terms of (1.12) drop out when δS0 is considerd. So the demands that δS0 = 0 and δS = 0 give the same information. A similar argument can be made, using higher dimensional versions of the fundamental theorem of calculus, to derive that divergence terms do not affect the equations of motion as well.

Another important quantity comes up when considering the time derivative of the Lagrangian dL dt = ∂L ∂qiq˙ i₊ ∂L ∂ ˙qiq¨ i₊∂L ∂t. (1.13)

After using (1.6) this equation becomes dL dt = ˙q i d dt ∂L ∂ ˙qi + ∂L ∂ ˙qiq¨ i₊∂L ∂t, (1.14)

the right hand side of this equation can be recognized as a total time derivative dL dt = d dt ˙ qi∂L ∂ ˙qi +∂L ∂t, (1.15)

rearranging this finally gives d dt ˙ qi∂L ∂ ˙qi − L = −∂L ∂t. (1.16)

This derivation shows that the quantity within the brackets is conserved if the Lagrangian is not explicitly dependent on time. This quantity is called the Hamiltonian.

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1.2 Hamiltonian mechanics

Mathematically the Hamiltonian is defined as the Legendre transformation of the La-grangian H = piq˙i− L (1.17) with pi being pi:= ∂L ∂ ˙qi . (1.18)

A Legendre transformation is a well-defined variable transformation from one set of variables to another. In the case of the Hamiltonian, ˙q is transformed to p, this makes the Hamiltonian a function that depends on q and p. This can be seen more directly by varying the Hamiltonian H with respect to q and ˙q,

δH = ˙qiδpi+ δ ˙qipi− δ ˙qi

∂L ∂ ˙qi − δq

i∂L

∂qi, (1.19)

the definition of pi reduces this to

δH = ˙qiδpi− δqi

∂L

∂qi. (1.20)

This makes it clear that any variation of ˙q only appears through p. So H is only explicitly dependent on q and p, allowing any Hamiltonian to be seen as a function H(q, p, t).

In a field theory the Hamilton formulation does not look much different. Again the big difference being that the Hamiltonian becomes a Hamiltonian density

H = ˙ΦΠ − L, (1.21)

with Π := ∂L/∂(∂0Φ) .

Aside from being a quantity that is conserved when the Lagrangian does not explicitly depend on time, the Hamiltonian can also be used to fully characterize the physical sys-tem. Equivalent equations to (1.6) can be derived that depend only on the Hamiltonian and its variables. Consider the total differential of the Lagrangian,

dL = ∂L ∂qidq i₊ ∂L ∂ ˙qid ˙q i₊∂L ∂tdt. (1.22)

The conjugate momenta can be used to rearrange this equation to d piq˙i− L = − ∂L ∂qidq i_{+ ˙}_qi_dp i− ∂L ∂tdt. (1.23)

The left hand side is just the differential of the Hamiltonian from which follows ∂H ∂qidq i₊ ∂H ∂pi dpi+ ∂H ∂tdt = − ∂L ∂qidq i_{+ ˙}_qi_dp i− ∂L ∂tdt. (1.24)

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This equation reduces to ∂H ∂qi = − ˙pi, ∂H ∂pi = ˙qi, ∂H ∂t = − ∂L ∂t, (1.25)

precisely when the E-L equations (1.6) hold. These are known as Hamilton’s equations and they change the problem of solving n second order PDE’s of the form (1.6) to solving 2n first order PDE’s. The benefit of working with Hamilton’s equations or Lagrange’s depends on the problem at hand.

The space of all the co¨ordinates and conjugate momenta is called the phase space. An important additional structure can be defined on it, the Poisson brackets. Say f and g are two functions defined on the phase space then the Poisson bracket of the two functions is defined as {f, g} = ∂f ∂qi ∂g ∂pi − ∂f ∂pi ∂g ∂qi. (1.26)

The Poisson brackets serve an important function in the Hamiltonian formalism for they generate the time evolution of phase space functions. Consider the function f (q, p, t), its time evolution can be written as

˙ f = ∂f ∂qiq˙ i₊ ∂f ∂pi ˙ pi+ ∂f ∂t. (1.27)

Substituting equations (1.25) we get ˙ f = ∂f ∂qi ∂H ∂pi − ∂f ∂pi ∂H ∂qi + ∂f ∂t (1.28) = {f, H} + ∂f ∂t. (1.29)

So the Poisson bracket of a function with the Hamiltonian defines the time evolution of the function on the phase space.

All these results can be summarized in the action functional S0[q, p] =

Z

( ˙qipi− H)dt, (1.30)

that is different from but equivalent to the original action (1.1). Hamilton’s equations and the time evolution of the system can be recovered from this action by using the regular variational methods.

An important thing to note here is that the momenta and coordinates were implicitly assumed to be independent. However this does not always have to be the case, in fact it will often happen that there are certain relations between them. When this happens we call a system constrained.

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1.3 Constraints

Lets say that there are constraints, or the momenta and coordinates are not fully inde-pendent thus there exist relations,

ϕk(qi, pj) = 0, (1.31)

between them. An obvious first question that should be asked is: Is there some condition that can be checked to see if a system is constrained or not?

To anwser this question we will first have another look at the E-L equations. As mentioned these equations determine the equations of motion of a system, by using that

∂L∂ ˙qi _{is a function of q and ˙}_{q equations (1.6) turn into}

¨ qj ∂ 2_L ∂ ˙qj_{∂ ˙}_qi + ˙q j ∂2L ∂qj_{∂ ˙}_qi = ∂L ∂qi, (1.32)

with the multivariable chain rule [10]. The form of this equation makes it immediatly clear that it has a unique solution for ¨qj in terms of ˙qj and qi only when ∂2L∂ ˙qj∂ ˙qi is invertible, or equivalently when

det ∂2L ∂ ˙qj_{∂ ˙}_qi 6= 0. (1.33)

When (1.33) is true one can use the techniques discussed in the previous section to find a set of functions that satisfy (1.6). The equations that are found are then guaranteed to be unique.

So what happens when ∂2L∂ ˙qj∂ ˙qi is not invertible? Firstly, when this happens there is not one solution to the E-L equations but an entire class of solutions to (1.6). Since the system is still a physical one, there can be only one time evolution. However every solution is just as valid a solution as the other, thus the difference between the solutions should have no effect on the equations of motion.

Secondly, the conjugate momentum p reduces the matrix ∂2L∂ ˙qj_{∂ ˙}_qi _{to ∂p}

i∂ ˙qj. So

we obtain a singularity condition on the derivative with respect to the velocities on p. Lets focus on the case where N = 1 to see how (1.31) comes about. In theory p is a function of q and ˙q, however if

∂p

∂ ˙q = 0 (1.34)

then p does not depend on ˙q, thus

pi = f (qi).

Therefore if the condition (1.33) does not hold the momenta and positions are not fully independent, as such there are some relations

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Thus the condition (1.34) implies that the system is constrained. The constraints deter-mined by this condition were called the primary constraints of the system by Dirac [11]. Primary because the equations of motion are not needed to derive them, they depend only on the form of the dynamical variables.

The constraints (1.31) define a subspace, called the constraint surface, in the original phase space but they impose no restriction on ˙q. So at first glance they introduce a problem. The dimensionality of the constraint surface is lower than the space defined by (qi, ˙qi). This has the effect that transformation between (qi, ˙qi) and (qi, pi) need

not be invertible. Invertibility can be recovered at the expense of adding additional coordinates. For simplicity lets assume the Hamiltonian is not explicitly dependent on time. The variation of H in terms of p and q is then

δH = ∂H

∂pδp + ∂H

∂q δq. (1.36)

There is another way to vary H. By using the definition (1.17) and varying ˙q and q, this resulted in (1.20). Of course it should not matter in what order one calculates the variation of the Hamiltonian. Thus these equations must be equal, leading to the vector equation ∂H ∂qi + ∂L ∂qi δqi+ ∂H ∂pn − ˙qi δpi= 0. (1.37)

This equation implies that

∂H ∂qi + ∂L ∂qi ∂H ∂pi − ˙q i ! , (1.38) is normal to δqi δpi . (1.39)

The interpretation of the space spanned by the vectors δqi and δpi is that of the space

tangent to the point (qi_{, p}

i), more intuitively as a plane intersecting the space only at

the point (qi, pi) and nowhere else. So (1.38) is a normal vector to (q, p) for every q and

p. Thus (1.38) forms a normal vectorfield to the solutions set of (1.31), this solution set is precisely the constraint surface. Now

∂ϕk ∂qi ∂ϕk ∂pi ! (1.40)

can be shown to be normal to (1.39) by taking the divergence ∂ϕk

∂qiδq

i₊∂ϕk

∂pj

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of (1.31). (1.40) is actually the unit normal vectorfield which is unique up to a sign. Thus (1.38) can be written in terms of (1.40) with certain scaling parameters ui, giving rise to the equations

˙ qi = ∂H ∂pi + uj∂ϕj ∂pi , (1.42) ˙ pi = ∂L ∂qi = − ∂H ∂qi − u j∂ϕj ∂qi. (1.43)

The first one is of particular importance since it allows the velocities to be recovered from the knowledge of the momenta and certain parameters ui. The addition of these parameters solves the problem of invertibility of the Legendre transformation. These equations can be seen as the generalisation of Hamilton’s equations, again these equations allow the general time evolution of a phase space function to be written in terms of the Poisson brackets

˙

f (q, p, t) = {f, H} + uj{f, ϕ_j} +∂f

∂t, (1.44)

but now with an extra term accounting for the constraints [10].

Just as in the unconstrained case there is an action functional that produces these equations via the variational method. That functional turns out to be

S = Z

( ˙qipi− H − uiϕi)dt. (1.45)

The extra parameters added now appear as Lagrange multipliers, that impose the pri-mary constraints. With this knowledge one can also first solve for the constraints reduc-ing the action to

Z

( ˙qipi− H)dt. (1.46)

The nomenclature of primary constraints imply that there is something more than just primary constraints. Indeed there are additional conditions called secondary constraints, they have a more physical origin. Secondary constraints come about because of the consistency condition, which states that all constraints must be preserved in time

˙

ϕi= {ϕi, H} + uj{ϕi, ϕj} = 0. (1.47)

Of course, this changes nothing if it manifestly holds but it is possible that it puts additional restrictions on uj. Adding to the list of constraints already found by (1.34). Some authors go even further, calling the constraints found by imposing the consistency condition on the secondary constraints tertiary and so on. We will not be doing so since the distinction between primary and secondary constraints is not that important when considering the duality we are interested in. There is however a classification that is quite significant: The idea of first and second class constraints [11].

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1.4 Classification of constraints

Say there is some complete set of constraints for a system {ϕi | i ∈ {1, . . . , m}}. A

function is called first class if its Poisson bracket with every constraint vanishes on the constraint surface2, or

{f, ϕ_i} = 0, ∀i. (1.48)

A constraint that is not first class is called second class. The important feature of first class constraints is that they generate gauge transformations.

To see this, say ϕi is first class. The general solution to the consistency condition

(1.47) for uj is then

uj = Uj+ vkV_kj, (1.49)

where Uj is a particular solution of equation (1.47) and vkV_kj is a linear combination of the most general solution of the homogeneous equation

Vj{ϕi, ϕj} = 0. (1.50)

This has the effect that the coeffcients vk are completely arbitrary.

Consider now a dynamical variable q at t0 and a moment later at t0+ t1 = δt. The

time evolution of q is governed by its Poisson bracket with the Hamiltonian

q(δt) = q0+ ˙qδt (1.51)

q(δt) = q0+ ({q, H} + U {q, ϕ} + v{q, V ϕ})δt. (1.52)

The choice of v was completely arbitrary and should have no effect on the physical system. So the difference in time evolution, corresponding to a different choice for v, becomes

∆q(δt) = δt(v − v0){q, V ϕ}. (1.53)

Now δt(v − v0) is an arbitrary function of time for the dynamical variables that has no effect on the dynamics. Thus (1.53) is a gauge transformation.

What to do then with the second class constraints? Obviously (1.50) does not hold anymore, so there need not be any arbitrary choice in the transformation generated by the second class constraints. Thus these transformations will not in general behave nicely on the phase space manifold. The way to deal with these constraints is through the introduction of the Dirac bracket. To see how we will first discuss an example.

Say we have a system of N canonical coordinates subject to the constraints q1 = 0

and p1 = 0. These constraints are obviously not first class since

{q1, p1} = 1. (1.54)

2

Dirac uses the notation ≈ to signify that a function is equal to zero on the constraint surface. We will just say when we are talking about the constraint surface.

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If we now restrict ourselves to the constraint surface, on this surface the constraints all equal zero, as such the Poisson brackets with the constraints should also be zero but this is not the case for (1.54). In this example the fix is easy, the constraints tell us that the first degrees of freedom are unnecessary and can be discarded by modifing the Poisson brackets to {f, g}∗ = N X n=2 ∂f ∂qn ∂g ∂pn − ∂g ∂qn ∂f ∂pn . (1.55)

The general case is defined as follows, say there is some set of second class constraints {χα}, then the Dirac bracket is defined as

{f, g}∗= {f, g} − {f, χα}Cαβ{χβ, g}, (1.56)

where

Cαβ = {ϕα, ϕβ}. (1.57)

All the equations of motion in the Hamilton formalism can now be expressed in terms of the Dirac bracket. The important property that the Dirac brackets have is that the transformation calculated in (1.53) remains formally the same. Furthermore the trans-formations generated by second class constraints become merely identities expressing some canonical variables in terms of the others. Thus the second class are in some way less fundamental. They can be removed, while first class constraints generate gauge transformations.

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2 Duality in Electromagnetism

2.1 Maxwell’s equations

The theory of electromagnetism is concerned with the interaction of electric and mag-netic fields. This connection was first postulated by Michael Faraday [12]. Inspired by Faraday, James Maxwell formalised the ideas first presented by Faraday into a set of equations [13]. In an attempt to understand the equations Maxwell wrote down, Oliver Heaviside simplified the equations to the four equations still known today [14].

∇ · B = 0, (2.1a) _{∇ × E = −}∂B

∂t , (2.1b)

∇ · E = 0, (2.1c) ∇ × B = ∂E

∂t. (2.1d)

These are Maxwell’s equations in vacuum, they constitute the equations of motion of the theory of Electromagnetism. Immediately from these equations the theory looks to be symmetric under the duality rotation

E0 B0 =cos ϕ − sin ϕ sin ϕ cos ϕ E B . (2.2)

This duality is what sparked the widespread search for dual structures in other physical theories. However the discussion of duality in EM is not concluded with this finding. For it can be shown that this dual nature is a consequence of a more fundamental duality structure of the action. To investigate this structure a suitable action must be constructed. To that end it is instructive to first cast equations (2.1a), (2.1b), (2.1c) and (2.1d) in their relativistic form.

So it is known that the electromagnetic potentials from the defining equations

E = −∇V −∂A

∂t , B = ∇ × A, (2.3)

form a four-vector

Aµ= (V, A), (2.4)

commonly known as the Maxwell field. With which the antisymmetric field strength tensor Fµν _{can be constructed as}

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and its dual as

?_Fµν ₌ 1

2ε

µναβ_F

αβ. (2.6)

From this definition the relations with the electric and magnetic field are derived as

Ei= F0i, (2.7)

Bi= −

1 2εijkF

jk_. _(2.8)

With these equations it becomes quite clear that the duality transformation (2.2) is equivalent to the rotation

F0µν ?_F0_µν =cos ϕ − sin ϕ sin ϕ cos ϕ Fµν ?_Fµν . (2.9)

Furthermore equations (2.7) and (2.8) reduce the equations of motion, from (2.1a) through (2.1d), too

∂µFµν = 0 (2.10)

∂µ?Fµν = 0. (2.11)

The action that generates these equations of motion turns out to be S[Aµ] =

Z −1

4FµνF

µν _d4_x. _(2.12)

To see that the theory of electromagnetism can be described by an duality symmetric action principle, we turn to the Hamiltonian.

2.2 Duality in the Hamiltonian action

The calculation of the Hamiltonian starts by considering the space and time like com-ponents of the electromagnetic tensor separately in a process called 3 + 1 splitting. So first consider the Lagrangian density

L = −1 4FµνF

µν_. _(2.13)

The terms that remain, after splitting this equation in time- and space-like components and by using the fact that Fµν is antisymmetric in its components, are

L = −1 2F0jF 0j₋1 4FijF ij _{= −}1 2(∂0Aj− ∂jA0)(∂0A j_{− ∂}j_A 0) − 1 4FijF ij _(2.14) = −1 2(∂0Aj∂0A j _{− ∂} 0Aj∂jA0− ∂jA0∂0Aj+ ∂jA0∂jA0) − 1 4FijF ij_. _(2.15)

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The next step is to choose dynamical variables and calculate their conjugate momenta. The most obvious choice for dynamical variables is

Ai, A0. (2.16)

The conjugate momenta are easily calculated. Notice that FijFij contains neither a ∂0Ai

nor a ∂0A0 term, in fact the entire expression contains no ∂0A0 term, making

∂L ∂(∂0A0)

= 0. (2.17)

This actually also the first primary constraint of the system. The other conjugate mo-mentum follows from direct calculation

pj = ∂L ∂(∂0Aj)

= ∂0Aj− ∂jA0 = −Ej (2.18)

The resulting Hamiltonian density H becomes

H = pjA˙j− L (2.19) = pj(pj + ∂jA0) + 1 2F0jF 0j ₊1 4FijF ij _(2.20) = 1 2p j_p j+ pj∂jA0+ 1 4FijF ij_. _(2.21)

When written in terms of the fields it already looks more invariant H = 1

2E

2₊1

2B

2_{− E · ∇V.} _(2.22)

It is easy to see that the first two terms of this equation do not change under transfor-mation (2.2). The last term can be dealt with by considering that equation (2.19) allows the action to be written as

S[Aµ] = Z dx0dx3 pjq˙j− 1 2p j_p j− 1 4FijF ij_{− p}j_∂ jA0, (2.23) = Z dx0dx3 pjq˙j− 1 2p j_p j− 1 4FijF ij_{− A} 0∂jpj. (2.24)

Where the last equality is attained by integration by parts and assuming the dynamical variables vanish at the boundary for each moment in time.

Now A0 can be identified as a Lagrange multiplier, which induces a constraint on the

system in the form of

∂jpj = 0, (2.25)

or in a more familiar form

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This constraint is first class, more importantly we can solve it to uncover a second potential that will end up being dual to the first. The general form of a function that satisfies (2.26) is

E = ∇ × A1, δA1i= ∂iϕ (2.27)

or

Ei= pi = εijk∂jA1k, (2.28)

since the divergence of a curl in R3 is always zero. This looks to have the exact same form as

B = ∇ × A2, δA2i= ∂iξ. (2.29)

As we can see the ambiguity in the definitions of A1 and A2, δA1 and δA2 have exactly

the same form, this motivates the duality rotation A∗ 2 A∗₁ =cos ϕ − sin ϕ sin ϕ cos ϕ A2 A1 . (2.30)

To see whether the action (2.24) is invariant under this duality rotation we first write it in terms of the potentials

S0[Aµ] = Z Ω (εijk∂jA2kA˙1i− 1 2∂jA 2i_∂j_A 2i− 1 2∂jA1i∂ j_A1i_)d4_x. _(2.31)

Note that the last two terms in this equation are manifestly duality invariant under transformation (2.30). What remains is to show that

Z

dx0dx3 εjik∂iA1kA˙2j (2.32)

remains unchanged under transformation (2.2). Considering this Z

dx0d3x εijk∂jA1kA˙2i, (2.33)

transforms into Z

dx0dx3εijk∂j(cos ϕA2k+ sin ϕA1k)(cos ϕ ˙A1i− sin ϕ ˙A2i)

= Z

dx0dx3εijk(cos2ϕ∂jA2kA˙1i− sin2ϕ∂jA1kA˙2i

+ cos ϕ sin ϕ∂jA1kA˙1i− cos ϕ sin ϕ∂jA2kA˙2i). (2.34)

The second term on the second line can be integrated by parts to Z

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Now by swapping indices i and k on the ε factor of the second term and afterwards renaming i back to k and vice versa we arrive at

Z

dx0d3x εijk∂jA1kA˙2i+ b.t. (2.36)

The remaining two terms can be shown to be zero modulo boundary terms. We will focus on one of them and omit the trigonometric factor, the same argument will apply for both. First we split the terms in two halves

Z dx0dx3 1 2εijk∂jA1kA˙1i+ 1 2εijk∂jA1kA˙1i , (2.37)

now we integrate the second term by parts Z dx0dx3 1 2εijk∂jA1kA˙1i+ 1 2εijk∂jA1iA˙1k + b.t = (2.38) Z dx0dx3 1 2εijk(∂jA1kA˙1i+ ∂jA1iA˙1k) + b.t. (2.39)

The part inside the parenthesis is symmetric with respect to i and k but it is multiplied by a totally antisymmetric tensor, thus zero. Of course the boundary term can be dropped by the assumption that the fields fall off rapidly at the boundary. Thus only the term (2.36) survives which is precisly the term before the transformation, making the Hamiltonian action principle duality invariant.

2.3 Source-full duality

The source-free EM theory is duality invariant, what about the source-full theory? We shall limit our discussion to the equations of motion. The Maxwell equation’s for the non-vacuum theory are

∇ · B = 0, (2.40a) ∇ × E = −∂B

∂t , (2.40b)

∇ · E = ρ_e, (2.40c) _{∇ × B = J +}∂E

∂t. (2.40d)

At first glance the answer to the question looks obvious: No, these equations are obvi-ously non invariant under transformation (2.2). This can be attributed to the apparent non existence of a magnetic charge, indeed an added magnetic charge density ρm to

equation (2.40a) and a current density −K to (2.40b) would make the theory invariant under (2.2). Provided the charge and current densities transform under duality in the same way.

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The duality invariance of Maxwell’s equations would then make it somewhat arbitrary to say wheter a particle is electrically charged but not magnetically. The only question that remains is whether all particles have the same ratio of magnetic charge and electric charge, if they do then one can always apply one duality rotation for all particles at the same time, so that the magnetic charge ρm = 0 and density K = 0, returning the

modified Maxwell equations to their regular form. The current experimental results from choosing the electric charge of an electron to be qe= −e and its magnetic charge qm = 0,

are that the proton will have qe= +e ± 10−20and qm = 0 ± 2 × 10−24. Thus the current

scientific conclusion is that all known particles either carry no magnetic charge or the ratio of electric charge to magnetic charge is that same for all of them. [15]

Why then is there still such a big interest in the discovery of magnetic monopoles? Examples being the MoEDAL experiment and an experiment using the ATLAS detector. Most of this interest stems from the seminal works of Dirac on the topic, in his paper he shows that the existence of magnetic monopoles would imply that the electric (and magnetic for that matter) charge is quantized [16]. What causes the electric charge to be quantized in multiples of ₃e is an interesting question. Charge carriers do not have to be quantized, the gravitational charge, or mass, for example does not appear to be quantized. Currently there is no other widely known mechanism for why there exists a fundamental charge quantum, other than the one Dirac proposed.

So let us consider the idea of a magnetic source and what it might imply for the theory of electromagnetism. Say there is a magnetic source with magnetic charge g, it is commonly known that the magnetic flux through the boundary of a certain volume S is given by

Φ = Z

∂S

BdS. (2.41)

This integral has the properties that, it gives the same values for all surfaces surrounding the source and it is independent of time. Implying that it can be interpreted as the magnetic charge inside the surface and this interpreted charge then must conserved. These considerations mirror the ideas that implied (2.40c) but then for magnetic charges, so they imply another Gauss law but then for the magnetic field

∇ · B = ρ_m. (2.42)

Another important property of the electric charges is that they satisfy a certain con-tinuity equation

∇ · J = ∂ρe

∂t . (2.43)

When there is a continuous flowing distribution J of charges. Meaning that any change in charge density over time in a region, is the result of the charges flowing out of that region. Now if there is some magnetic current K and we demand that magnetic charges also satisfy a continuity equation then the only way to modify Maxwell’s equations happens to be

∇ × E = −K − ∂B

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if we do not want the behaviour of electrodynamics to change in the absence of magnetic charges. The last thing to wonder is whether there is a magnetic analogue to the Lorentz force

F = q(E + v × B). (2.45)

A symmetry argument would say that there is and that its form has to be

F = g(B − v × E). (2.46)

A more formal derivation of this law exists but would go beyond the scope of this discussion [17].

Interestingly the energy and momentum densities do not change when allowing mag-netic charges. The fact that this holds for energy is not really surprising. The energy stored in the electric field is derived by considering the amount of energy it takes to assemble an electric charge configuration. The energy density stored in the electric field is then calculated to be

ε = 1 2E

2_. _(2.47)

The magnetic component is classically computed by considering the amount of work required to get currents moving. However the derivation becomes a lot easier when magnetic charges are allowed in the theory, because one can just mirror the process for electric charges. The symmetry between electric and magnetic charges then almost immediately forces the same form of the magnetic energy stored in the fields, making the total energy density

ε = 1

2 E

2_{+ B}2_. _(2.48)

The momentum, on the other hand, is a different story since there is an entirely new force to consider. In general the force per unit volume caused by the electromagnetic fields is

f = ρe(E + v × B) + ρm(B − v × E) (2.49)

= ρeE + J × B + ρmB − K × E. (2.50)

By solving for J , K ρe and ρm with the modified Maxwell’s equations this expression

can be written in terms of the fields alone

f = (∇ · E)E + (∇ · B)B − E × ∇ × E − B × ∇ × B (2.51)

−∂E

∂t × B + ∂B

∂t × E. (2.52)

The the product rule

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and the standard product rule for the time derivative can further rearrange this equation to f = [(∇ · E)E + (E · ∇)E] + [(∇ · B)B + (B · ∇)B] −1 2∇ E 2_{+ B}2₋ ∂ ∂t(E × B). (2.54)

With the introduction of the Maxwell stress tensor Tij := EiEj− 1 2δijE 2_{+ B} iBj− 1 2δijB 2 _(2.55)

and the Poynting vector

S = E × B, (2.56)

the expression finally simplifies to

f = ∇ · ~T − ∂S

∂t. (2.57)

Newton’s second law gives the interpretation of this expression since dp dt = Z f dV = I S T · da − d dt Z V SdV. (2.58)

The second term on the right hand side of (2.58) can now be interpreted as the momen-tum in the fields flowing through a certain volume V . The first term is the momenmomen-tum flowing out of that same volume through the surface of V . The interpretation of what can be thought of as momentum stored in the fields happens to be the same as in the classical case.

These changes to Maxwell’s equations do come with a cost. They make it impossible to write the fields as global potentials since the domain of the potentials change from R3 to R3\ {0}, making Poincare’s lemma non-applicable. Dirac proposed a solution to this problem by introducing an object that is now called a Dirac string. He noticed that the magnetic field corresponding to a pure magnetic monopole

Bmono =

g

4πr3r, (2.59)

can be derived from the potential

A = g 4π    y r(r−z) −_r(r−z)x 0   . (2.60)

The problem with this potential is that it is singular along the entire positive z-axis. This singularity along the z-axis does not appear in the monopole field. Upon closer inspection this vector potential corresponds to the potential of an idealized semi-infinite

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soleniod. With this insight Dirac constructed an action principle that, provided that any charged particle never crosses the string, produced the modified Maxwell’s equations as stated before.

Now lets see how this leads to charge quantization, consider a system with a pure electric charge and a pure magnetic charge at some distance apart. If the particles are placed at rest they stay at rest because of (2.49), there is however a non-zero amount of momentum stored in the fields since E ×B is clearly non-zero. Consequently the angular momentum density ` := r × S is non-zero as well. With the standard expressions for point particle fields the total angular momentum can be calculated

Lx= 0, Ly = 0, Lz= −

gq

4π, (2.61)

here the magnetic charge is placed at the origin and the the positive z-axis points to-ward the electric particle [17]. If we treat this system semi classical then the angular momentum should be quantized as n₂~. Thus

gq 4π =

n

2~ (2.62)

should hold, as a result the magnetic and electric charge must be quantized. Note as well that the total angular momentum is independent of the separation between the particles. A single magnetic monopole somewhere on the other side of the universe would be enough to quantize the electric charge.

The relationship (2.62) also plays a role in quantum field theory. If we define the electrodynamic coupling constants as

α = e

2

4π~, β = g2

4π~, (2.63)

we see using (2.62) that β ∝ α−1. So the fine structure constant would be transformed from 1/137 to 137. The duality thus exchanges the weak and strong coupling regimes.

2.4 Comments on quantum field theory

The example of source-full electrodynamics shows that quantum mechanics can add to the discussion of duality. In fact, duality plays quite an important role in the modern quantum theories. For example it is used in the study of non-perturbative properties of certain supersymmetric Yang-Mills theories [18, 19]. To illustrate this, we will give a small overview of some interesting findings of a quantum treatment of duality. These findings are by no means elementary so this will not be a detailed discussion on the topic and further reading is supplemented when needed.

In the language of differential geometry, electrodynamics is concerned with an 1-form, or alternating 1-tensor field A over a four-manifold X. A defines a two-form F = dA, where d is the exterior derivative operator. The equations of motion that A is supposed to obey are

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where ? denotes the Hodge star operator. F is known as a exact form, because of its definition as the exterior derivative. A property of d is that d2 = 0 as such for any exact form α, dα = 0 will hold. Thus F will also satisfy the identity dF = 0, called the Bianchi identity. The duality is defined as the exchange of the equations of motion and the Bianchi identity. Hodge theory is largely concerned with the simultaneous solutions of these two equations [1].

It turns out that quantum mechanically this duality can also be formulated on a the level of the action. In quantum field theory the fundamental quantity defined by the action is the partition function

Z = Z

DAeiS[A]_. _(2.65)

Where S is the classical action of a theory. In the case of electrodynamics, the action S is the standard Maxwell action with an added term that does not effect the classical equations of motion, given by

S[A] = Z X 1 2e2F ∧ ?_{F −} iθ 8π2F ∧ F . (2.66)

Quantum mechanically the so called θ-term does have an effect. If X = R4 there is a way, described in [20], to write (2.65) as

Z = Z X DF DBexp −S[A] + i Z X B ∧ dF . (2.67)

The B that is introduced is another 1-form. When the integration over F is carried out the partition function will take the form

Z = Z

DBeiS[B]. (2.68)

With the new 1-form B, a new 2-form G = dB and new parameters ˆθ, ˆe, then the action will have the form

S[B] = Z X 1 2ˆe2G ∧ ?_{G −} iˆθ 8π2G ∧ G ! . (2.69)

The new parameters are defined by the equation ˆ θ 2π + 2πi ˆ e2 = − θ 2π+ 2πi e2 −1 . (2.70)

These parameters form the coupling constants in quantum field theory. If we name the left hand side of equation (2.70) τ−1, then we can see that this change of variables exchange τ → −τ−1. So just like in the previous section we see that duality acts as a strong/weak exchange in coupling constants.

The review here was done on the space R4 and it turns out that classically this type of duality does not appear on manifolds topologically more complex than R4 [20]. This is what makes quantum field theory interesting, in quantum field theory duality is not restricted to topologically trivial manifolds [21].

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3 Linearized Gravity

3.1 Introduction

The next example that is interesting to look at is that of linearized gravity. It offers the first introduction with a theory of gravity. That being said, it is not the main focus of this thesis and will serve more as a way to introduce the concepts of a theory of gravity in a way that remains in theme with the rest of this thesis. To that end, major calculations will be omitted. The reader is pointed to the work of Henneaux and Teitelboim to see the full justification of the expressions in this chapter [3].

A linear theory of gravity is concerned with a spacetime metric that is only a small symmetric pertubation away from the flat Minkowski metric

gµν = ηµν+ hµν, (3.1)

the metric signature is taken to be (−1, 1, 1, 1). The result of hµν being small is that any terms in the equations of motion that are higher than first order in hµν can be omitted, but quadratic terms are kept in the action. Furthermore raising and lowering of indices can be done with the Minkowski metric, this has the effect that raising or lowering spatial indices can be done for free but temporal indices cost a minus sign. The last important consequence of the linear metric is that covariant derivatives are reduced to partial derivatives.

An important quantity constructed from the metric is the Riemann tensor Rµνρσ, it indicates the curvature of the space-time under consideration. The formal definition of the Riemann tensor is

Rρσµν = ∂µΓρνσ− ∂νΓρµσ+ ΓρµλΓλνσ− ΓρνλΓλµσ, (3.2)

where the Γ’s are the Christoffel symbols. However, the formal definition is less impor-tant for our purposes than the properties it obeys, the algebraic ones are:

Rµνρσ = −Rνµρσ = −Rµνσρ, (3.3)

and

R_[µνρ]σ = 0, (3.4)

where the brackets mean the complete antisymmetrization of the indices within the brackets. It also obeys one differential identity, called the Bianchi identity

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Where the comma stands for the index of a partial derivative. These properties turn out to have the effect that there is a quite simple functional form of the linearized Riemann tensor in terms of the metric pertubation

Rµνρσ = −

1

2[∂µ∂ρhνσ+ ∂ν∂σhµρ− ∂µ∂σhνρ− ∂ν∂ρhµσ]. (3.6)

3.2 Duality in linearized gravity

The action principle that is commonly used in general relativity is the Einstein-Hilbert action S[gµν] = 1 2 Z Ω d4xR√−g, (3.7)

here R = Rµνµν. The equations of motion calculated from this action are

R_µν := R_µρνρ= 0. (3.8)

Which are the vacuum version of the equations of motion for general relativity, known as the Einstein equations.

In its linear form the action (3.7) reduces to S[hµν] = − 1 4 Z Ω d4x(∂ρhµν∂ρhµν− 2∂µhµν∂ρhρν+ 2∂µhρρ∂νhµν − ∂µhρρ∂µhνν), (3.9)

known as the Pauli-Fierz action. A dual structure can be made by taking what is known as the Hodge dual of the Riemann tensor

Sµνρσ =

1

2εµναβR

αβ

ρσ. (3.10)

One can show that S obeys the same algebraic and differential identities as R, implying that S has the same functional form as R in terms of some different symmetric metric pertubation fµν. Moreover, S even has a similar equation of motion associated with

it as a result of the Bianchi identity (3.5), Sµν = 0. Thus there are two equivalent

ways of writing the theory one defined by hµν, an other one defined by fµν and duality

transformation ?R µνρσ_[h] Sµνρσ_{[f ]} = 0 1 −1 0 Rµνρσ_[h] Sµνρσ_{[f ]} , (3.11)

of R and S relating the two.

So the first hints of a dual structure are there on the level of the equations of motion. We must know ask ourselves whether there is a formulation of the action principle equivalent to (3.9) in which this duality emerges as a symmetry. With the first chapter in mind, the natural next step is to go over to the Hamiltonian action by taking hij

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as canonical co¨ordinates and calculating πij := ∂L.∂ ˙hij. In doing so one will stumble

upon the two first class constraints

ϕ := ∂i∂jhij − ∆h = 0 (3.12)

and

ϕi:= ∂jπij = 0. (3.13)

The Hamiltonian action now takes the form S[hij, πij] = Z Ω d4x(πij˙hij− H − h00ϕ − h0iϕi), (3.14) with H = πijπij− 1 2π 2₊1 4∂ k_hij_∂ khij− 1 2∂ih ij_∂ khkj+ 1 2∂ i_h∂j_h ij− 1 4∂ i_h∂ ih. (3.15)

The constraints are solved to establish the duality invariance of this action. First (3.12), the general solution of a constraint of this type takes the form

hij = εikl∂kΦlj+ εjkl∂kΦli, (3.16)

up to the transformation

δΦij = ∂iξj+ ∂jξi+ δijξ. (3.17)

The second constraint has a general solution of the form

πij = εiklεjmn∂k∂mPln, (3.18)

also defined up to the transformation

δPij = ∂iζj+ ∂jζi+ δijζ. (3.19)

When entered into the Hamiltonian we get πijπij − 1 2π 2_{= ∆P} ij∆Pij + 1 2(∂ i_∂ jPij)2+ ∂i∂jPij∆P − 2∂i∂jPjk∂i∂lPlk− 1 2(∆P ) 2_, (3.20) up to some total derivative terms. Interestingly, the other part of the Hamiltonian

1 4∂ k_hij_∂ khij− 1 2∂ih ij_∂ khkj+ 1 2∂ i_h∂j_h ij− 1 4∂ i_h∂ ih = ∆Φij∆Φij + 1 2(∂ i_∂ jΦij)2+ ∂i∂jΦij∆Φ − 2∂i∂jΦjk∂i∂lΦlk− 1 2(∆Φ) 2_, _(3.21)

takes on exactly the same form, again up to total derivatives, making the Hamiltonian invariant under the rotation of the two potentials. The kinetic term

Z

dtd3x(πij˙hij) = 2

Z

dtd3εjkl(∂m∂i∂kPml− ∆∂kPil) ˙Φij, (3.22)

is also invariant under the duality transformation after repeated integration by parts. With this last term the dual symmetry of the action (3.14) is established.

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4 Linearized Conformal Gravity

4.1 Introduction

Conformal gravity is a theory of gravity in which the fundamental object is the Weyl tensor Wµνρσ. It enjoys the properties of being invariant under conformal rescaling and diffeomorphisms and contains the vacuum solutions of general relativity [22]. The down-side of conformal gravity is that it is a fourth-order theory, meaning that the equations of motion contain four time derivatives of the metric. Fourth order theories generally have some problems with the Hamiltonian being unbounded and at the quantum level, being renormalizable but containing unphysical ghosts [23].

For the purpose of duality, the motivation to investigate this theory is that the action principle that governs conformal gravity has a very similar structure to electrodynamics. The usually used action principle for conformal gravity

S = −1 4

Z

Ω

d4x√−gW_µνρσWµνρσ_, _(4.1)

is quadratic in the the Weyl tensor, just like electrodynamics is quadratic in the field strength tensor.

The Weyl tensor is defined as the trace free part of the Riemann tensor Rµνρσ,

Wµνρσ := Rµνρσ− 2(gµ[ρSσ]ν− gν[ρSσ]ν), (4.2)

with Sµν being the Schouten tensor

Sµν =

1

2(Rµν− 1

6gµνR). (4.3)

By definition the Weyl tensor is traceless, meaning

Wµν ρ_µ = 0, (4.4)

furthermore the Weyl tensor is, as mentioned, invariant under conformal rescaling of the metric

gµν → gµν0 = Ω2gµν =⇒ W

0_µνρσ

= Wµνρσ. (4.5)

As well as under diffeomorphisms

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These are the symmetries of the action (4.1). It also obeys the symmetry properties W_µνρσ = W_[µν][ρσ]= Wρσµν, (4.7)

W_[µνρ]σ = 0, (4.8)

∂µWµν[ρσ,α]= 0, (4.9)

because of the way it is constructed in (4.2) through the Riemann tensor and its con-tractions.

The equations of motion of conformal gravity derived via the variational method applied to (4.1) are

2∇ν∇σWµνρσ− WµνρσRρσ= 0, (4.10)

known as the Bach equations. The theory we are interested in is the linearized version of conformal gravity which, just like in regular gravity, is concerned with linear pertubations of the Minkowsky metric gµν = ηµν+ hµν. The equations of motion reduce to

∂µ∂ρWµνρσ = 0, (4.11)

in linearized conformal gravity. Furthermore as a consequence of (4.9), W also obeys

∂µ∂ρ?Wµνρσ = 0. (4.12)

It is in these equations that we find our first clue that this theory might contain some dual structure.

4.2 Duality of the equations of motion

The equations of motion can be cast in a twisted self dual form, just like in linearized gravity [3]. The Hodge dual of the Weyl tensor

H_µνρσ =?W_µνρσ = 1 2ε

αβ

µν Wαβρσ, (4.13)

satisfies the same symmetry properties as the Weyl tensor. First

H_µνρσ = −Hνµρσ (4.14) because Hµνρσ = 1 2ε αβ µν Wαβρσ = − 1 2ε αβ νµ Wαβρσ = −Hνµρσ, (4.15) and Hµνρσ = −Hµνσρ (4.16)

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because Hµνρσ = 1 2ε αβ µν Wαβρσ = − 1 2ε αβ νµ Wαβσρ = −Hµνσρ. (4.17)

Secondly, the antisymmetrization of the first three indices vanishes

H_[µνρ]σ = εµνργH_µνρσ = 0 (4.18)

with the definition of the Hodge dual (4.13) this becomes 1 2ε µνργ_ε αβ µν Wαβρσ= (δραδγβ − δρβδαγ)Wαβρσ (4.19) = δραδγβW_αβρσ− δρβδαγW_αβρσ (4.20) = δραW_{α ρσ}γ − δρβWγ_βρσ = Wργ_ρσ − Wγρ_ρσ = 0. (4.21)

Where the last equality holds because the Weyl tensor is tracefree.

The Hodge dual also obeys the same differential equations since (4.9) holds. Thus H has the same functional form in terms of some different metric fµν. So there are two

constructions of W and H, again one in terms of hµν and the other in terms of fµν. The

transformation that relates the two is ?W µνρσ_[h] Hµνρσ_{[f ]} = 0 1 −1 0 Wµνρσ_[h] Hµνρσ_{[f ]} . (4.22)

We see now that taking the divergence twice on both sides we immediatly get the equa-tions of motion for W[h] and H[f ] because of the Bianchi identity (4.12) associated with both of them.

However, the addition of the Hodge dual has caused the equations (4.22) to be overde-termined. Some of the equations can be constructed from the others, specifically the first line is simply the Hodge dual of the second line. Thus the second line can be discarded keeping only

?_W0i0j _{= H}0i0j_, _(4.23)

?_W0ijk _{= H}0ijk_, _(4.24)

and

?_Wijkl _{= H}ijkl_. _(4.25)

Moreover, the last equation can be recovered from the second one:

?_Wijkl_{= H}ijkl_, _(4.26) εij0mW_0mkl= Hijkl ⇐⇒ W0mkl = − 1 2ε 0m ij Hijkl= −?H0mkl. (4.27)

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If we define the electric and magnetic component for the Weyl tensor as E_mn[h] = W0m0n, (4.28) Bmn[h] = − 1 2εnpqW pq 0m . (4.29)

From their definition we can see that they are symmetric and traceless. Similarly we define E and B for H with respect to the metric fµν. Then the equations (4.22) can be

reduced to ?E ij_[h] Eij_{[f ]} = 0 1 −1 0 Bij_[h] Bij_{[f ]} . (4.30)

This shows that the equations (4.30) generate the entire set of equations (4.22). Thus linear conformal gravity displays a dual structure on the level of the equations of motion. the question still stands whether this is a result of a deeper duality on the level of an action principle. To answer that question we turn to the Hamiltonian formalism.

4.3 Hamiltonian formulation

The linearized action takes the form S[hµν] = − 1 4 Z Ω d4xWµνρσWµνρσ. (4.31)

The Lagrangian density subject to 3 + 1 splitting takes the form

WµνρσWµνρσ = 4W0i0jW0i0j+ 4W0ijkW0ijk+ WijklWijkl. (4.32)

This is due to the symmetry properties discussed in the previous section. There is one more thing that can be simplified since

W_ijkl= ε0mijε0nklW0m0n, (4.33)

thus (4.32) can be reduced to

W_µνρσWµνρσ _{= 8W}

0i0jW0i0j + 4W0ijkW0ijk. (4.34)

Now there is a problem with this theory and that is, that it is a fourth order theory meaning that the equations of motion can contain up to four time derivatives of the metric. These types of theories don’t in general lend themselves well to be written in the Hamiltonian form. The regular choice of canonical variables hij will not suffice. The

discussion in chapter one even assumed that the Lagrangian depended only on first order time derivatives of the canonical variables. The workaround, known as the Ostrogradsky method, is to pick a set of variables that depend on the time derivatives of the metric, so that the final equations contain only second order time derivatives of the canonical

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variables. A natural object that satisfies these conditions is the extrinsic curvature Kij

which takes the form

Kij =

1

2( ˙hij− ∂ih0j− ∂jh0i), (4.35) in the linear case. Thus we choose hij and Kij as canonical variables. We will treat

them as independent and later add a constraint to compensate for the fact that they are not.

The next step is to determine the conjugate momenta, so (4.32) would need to be written in terms of the extrinsic curvature. First the linear Riemann tensor in terms of the metric is

Rµνρσ = −

1

2[∂µ∂ρhνσ+ ∂ν∂σhνρ− ∂µ∂σhνρ− ∂ν∂ρhµσ], (4.36) the components of interest Rijkl, R0i0j and R0ijk are

Rijkl= − 1 2[∂i∂khjl+ ∂j∂lhik− ∂i∂lhjk− ∂j∂khil] (4.37) R_0i0j = − ˙Kij − 1 2∂i∂jh00 (4.38) and R_0ijk = ∂kKij− ∂jKik (4.39)

in terms of the extrinsic curvature and metric. The relevant components of the Ricci tensor are R00= R0i0i= − ˙K − 1 2∆h00 (4.40) R0i= ∂kKik− ∂iK (4.41) Rij = − ˙Kij − 1 2∂i∂jh00+ R k ikj . (4.42)

Lastly, the Ricci scalar

R = 2 ˙K + ∆h00+ Rijij. (4.43)

By combining equations (4.2) and (4.3), the definition of the linear Weyl tensor reads W_µνρσ = Rµνρσ− (ηµ[ρRσ]ν − ην[ρRσ]µ) +

1

3ηµ[ρησ]νR. (4.44) From this definition we can start by filling in the terms in (4.32), so first

W_0i0j = 1 2(R0i0j+ R m imj ) − 1 6δij(R mn mn + R0m0m) (4.45) = 1 2(− ˙Kij− 1 2∂i∂jh00+ R m imj ) − 1 6δij(− ˙K − 1 2∆h00+ R mn mn ) (4.46)

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giving

W_0i0jW0i0j = 1 4[ ˙KijK˙

ij _{+ R} m

imj Rinjn+ ˙Kij∂i∂jh00− 2 ˙KijRimjm− ∂i∂jh00Rimjm]

− 1 12[ ˙K 2_{+ R} ij ij R mn mn + ˙K∆h00− 2 ˙KR_ijij − ∆h00Rmnmn] + 1 24∆h00∆h00 (4.47) and secondly W_0ijk= ∂kKij− ∂jKik+ 1 2(δij(∂ l_K lk− ∂kK) − δik(∂lKjl− ∂jK)) (4.48) giving W_0ijkW₀ijk= 2∂jKik∂jKik+ 2∂kK∂lKkl− ∂lK∂lK − 3∂lKkl∂mKkm. (4.49)

The Lagrangian density now becomes L = −α[1

2( ˙Kij ˙

Kij + R_imjmRinj_n+ ˙Kij∂i∂jh00− 2 ˙KijRimjm− ∂i∂jh00Rimjm)

−1 6( ˙K 2_{+ R} ij ij Rmnmn+ ˙K∆h00− 2 ˙KRijij − ∆h00Rmnmn) + 1 12∆h00∆h00 − W0ijkW0ijk]. (4.50)

Noting that (4.49) contains no terms explicit in ˙Kij, we calculate the conjugate

momen-tum Πij = ∂L ∂ ˙Kij = −[ ˙Kij +1 2∂ i_∂j_h 00− Rimjm− 1 6∆h00δ ij ₊1 3R mn mn δij − 1 3 ˙ Kδij]. (4.51)

Immediatly from this definition it becomes clear that the momentum is tracefree

Π := Π_ii = 0. (4.52)

This actually serves as a primary constraint. We are now in a place to introduce the Hamiltonian

H = ΠijK˙ij− L. (4.53)

Of course the Hamiltonian should be a function of Πij and Kij, so we must eliminate all

the terms dependent on ˙Kij using (4.51):

ΠijK˙ij = Πij[−Πij − 1 2∂i∂jh 00_{+ R} m imj + 1 3δijK +˙ 1 6δij∆h00− 1 3δijR mn mm ] = −ΠijΠij− 1 2Π ij_∂ i∂jh00+ ΠijRimjm (4.54)

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1 2K˙ijK˙ ij ₌ 1 2[ΠijΠ ij _{+ R} m imj Rinjn+ 1 3K˙ 2₋1 3(R mn mn )2+ Πij∂i∂jh00− 2ΠijRimjm − ∂i∂jh00Rimjm+ 1 6(∆h00) 2₊1 3R mn mn ∆h00] (4.55) 1 2 ˙ Kij∂i∂jh00= 1 2[−Π ij_∂ i∂jh00− 1 3(∆h00) 2_{+ R} m imj ∂i∂jh00+ 1 3 ˙ K∆h00− 1 3R mn mn ∆h00] (4.56) − ˙KijRimjm= ΠijRimjm+ 1 2∂i∂jh00R imj m−(Rimjm)2− 1 3 ˙ KR_mnmn−1 6∆h00R mn mn + 1 3(R mn mn)2. (4.57) The Hamiltonian reduces to

H = −1 2ΠijΠ

ij₋ 1

2∂i∂jΠ

ij_h

00+ RimjmΠij − W0ijkW0ijk. (4.58)

Up untill this point we have treated Kij and hij as independent variables, but they are

clearly not due to (4.35). To account for this we must introduce the additional constraint ϕ := ˙hij− ∂ih0j− ∂jh0i− 2Kij = 0. (4.59)

Summing all this up we can finally write down the Hamiltonian action principle

S[hij, pij, Kij, Πij] = Z d4x[ΠijK˙ij+ 1 2ΠijΠ ij₊1 2Π ij_∂ i∂jh00− ΠijRimjm + 2∂jKik∂jKik+ 2∂kK∂lKkl− ∂lK∂lK − 3∂lKkl∂mKkm + λij( ˙hij − ∂jh0i− ∂ih0j− 2Kij)]. (4.60)

First off, the Lagrange multiplier λij can now be associated with the conjugate mo-mentum of hij by λij = πij. Subsequently the terms h0j and h0i can be interpreted

as Lagrange multipliers, after integration by parts. They in turn enforce the constraint ∂jπij = 0.

Secondly, the term 1₂Πij∂i∂jh00 can also be written as a Lagrange multiplier after

integration by parts. Imposing the constraint

∂i∂jΠij = 0. (4.61)

And lastly, we can now find an additional constraint stemming from the consistency condition applied to (4.52),

{Π, H} + {Π, ϕ} = {Π, 2πij_K

ij} = 0, (4.62)

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4.4 Duality

So with everything laid bare, we can now start to work towards showing that the ac-tion (4.60) contains a duality symmetry. The general soluac-tion of the constraint (4.61), provided that Π = 0, is

Πij = εimn∂mψnj+ εjmn∂mψni, (4.63)

for some symmetric potential ψij. The variation

δψij = ∂i∂jξ + δijθ, (4.64)

turns out to have the same form as δKij which suggest that these two constructions can

be treated on equal footing. We shall rename ψij to ˜Kij to keep this in mind.

The next step is to substitute this into action (4.60). We shall do so term by term starting with

1 2Π

ij_Π

ij = 2∂jK˜ik∂jK˜ik+ 2∂kK∂˜ lK˜kl− ∂lK∂˜ lK − 3∂˜ lK˜kl∂mK˜km. (4.65)

This term is exactly dual to (4.49), establishing even further the idea that K and ˜K should be rotated into each other. The second term to check is the kinetic term

ΠijK˙ij = εimn∂mK˜njK˙ij + εjmn∂mK˜niK˙ij

= 2εimn∂mK˜njK˙ij (4.66)

which gets transformed to

−2εimn∂mKnjK˙˜ij. (4.67)

This term can be calculated to be equal to (4.66), modulo some boundary terms, in much the same way as the kinetic term in (2.32).

This is where our luck runs dry, the terms −PijR m

imj and −2pijKij are not manifestly

invariant under this transformation. There is still an unused piece of information: The constraint ∂jπij and the fact that πij is symmetric. If we solve for this we get

πij = εimn∂mωnj+ εjmn∂mωni. (4.68)

From the fact that πij is symmetric and the constraint on πij, we get that ∂iπij = 0.

This implies

∂iεjmn∂mωni = 0. (4.69)

Here we can find the solution, a specific choice of ω satisfying (4.69) will complete the duality. That choice is Rij[f ] = Rimjm[f ] of a second dual metric fµν. From the form of

(4.37) we then see that

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So condition (4.69) is satisfied, thus a correct choice for πij is πij = −1 2[εimn∂ m_Rnj_{[f ] + ε} jmn∂mRni[f ]], (4.71) this transforms −2πijKij to −2πijKij = 2εimn∂mRnj[f ]Kij, (4.72)

which can be written as

2εimn∂mKijRnj[f ] (4.73)

by using integration by parts and renaming indices, again modulo boundary terms. So we now see that

−PijR_imjm[h] = −2εimn∂mKR˜ ij[h], (4.74)

gets rotated into (4.73), provided

hij → fij

fij → −hij. (4.75)

This does come with a new problem, the term πij˙hij = − ˙hijεimn∂mRnj[f ], that was

prieviously unaffected, gets transformed into − ˙fijεimn∂mRnj[h]. Which can be shown

to be invariant after repeated integrations by parts, − ˙fijεimn∂mRnj[h] = −

1

2f˙ijεimn∂m(∂l∂lhnj− ∂j∂khkn). (4.76) If we apply integration by parts first with respect to time on ˙f we get

−1

2fijεimn∂m(∂l∂l˙hnj− ∂j∂k˙hkn). (4.77) Now we it three times with respect to l twice and then once with respect to m for the first term to get

−1

2εimn∂m∂l∂lfij˙hnj+ 1

2εimnfij∂m∂j∂k˙hkn. (4.78) For the second term we do it with respect to m, j and k then we arrive at

−1 2εimn∂m∂l∂lfij˙hnj+ 1 2εimn∂m∂j∂kfij˙hkn 1 2εimn˙hij∂m(∂l∂lfnj− ∂j∂kfkn) = − ˙hijεimn∂mRnj[f ], (4.79) the starting point modulo boundary terms. This completes the duality invariance.

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So we introduced a second dual symmetric metric fµν, this allows us to interpret ˜K

as the extrinsic curvature of this second metric by introducing the constraint term qij( ˙fij− ∂if0j− ∂jf0i− 2 ˜Kij), (4.80)

to the action. Now qij is the conjugate momentum associated with fij, and it is subject

to the same constraints as πij. If we now repeat the process of solving these constraints for qij and pick as ωij = Rij[h], then the action can be written in its final duality symmetric form S[hij, fij, Kij, ˜Kij, ] = Z d4x[2εimn∂mK˜jnK˙ij + 4 ˙hijεimn∂mRnj[f ] − H] (4.81) where − H = 2∂_jKik∂jKik+ 2∂kK∂lKkl− ∂lK∂lK − 3∂lKkl∂mKkm + 2∂jK˜ik∂jK˜ik+ 2∂kK∂˜ lK˜kl− ∂lK∂˜ lK − 3∂˜ lK˜kl∂mK˜km − 2ε_imnK˜njRij[h] + 2εimnKnjRij[f ]. (4.82)

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5 Conclusion

In this thesis we have added linearized conformal gravity to the list of theories that can be described by a manifest duality symmetric action principle. The price to pay for manifest duality symmetry seems to be manifest space-time covariance. This is because the way that we introduce the duality via the Hamiltonian formulation, duality symmetry being a hidden symmetry in the Lagrangian action principle. We solve the constraints of the system to introduce potentials, under the particular choice (4.71) of the form of these potentials, they dually rotate into each other. It would be interesting to see whether this is possible without the assumption. Another interesting topic would be to see whether it is possible to retain covariance along side manifest dual symmetry. Especially in the light of research done on the theory of electromagnetism, where the duality symmetric action was put in an covariant form [24] at the expense of introducing auxiliary fields in a non polynomial way. All the theories looked at here are linear theories, this might suggest that duality only happens on the linearized level. However certain non linear theories exist that exhibit dual structures, Born-Infeld theory is one for example [25]. If there exists non linear version of conformal gravity that is duality symmetric remains to be seen however.

In any case the duality structure that conformal gravity turns out to obey is an interesting one. It appears, as seen, in multiple field theories but its origin and physical consequences remain unknown. Even in the case of electrodynamics, where the theory can be extended in a way consistent with duality by introducing magnetic monopoles. In this case, although one can derive as a physical consequence the quantization of electric charge, the analysis is based on the introduction of a yet undiscovered object. There is one interesting development in the field that tries to tackle the question of the physical implications of duality. A recent paper showed that duality invariance implied Poincare invariance [26], suggesting somehow that duality symmetry is a more fundamental structure than space-time covariance. With the popularity of theories in which space-time is more of an emergent quantity, it would be interesting to see wheter these ideas can be applied.

This all kind of begs the question: Why does duality appear in so many theories? It seems that we are even further away from answering this question than the question on the physical consequences that duality is supposed to have. For even Feynmann once said that ’why’ questions are amongst the hardest ones to answer in physics. Since they probe not the physical quantities in the universe but the structure of the universe itself [27]. So currently we are still only able to philosophize about the reason for the appearance of duality.

Electromagnetic Duality in Linearized Conformal Gravity