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Helicity and duality symmetry in light matter interactions: Theory and applications
Ivan Fernandez i Corbaton
Thesis accepted by Macquarie University
for the degree of
Doctor of Philosophy
Department of Physics and Astronomy
Except where acknowledged in the customary manner, the material presented in this thesis is original, to the best of my knowledge, and has not been submitted in whole or part for a degree in any university.
Ivan Fernandez i Corbaton
Acknowledgments
First and foremost I want to thank my wife Magda for agreeing to embark in an adventure that was bound to change both our lives. It indeed has. Her support, in many different ways, has been crucial.
I also want to thank my mother, Neus, for her amazing encouragement and support during all the years that I have been studying, working, and again studying in different countries; some of them quite far away from our home town of Almacelles (Catalonia). My brother Darius, and my extended family have also always been supportive of my choices. I particularly want to mention Julia, my goddaughter, to whom I owe a few steak dinners. A debt which I fully intend to settle.
I have the fortune to have a few lifelong friends. I would trust and help them with anything. Every time that I see them I can feel my bond with them growing stronger, overcoming the effects of large spatiotemporal distances.
I have made new friends in Macquarie University, to whom I wish the best for the future. I will make special mention of Xavier ZambranaPuyalto, Nora Tischler and Mauro Cirio. I always enjoy talking with them about physics, conversations that have no doubt contributed to my thesis. I also very much enjoy our conversations about (so many!) other matters.
I also would like to thank my friend and long time colleague Srikant Jayaraman. I learned a lot from Srikant during the years that we worked together in Qualcomm. Also, Srikant was the one that put the idea of a PhD in my head.
Last, but not least by any measure, I thank my advisor A. Prof. Gabriel MolinaTerriza. His guidance, insights, flexibility and wide field of view have made this thesis possible. I want to particularly thank him for taking a risk with me. He took an electrical engineer as his student and approved of him venturing in uncharted territory quite early on. Judging from my short experience, Gabriel seems to me the prototype of an all around physicist, blurring the divide between experimentalists and theorists. I have greatly benefited from his world class knowledge of both disciplines.
Agraïments
Primer de tot, vull agrairli a la meva dona Magda que accedís a començar una aventura que havia de canviar les nostres vides. Com efectivament ha passat. El seu suport, de molts tipus, ha estat crucial.
També vull donarli les gràcies a la meva mare Neus pel seu suport i encoratjament durant tots els anys en que he estat estudiant, treballant i estudiant altre cop a diferents països; alguns d’ells molt lluny del nostre poble d’Almacelles. El meu germà Dario i la resta de la meva família també han estat sempre comprensius envers les meves anades i vingudes. Vull mencionar particularment la meva fillola Júlia, a qui dec alguns sopars. Un deute el qual tinc la intenció de saldar.
Tinc la sort de tenir alguns amics de per vida, als quals confiaria qualsevol cosa. Cada cop que els veig sento que la meva connexió amb ells es fa més forta, superant els efectes deguts a grans distàncies espaitemporals.
He fet alguns amics nous a Macquarie University, als quals desitjo el millor per al futur. Mencionaré en particular en Xavier ZambranaPuyalto, la Nora Tischler i en Mauro Cirio. Sempre disfruto parlant amb ells de física, unes conversacions que han contribuït a la meva tesi. També disfruto de les conversacions amb ells sobre (tants!) altres temes.
També vull donarli les gràcies al meu amic, i company de feina durant molt de temps, Srikant Jayaraman. Vaig apendre molt d’en Srikant durant els anys en que vàrem treballar junts a Qualcomm. A més a més, ell fou el que em va ficar al cap la idea de fer un doctorat.
Per últim, però no pas pel que fa a la importància, vull donar les gràcies al meu director de tesi, el professor associat Gabriel MolinaTerriza. Les seves indicacions, visió i flexibilitat han fet possible aquesta tesi. Volia agraïrli especialment haver pres riscos amb mi. Va acceptar com a estudiant un enginyer en telecomunicacions i va deixar que s’endinsés en territori desconegut molt ràpidament. Desde la meva curta experiència en física, en Gabriel em sembla el prototip del físic complet, que esborrona la línia que divideix els experimentalistes dels teòrics. M’he beneficiat enormement del seu coneixement, de nivell mudial, de les dues disciplines.
Publication list
References
 [FCTMT11] Ivan FernandezCorbaton, Nora Tischler, and Gabriel MolinaTerriza. Scattering in multilayered structures: Diffraction from a nanohole. Phys. Rev. A, 84:053821, Nov 2011.
 [FCZPMT12] Ivan FernandezCorbaton, Xavier ZambranaPuyalto, and Gabriel MolinaTerriza. Helicity and angular momentum: A symmetrybased framework for the study of lightmatter interactions. Phys. Rev. A, 86(4):042103, October 2012.
 [FCZPT13] Ivan FernandezCorbaton, Xavier ZambranaPuyalto, Nora Tischler, Xavier Vidal, Mathieu L. Juan, and Gabriel MolinaTerriza. Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwellâs equations. Phys. Rev. Lett., 111(6):060401, August 2013.
 [FCVTMT13] Ivan FernandezCorbaton, Xavier Vidal, Nora Tischler, and Gabriel MolinaTerriza. Necessary symmetry conditions for the rotation of light. J. Chem. Phys., 138(21):214311–214311–7, June 2013.
 [ZPFCJ13] X. ZambranaPuyalto, I. FernandezCorbaton, M. L. Juan, X. Vidal, and G. MolinaTerriza. Duality symmetry and kerker conditions. Opt. Lett., 38(11):1857–1859, June 2013.
 [FCMT13] Ivan FernandezCorbaton and Gabriel MolinaTerriza. Role of duality symmetry in transformation optics. Phys. Rev. B, 88(8):085111, August 2013.
 [FC13] Ivan FernandezCorbaton. Forward and backward helicity scattering coefficients for systems with discrete rotational symmetry. Optics Express, 21(24):29885–29893, December 2013.
 [FCMT14] Ivan FernandezCorbaton and Gabriel MolinaTerriza. Introduction to helicity and electromagnetic duality transformations in optics. Book Chapter, Handbook of Photonics, Wiley, Accepted for publication, 2014.
 [FCZPMT13] Ivan FernandezCorbaton, Xavier ZambranaPuyalto, and Gabriel MolinaTerriza. On the transformations generated by the electromagnetic spin and orbital angular momentum operators. Under review, arXiv eprint 1308.1729, August 2013.
 [TFCZP14] Nora Tischler, Ivan FernandezCorbaton, Xavier ZambranaPuyalto, Alexander Minovich, Xavier Vidal, Mathieu L. Juan, and Gabriel MolinaTerriza. Experimental control of optical helicity in nanophotonics. Light Sci Appl, 3(6):e183, June 2014.
 [TJFC13] Nora Tischler, Mathieu L. Juan, Ivan FernandezCorbaton, Xavier ZambranaPuyalto, Xavier Vidal, and Gabriel MolinaTerriza. Topologically robust optical position sensing. Under review, 2013.
Abstract
The understanding of the interaction between electromagnetic radiation and matter has played a crucial role in our technological development. Solar cells, the internet, cell phones, GPS and Xrays are examples of it. In all likelihood this role will continue as we strive to build better solar cells, millimeter sized laboratories and more sensitive medical imaging systems, among other things. Many of these new applications are stretching the capabilities of the tools that we use for studying and engineering the interaction of electromagnetic radiation and matter. This is particularly true at the meso, nano and microscales.
My thesis is an attempt to build a new tool for studying, understanding and engineering the interaction of electromagnetic radiation with material systems. The strategy that I have followed is to approach interaction problems from the point of view of symmetries and conservation laws. The main novelty is the systematic use of the electromagnetic duality symmetry and its conserved quantity, the electromagnetic helicity. Their use allows to treat the electromagnetic polarization degrees of freedom in a straightforward way and makes the framework useful in practice.
Since the tool is based on symmetries, the results obtained with it are very general. In particular, they are often independent of the electromagnetic size of the scatterers. On the other hand, they are often mostly qualitative. When additional quantitative results are required, more work needs to be done after the symmetry analysis. Nevertheless, one then faces the task armed with a fundamental understanding of the problem.
In my thesis, I first develop the theoretical basis and tools for the use of helicity and duality in the study, understanding and engineering of interactions between electromagnetic radiation and material systems. Then, within the general framework of symmetries and conservation laws, I apply the theoretical results to several different problems: Optical activity, zero backscattering, metamaterials for transformation optics and nanophotonics phenomena involving the electromagnetic angular momentum. I will show that the tool provides new insights and design guidelines in all these cases.
Preface
I have tried to write my thesis so that it may be useful for as many people as possible. In doing so, I have included material that will appear unnecessary to readers that are familiar with the study of symmetry transformations and Hilbert spaces. Such material is contained mostly in the background chapter (Chap. A) and the first section of the theory chapter (Sec. B.1). I hope that those parts will allow the readers that are not familiar with their content to better judge whether the core parts of my thesis, namely the rest of the theory chapter and the application chapters, are of any use to them.
Later in this preface I give a short account of how my research changed from its initial direction to the development of a tool for the study of the interaction between electromagnetic radiation and matter. Chapter A is the background and introduction chapter. Chapter B contains the theoretical basis and tools for the use of helicity and duality in the study, understanding and engineering of the interaction between electromagnetic radiation and material systems. Subsequent chapters contain applications of the theoretical framework that have lead to new insights in phenomena related to angular momentum (Chap. C), zero backscattering (Chap. D), molecular optical activity (Chap. E) and metamaterials for transformation optics (Chaps. F). Chapter G contains a summary of the main contributions to the field contained in this thesis, conclusions and outlook. The most relevant publications are attached at the end.
I sincerely hope that you find some of the contents of my thesis useful for your work.
Why develop a new tool?
My research was not initially aimed at the development of a new framework for the study of light matter interactions. The following is an account of how the subject of my thesis changed. I include it here because the process may interest some readers. In short, what happened was that the analysis of some numerical and experimental results revealed inconsistencies in the stateofthe art theoretical explanation of those results. Those inconsistencies were completely solved using the point of view of symmetries and conservation laws and, in particular, the electromagnetic duality symmetry and helicity conservation. Motivated by this initial success, my advisor and I agreed to redirect my research.
The initial title of my thesis was “Exploring the limits of spatially entangled photons”. I was supposed to study the properties of photons entangled in their momentum degrees of freedom. To start my project, my advisor A. Prof. Gabriel MolinaTerriza suggested to get hold of the radiation diagram of a nanohole in a metallic film. The idea was to explore the interaction of momentum entangled photons interacting with subwavelength size structures. The classical radiation diagram of the nanohole was therefore a necessary first step. The high degree of symmetry of the system suggested the existence of an analytical expression for its radiation diagram. Such expression does not currently exist. Even though the literature about interaction of light with nanoapertures is vast ([1, 2, 3, 4] and references therein), there is no exact analytical solution for the radiation diagram of a nanohole in a metallic film. Had there been one, my thesis would be very different.
The next best thing was a numerical approach. None of the existing techniques (notably [5, 6]) seemed to provide what I was after, i.e., the plane wave decomposition of the field scattered by the nanohole upon excitation by an arbitrary plane wave. I decided to try to develop a method and succeeded in devising a suitable semianalytical approach. The technique allows to obtain the plane wave decomposition of the field scattered by objects embedded in a planar multilayer structure under general illumination [7]. After I wrote and tested the code to implement the technique, my advisor suggested to illuminate the hole with modes of different polarization and spatial phase dependence. These are modes whose expression in the collimated limit^{1}^{1}1The optical axis being the axis. is dominated by a term to or , where is an integer, and and are the left and right circular polarization vectors, respectively. An phase dependence implies the existence of a phase singularity with zero intensity and topological charge in the center . These modes are the well known “doughnut” beams.
With additional simulation code, I modeled the focusing of the input mode, its interaction with the nanohole (using its radiation diagram) and the action of a collection objective on the transmitted light. Figure 17 shows exemplary results of the amplitude and phase of the two circular polarizations at the output. The results show polarization conversion^{2}^{2}2The percentage of conversion is not relevant for this discussion.. Crucially, an invariant quantity is found by assigning the value +1 to and to , and summing it to the azimuthal phase number . This sum is preserved: Its value is the same for the input and the two output polarizations.
It was then time to try to observe this preservation effect in the laboratory. My colleague Xavier ZambranaPuyalto and I worked in the experimental setup for a few weeks. It was the first serious contact with an optics laboratory for both of us. Figure 21 is the schematic representation of the setup and Figs. 22 and 25 show pictures of parts of the actual setup.


The aim of the experiment was to observe the signature of a phase singularity of charge 2 in the CCD camera when the analyzing polarizer was set to select the polarization opposite to the one carried by the input Gaussian beam. This corresponds to the simulated cases in Fig. 17(c)(d)(g)(h). After a few discouraging days, it turned out that the lack of results was due to a faulty servo in the nanopositioning stage (Fig. 25(b)). The sample was moving too much. Turning it off immediately produced an image with two intensity nulls, like the one in Fig. 21(c). This is the intensity signature of a charge 2 phase singularity after splitting into two charge 1 singularities because of noise, something that higher order singularities tend to do [8]. It was an exciting moment.
Simulation and experiment were in agreement. What about the theory? The literature did offer an explanation of the results based on the separation of the electromagnetic angular momentum into the spin and orbital angular momenta [9, 10, 11]. In such explanation, is associated with circular polarization and takes values 1 for and 1 for . is associated with the azimuthal phase dependence and takes the value . The explanation sustains that, while the total angular momentum has to be preserved due to cylindrical symmetry, there is a transfer between spin and orbital angular momentum in the interaction. For example, a input beam originates two outputs, one with the same values and another with . In the latter beam, the value of has to decrease by 2 units in order to compensate the increase of by 2 units. All the examples in Fig. 17 and the experimental results fit this explanation. Nevertheless, there are several problems with it.
First of all, there was no single explanation for the actual cause of spin to orbital angular momentum transfer. It seemed to happen in the interaction of focused beams with nanoapertures [9, 10] and with semiconductor microcavities [12], during the focusing itself [13], and also for a collimated beam in inhomogeneous and anisotropic media [14]. The question of why it happened does not have a single answer in this framework.
Then, there is authoritative literature against the separate consideration of and . For example, on page 50 of CohenTannoudji et. al’s Photons and atoms [15] we read: “Let us show that and are not separately physically observable as is.” Also, in Sec. 16 of the fourth volume of the Landau and Lifshiftz course of theoretical physics Quantum Electrodynamics [16] it says that: “In the relativistic theory the orbital angular momentum and the spin of a moving particle are not separately conserved. Only the total angular momentum is. The component of the spin in any fixed direction (taken as the axis) is therefore not conserved and cannot be used to enumerate the polarization (spin) states of the moving particle.”
This, to me, was enough evidence to discard the explanation for the experimental and numerical results. The results clearly showed that something was changing. A portion of the beam changed into a very different kind of beam. What was then changing? The answer is also in [16], only one paragraph below the one I have copied above:
”The component of the spin in the direction of the momentum is conserved, however: since the product is equal to the conserved product . This quantity is called helicity; […]. Its eigenvalues will be denoted by () and states of a particle having definite values of will be called helicity states.”
Helicity is therefore an observable quantity. Could it be that it was the one changing? The answer is yes. An analysis of the experiment by means of helicity states showed that the results were consistent with helicity changes in the interaction with the nanohole. It also explained other instances of “spin to orbital angular momentum transfer” (see Chap. C). After understanding what was happening, the question was why was helicity changing? The answer is: Because electromagnetic duality symmetry was broken by the sample. The results could be explained by quite simple considerations of symmetries and conserved quantities in the system [17]. Helicity is expected to change in such a setup in the same way that a non cylindrically symmetric target is expected to change angular momentum: The scattering breaks the symmetry associated with the corresponding conservation law.
This first example where the symmetry approach using helicity and duality allowed to gain new insight was an encouraging sign: Maybe it would also be useful in other problems. That was the point where my thesis changed definitively. It turned into the development of a framework based on symmetries and conservation laws for the study of interactions of electromagnetic radiation with matter. Helicity and duality ended up playing a crucial role in it.
The framework is proposed in [17], where it is used to clarify the “spin to orbital angular momentum transfer” explanation. The discussion about helicity preservation and duality symmetry in the presence of matter is in [18] and also in [19]. The framework has indeed produced results in different areas: Optical activity [20], metamaterials [19], zero backscattering [21] and nanophotonics [22].
List of Figures
 a
 a
 22 Picture of the laboratory setup
 a
 A.1 Different transformations applied to a cone.
 A.2 Non commutative transformations.
 B.1 Electromagnetic scattering.
 B.2 Angular momentum versus helicity: Turning versus twisting.
 B.3 Operational definition of helicity.
 B.4 Construction of Bessel beams of well defined helicity.
 B.5 Scattering problem.
 B.6 Helicity preserving versus helicity nonpreserving scatterers.
 B.7 Independence of helicity preservation from geometry.
 B.8 The symmetries of a cone.
 a
 C.5 Hilbert space of transverse Maxwell fields.
 C.6 Setup analysis: Preparation.
 C.7 Setup analysis: Focusing
 C.8 Polarization intensities for small and large transverse momenta.
 C.9 Setup analysis: Interaction with the nanohole
 C.10 Setup analysis: End to end.
 C.11 Scattering off a cone.
 C.12 The action of .
 C.13 Example of using in light matter interactions: Waveguide.
 D.1 Systems with discrete rotational symmetries .
 D.2 Forward and backward scattering.
 D.3 Zero backscattering.
 E.1 Polarization rotation in a general scattering direction.
 E.2 Generalization of linear polarization rotation.
 E.3 Polarization rotation in forward scattering.
 a
 E.7 Measurements at 90 degree scattering off a maltose solution.
 a
 E.11 Faraday effect versus optical activity.
 a
List of Tables
 B.1 Dual correspondences of vectors and operators.
 B.2 Commutation rules for generators and discrete transformations.
 B.3 Generators in the coordinate representation.
 B.4 Transformation properties of plane wave vectors in .
 D.1 Plane waves of well defined helicity in the coordinate representation.
 E.1 Maltose forward scattering polarimetric measurements.
Contents
 Acknowledgments
 Agraïments
 Publication list
 Abstract
 Preface
 A Background
 B Theory
 A Duality symmetry at media boundaries
 B Multilayered systems and Mie scattering
 C SAM and OAM: A symmetry perspective
 D Forward and backward scattering
 E Optical activity
 F Duality in transformation optics
 G Conclusion
Appendix A Background
La pregunta ha de ser clara, i ha de permetre una resposta explícita.
The question has to be clear, and has to allow an explicit answer.Carme Forcadell (catalan political activist)
We have cell phones, GPS, high speed internet, Xrays for medical diagnose, radiotherapy, the microwave oven, 3D movies, solar cells … . We have been able to develop these technologies because we understand fairly well how electromagnetic radiation interacts with matter.
We want more efficient solar cells, nanomachines able to seek and destroy cancer cells from inside our bodies, millimeter sized laboratories that use only a nanoliter of blood, and the means to manipulate electromagnetic radiation at will. These are some of the current research areas that push our understanding of light matter interactions.
My thesis is an attempt to build a tool for studying, understanding and engineering light matter interactions. The basis of the tool is a concept that is central in physics: Symmetry. In the first part of this chapter I will go over the concepts of symmetry, invariance and conservation laws. I will make no attempt to be either exhaustive or formally rigorous. My only aim is to provide a simple introduction to the ideas, language and notation that I will be using in the next chapters. These ideas are, roughly, that:

We can apply transformations to a physical system.

If a transformation leaves a system unchanged, it implies that a particular property of external objects interacting with the system does not change either. It is preserved by the system.
In the second part of the chapter, I will introduce the two main characters of this thesis and provide some background information on them. They are a transformation for electromagnetic fields called electromagnetic duality and the property that is preserved by systems that are unchanged by the duality transformation: Helicity. Both of them together allow to study light matter interactions by means of symmetries and conserved quantities in a relatively simple way.
a.1 Symmetry, invariance and conservation laws
Look at the cone in Fig. A.1(0). The other four subfigures show the cone after a transformation has been applied to it.

(a): Rotation along the axis by an arbitrary angle.

(b): Mirror reflection across the plane.

(c): Rotation of 90 degrees ( radians) along the axis.

(d): Mirror reflection across the plane.
We immediately see that the first two transformations have no effect on the cone: After the transformation, the cone looks exactly as it did before in Fig. A.1(0). The second two transformations have a distinguishable effect on the cone. After the transformation, the cone looks different than before it. We say that the cone is invariant under the first two transformations and is not invariant under the second two. We also say that the cone has rotational symmetry along the axis and mirror symmetry across the plane and that it breaks rotational symmetry along the axis and mirror symmetry across the plane.
Similarly simple ideas about transformations and symmetries are nowadays regarded as the most fundamental basis of our description of Nature [23]. From the standard model of particle physics, through the study of atoms, molecules and crystals to the movement of the stars: Symmetry is the concept that allows us to study and attempt to understand these systems.
According to David Gross [23] “Einstein’s great advance in 1905 was to put symmetry first, to regard the symmetry principle as the primary feature of nature that constrains the allowable dynamical laws”. Einstein brought the concepts of symmetry and invariance to the forefront of theoretical physics. First with his special theory of relativity [24], which is based on the invariant validity of physical law upon changes of inertial reference frame. Later, with his general theory of relativity: The idea that the laws of physics are invariant under changes of spacetime coordinates resulted in our best model of space and time. Many spectacular predictions originated from it and have been experimentally verified. For example that the measure of time is affected by the presence of massive objects or that the light of the stars bends around the sun, to name just two. Understanding the first one allows satellite navigation systems like GPS to be as precise as they are.
Just a bit later in the century, in 1939, Wigner proposed to define an elementary particle as an “object” with some properties, like mass, that are invariant under a particular set of transformations [25]. His idea is the theoretical cornerstone of the current standard model of elementary particles, and the extensions under consideration like supersymmetry, string theory or Mtheory. Wigner did not stop there, and used the concept of symmetry to formalize our modern understanding of atoms [26].
Many areas have followed suite in employing the concept of symmetry and the mathematical branch which allows its formalization: Group theory [27]. This has brought uncountable advances. For example, we understand the Higgs boson, the spin of the electron, the atomic Zeeman and Stark effects, the rules for exciting a molecule with light and the behavior of matter waves in crystals thanks mainly to the study of symmetry.
Having briefly highlighted the importance of symmetry in modern physics, let us go back to the example of the cone and discuss two important concepts related to transformations: Commutativity and the difference between discrete and continuous transformations. In Fig. A.2, two successive transformations are applied to the cone: Rotation by along the axis () and rotation by along the axis (). The difference between the upper and lower rows in the figure is only the order in which the two rotations are applied. Figs. A.2(a2) and (b2) clearly show that the end result is not the same. The order of application matters. If, in order to allow a more formal discussion, we take the convention that means that we first apply and then we apply to the result of the first transformation, what the figure shows is that:
(A.1) 
where the commutator between two transformations is defined. When the commutator is not zero, as for the rotations in the figure, we say that the two transformations do not commute. When the commutator is zero, the order of application of and does not matter, we say that the transformations commute. For example, spatial translations in a plane always commute: if in a flat football field you run straight for forty meters and then turn left and run for another ten meters you will meet a team mate of yours having started from the same initial point and first turned left and run for ten meters, and then turn right and run for forty meters.
Let us go back to Fig. A.1 where I applied rotations and mirror reflections to the cone. There is a fundamental difference between these two transformations. A rotation is a continuous transformation. After choosing an axis, we can rotate the cone by infinitely many different angles which we can select from a continuous interval extending from to . A mirror reflection is a discrete transformation. We either do the reflection or not. In this case, the choice that we have is only binary.
The continuous nature of rotations allows to subdivide them into an ever growing number of successive rotations by an ever smaller angle. For example:
(A.2) 
and so on. When the angle of the constitutive rotations gets infinitely small, there is an infinite number of rotations, and each of them is just an infinitesimal perturbation on the identity transformation (whose action is to leave any object unchanged):
(A.3) 
The second equality follows from the definition of the exponential. The reason why we write the perturbation as instead of is not really important. What is important is that this object, , is the only thing that is needed in order to manufacture a rotation along the axis by an arbitrary angle. We say that is the generator of rotations along the axis. We also say that is the component of angular momentum.
Every continuous transformation has a generator. For example, linear momentum generates translations in space and the energy operator generates translations in time. The generators are also operators. The consideration of how objects transform under the action of a generator and its corresponding transformation before and after interacting with another system leads me to one of the most profound consequences of invariance: Conservation laws.
For this task, instead of the cone, I need to consider abstract mathematical objects, which we will call vectors. These vectors “live” in some abstract space . The properties of and the vectors in it are much like the properties of the three dimensional space with coordinates and the three dimensional vectors which “live” in it. Adding two vectors from results in another vector in , as does multiplying a vector by a number. These are the properties of a linear vector space. We can also transform vectors like I did with the cone. In linear vector spaces the transformations are represented^{1}^{1}1Transformations are abstract entities. In order to apply them to concrete mathematical objects, they have to be mapped into appropriate operators for those objects. For example: We can rotate an electron and we can rotate a Higgs boson. The rotation operator for a Dirac four spinor is formally very different from that which rotates a scalar. The representation is different but the essence of the transformation is the same. Unfortunately, the formula operator that represents transformation X is too long and I will just use transformation X when I mean the former. by operators. They take the vectors in and map them back to possibly different vectors in . For an original vector , an operator and a resulting vector we write:
(A.4) 
In general, we would end up with a vector with no obvious relationship with , but, given an operator that represents a transformation, some of the vectors in are special with respect to it in that they transform very simply by just acquiring a phase. Take for example . There exists a set of vectors such that [27, Chap. 7]:
(A.5) 
where is an integer. If we make the angle infinitesimally small , we can figure out how do the transform under the action of the generator of rotations :
(A.6) 
When the vector resulting from the action of an operator on an initial vector is for a scalar , we say that is an eigenvector or eigenstate of with eigenvalue . In the above case, we say that the are the eigenvectors of with eigenvalues equal to . We also say that has a well defined, sharp, or definite angular momentum equal to . A counter example is, for instance , which is a linear combination of two eigenvectors with different eigenvalue: Its angular momentum is not well defined.
This formalism can be used to study interaction problems in physical situations. I will discuss this point at large in Chap. B. For example, an electron travels towards a sample, interacts with it, and as a result of the interaction some properties of the electron, like its momentum, change. Roughly speaking, in Eq. A.4, represents the electron before the interaction, the action of the sample and the electron after the interaction. Assume that commutes with , that is . This implies that the operator is invariant under rotations along the axis. It also implies that the sample that represents exhibits such invariance as well.
To discuss conservation laws, let us examine the properties of after the action of . Before the action of , the had very simple and definite transformation properties under rotations (A.5). Did the action of change this? In other words: are the still eigenvectors of with eigenvalue ? They are, because:
(A.7) 
where the first equality follows from the assumption that and the second one from (A.5). So, after interaction with a system which is invariant under rotations, an initial eigenstate of is still an eigenstate of with the same eigenvalue as before. The same can be said about : After the interaction, each eigenstate of is still an eigenstate of with the same eigenvalue that it had before the interaction. This is the expression of a conservation law: We say that a system that commutes with a transformation is invariant under and it preserves the eigenstates of the symmetry transformation and its generator. Please refer to [28, Sec. 4.1] for a rigorous discussion. For discrete transformations, like mirror reflections, a conservation law is expressed as the preservation of the eigenstates can corresponding eigenvalues of the transformation in question. Conversely, when an eigenstate of a transformation interacts with a system which is not invariant under such transformation, the resulting state is, in general, a linear superposition containing not only the original but also all the other eigenstates,
(A.8) 
The transformation properties of under the original transformation are not simple anymore.
The original result on invariance and conservation laws is due to Emily Nöther [29], who in 1918 derived her celebrated theorem. Using the original form of the theorem and its posterior extensions, each invariance of the time evolution equations of a system can be linked to a conservation law. In the previous discussion, I have hidden the evolution of time by using terms like before the action and after the interaction. This trick is the basis of the study of collisions: Scattering theory [30, Chap. 3], [31, Chap. XIX]. Picture an object traveling on a collision path towards a target. As mentioned before, the action of the target on the object is modeled by a scattering operator acting on the vector space where each vector represent a different state of the object. The before the action initial state is taken to be the state of the object an “infinite” time before the collision, and the after the interaction final state represents the object an “infinite” time after the collision. The idea is that much before or much after the collision, the object can be described ignoring the presence of the target. From a practical point of view, this is a useful simplification and, among many other important applications, is used in spectroscopy or high energy particle accelerators. From a theoretical point of view, it may be argued that the exact details of what happens during the collision of, for example, two protons traveling at 99.99% of the speed of light are unknown or even outside the domain of applicability of the theories that we have [30, Chap. 12]. When it suffices for one’s purposes, the scattering picture is quite convenient, and much can be learned from the study of the symmetries of . Nevertheless, a statement like the scattering operator commutes with transformation is related to Nöther’s theorem because the definition of involves the time evolution operator of the system [30, Chap. 3.2]. Ultimately, such a statement reflects the invariance of the dynamical equations upon transformation with .
What does invariance of the dynamical equations mean? Consider the following time evolution equations for the position of a pointlike object A in a two dimensional world interacting with another object B:
(A.9) 
Let me now transform them using the spatial inversion . The transformation of the equations, which you can think of as a change of basis, goes like this:
(A.10) 
The form of the evolution equations for the variables is exactly the same as that of the evolution equations for the variables. This is what invariance of the dynamical equations means. This does not happen for every transformation. Let me now take a rotation , repeat the above steps and reach:
(A.11) 
to see a counter example.
These straightforward arguments allow us to categorically affirm that object B is not rotationally invariant but has spatial inversion symmetry … and we do not even know what object B looks like !
Besides being the tell tale sign of symmetry, this form invariance of the equations has a very practical application. If a system of equations is invariant under a given transformation, and you know one solution of the system, you can produce a new solution by transforming the one you know with the said transformation. A little reflection on the fact that the equations look the same for both the original and the transformed variables should convince you of the general validity of the previous statement. You can see explicitly what this means by assuming that you have a couple of functions which solve Eq. (A.9) and verifying that is also a solution of (A.9). You just doubled the number of solutions at your disposal. Try to produce a new solution by rotating the initial one. It does not work for an arbitrary angle, reflecting what we learned from (A.11). It works for . The reason is that, in this case, a rotation by results in .
In summary, the existence or breaking (lack of) of a symmetry allows us to know whether or not a system preserves the eigenstates of the transformation in question. For continuous transformations it implies the preservation of the eigenstates and corresponding eigenvaluesand corresponding eigenvalues of its generator as well. We can also use the knowledge that a symmetry exist to produce new solutions of a problem from known ones. Symmetry, invariance and conservation laws have many more theoretical and practical implications and applications. The excellent text by WuKi Tung [27] has allowed me to scratch the surface of this beautiful subject.
In my thesis, symmetries and conservation laws are mostly used as a tool to study, predict and understand the results of light matter interactions. Sometimes, careful considerations of the symmetries and conserved quantities is all that is needed in order to pinpoint the fundamental reason of some “mysterious” effect (Sec. D.3), make new experimental predictions about light scattering (Chaps. D, E) or to be able to isolate the actual reasons for observations that lacked a consistent explanation until then (Chap. C). In other cases, it gives a very solid theoretical stepping stone for the posterior investigation of the details of the problem with analytical, numerical or experimental techniques (Chaps. C, F) . Also, analyzing the symmetries of the equations can provide valuable design guidelines and constraints (Chap. F).
When faced with a new problem, Wigner taught us that it pays off to consider its symmetries first: I hope that, after reading the applications chapters, you will agree that this is indeed a fruitful approach to the study of light matter interactions.
Let me now introduce a continuous transformation for electromagnetic fields, duality, and its generator, helicity. These two are the main characters in my thesis. Thanks to them, the study of light matter interactions using symmetries and conservation laws can be made relatively simple.
a.2 Duality and its generator: Helicity
The transformations that I have considered in the previous section are geometrical: they rotate, invert and translate an object in three dimensional space. These transformations are easy to understand because we ourselves inhabit the same space in which they act. When we also include the time axis, new transformations are possible in this fourth dimensional space that did not exist in three dimensions. The action of some of them is also easy to grasp. For example, we can imagine the effects of time going backwards quite easily. Others pose a much bigger challenge to our imagination: Can you imagine the effect of a rotation along an axis which, instead of being perpendicular to a spatial plane, is perpendicular to the plane formed by the spatial axis and the time axis? This is not easy to do. And then, there are even more abstract kinds of transformations. They do not act on space or time coordinates but in the extra dimensions that the experimental observations of the last century have forced us to include in the models. For example, transformations of the spin of the electron, “rotations” that turn quarks of one kind onto quarks of another kind or the exchange of the labels of the two photons in a two photon state. These transformations, their generators (for the continuous ones), and the abstract spaces in which they act are crucial in physics. Electromagnetic duality is one of these more “strange” transformations which does not act on space or time but on a different, more abstract, dimension of electromagnetism.
Here is the action of duality on the electric and magnetic fields in free space [32], [33, Eq. 6.151]:
(A.12) 
where is a real angle, and are the vacuum permittivity and permeability constants. Duality first appeared in print more than a century ago [34], which probably makes it the first of this kind of abstract transformations to be considered. From the beginning, it was seen to be a symmetry of the free space Maxwell’s equations^{2}^{2}2Note that (A.13) are dynamical evolution equations for and plus the extra conditions .:
(A.13) 
The form of (A.13) is invariant under the transformation (A.12). As we discussed in the previous section, this form invariance means that if the electromagnetic field is a solution of the free space Maxwell’s equations, then the field is also a solution for any value of . This mixing of electricity and magnetism is inherent in the duality transformation. In 1968 Zwanziger [32] studied the duality transformation in the context of a quantum field theory with both electric and magnetic charges. He used a generalization of the microscopic Maxwell’s equations which included magnetic () sources in addition to the common electric () sources. The equations contain the two types of scalar charge densities and vector current densities , related by .
(A.14) 
Zwanziger studied the transformation properties of (A.14) under the simultaneous action of the duality transformation of the fields in (A.12) and the following similar extra transformation of the sources:
(A.15) 
He found that the whole electrodynamic theory is invariant under simultaneous action of the two transformations. He called it the chiral equivalence theorem. In the next chapter, I will discuss what it actually means to perform the extra source transformation and what type of questions can and cannot be addressed when using it. As an advance, let me say that this thesis is about duality symmetry without using the extra source transformation. I will show that, under certain conditions, material systems can be dual symmetric without the extra source transformation. By a dual symmetric system I mean one whose electromagnetic equations are invariant under the transformation (A.12) alone.
In the same paper, Zwanziger showed that helicity was the generator of the duality transformation. Just three years earlier, Calkin [35] had proved the same result for the source free equations. What Calkin and Zwanziger showed is that helicity is the generator of duality in the same sense that angular momentum is the generator of rotations or linear momentum is the generator of translations. The helicity operator is defined [27, Sec. 8.4.1] as the projection of the total angular momentum onto the linear momentum direction, i.e. . Curiously, neither Calkin nor Zwanziger used the name “helicity” in their seminal papers nor the definition of helicity that I just wrote. On top of this, they worked in different vector spaces and reached two different formal expressions for the generator of duality ([35, expr. 18],[32, expr. 2.17]). Possibly because of this heterogeneity of the initial treatment and naming, the connection between helicity and duality has been reported several times. Here are the ones that I am aware of [36, 37, 38, 39].
a.2.1 Using duality
The exploration of electromagnetic duality has turned out to be a fruitful theoretical endeavor in fundamental physics.
Zwanziger [32] and Schwinger [40] used the invariance of electromagnetism under simultaneous application of (A.12) and (A.15) to refine the famous argument by Dirac on electric charge quantization [41]. Dirac showed that the mere existence of a single magnetic monopole (a particle with magnetic charge) implies the quantization of electric charge. Nobody knows why, in isolated particles, electric charge is only observed in quantized multiples of the charge of the electron. This empirical truth is one of the remaining mysteries in physics. Finding a magnetic monopole would solve it.
In an application to a different field, the magnetic monopole idea and a generalization of electromagnetic duality allowed Montonen and Olive to formulate their celebrated conjectured unification of supersymmetry and the theories of elementary particle interactions [42].
In a more practical sense, the duality symmetry has been skillfully exploited for the study of objects with zero back scattering in the context of light matter interactions: The authors in [43, 44] find that when a plane wave impinges on an scatterer with both duality symmetry and discrete rotational symmetry along the axis defined by the momentum of the plane wave, there is no energy reflected in the backwards direction. I shall show in Chap. D that when the other piece of the conservation law, helicity, is taken into account, those results can be fully understood using only symmetries and conservation laws. Additionally, the consideration of helicity will allow me to show that any dual symmetric object with a discrete rotational symmetry for will exhibit zero backscattering.
a.2.2 Using helicity
The helicity operator does not only act in vector spaces representing electromagnetism. Helicity is routinely used in particle and high energy physics. Whether we are talking about photons, electrons, neutrinos, or many of the other inhabitants of the zoo, helicity has a well defined meaning. Its importance is apparent from the get go since helicity is the operator that is typically chosen to represent the internal degrees of freedom of elementary particles, often called spin or polarization degrees of freedom. For example, an electron state or a photon state can be specified by fixing its four momentum and helicity [27, Sec. 10.4],[30, Chap. 2]. In the analysis of high energy collisions, helicity allows a unified treatment of massive and massless particles [45].
For massless particles like the photon, helicity is quite special: It can only take two different values ( for the photon, for the graviton), and is a Poincaré invariant of the field. That is, helicity is invariant under space and time translations, spatial rotations and Lorentz boosts. Note that this is not the case for other properties like, for instance, momentum. A rotation changes the momentum vector. It is also different from the helicity of massive particles. The helicity of a massive particle takes different values, where is the intrinsic spin of the particle, and it is not a Poincare invariant because a boost mixes the different helicities [30, Chap. 2],[27, secs. 10.4.210.4.4]. Take for instance an electron with helicity equal to in some reference frame. Since , I can always “boost myself” along the electron’s propagation direction until I advance it to a reference frame where , and the sign of helicity will flip and become . This argument does not work for massless particles: A photon, for example, cannot be advanced. In some sense, two photons of different helicity can be seen as two different elementary particles. The different helicity value denotes a different representation of the Poincare group, which is the criteria used to define an elementary particle [27, Chap. 10]. Why should we then say that the two photons of different helicity are the same particle? Here is why: If the spatial inversion (parity) transformation is added to the Poincare group, the two photons of different helicity merge into one particle with two possible helicities because the parity operator () interchanges the two helicity eigenstates :
(A.16) 
Even though parity is not a symmetry of Nature [46, 2.6, 2.7], it is considered a symmetry in electromagnetism [33, 6.10]. The photon must hence be a single particle with two possible helicities in order to keep up with the action of . Consequently, the electromagnetic field has two possible helicities states.
All this may sound too abstract and separated from light matter interactions. It is not. Chapters D, E and F contain practical engineering guidelines for building zero backscattering objects, polarization rotation objects and metamaterials for transformation optics. These guidelines are obtained by symmetry considerations involving helicity and duality.
Helicity has been exploited by Prof. Iwo BialynickiBirula in his works on theoretical electromagnetism. While showing us how to construct a proper real space wave function for the photon [47, 48], he makes use of the division of the field in its two helicity components, studies the condition of helicity preservation in inhomogeneous and isotropic media and realizes that helicity is conserved in a general gravitational field. All these ideas are indispensable in my thesis. I also make extensive use of the RiemannSilberstein formulation of electromagnetic fields that Prof. I. BialynickiBirula [47, 48] together with Prof. Z. BialynickaBirula promote [49]. I had the pleasure and honor of meeting and interacting with both of them during their month long visit to the group of my advisor at Macquarie University (Sydney).
I want to finish this chapter by stating that duality is almost always broken and helicity is almost never preserved in light matter interactions. You may consequently think that this conservation law is almost surely useless. But, if you keep reading, I can hopefully show you that it is not. For now let me say two things. First, broken symmetries can be as crucial for our understanding as unbroken ones: One example of this is the Higgs mechanism. Knowing that duality is broken and that helicity is expected to change can help you understand something about your observations in the laboratory. And second, very interesting phenomena happen when duality is an actual symmetry of the system and helicity is preserved either naturally or as a result of engineering.
Appendix B Theory of helicity and duality symmetry in light matter interactions
In fact the natural correspondence between the basis vectors of unitary irreducible representations of the Poincaré group and quantum mechanical states of elementary physical systems stands out as one of the remarkable monuments to unity between mathematics and physics.
Wu Ki Tung, “Group Theory in Physics”
In this chapter, I aim to establish the theoretical foundations for the use of helicity and duality in the study of light matter interactions. I also develop the tools that I will use in the application chapters.
Sections B.1.1 and B.1.2 contain a collection of results on Hilbert spaces and transformation properties which are needed for this chapter. They are taken from [31, Chaps. V, VII and VIII], [28, Chaps. 1] and [27]. These results are presented in a way that I believe most appropriate for my thesis. Section B.1.1 also contains examples which are useful for understanding the use of symmetries and conservation laws in the study of scattering problems. The rest of the sections contain original material, to the best of my knowledge and except for the cited work.
b.1 General framework
Take a look at Fig. B.1. An incident electromagnetic field impinges onto a material scatterer . As a result of the interaction between the field and the material system, a scattered field is produced. There are many reasons why this problem is interesting. For example, the scattered field contains information about which may be used in microscopy to image the system, or in spectroscopy to analyze its components. We may also want to engineer the object so that is some desired function of like in transformation optics [50].
The underlying theme in my thesis is to study this problem by means of symmetries and conservation laws. In order to be able to do this, I will need to introduce some formal machinery, namely that of Hilbert spaces. As I will show you later, both and can be modeled by vectors in a Hilbert space (represented by ) and the action of the object by means of a linear operator in such space. This is the main assumption/restriction in my thesis: The action of is linear. This means that the operator meets:
(B.1) 
The symmetries of , which are reflected in the properties of , will play a central role through their corresponding conservation laws. As vectors in a Hilbert space, the fields can be expanded in a basis of the space. Each vector of a basis has four numbers that identify it. These numbers are the eigenvalues of at least four independent commuting operators. The symmetries of and their corresponding conservation laws will tell us which of those numbers are going to be maintained between input and output and which ones are allowed to change. Helicity will enter the picture as one of the four numbers, the one that labels the polarization of the field. The polarization is the nonscalar degree of freedom that a scalar field like for instance temperature does not have.
b.1.1 Hilbert spaces, vector basis, transformations and operators
A Hilbert space is a linear vector space with an inner product having certain properties [31, Chap. V §2]. In a linear vector space, the addition of any two vectors of the space results in a vector that also belongs to the linear vector space. The same is true for the multiplication of a vector by a number . The inner product is a function which takes two vectors and produces a number :
(B.2) 
What is ? If belongs to a vector space , is a member of the vector space dual to . In my thesis, this is an unfortunate naming coincidence. The word “dual” here does not have anything to do with the duality transformation. To avoid confusion, and following Sakurai [28, Chap. 1.2], I will use the acronym DC for this dual correspondent between vector spaces. Operators also have DC versions. Tab. B.1 contains some of the correspondences between some objects and their DC version.
Vector space  DC vector space 

When , we say that the two vectors are orthogonal. In a Hilbert space () we have the concept of mutually orthogonal basis: A set of orthogonal vectors in which can produce any vector in by linear combinations, like:
(B.3) 
where the numbers are . The different basis vectors are mutually orthogonal:
(B.4) 
If they are scaled so that , we say that they form an orthonormal basis.
There are many different orthonormal bases that expand (can produce any vector in) a given Hilbert space. As I will show you later, when the vectors of the space represent electromagnetic modes, its basis have one characteristic in common. Each of their vectors is uniquely identified by four numbers. These four numbers are the eigenvalues of four independent commuting operators. The symbolic index contains then four numbers. Some of those numbers may take continuous values and others may take discrete values. The summation over symbol () in (B.3) is a short hand notation for whatever discrete summations or continuous integrals are needed, for example it may mean
(B.5) 
or
(B.6) 
etc.
The difference between continuous and discrete indexes is subtle and complex. For starters, vectors with a continuous index have an infinite norm: . This means that they do not belong to the Hilbert space since one of the requirements for membership is finite norm. Please refer to [31, Chap. V §8; Chap. VII §4,§9] for the discussion about why it is nonetheless possible to use these outsiders as members of a basis for expanding proper vectors in the Hilbert space. The discussion makes use of the theory of distributions. For practical purposes we may say that they have the same properties (B.3)(B.4) as the basis vectors of finite norm, but, in order to express their orthogonality properties, one has to substitute the Kronecker deltas ( if , and ) by the Dirac delta distributions . I will not discuss this anymore after this paragraph^{1}^{1}1If you do not want to read the sections from Messiah’s book that I have indicated, but feel a bit uneasy, here is a quick nonrigorous palliative: In an expansion with , the infinite norm “vectors” will be multiplied by the corresponding infinitesimally small differential , and so an infinite sum of infinite norm “vectors” can result in a finite norm vector: A proper member of the Hilbert space.. This serious complication is worth it because of the very simple properties that some of those outsiders have under relevant transformations. From now on, I will call vectors both the proper members of the Hilbert space and the infinite norm ones.
Same as vectors, linear operators acting in can be expanded using other operators which can be built from the vectors of a basis and their DC versions. Since any vector in can be expanded by a chosen basis , a linear operator can be expanded as:
(B.7) 
where the are numbers. By appropriately choosing you can produce any linear map from to .
While I go over all this machinery, I want to give some examples of its use. The machinery is general, but in order not to use abstract examples, I will, from now on, implicitly assume the context of electromagnetic scattering: An input vector produces an output vector after the interaction with the object which is represented by the linear operator (see Sec. B.6).
(B.8) 
The numbers on top of the equal signs are the references to the information (previous expressions, tables, etc …) needed to take the corresponding step. I will use this notation quite often. For example, above means that the equality follows because the and indexes collapse due to Eq. (B.4): unless .
By construction, the set of all the completely define the operator . To recover one of them, say we do:
(B.9) 
The are called the matrix elements of the operator between states of the basis.
In chapter A, we saw that symmetry transformations are represented by operators that act on vectors. They also act on other operators. Here is how the operator is transformed by an operator representing a given transformation [27, Chap. 1.1]:
(B.10) 
needs to be invertible. This is not a problem since any transformation that leaves a physical system invariant must be represented by a unitary operator [31, Chap. XV §1], [27, Sec. 3.3]. A unitary operator meets
(B.11) 
where is the identity operator. So, any operator susceptible to represent a symmetry of the system has an inverse, and the inverse is its hermitian adjoint. This is the case for unitary transformations like translations, rotations, parity, duality, time reversal and Lorentz boosts. It is also the case for time inversion, which is antilinear and antiunitary. For operators representing symmetry transformations, I will often use instead of .
In both unitary and antiunitary cases, invariance of under transformation means then
(B.12) 
which is also equivalent to saying that and commute:
(B.13) 
Equations (B.10)(B.12), together with Tab. B.1 and the associativity of the product of operators
(B.14) 
is all we need to start exploiting the consequences of symmetry on the scattering operator . The methodology is general but, to make the discussion less abstract, I will give examples with specific transformations.
Let us say that the object is, like the cone in Fig. A.1, invariant under rotations along the axis. What this means for its scattering operator is that . Let me we expand the incoming field in Fig. B.1 in a basis where one of the four indexes in is the eigenvalue of . Without explicitly writing the other three indexes, we have that:
(B.15) 
Since must be true for all , this means that unless . This is the manifestation of invariance under symmetry transformations (Sec. A.1): Eigenstates of before the interaction are still eigenstates of with the original eigenvalue after the interaction. The number of coefficients needed to describe is drastically reduced if we choose our basis according to the symmetries of the system. This is going to be a recurring theme.
Imagine now that the scatterer does not have the full cylindrical symmetry but is only invariant under discrete rotations with angles for integer . For example, for , this is the symmetry of a square prism. Repeating the above steps leads to the conclusion that unless for integer .
Let us now switch to translations. Consider an infinite slab of material parallel to the plane. Because of the invariance of the infinite wall to any transverse translation , the matrix elements of between plane waves with different components of momentum parallel to the wall must vanish.
(B.16) 
In a three dimensional lattice, for instance a cubic lattice of ions, discrete translations can be used in much the same way as we have used discrete rotations above to figure out which are the momenta that are invariant upon scattering through a lattice. In crystallography, these momenta form what is called the reciprocal lattice.
These simple examples show the importance of taking into account the symmetries of the problem when choosing the working basis in .
Enough for now on the effect of symmetries in scattering. Section (B.8) contains a complete example which better shows the capabilities of the technique. Helicity and duality are needed for it and for all the other applications in the later chapters. We will also need to know how different transformations affect helicity.
b.1.2 Transformation properties of helicity and other generators
In this section, I provide a table with the commutation properties of the helicity operator [27, Chap. 8.4.1], some of the other generators and two discrete transformations, spatial inversion (parity, ) and time inversion (). Time inversion is the only antiunitary transformation that appears in my thesis.
In the table, I use the anticommutator between two operators
(B.17) 
If we say that and anticommute. If and has an inverse, then .
From the commutation properties of the generators one can obtain those of the generated transformations following a few rules.

If two generators commute, their generated transformations also commute, and each of the generators commutes with the transformation generated by the other generator.

If is a generator and a unitary discrete transformation


If is a generator and an antiunitary discrete transformation

For example, in the table we see that parity anticommutes with momentum and hence flips its sign. This means that when transforming a translation with the parity operator, the result is a translation in the opposite direction, which makes intuitive sense: .
Since and , and if many cells in Tab. B.2 are empty to avoid redundant entries.
Helicity Energy Momentum Parity H
The helicity operator commutes with rotations, translations and time inversion. It anticommutes with parity, which flips the helicity. On the other hand, parity commutes with angular momentum. This difference in the behavior under parity between angular momentum and helicity is worth discussing a bit more. It is one of the differences between turning and twisting.
b.1.3 Turning versus twisting
An ice skater spinning around in a fixed position and a spinning top after a skilled kid pulls out the cord: These systems are turning. What an ant does if it wants to walk along a wineopener or the movement of a screwdriver when you tighten or loosen a screw. This is twisting.
We appreciate the difference intuitively. In order for me to turn I only need to rotate (), but, if I want to twist, I need to rotate () and advance () at the same time. We also see intuitively that there are two possible kinds of twist, lefthanded and righthanded. Helicity describes the sense of twist. Its name is quite appropriate in relation with a helix.
The transformation properties of turns and twists are quite different, and correspond to those of and . Turns can change upon rotation while twists do not. In the formal language: the components of do not commute with each other but they all commute with (Tab. B.2). Imagine that the spinning ice skater is able to do a back flip and start spinning on her head. If you have good spatial intuition you will realize that now she is spinning on the sense opposite to the one before the pirouette. On the other hand, turning a wineopener on its head does not change its sense of twist. Now, if the ice was clear enough for you to see the ice skater reflected on it while she turns, you would see the “mirrored” ice skater turning in the same sense as the real person. Taking your wineopenerant system next to a mirror shows that the sense of twisting changes … no matter how the mirror and the wine opener are oriented relative to each other. In the formal language: Any inversion of coordinates flips helicity while it does not necessarily change angular momentum.
Note that a mirror reflection across a plane perpendicular to axis , , can be written as parity times a rotation of 180 degrees along . The order does not matter because rotations and parity commute:
(B.18) 
The transformation properties of and under mirror reflections can now be worked out using (B.18) and Tab. B.3. Fig. B.2 illustrates the differences between the transformation properties of helicity and those of angular momentum.
b.2 The Hilbert space of transverse Maxwell fields
Consider the source free Maxwell equations for an infinite homogeneous and isotropic medium with scalar electric and magnetic constants and :
(B.19) 
These equations are linear and homogeneous in . It follows that adding two of their solutions and produces another valid solution. The same is true for multiplying a solution by a number. The set of all solutions of (B.19) is hence a linear vector space. Together with an inner product between two solutions and , which I will discuss shortly, they form a Hilbert space [51, Chap. 13.3], which I will call