aa r X i v : . [ phy s i c s . e d - ph ] S e p Teaching gauge theory to first year students.
N.-E. Bomark
Institute for natural science, University of Agder, Universitetsveien 25, 4630Kristiansand, NorwayE-mail: [email protected]
Abstract.
One of the biggest revelations of 20th century physics, is virtually unheard of outsidethe inner circles of particle physics. This is the gauge theory, the foundation for howall physical interactions are described and a guiding principle for almost all work onnew physics theories. Is it not our duty as physicists to try and spread this knowledgeto a wider audience?Here, two simple gauge theory models are presented that should be understandablewithout any advanced mathematics or physics and it is demonstrated how they canbe used to show how gauge symmetries are used to construct the standard model ofparticle physics. This is also used to describe the real reason we need the Higgs field.Though these concepts are complicated and abstract, it seems possible for atleast first year students to understand the main ideas. Since they typically are veryinterested in cutting edge physics, they do appreciate the effort and enjoy the moredetail insight into modern particle physics. These results are certainly encouragingmore efforts in this direction.
Keywords : particle physics, gauge theory, Higgs mechanism, didactics eaching gauge theory to first year students.
1. Introduction
Anyone reading popular science have heard something about quantum physics. Thereis no shortage of attempts to popularise the intricacies of both quantum mechanics and(perhaps to a lesser degree) quantum field theory.However, when modern physics theories are formulated, perhaps the most crucialingredient is virtually unheard of outside the particle physics community. This is thegauge theory. Without gauge theory the standard model of particle physics cannot beformulated and it is fundamentally impossible to fully understand the role of the Higgsfield without some understanding of the role of gauge symmetries.Why is this the case?This question is hard to answer, gauge theory does not appear any more complicatedto understand than quantum physics, however, it may be more abstract and it does nothave large philosophical implications for how we see the world like quantum physics does.Therefore one could argue that teaching gauge theory is a waste since essentially no-onewill have any use of it. On the other hand, we are talking about the most important(alongside quantum field theory) theoretical development in our understanding of theworld since quantum mechanics. Is it acceptable to restrict this knowledge to the fewwho venture into the quite challenging field of advanced quantum field theory?Just as Shakespeare and Beethoven are considered expected knowledge among theeducated populace, it is reasonable to expect knowledgeable people and especiallyphysics students and teachers to have some idea about the basis of our currentunderstanding of nature, and for this gauge theory is unavoidable.This may sound all noble, but if this is to actually happen, we must have ways ofexplaining gauge theory in a not too mathematical manner. In the following paper wewill explore two models of gauge theories that are designed to introduce the conceptwithout the level of abstraction and mathematical complication, present in the particlephysics models. Neither of these models were first invented by the author, the time-zone model was found at [1] and has been elaborated by the author, while the economicmodel was published by Maldacena [2] after [3] pointed it out, this model has beenreformulated to more directly demonstrate the role of symmetries in gauge theory.The topics presented here are being used to convey an understanding of particlephysics to first year students, in a course on particle physics and cosmology. In thatcourse gauge theory is embedded in a wider discussion about what quantum field theoryis and it is presented including some more mathematical detail than given here. In thispaper the topics are presented as mathematically minimal as possible in order to beaccessible to as wide an audience as possible.In the following we will start with describing what we mean by a symmetry, thisis important since the symmetry concept we need is more abstract than most studentsare used to. In section 3 we introduce the two gauge theory models and explain howgauge theory constitutes the bases for modern physics. The Higgs field is discussed insection 4, both why we need it, what it does and how it is connected to gauge theory. eaching gauge theory to first year students.
2. Symmetries
Although the concept of symmetry is familiar to most people, we need to introducethis concept carefully since its mathematical definition is not that obvious and thesymmetries we will encounter are rather abstract. The understanding of symmetriesmay be one of the biggest obstacles in understanding gauge theory.Mathematically a symmetry is some transformation that leaves the object we arestudying unchanged or invariant as well usually express it. In physics the object toremain invariant is usually the laws of physics. Though this definition is technicallyequivalent with our every day intuition of what a symmetry is, it is formulated in arather more abstract fashion and thus takes some time getting used to.To gain experience with this symmetry concept, it is necessary with some examples,unfortunately there are not very many mathematically simple ones. One case is atranslational symmetry where a spacial coordinate, x , is transformed into, x → x ′ = x + ∆ x where ∆ x is some constant. It can be shown that this is a symmetry toNewton’s second law (neither forces nor acceleration are affected by this since ∆ x isa constant and therefore disappear in the derivative). One can also show that Newton’ssecond law respects a rotational symmetry, but that gets more technical since we needa mathematical description of rotations which requires some linear algebra.One can also point out that electromagnetism is invariant if the electric potential, V , is shifted by any constant value, V → V ′ = V + V , where V is any constantnumber. This should already be known since it is just saying that the level of zeropotential can be chosen arbitrarily. This is a nice example since it is part of thegauge symmetry of electromagnetism. For more advanced students one can demonstratethat V → V ′ = V + dφ ( ~x,t ) dt together with ~A → ~A ′ = ~A − ∇ φ ( ~x, t ) is a symmetry ofMaxwell’s equations for all functions φ ( ~x, t ) , which is the full gauge symmetry for theelectromagnetic field, see Appendix B for more details.
3. Gauge theories
The key feature to gauge theories, is the concept of a local symmetry. With this wemean that the mathematical transformation that defines the symmetry may be applieddifferently in different points in space. For example, if we have a local rotationalsymmetry, we rotate with a somewhat different angle in different points.To illustrate how a local symmetry leads to interactions, we shall look at twoexamples where the symmetry is more intuitive than is the case in our particle physicsmodels. Hopefully these models can create an intuition about gauge theory that can betransferred to modern physics theories. eaching gauge theory to first year students. Suppose we try to measure the path of a ball that has been thrown through the air.When doing so it does not matter how we have set the clock we use when measuringwhen the ball is at different positions. We can set the clock forward with 1 hour or backwith 15 minutes and nothing will change in our results because we only care about timedifferences.If you now recall our definition of a symmetry, you recognize that this is exactlythat; since we can reset our clock as we like without the results being affected, we havea symmetry that we can call a time-zone symmetry.Let us now assume that we have one clock in each point in space and we use thatclock to measure when the ball is at that point. If all clocks are synchronized this is nodifferent from the previous situation, we can still reset all clocks as we like as long aswe set all of them the same way.To make this a gauge theory we need to make this symmetry local. This meanswe insist on being allowed to reset all the clocks individually, i.e., they will no longeruse the same time-zone. Of course, this does not work right away; if all clocks are setrandomly with respect to each other the time we measure the ball takes between twopoints has no meaning; how much of this time difference is just due to the clocks beingset differently?It does not take too much imagination to solve this, all we need is to keep trackof how differently the clocks are set. For this we need to introduce a function thatwe can call ~A ( x, y, z ) that tells us the difference between neighbouring clocks. Whenwe reset the clocks we update ~A ( x, y, z ) with the difference in how much we reset theneighbouring clocks in that point in space, ~A ( x, y, z ) is a vector because it needs to tellus the difference between the clocks in both the x , y and z direction.We see that by insisting that our system is invariant under this local version of thesymmetry we are forced to introduce a vector-field, ~A ( x, y, z ) . This field we can call agauge field and we say that we have gauged the symmetry.If the gauge field, ~A ( x, y, z ) , is allowed to be dynamical, a fluctuation in this fieldwill act as a force on our ball. To see this imagine that when moving in a certaindirection in some region, the gauge field tells us the clocks are set progressively forward;this will make it look like the ball is moving faster since the time differences will bemeasured as smaller than they actually are. This fluctuation in the gauge field thusgives us an acceleration of the ball, in other words it acts on it with a force.In summary, by making our time-zone symmetry local, i.e., allowing different time-zones in different points in space, we were forced to introduce a gauge field and thatfield, when allowed to be dynamical, gives us forces. The point of gauge theory is to uselocal symmetries to introduce forces, or more generally, interactions into our theories. eaching gauge theory to first year students. In principle it is possible to rescale all monetary values with any factor and nothingin the economy would change. If we for example multiply all prices with 1 000 000everything will look very expensive, but if we at the same time multiply all salaries,savings and loans with the same number nothing really changed; you can still buy theexact same things for your salary. Of course, in the real world it is not that simple, thereis psychology involved and people have savings in their mattress, but we will ignore thatand pretend these rescalings work perfectly.Again we have a mathematical transformation that leaves everything unchanged,in other words, a symmetry.The world we live in consists of a number of different countries that all scale theireconomy differently. How then should we know how to deal with my Norwegian moneywhen I come to Sweden?We need a way of keeping track of the differences in scaling between the differentcountries. This is exactly (at least in our ideal economy) what the exchange rates arefor. If we also rescale the exchange rates when we rescale everything else in a countrywe can without problem allow every country to maintain whatever scaling they wantindependently of all other countries.With the exchange rates in place we see that our scaling symmetry is now local,we can scale the economy of every country independently of the other countries and theexchange rates is our gauge field that allow this local symmetry.Let us now imagine that the exchange rates fluctuate and perhaps we notice that ifI start with 100 Norwegian kroner, if I go to Sweden I get 110 Swedish kroner for those.If I then take my 110 Swedish kroner and go to Denmark I get only 88 Danish kroner,but for those I get 104.5 Norwegian kroner. This means that I have more money than Istarted with and if this is the case, investors will travel this circle over and over again tomake money. Like with the time-zone model the fluctuation in the gauge field producesa force that makes investors move around in this circle.In this example we can also see that the interaction goes both ways, we know fromthe real world that movements of money between countries will change the exchangerates. In other words, the gauge field (exchange rates) affect the money and the moneyaffect the gauge field. This bidirectionality of interactions is a general feature of physicstheories; if something affects something else, the second thing also affects the first.
When we summarise the above two models of gauge theories, we see that the main pointis to take a global symmetry and make it local, i.e., allow different transformationsin different points in space, and then, to ensure the theory to respect this localsymmetry, we need to introduce gauge fields that related the symmetry transformationsin neighbouring points. Allowing the gauge fields to be dynamic means we getinteractions between the gauge fields and the other components in our theory. eaching gauge theory to first year students. ‡ corresponding to charged particles with an arbitrarycomplex phase § . This symmetry is harder to visualise but is similar to the time zonesymmetry in that the transformation is parametrised by an angle; reading a clock canalso be viewed as measuring an angle.In making this symmetry local we get a gauge field, usually written A µ , that turnsout to be the electromagnetic field and we have indeed deduced electromagnetism. Thisis actually rather remarkable, it took physicists almost 100 years to get the theory ofelectromagnetism right and here we can deduce it in a few lines of algebra from justmaking this phase symmetry local!Today all interactions are described by gauge theories. As a matter of fact, thisis the only way we found to describe the forces we see in nature in a mathematicallyconsistent way k . Let us see which symmetries are used for the various interactions. Electromagnetism — as already mentioned a phase rotation for all electrically chargedfields. This gives us the mathematically simplest gauge theory. The gauge field is theelectromagnetic field whose quanta are called photons. For reference, this theory ispresented in some mathematical detail in Appendix B.
Weak nuclear force — here we have a more complicated symmetry that we cansee as two independent phase rotations, one for electrons and down quarks and onefor neutrinos and up quarks, and one transformation that transforms electrons intoneutrinos and down quarks into up quarks and vice verse. Since there are threecomponents to this symmetry, we get three gauge fields, the W + , W − and Z . Strong nuclear force — again a more complicated symmetry, this time it transformsthe quarks of different colour charge into each other and rotates their phases. Thoughnot obvious, it turns out that this contains eight independent transformations and hencewe get eight gauge fields called gluons.
Gravity — though not part of particle physics since we have not managed to writedown a mathematically consistent quantum field theory for it, gravity is actually also ‡ Despite its name, particle physics does not foremost deal with particles, but with fields. We have forexample a field that represents all electrons and positrons in the universe. This means all fields haveparticles associated with them and all particles have corresponding fields. § This means multiplying the fields with a factor e iθ , which is the same as rotating in the complexplane with an angle θ . This is described in some mathematical detail in Appendix B. k This may sound surprising, why should mathematical consistency be hard to achieve?Because when one field interacts with another, the properties of the first changes as a result of theinteraction. In some cases these changes lead to probabilities being larger than one and hence thetheory being mathematically inconsistent. Gauge symmetries prevent this from happening. eaching gauge theory to first year students.
7a gauge theory. The symmetries here are more intuitive since they are symmetries inour physical space-time; these include rotational invariance (the laws of physics arethe same in all directions), translational invariance (the laws of physics are the sameeverywhere) and Lorentz invariance (the laws of physics are the same in all inertialcoordinate-systems). The gauge fields here are basically the curved spacetime itself andif we manage to quantize this its quanta would be gravitons.We see that gauge theories are literally the foundation of our whole understandingof physics; in the vocabulary of contemporary physicists an interaction or theory isdefined by which gauge symmetries it is based on. This is why it is such a shame thatknowledge about gauge theory is limited to those few with Master degrees in particlephysics.
4. The Higgs mechanism
One of the most important events of the 21st century took place at CERN in 2012 whenthe Higgs boson was discovered. However, to understand why this was so importantis not easy. Most popular accounts say something about explaining why particles havemass or that Higgs gives mass to other particles.Though it is true that the other particles of the standard model get their massesfrom interacting with the Higgs, this does not really get to the core of the issue. Wouldit not be easier to just do what we always have done and treat mass as an intrinsicproperty that just happen to have the values we measure?One could object it would be more satisfying to understand where those valuescome from. However, the Higgs mechanism does not help in that, it does not explainany deeper meaning of the concept of mass and it does not tell us what the valuesof the various particle masses should be, they are still just numbers we take frommeasurements. As a matter of fact, if we could keep using the measured mass-valueswithout invoking the Higgs, we would.What is then the purpose of the Higgs field?It is all about the weak nuclear force. We can immediately see that there issomething odd with it; we learned that it is based on a gauge symmetry that amongstother things transforms electrons into neutrinos and vice verse. But how can thatpossibly be a symmetry? If all electrons in our atoms were replaced by neutrinos,things would certainly change a lot. Something is wrong here. Contrast this with thestrong nuclear force that transform for instance red upquarks to blue upquarks; thereis a reason they are both called upquarks, they are identical except for the colorchargeso swapping them makes no difference whatsoever.There are other issues with the weak force as well, the gauge fields W + , W − and Z are known to have large masses; this is why they have so little impact on our lives, theycause β -decay and that is essentially it. However, massive gauge fields are problematic,it turns out they break the gauge symmetry (this is not obvious, some details are givenin the end of Appendix B). eaching gauge theory to first year students. The trick is to make it look like the symmetry is broken without actually breaking it.This is not as strange as it first may sound, we have plenty of everyday examplesof this happening. The most obvious is the direction down; we know that the laws ofphysics are the same in all directions, but since we happen to live on a big planet itseems that the direction down is special. We can say that the symmetry between alldirections is spontaneously broken by the gravitational field of the Earth. The laws ofphysics still respect this symmetry, but here on the Earth’s surface we do not really seeit. A more direct analogy is a ferromagnetic material where at high temperature allthe little magnets (spin of the electrons) are pointing in random directions and hence alldirections look the same. At low temperature the magnets arrange themselves so thatthey all point in the same direction. This direction now looks special and the directionalsymmetry seems to be broken, but it is only the lowest energy state that breaks thesymmetry, it is still present in the fundamental equations governing the system.We need to do the same with the gauge symmetry of the weak nuclear force. Todo this we need a scalar field (the Higgs field) and then we give it a potential energysuch that the lowest energy does not happen in a symmetric point. Such a potential isdepicted in figure 1.It is not easy to comprehend what it means for a symmetry to be spontaneouslybroken; in exactly what sense is it broken?We can understand this intuitively by looking at figure 1, the potential isrotationally symmetric which here means we have a rotational symmetry. It shouldbe noted that this rotational symmetry does not rotate real space, but field space, thespace where the Higgs field lives, notice that the axes display the real and imaginarypart of the Higgs field. This symmetry is in fact the type of symmetry responsible forelectromagnetism, i.e., a phase rotation symmetry. Why not use the actual symmetryof the weak force? Because then the figure would need to be five dimensional.A state of lowest energy (the vacuum state) will sit at the bottom of the potentialand from there it does not look rotationally symmetric any more (the bottom is in avalley with two flat directions and two walls). In more technical terms we can say thatif we rotated everything around the vacuum state, the theory is no longer invariant.If we on the other hand were to rotate everything including the vacuum state aroundthe origin, the theory is invariant and therefore the symmetry is still there, it just doesnot look like that from our vacuum state. This is what it means to be spontaneouslybroken. eaching gauge theory to first year students. −1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.500.20.40.60.81 Re( φ )Im( φ )V( φ ) Figure 1.
A rotationally symmetric potential, with a minimum that spontaneouslybreaks the symmetry. In the vacuum state at the bottom of the potential, we do notsee the rotational symmetry, things do not look the same in all directions from downthere.
In short, the Higgs field is included in the standard model to spontaneously break thegauge symmetry of the weak nuclear force. In doing so it turns out the Higgs gives massto the W + , W − and Z bosons, the masses that would otherwise be forbidden by thegauge symmetry. This is the main reason the Higgs is needed.The masses of other particles are more complicated, it turns out that also all thefermions that interact with the weak nuclear force (i.e., all known fermions) need theHiggs field to get their masses. This fact is far from trivial ¶ , but it means all the massespresent in the standard model comes from the Higgs field + . As a consequence, howmassive a particle is is determined by how much it interacts with the Higgs field andall masses are proportional to the absolute value of the Higgs field in the vacuum state,i.e., the radius of the circle at the bottom of the potential in figure 1.We now see why everyone keep talking about the Higgs giving mass to particles, ¶ Fermion masses mix left and right chiral states, but the weak force only sees left chiral states andtherefore forbids those masses. It is hard to formulate this in a simple way. + As is often pointed out; the mass of the proton does not, it mostly comes from the strong nuclearforce, but we talk about fundamental particles here. eaching gauge theory to first year students.
5. Some observations from teaching gauge theory
These models have been used to teach gauge theory to first year physics students ina course in particle physics and cosmology at the University of Agder in Kristiansand,Norway. It was first used in 2016 and the curriculum has been developed since then. Togauge the reception of this, the students of 2020 answered some questions about theirexperience with these concepts and here we present the results of the enquiry.The students answered a short electronic questionnaire about their thoughts aboutlearning about gauge theory and spontaneous symmetry breaking. These questionswere not designed to test their understanding of gauge theory or if they really see theunderlying structure of the standard model, that would require a much more elaborateinvestigation that the circumstances did not allow.The question that this investigation tries to answer is:
What are the students thoughts on whether teaching gauge theory on this level isworthwhile?5.1. The students
All the students were at the end of a one year physics program. This one year is allthe physics they have studied at university and include mechanics, thermodynamics,electromagnetism, quantum physics, wave mechanics and the particle physics andcosmology course under discussion here. Some of the students are to become highschool teachers in mathematics and physics, some take this as part in their bachelor eaching gauge theory to first year students.
To ensure as high degree of participation as possible the questionnaire was kept veryshort with just five questions of which four were multiple choice. All the questions aswell as the rest of the course were given in Norwegian, so in the following, replies willbe translated to English but the original Norwegian text will also be given for reference.All the questions, both in original and translated are given in Appendix A. Sincethe number of questions and student replies are rather limited, the data analysis isrestricted to showing the distribution of answers on the multiple choice ones and quotethe text replies to the last question.This questionnaire was given after the exam in the course and the students wereexplicitly encouraged to answer what they think rather than what they think the authorwanted to hear.Though most questions were designed to only find out what the students thinkabout the topic rather than how much they understand of it, question 3 is an attemptto check that they gathered what the purpose of the Higgs mechanism is, this is doneso that the replies about the purposefulness of teaching about it can be trusted; if thestudents have misunderstood what the purpose of the Higgs mechanism is, one cannottake comments about whether it is worth teaching too seriously. It is, though, a valid eaching gauge theory to first year students.
Table 1.
Summary of the replies to the multiple choice questions. Note that onquestion 1. more than one reply was allowed, hence the larger count. question as to what extent question 3 succeeds in this.
The replies to the multiple choice questions are summarised in table 1. We see thatthe answers are very well aligned among the students and only a few have differingopinions. The replies are also encouragingly positive, though perhaps one should bearin mind that not everyone answered and it is possible that those who did not answerare less positive. It is also possible that it feels easier to give a positive reply if onedoes not have very strong feelings against that. Therefore one should perhaps be a littlecareful in interpreting the result, however we do not have any concrete reasons why thenon-responding students should be less positive than the responding nor the respondingones less positive than they express, so this is indeed a positive result.Let us look at each question a bit closer, starting from the beginning with question1. The purpose of this question was to see if the students think learning gauge theorymakes the standard model more of a unity rather than just a collection of particles.From the replies they seem to agree that it does, hence this argument for teachingthe subject might have some merit, though one student thinks it is hard without more eaching gauge theory to first year students.
Yes, I think it was especially exciting with Higgs. I had heard a lot about “theGod particle” when it was discovered, but had not studied it in depth.
This illustrates a point, many physics students are very interested in physics and readpopular science about it. It therefore makes sense that they also find it interesting learnmore about these things, even if it is not directly useful from a practical point of view.The God particle is a nickname for the Higgs which is popular in the media, but neverused by particle physicists.absolutt riktig absolutely the right thing to do
Clearly positive response. eaching gauge theory to first year students.
Yes, it is complicated and require some extrawork, but it pays of in the long run.
It is nice to see that something being difficult is not necessarily seen as negative bystudents. Working hard to learn complicated things have its benefits by itself.Jeg har personlig vært interessert i fysikk i mange år, men aldri hattnoe bedre forståelse enn den lille man klarer å opparbeide seg gjennompopulærvitenskapen. Å få lære om Gauge-teori og Higgs-feltet, selv om vi harfysikk på et relativt lavt nivå, har gitt meg et annet syn på hva partikkelfysikker. Jeg synes det er viktig å få lære litt om disse tingene for å få et riktigere bildeav partikkelfysikk og av universets utvikling. Jeg synes også det er god moralsånn vitenskaplig å tenke at feltet ikke er helt låst for de aller flinkeste, men atalle med interesse for faget kan dra noe nytte ut av forenklede forklaringer. Selvom jeg ikke skal studere mer fysikk kommer jeg til å fortsette å prøve å forståpartikkelfysikken bedre, og jeg tror også jeg ville synes det var kjedelig om jegikke fikk en oversikt over hva man kan dykke ned i.
I have personally beeninterested in physics for many years, but never had any better understandingthan the little you can get from popular science. To learn about gauge theoryand the Higgs field, even though we have physics at a rather low level, have givenme a different view on what particle physics is. I think it is important to learnsomething about these things to get a more correct picture of particle physicsand the history of the universe. I also think it is good moral, scientifically, tothink that the field is not locked for the very best, but that everyone with aninterest for the subject can benefit from simplified explanations. Even though Iwill not study more physics, I will continue to try to understand particle physicsbetter, and I also think I would find it sad if I did not get an overview of whatone can dive into.
This is as if I would have written it myself; it is very satisfying that a student practicallygives the exact arguments that motivated this effort in the first place.Ja det gir mening, men kan være litt forvirrende å tenke seg hvilke oppgaversom kan være aktuelle for temaet.
Yes it makes sense, but can be a bit confusingto imagine what questions can be relevant for the subject.
This statement mostly concerns the exam in the course, and it is true that it is hard tomake good exam question on gauge theory at this level, but that is a different questionas to whether we should include it in the curriculum.
All in all, the students agree that gauge theory and the Higgs mechanism are interestingtopics that they want to learn about. It is hard to learn, but that should not discourageus from trying. eaching gauge theory to first year students.
6. Conclusions
In this work it has been demonstrated how the concept of gauge theory and reasonsbehind the Higgs mechanism can be described with relatively little mathematicalcomplexity. This opens the door for a broader audience to get a more firm understandingof the theoretical underpinnings of modern physics.Though these concepts are far from simple, it is fully possible to teach to first yearbachelor students and possibly also earlier than that if required. This is of course alsoof interest to the scientifically interested populace outside the universities.A small questionnaire among the students exposed to this teaching, indicates asignificant interest in this and in general a positive reception. We can take this as aconfirmation that this sort of effort is appreciated and these simple models can hopefullyserve as part of an increased effort by the particle physics community to give theinterested audience a more detailed and correct description of what we are doing.
Appendix A. The questionnaire
Here follow the questions asked to the students. Since they were given in Norwegian,the original Norwegian texts is given together with an English translation in italics. Thefirst four questions are multiple choice where the first allow several options while theother three only allow one option to be marked. The last question is an open one wherethe students were allowed to write down their own thoughts.
1. Hva syns du om å lære om gauge teori? Flere svar mulige.
What do youthink about learning gauge theory? Multiple answers possible. (cid:3)
Det gjør det mulig å se helheten i standardmodellen.
It makes it possible to see thewhole picture of the standard model. (cid:3)
Det er interessant å få en innblikk i hvordan moderne fysikk-teorier er bygget opp.
Itis interesting to get some insight into how modern physics theories are constructed. (cid:3)
Det blir mest forvirrende siden vi ikke har nok bakgrunnskunnskap til å forstå detordentlig.
It gets mostly confusing since we do not have the background knowledgerequired to understand it properly. (cid:3)
Det finnes mer fornuftige ting å bruke tiden på.
There are better ways to use thetime. (cid:3)
Det er vanskelig å se poenget med det.
It is hard to see the point.eaching gauge theory to first year students.
2. Hva syns du om våre modeller for gauge-teorier; eksemplet medforskjellige tidssoner og den økonomiske modellen?
What do you think aboutour models for gauge theories; the example with different time-zones and the economicmodel? ◦ Jeg likte tidssonen best.
I liked the time-zone model best. ◦ Jeg likte den økonomiske modellen best.
I liked the economic model best. ◦ Begge modellene er gode og det er greit med flere eksempler.
Both models are goodand it is good to have several examples. ◦ De kompletterer hverandre godt.
They complement each other well. ◦ Jeg skjønte aldri helt hva vi skal lære fra noen av dem.
I never understood what wewere supposed to learn from any of them. ◦ Det går greit å forstå modellene, men jeg skjønner ikke hvordan det er relatert tilpartikkelfysikk.
It is okay to understand the models, but I do not understand howthey are related to particle physics. ◦ Jeg syns det går like greit å forstå gauge-teori uten de modellene.
I think it is justas easy to understand gauge theory without those models.
3. Syns du at du fått en forståelse for hvorfor Higgs er så viktig forstandardmodellen?
Do you think you have understood why the Higgs is so importantfor the standard model? ◦ Ja, vi kan ikke formulere standardmodellen uten spontant symmetribrudd.
Yes, wecannot formulate the standard model without spontaneous symmetry breaking. ◦ Det er jo ikke rart at folk spekulert i hva masse er og hvorfor elektroner er så myelettere enn protoner.
It is not strange that people ponder about what mass is andwhy the electron is lighter than the proton. ◦ Jeg skjønner ikke helt hvorfor man ikke bare kan si at partiklene har de masser dehar.
I do not really understand why we cannot just say that the particles have themasses they have. ◦ Jeg skjønte aldri helt hvorfor vi trengte Higgs.
I never really understood why weneed the Higgs. ◦ Konseptet spontant symmetribrudd var vanskelig å forstå.
The concept ofspontaneous symmetry breaking was hard to understand. ◦ Nei, denne delen skjønte jeg ikke helt.
No, this part I did not really understood.
4. Har din forståelse av hva moderne partikkelfysikk dreier seg om endretseg som følge av undervisningen i gauge-teori og Higgs-mekanismen?
Haveyour understanding of what modern particle physics is about, changed as a consequenceof the teaching of gauge theory and the Higgs mechanism? ◦ Ja, jeg har nå en mye klarere bild av hva det dreier seg om.
Yes, I now have amuch clearer picture of what it is about.eaching gauge theory to first year students. ◦ Ja, jeg har fått innblikk i en fysikk-verden som jeg ikke visste eksisterte.
Yes, Ihave gotten insight into a physics-world I did not know existed. ◦ Jeg hadde egentlig ikke noen ide om hva partikkelfysikk er, nå skjønner jeg noe avhva det dreier seg om.
I did not really have any idea what particle physics is, nowI have some understanding of what it is about. ◦ Jeg visste veldig lite om partikkelfysikk, og er fortsatt ganske usikker på hva detegentlig dreier seg om.
I knew very little about particle physics, and I am stilluncertain about what it is. ◦ Nei, Ikke noe særlig.
No, not particularly. ◦ Nei, men detaljene er noe klarere.
No, but the details are somewhat clearer. ◦ Nei, jeg føler jeg visste ganske godt hva partikkelfysikk er allerede før kurset.
No,I feel I knew quite well before the course what particle physics is. ◦ Ja, litt.
Yes, a little
5. Syns du det gir mening å lære om disse tingene (gauge-teori og Higgs) pådette nivået?
Do you think it makes sense to learn about these things (gauge theoryand the Higgs) at this level?
Appendix B. QED
For reference and more advanced students, let us deduce the Lagrangian for QuantumElectroDynamics (QED). In the following the reader is assumed to be familiar withfour-vector notation and Lagrangians.We start with a free fermion field ψ . The free field Lagrangian is, L = ¯ ψ ( iγ µ ∂ µ − m ) ψ, (B.1)where γ µ are the gamma matrices, for our purposes they can be considered someconstants, ∂ µ represents derivatives in space and time and m is the mass of the fermions.This Lagrangian has a global symmetry ψ → ψ ′ = e iφ ψ . Since ¯ ψ is a kind of complexconjugate, it must transform like, ¯ ψ → ¯ ψ ′ = e − iφ ¯ ψ , and then the two exponentials cancelin the Lagrangian, which therefore remains invariant.Let us try a local version, ψ → ψ ′ = e iφ ( ~x,t ) ψ . This no longer leaves the Lagrangianunchanged since we cannot take e iφ ( ~x,t ) through the derivative ∂ µ , we must use the chainrule. Doing so yields, L → L ′ = ¯ ψe − iφ ( ~x,t ) ( iγ µ ∂ µ − m ) e iφ ( ~x,t ) ψ == ¯ ψ ( − γ µ ∂ µ φ ( ~x, t ) + iγ µ ∂ µ − m ) ψ = L . (B.2)To restore the invariance we must change the partial derivative to a covariantderivative D µ ≡ ∂ µ + ieA µ , where e is the elementary charge and A µ is a vector- eaching gauge theory to first year students. A µ → A ′ µ = A µ − e ∂ µ φ ( ~x, t ) , under these symmetrytransformations. One can then show that, L = ¯ ψ ( iγ µ ∂ µ − γ µ eA µ − m ) ψ − F µν F µν , (B.3)where F µν ≡ ∂ µ A ν − ∂ ν A µ is a kind of kinetic energy to the field A µ , is invariant underthis gauge symmetry.It turns out that (B.3) describes all we know about electromagnetism. We havededuced one of the most important theories in human history with nothing more thaninsisting that the phase symmetry, ψ → ψ ′ = e iφ ψ , should become a local symmetry.To see why a mass term for the gauge field is problematic, let us look at such a term(we do not want a mass term for the photon, but we use this as a simple example toillustrate the point). Such a term has the form m A µ A µ , and we see that this cannotin general be invariant under the symmetry transformation A µ → A ′ µ = A µ − e ∂ µ φ ( ~x, t ) ,i.e., m A µ A µ = m A ′ µ A ′ µ . That is why gauge fields cannot have mass without thegauge symmetry being spontaneously broken. References [1] everything2.com/title/gauge+theory[2] Maldacena J, 2016
Eur. J. Phys. no.1, 015802 [1410.6753].[3] Young K, 1999 Am. J. Phys.67