AActive Learning in a Graduate Quantum Field Theory Course
G. Peter Lepage ∗ Department of Physics, Cornell University, Ithaca, NY, 14853 (Dated: December 8, 2020)This article describes how the author successfully adapted techniques drawn from the literature on activelearning for use in a graduate-level course on quantum field theory. Students completed readings and onlinequestions ahead of each class and spent class time working through problems that required them to practice thedecisions and skills typical of a theoretical physicist. The instructor monitored these activities and regularlyprovided timely feedback to guide their thinking. Instructor-student interactions and student enthusiasm weresimilar to that encountered in one-on-one discussions with advanced graduate students. Course coverage wasnot compromised. The teaching techniques described here are well suited to other advanced courses.
I. INTRODUCTION
Techniques drawn from the extensive research literature onactive learning are revolutionizing introductory courses inphysics and other subjects. These techniques emphasize farmore student-student and student-instructor interaction in theclassroom than in more traditional formats, with much morefine-grained, real-time assessment (multiple times in eachclass) to help students evaluate their own understanding of thematerial, and to help instructors evaluate student understand-ing and provide targeted feedback. The goals of such instruc-tion are less about the acquisition of facts, and more about“deliberate practice” of expert thinking and performance.
Most physics education research has focused on lower-levelcourses, but research shows that active-learning designs arealso effective for upper-level courses, provided those designsare modified to reinforce the specific expert practices that un-derly the course material. In this paper we describe how thisapproach was adapted by the author for a second-year gradu-ate course on quantum field theory for particle physics.Active learning is well suited to graduate courses, wherestudents must absorb a lot of technical material and learnhow to manipulate it like an expert (since they aspire to be-come one). Much of what they must learn is about decisions:what to pay attention to, which tool to use, what strategyto adopt, how and when to test a potential solution, and soon. Active-learning designs give students far more practicemaking such decisions than traditional lecture formats. Theyalso give instructors vastly more information about what theirstudents are thinking, because the students are working care-fully crafted problems in class, with the instructor wanderingfrom student to student throughout. The instructor is likelyto have direct interactions with every student in the class dur-ing each meeting time. The course becomes a long discussionbetween instructor and students — very similar in character tothe one-on-one meetings Ph.D. advisors have with their ad-visees, at their office blackboards. This kind of coaching inexpert thinking makes the learning process much more effi-cient and effective for students.Active-learning designs also provide invaluable feedback toinstructors about the actual obstacles faced by their students,as opposed to what the instructor imagines are obstacles —there is much more information than can be gleaned just fromhomework assignments and exams. This feedback allows the instructor to make real-time adjustments to the materials usedinside and outside of class, based upon substantial amounts ofdata. It also means that active-learning courses can improvesignificantly after being taught once or twice.Here we describe an implementation for active learning ina particular graduate-level course. Our discussion is not in-tended as physics education research; rather it is designed asan example for colleagues who might wish to experiment withactive learning in their advanced courses. In Section II, we de-scribe the goals and structure of our course. In Section III, weaddress the most challenging aspect for the instructor, which isthe creation of in-class activities for the students. We give sev-eral examples of such activities, relating them to the researchliterature on teaching. Finally, in Section IV, we summarizeour experiences teaching this course.
II. COURSE STRUCTURE
The course was introductory quantum field theory for par-ticle physicists, usually taken by a mixture of first-year andsecond-year grad students and a very small number of ad-vanced undergrads. Class size averaged around 20 people inall. It met twice a week for 70 minutes in a classroom withseparate tables for groups of four or five students each, sur-rounded by blackboards. This course was taught four times infour consecutive years.The instructor began the course with an explanation of thecourse format and why it was being used. The students weretold that this was a “reading course,” and that as practicingphysicists it would be important to know how to learn ontheir own by reading scientific literature. They were told thatthere was much in the course that they would learn just fromthe readings, material that would not be repeated in class butwhich they were responsible for. Class time was reservedfor issues raised in the readings that were particularly sub-tle or important, or easily misunderstood. These would beaddressed collaboratively.The following were the main components of the course:•
Small number of key themes:
The course had a co-herent story line throughout, with a small number ofideas/questions appearing again and again in differentparts of the course: for example, Where do quantumfield theories come from? What information goes into a r X i v : . [ phy s i c s . e d - ph ] D ec the design of a field theory? How do we extract experi-mentally useful information from such a theory? Whattechniques are available, how do we use them, and whatare their limitations? How do we use symmetries to ex-tract useful information?• Assigned reading for every class meeting:
Studentswere assigned about 10 pages of reading per class meet-ing, which were mostly from a textbook but in someparts of the course ventured into the research litera-ture. The lecturer posted the reading assignments onlineseveral days before class, introducing each assignmentwith a paragraph or two of text that highlighted whatwas important in the reading and how it related to thecourse as a whole.• Online questions about reading:
Students were askedto answer two to four questions on the reading and posttheir answers online the night before class, for exam-ple, by uploading cellphone photos of their handwrit-ten answers. The questions took students roughly 10–30 minutes to finish, after doing the reading. It tookthe instructor about 30 minutes to go through the stu-dents’ answers each time, and their answers would usu-ally affect what was done in class the next day. Thestudents were told that the questions were the sort ofquestions that professional physicists, such as the in-structor, would ask themselves as they read these ma-terials, to test their understanding. Sometimes a ques-tion would focus on a detail in a particular paragraphor equation, when it was important that everyone lookclosely at that paragraph/equation. Other times a ques-tion might prepare students for a discussion planned forthe next class. The last question was always a variationon “What about the readings needs more explanation?”This was frequently illuminating for the instructor, evenafter teaching the course multiple times.•
In-class discussion of reading:
The instructor startedeach class by commenting on students’ answers to theonline questions, usually referencing specific answers(anonymously). This was an opportunity to straightenout misunderstandings and/or to set up activities for therest of the class period. It also underscored for the stu-dents that the reading and online questions were essen-tial components of the course, fully integrated with thein-class activities.•
In-class worksheets:
The instructor next had the classwork two to four problems on worksheets that werehanded out. (Sample problems are discussed in Sec-tion III.) The class worked on one problem at a time.Typically the instructor introduced the problem with afew minutes of lecturing, and then had students work onit. They first worked individually, and then discussedtheir results with their group of four or so studentsseated at a table. While this went on, the instructor cir-culated through the room, looking at the students’ work(on their worksheets) and possibly discussing it withthem. The instructor frequently failed to get through all of the prepared problems; the remaining problemsoften ended up on homework or as online questions.The instructor stopped each exercise once most studentshad completed most or all of the problem. He thendescribed how an expert would approach the problem,usually referencing work the students had done individ-ually and sometimes calling on individuals or groupsto explain. In all, the instructor lectured for as muchas 50% of the time in class, depending on the topic.Some lecturing is essential to active learning, butmost of the lecturing came in pieces after students hadtried out the concepts themselves on the worksheets —they had to earn the lecture. Students handed in their(signed) worksheets at the end of class, to claim par-ticipation credit for their work; the worksheets werereturned at the next meeting. The instructor wouldsometimes examine the worksheets before they were re-turned, but this was usually unnecessary because he hadseen them while they were being filled out (and he wascirculating).•
Worksheet solutions:
After every class the instructorposted his solutions to the worksheet problems online.These functioned as lecture notes for the course.•
Homework and solutions:
Students were assigned sub-stantial homework assignments every one to two weeks,as in a typical graduate course. These were followed bydetailed solutions, posted online by the instructor theday after the homework was handed in (while studentsstill remembered the assignment). The solutions pro-vided the instructor an opportunity to explain how thehomework exercises related to the larger themes of thecourse.The grading scheme for the course was designed to providefeedback to students on their performance and to signal whichactivities were important. Students received feedback soon orimmediately after they completed the online questions and in-class worksheets, since the instructor discussed these in class.They were made accountable for this work, by handing it into the instructor, and they received participation credit (15%)for it in their final grade; it was not graded for correctness.The bulk of the final grade was awarded for correct answerson the homework (60%) and on a take-home, open-book finalexam (25%).
III. IN-CLASS ACTIVITIES
A lecture in a traditional course is usually built around twoor three key ideas that the instructor wants students to takeaway from the class. In an active learning class, the instructordesigns two or three activities around the same ideas. Theactivities are designed to help students understand what issueis resolved by each idea and how it is resolved — for example,how time-ordered operators allow us to extract physics (i.e., S -matrix elements via the LSZ Theorem) from quantum fieldtheories. Importantly, these activities also allow students to tryout ideas with the instructor present, to give them immediatefeedback on whether they are getting the point.There is much research showing that student learning isgreatly enhanced, often by factors of two or three (as mea-sured in diagnostic tests), if students struggle with a problem before they are told how to solve it. There is also evidencethat students who have learned concepts this way are betterable to transfer those concepts to novel contexts.
So whenteaching a key idea, an instructor should not start by explain-ing the idea and then giving students examples to practice on.Instead, the instructor seeks to do the reverse: first give stu-dents an example that challenges them to discover the key ideathemselves, and then give them a mini-lecture that shows themhow an expert would organize the problem. This suggests astrategy for selecting problems: decide which topics warranta mini-lecture, and then design a student activity that leadsinto each mini-lecture.Research also shows that there is great advantage in havingstudents work problems in small groups (3 or 4 people). Stu-dents learn at least as much from discussing a problem withother students as they do from hearing the instructor lecture onthe problem, even when everyone in the group is wrong. Group work benefits weaker and stronger students alike, un-like lecturing which appears to be particularly ineffective forstronger students. But the combination of small group dis-cussions followed by a mini-lecture is substantially more ef-fective for everyone than either separately. The unexpected power of student-student social learning isa major result from education research. This works partly be-cause groups provide feedback for students when the instruc-tor is elsewhere, but it is also because the brain works differ-ently when teaching or critiquing a peer than when thinkingabout a topic alone. Critiquing is a particularly important ex-pert skill that students typically have little experience with.Working in groups also helps keep individuals from gettingstuck and makes it easier for the instructor to monitor studentprogress (e.g., by listening in on their conversations).The following sections describe sample activities used inthe quantum field theory course.
A. First Problem
The first activity in the course occurred in the first meeting,before students had done any reading. Their exercise con-cerned the origins of the wave equation that describes wavesin an infinite stretched string. They were to derive the waveequation “the way a theoretical physicist would derive it,” us-ing symmetries (but not using Newton’s laws). The instructordid the first part. He used locality and translation invariance(in x and t ), together with the assumption of small amplitudes,to argue that the general form of the equation describing dis- placements y of the string from equilibrium is b + b y + b ∂y∂x + b ∂y∂t + b ∂ y∂x + b ∂ y∂x∂t + b ∂ y∂t + b ∂ y∂x + · · · (1)with derivative terms of arbitrarily high order. There are alsoterms nonlinear in y and its derivatives, but these were ne-glected initially. The students were then asked to work ingroups on the following questions: Question 1:
Identify symmetries of the stretchedstring that remove further terms from the gen-eral equation (above). There are at least four,although you may not need all of them. Char-acterize each symmetry using the “If y ( x, t ) is asolution, so is. . . ” formulation. Also write a sen-tence describing the symmetry. Question 2:
The wave equation is buried in thelist of possible terms. In what limit(s) would thetwo wave-equation terms dominate the others inthis list? Are there restrictions on the relative signof the the two wave-equation terms?
Question 3:
What are the most important cor-rections to the wave equation given your approx-imation(s) (and consistent with all of your sym-metries).The second and third questions were there to challenge groupswho got through the first question more quickly than the restof the class.Most groups came up with time-reversal symmetry (if y ( x, t ) is a solution, so is y ( x, − t ) ), which rules out termswith an odd number of derivatives ∂/∂t , and space-reflectionsymmetry or parity, which rules out terms with an odd numberof derivatives ∂/∂x . Usually only one or two groups wouldthink of y -displacement symmetry (if y ( x, t ) is a solution sois y ( x, t ) + ∆ where ∆ is a constant). This symmetry, whichrules out the b y term, is a classical example of spontaneoussymmetry breaking and Goldstone’s Theorem, which playeda significant role later in the course. The constant b = 0 because y ( x, t ) = 0 is the equilibrium solution, but can alsobe ruled out by a y → − y symmetry which many groupsproposed. Occasionally groups would suggest more complexsymmetries, like variations on a dilatation symmetry.Symmetry and stability arguments reduce the general equa-tion to something that can be written c ∂ y∂t = ∂ y∂x + g a ∂ y∂x + g a ∂ y∂x + · · · (2)where c is the wave speed, a is a length that is somehow re-lated to physical characteristics of the string (e.g., the string’sdiameter), and the g n are dimensionless. The second ques-tion asks under what circumstances the a n terms in this equa-tion can be neglected. Most groups found this difficult to an-swer. The instructor helped individual groups move forwardby suggesting that they write down solutions of the wave equa-tion and substitute them into the equation. Eventually theyfigured out that the a n terms are suppressed, by ( a/λ ) n , forsolutions y ( x, t ) that involve only long wavelengths λ (cid:29) a —leaving behind the wave equation.The wave equation is not exact in this analysis, but hasa hierarchy of correction terms. Terms with many powersof a are more suppressed than terms with few or no pow-ers. The number of powers is determined by the dimension of ∂ n y/∂x n . The third question focuses attention on the leadingcorrection, whose form is completely specified by symmetriesup to a single unknown constant g a .This analysis shows how the wave equation (with its correc-tions) emerges from very general assumptions about localityand symmetries, as a low-amplitude, long- wavelength, low-frequency approximation to other arbitrary dynamics. Theseproperties are not uncommon, and this is why the wave equa-tion arises in so many different contexts. It represents the uni-versal behavior of systems with these (quite generic) charac-teristics.These ideas carry over to quantum field theory as was em-phasized by closely related activities in the subsequent daysand weeks:• Students were asked to derive the Lagrangian, includ-ing higher-dimension correction terms, for a scalar fieldtheory using Lorentz invariance, parity symmetry, etc.,together with a low-energy/momentum expansion. Thisyields the Lagrangian used to introduce quantum fieldtheory in the early weeks of the course.• Dimensional analysis of the scalar theory indicates thatthe scalar particle’s mass is naturally very large —so large that a low-energy expansion is not possible,thereby invalidating the Lagrangian’s derivation. Stu-dents were asked what symmetry could be used to ad-dress this issue and what the resulting Lagrangian is.That symmetry is φ → φ + ∆ , in analogy to the waveequation symmetry that gets rid of the b y term. Massterms violate this symmetry, so the resulting theory de-scribes massless scalar particles. This theory is a use-ful prototype for the chiral theory that describes low-energy pions, and also the theory that describes the (hy-pothetical) axion particle. It is also the first timestudents confront the “mass problem” in quantum fieldtheory, which is a primary driver for current experimen-tal work in particle physics.• Students were asked to use gauge symmetries, Lorentzinvariance, etc. to derive the QED Lagrangian, includ-ing corrections, first for charged scalar particles, thenfor neutral scalars, and later for charged and neutralspin- / particles. These Lagrangians were central toall discussions in the last half of the course. The theo-ries describing the QED interactions of neutral particlesare not in standard textbooks, which makes them par-ticularly useful for activities.The ideas and techniques applied in the wave equationproblem, on day one, are fundamental to the modern under- Real scalar field: ! ∂ + m " φ = 0 = ⇒ φ ( x ) = d p (2 π ) ! a p e − ipx + a † p e ipx " Complex scalar field: ! ∂ + m " φ = 0 = ⇒ φ ( x ) = d p (2 π ) ! a p e − ipx + b † p e ipx " Complex non-rel. field: $ i∂ t + ∇ m % ψ = 0 = ⇒ ψ ( x ) = d p (2 π ) a p e − ipx Complex Dirac field: ! i∂γ + m " ψ = 0 = ⇒ ψ ( x ) = & s d p (2 π ) ! a s p u s ( p )e − ipx + b † s p v s ( p )e ipx " FIG. 1. Contrasting cases illustrating when a quantum theory re-quires a new anti-particle, distinct from the particle. standing of the Standard Model of particle physics as a low-energy/momentum effective field theory. Introducing theseideas early on gave students an opportunity to play with thembefore they had to grapple with the complexities of quantizedfields, Dirac equations, renormalization, and all the other top-ics in the course. Revisiting them repeatedly throughout thecourse gave students an opportunity to practice the ideas indifferent contexts, thereby exercising and updating the men-tal maps they use to organize material from the course. Many students had heard of things like spontaneous symmetrybreaking and axions. Having these appear early in the coursehelped reinforce the course’s relevance to their larger interests.
B. Contrasting Cases The following example illustrates the use of contrastingcases for learning general principles.
Anti-particles ap-peared early in the course. Rather than explaining the rulesgoverning the introduction of anti-particles, the instructorgave the students several examples or cases and challengedthem to find the rules:
Question:
One of the early triumphs of rela-tivistic quantum mechanics was its prediction ofthe existence of anti-particles. Consider the fourtypes of (non-interacting) quantum field theoryshown in Fig. 1. The second and fourth theoriesrequire new anti-particles, distinct from the origi-nal particles. The others do not. What is the gen-eral rule that determines whether there is a newanti-particle?The students had read about the first two cases in the textbook,and were familiar with the third case from previous work-sheet exercises. Given only the first two cases, one might con-clude that the general rule is that complex-valued fields have L = ψ ! i∂γ − m " ψ + gφψψ + ( ∂φ ) − λ φ L = ψ ! i∂γ − m " ψ + gφψγ ψ + ( ∂φ ) − λ φ L = ψ ! i∂γ − m " ψ + gφψγ ψ + ( ∂φ ) + h φ − λ φ FIG. 2. Lagrangians that illustrate the systematics of parity invari-ance. new anti-particles. That idea is undermined, however, by thethird example, which has a complex field but no anti-particle.Maybe the third case is special because its field equation is lin-ear in ∂ t , unlike the first two which are quadratic. The fourthcase, however, also has a field equation that is linear in ∂ t , butrequires a new anti-particle. The actual rule has two parts: one needs to determineboth whether there are negative energy solutions to the fieldequations, and, if so, whether the number of degrees of free-dom encoded in the field requires independent Fourier coeffi-cients for the positive and negative-energy terms. . In prac-tice, many students managed to understand the relevance ofthe number of degrees of freedom, but few paid attention tothe need (or not) for negative-energy states, despite this be-ing the starting point for most elementary discussions of anti-particles — this was prior knowledge that needed to be acti-vated.As discussed above, having students struggle to find a ruleis far more effective than telling them the rule and havingthem practice on examples. The contrasting cases werecarefully selected to make it difficult for students to fastenon a simpler rule by thinking too narrowly about the possibili-ties. Each of the surface features mentioned above — complexversus real, first-order versus second-order, scalar versus ma-trix — is relevant to the correct rule, but none is definitive onits own. A major point of the exercise is to get students tothink through how these surface features relate to the real is-sues. This illustrates how the same principles of learning (e.g.,distinguishing key underlying structures from surface featureswhen approaching a new problem) apply across the physicscurriculum from introductory mechanics to quantum field the-ory.
C. Multiple-Choice Review Research strongly suggests that review activities have amuch larger impact on student learning than review lectures,even if students have trouble recalling the earlier lessons.
This is especially true of in-class activities since they forcestudents to rely upon their memories rather than texts or notes.Getting information into the brain is relatively easy; it is get-ting it out again that is difficult and requires effortful practice.The interconnections in the brain that allow for retrieval arestrengthened by such practice, just as muscles respond to ex-ercise — the brain is physically modified. This problem was given as an in-class activity to review how parity is expressed in quantum field theories:
Question:
Consider a spin-1/2 spinor field ψ coupled to a real-valued spin-0 field φ in each ofthe three Lagrangians in Fig. 2. For each caseindicate whether the Lagrangian has a symmetryunder a parity transformation, remembering that ψ ( x , t ) → γ ψ ( − x , t ) under parity. Choose oneof:a) Yesb) Noc) It depends on whether φ is a pseudo-scalaror a scalar.The techniques used to analyze spin-1/2 parity are mechani-cal, and so were relegated to out-of-class reading and home-work. The review activity, coming after the reading, providedan opportunity to discuss the techniques in class to whateverextent made sense given how the students, individually andin their groups, approached the problem.An additional reason for this problem was a widespread andsurprisingly durable misconception that was apparent everyyear in teaching this course. Students tend to think of a field’sbehavior under parity (i.e., whether it is a pseudo-scalar or ascalar) as a property intrinsic to the field. In fact, it is a prop-erty of the Lagrangian of the theory: the field’s transformationunder parity is whatever it takes (if anything) to make its La-grangian invariant when the spatial coordinates are reflectedthrough the origin ( x → − x ).The correct answers for this problem are a), a), and b). Despite earlier readings and practice problems, and explicitdiscussions in earlier classes, a large fraction of the classchose c), c), and c), and almost no one had a perfect score.Answer c) is nonsense because it ignores the underlying La-grangian. Multiple-choice problems can help students appre-ciate that they harbor misconceptions because the problemscan be constructed — here, by including option c) — to guar-antee that everyone engages with the misconception.Multiple-choice problems also leave no question in the stu-dents’ minds about whether they understand the underlyingidea — if they chose c) they didn’t understand. It catches theirattention. One student leaving class was overheard to com-ment wryly: “Three out of three wrong. This is the worstphysics quiz result I have ever received.”
D. Other Strategies
A variety of approaches were used to generate problems forin-class group work, always with the requirement that theywould engage students in the skills and reasoning processes atheoretical physicist would routinely use in their work:• Immediately after students read about a standard tech-nique, they would often be asked to apply the sametechnique to a different field theory, one not discussedin the text. For example, such methods as canonicalquantization of fields, and the Schwinger-Dyson equa-tion for generating perturbation theory were introducedin the text using a relativistic scalar field theory. Thestudents were asked to apply these methods to a non-relativistic theory, which is sufficiently different to benontrivial but simple enough to be doable in class.• In-class exercises were useful for giving students prac-tice with the mechanics of new techniques. Theirprogress can be slowed by trivial mistakes and mis-understandings that are readily resolved in the class-room. For example, when first assembling a Feynman-diagram contribution that involves spin- / particles,students frequently wrote things like ( /k + m ) ieγ µ u ( p ) u ( q ) ieγ µ (3)rather than u ( p ) ieγ µ ( /k + m ) ieγ µ u ( q ) . (4)They were forgetting that γ µ , /k , and m all represent × matrices, while u ( q ) and u ( p ) are column androw 4-vectors, respectively — so the order in which youwrite them down matters. That Eq. (3) is nonsense isobvious to an expert, but less so to a novice. Such sim-ple misunderstandings are addressed very efficiently inthe classroom, where an expert is close at hand.• In-class activities are particularly well suited to con-necting course content with experimental results be-cause groups are more likely to succeed in making theconnections than individuals. Also it creates an oppor-tunity for the instructor to elaborate on the larger signfi-cance of the results immediately after the students haveengaged with them. Late in the course, for example, thestudents worked through the consequences of sponta-neous symmetry breaking for a simple Yukawa theorywhere a spin- / particles is coupled to a neutral scalar.They were then asked how their analysis was relatedto a famous plot from CERN which shows the quark-Higgs coupling as a function of quark mass. The modelthe students analyzed is a toy model, not the real Stan-dard Model Lagrangian, and so was sufficiently simpleto be studied in class. It nevertheless exhibits the correctrelationship between the coupling and quark mass, andso provides a qualitative explanation of the experimen-tal data. This experimental result was important evi-dence, when it came out, that quark masses (and proba-bly all other masses) come from spontaneous symmetrybreaking — a major advance.• To test their understanding of a derivation in a readingassignment, students were sometimes asked to explainprecisely what goes wrong with the derivation if oneof the assumptions is changed. For example, studentswere asked how the derivation of the conserved energy-momentum tensor for a scalar theory was invalidatedif the particle’s mass depended on time. A variationon this assignment would be to ask students to invent a Lagrangian for which the energy-momentum tensor isnot conserved.• Students were asked to connect their current work toideas from previous weeks. Again this works particu-larly well as an in-class activity because of the groupsand timely instructor feedback. When deriving rep-resentations of the Poincar´e group for spin- / parti-cles, for example, students were asked what equationfor massive spin- particles corresponds to the Diracequation. The corresponding equation for spin-1 is ∂ µ A µ = 0 . These equations are needed because thefields in both cases have more components than thereare spin states. This connection is not obvious to moststudents, but reflects a fundamental tension in quantumfield theory between locality and Lorentz invariance.An expert’s conceptual map of the subject is built outof these kinds of insights. IV. RESULTS AND CONCLUSIONS
Students were generally open to active learning, though ittook time for some to get used to the group work. Atten-dance in class was nearly perfect (95%), despite the 8:30 amstart time. End-of-semester student evaluations of the coursewere mediocre (3.7/5) in the first year, but improved substan-tially as the course evolved, until they were as positive asthey could be (5/5) in the last year. The instructor circulatedthroughout the classroom continuously while students workedon activities, helping groups both with the physics and withgroup dynamics. This informal contact meant that discussionsfollowing an activity were far more lively than in a traditionallecture, with a much larger fraction of the class participating.There was also substantially more participation in after-classdiscussions and office hours than in earlier courses taught tra-ditionally by the instructor.One byproduct of the larger role for students in the class-room was that teaching the course never became stale or bor-ing for the instructor, no matter how often he taught it —each new group of students brought fresh perspectives andraised new issues that often changed the course of class meet-ings. It also gave the instructor more insight into the obsta-cles encountered by students. Some of these involved incor-rect preconceptions about quantum field theory, as discussedearlier. Others involved tools they had been taught in pre-vious courses. While they understood the physics behindthese tools, they had insufficient practice with them to attainfluency. This lack of fluency was evident in the in-class ac-tivities and led to changes in their designs in subsequent years,as well as to new online questions and homework problemsthat provided extra practice.Course coverage was not compromised by active learning.This was because straightforward topics were relegated to thereading assignments and not covered in class. Students didthe reading, as was evident from their responses to the readingquestions submitted online before each class meeting — overthe semester, students earned on average 96% of the avail-able participation credit for the online questions. The mate-rial covered by the course was fairly standard for an introduc-tory quantum field theory course, covering roughly the first200–250 pages of standard texts, together with a discussionof renormalization theory through leading-log order built onmaterial from Ref. 23. Homework assignments and the fi-nal exam were very similar to what is used in conventionalcourses, and the students did well on both (average and stan-dard deviation on the final were 72% and 13%, respectively).Moving topics to the reading assignments also freed up timeto talk about the course and the current state of quantum fieldtheory and particle physics. Quantum field theory textbooks,for example, are highly optimized in their choice of topics,notation, and emphasis for applications of weak-coupling per-turbation theory (Feynman diagrams). This reflects the pre-dominance of perturbation theory in the history of the subject.Nonperturbative techniques, like the numerical methods usedin QCD simulations or the mathematical methods used in con-formal field theory, have become far more important in recentdecades. It was a useful (metacognitive) exercise to imaginehow textbooks and courses might be different if they were op-timized for nonperturbative analyses. The history of quantum field theory is itself illuminatingabout the importance of practice with expert feedback whenlearning a hard subject. David Kaiser, in his book
Draw-ing Theories Apart , traces the dispersion of Feynman-diagramtechniques through the theoretical physics community in thefirst several years after Feynman invented them. Kaiser findsthat the only theorists who adopted Feynman diagrams in thatperiod were those who had had extended personal contactwith Feynman or Feynman’s former Cornell colleague Free-man Dyson, or with one of their postdocs or graduate students.This is despite the fact that Feynman’s techniques were dra-matically more efficient than the alternatives, and were welldocumented in the literature. Kaiser summarizes: “No uncrossable, epistemic barriers separated di-agram users from nonusers. Physicists could cer- tainly learn something about the diagrams fromthe published literature or from Dyson’s unpub-lished lecture notes. Yet reading texts is notthe same as using tools. Long after textual in-structions became widely available, almost noone used the diagrams in actual calculations uponlearning about them from texts alone. . . Feynmandiagrams are practices, and as such they must be practiced . . . ”And that practice entailed “personal contact and sustained,face-to-face training” from people who had already masteredthe techniques.The main goal of a graduate quantum field theory course isto make novice quantum field theorists more expert. The ex-pert knows more facts about quantum field theory, but, moreimportantly, also knows how to organize and connect thosefacts much more effectively. The expert knows how to choosethe right tool for a new problem, how to evaluate and test so-lutions, and how to select/modify a problem so it is relevantand solvable. Short of one-on-one tutoring, active learning of-fers an instructor the most powerful tools available for helpingnovices become expert at such skills, since it allows studentsto practice the skills (in the classroom) with an expert closeby. As discussed above, textbooks are poor substitutes forthis kind of training; so are conventional lectures (in personor online) for much the same reason — little or no real-timefeedback to students. Student engagement and enthusiasm aremuch higher in an active learning class. And the constant,close interaction with students makes active-learning classesendlessly stimulating and enjoyable for the instructor. ACKNOWLEDGMENTS
We thank Carl Wieman for his numerous comments on themanuscript and for years of advice on all matters connectedwith active learning. ∗ [email protected] D. L. Schwartz, J. M. Tsang, and K. P. Blair,
The ABCs of HowWe Learn, J. D. Bransford, A. L. Brown, and R. R. Cocking,
How PeopleLearn; Brain, Mind, Experience, and School, expanded edition(NAS Press, Washington, 2000). J. M. Lang,
Small Teaching: Everyday Lessons from the Scienceof Learning, The website has an extensivecollection of resources for instructors, including many two-page“how-to” documents on active learning. It also has a curated col-lection of education research articles relevant to university-levelteaching. For example: L. Deslaurier, E. Schelew, and C. Wieman, “Im-proved Learning in a Large-Enrollment Physics Class,” Science , 862–864 (2011). S. Freeman, S. L. Eddy, M. McDonough, M. K. Smith, N. Oko-roafor, H. Jordt, and M. P. Wenderoth, “Active learning increases student performance in science, engineering, and mathematics,”PNAS (23), 8410-8415 (2014). C. Wieman, “Expertise in University Teaching & the Implicationsfor Teaching Effectiveness, Evaluation & Training,” Daedalus (4), 47–78 (2019). K. A. Ericsson, R. Th. Krampe, and C. Tesch-R¨omer, “The Roleof Deliberate Practice in the Acquisition of Expert Performance,”Psych. Rev. (3), 363–406 (1993). K. A. Ericsson, R. R. Hoffman, A. Kozbelt, and A. M. Williams(editors),
The Cambridge Handbook of Expertise and Expert Per-formance, For example: D. J. Jones, K. W. Madison, and C. E. Wieman,“Transforming a 4th year Modern Optics Course Using a Delib-erate Practice Framework,” Phys. Rev. ST Phys. Educ. Res. ,020108 (2015). The textbooks used in this course were: M. E. Peskin and D. V.Schroeder,
An Introduction to Quantum Field Theory,
Quantum
Field Theory and the Standard Model, Several of the approaches described in Sec. III D are also usefulfor formulating online questions. D. L. Schwartz and J. D. Bransford, “A Time for Telling,” Cogni-tion and Instruction (4), 475–522 (1998). For a short overview, see Chapter J in Ref. 1. More than two or three ideas per lecture overloads students’ short-term memories, leading to poor retention. Overloading is surpris-ingly easy; see for example: L. McDonnell, M. K. Barker, C.Wieman, “Concepts first, jargon second improves student artic-ulation of understanding,” Mol. Bio. Ed. (1), 12–19 (2015);and R. E. Mayer, E. Griffith, I. T. N. Jurkowitz, and D. Rothman,“Increased Interestingness of Extraneous Details in a MultimediaScience Presentation Leads to Decreased Learning,” J. of Exp.Psych. (4), 329—339 (2008). For a review, see: J. D. Bransford and D. L. Schwartz, “Rethink-ing Transfer: A Simple Proposal With Multiple Implications,”Rev. Res. Ed. , 61–100 (1999). D. L. Schwartz and T. Martin, “Inventing to Prepare for FutureLearning: The Hidden Efficiency of Encouraging Original Stu-dent Production in Statistics Instruction,”, Cognition and Instruc-tion (2), 129–184 (2004). For example: M. K. Smith, W. B. Wood, K. Krauter, and J. K.Knight, “Combining Peer Discussion with Instructor ExplanationIncreases Student Learning from In-Class Concept Questions,”CBE-Life Sci. Ed. (1), 55–63 (2011). M. K. Smith, W. B. Wood, W. K. Adams, C. Wieman, J. K.Knight, N ˙Guild, and T. T. Su, “Why Peer Discussion ImprovesStudent Performance on In-Class Concept Questions,” Science , 122–124 (2009). There is no need for terms involving extra time derivatives, like ∂ y/∂t , since these can be replaced by terms with only spa-tial derivatives by using the wave equation (for solutions). Thisis closely related to the topic of redundant operators in a quantumfield theory. Scalar field theory makes sense as an effective field theory witha finite ultraviolet cutoff Λ of order the scale at which newphysics, beyond the scalar theory, appears. Contributions fromhigh-dimension operators (e.g., gφ / Λ ) are normally suppressedin applications by powers of p/ Λ where p is a momentum typicalof the application. The natural size for the mass term, however,is Λ φ , which means that: the φ particle’s mass is of order Λ , p/ Λ ≈ , and high-dimension operators are not suppressed. Alow-3-momentum expansion is still possible, leading to a non-relativistic effective field theory. The particles are also weakly interacting because the leading in-teraction has dimension eight and so is suppressed by / Λ : L int = − g ( ∂φ · ∂φ ) / Λ . G. P. Lepage, “What is Renormalization?”, in
Actions to Answers ,edited by T. Degrand and D. Toussaint, (World Scientific, Singa-pore, 1989) [arXiv:hep-ph/0506330]. One of the most important differences between experts andnovices is in their mental organizations of subject material. SeeChapters 2 and 3 in Ref. 2 for a discussion of expert-novice differ-ences and their relevance to teaching and learning. For a succinctsummary, with annotated references to the research literature, see:W. Adams, C. Wieman, and D. L. Schwartz, “Teaching ExpertThinking,” at . This subsection is adapted from a supplement (written by the au-thor) for Ref. 7. D. L. Schwartz, C. C. Chase, M. A. Oppezzo, and D. B. Chin,“Practicing Versus Inventing With Contrasting Cases: The Effects of Telling First on Learning and Transfer,” J. Ed. Psych. (4),759–775 (2011). See Chapter C in Ref. 1. Another example that could have been included is the Majoranaspinor field, which is a relativistic theory with a complex field anda first-order field equation, but no new anti-particle. Students weregiven an opportunity to revisit the thinking used in the presentproblem when they were analyzing Majorana fields for a home-work problem several weeks later. Relativistic theories typically have both positive-energy andnegative-energy solutions, because E = p + m has two so-lutions. Such theories need two terms in the Fourier expansion: exp( − ipx ) for the positive-energy solutions, and exp( ipx ) forthe negative-energy solutions, which are associated with the anti-particles. Non-relativistic theories have only positive-energy solu-tions (because E = p / m has only one solution), and so onlyone term. The last case, like the third case, is linear in i∂ t ↔ E ,but it is a matrix equation whose solution leads immediately to E = p + m . For the real scalar field, the coefficient of the negative-energy so-lution must be the conjugate of the coefficient of the positive-energy solution, so that the field is real-valued. This is an examplewhere a particle is its own anti-particle; there is no additional par-ticle. The complex field contains twice as much information, be-cause it has a real and imaginary part, and so needs twice as manyFourier coefficients: a p and b † p , where the first is associated withthe original particle, and the second with the new anti-particle. See Chapter 1 in Ref. 3. For an elaboration on in-class review activities, see: E. J.Maxwell, L. McDonnell, and C. Wieman, “An Improved De-sign for In-Class Review,” J. of Coll. Sci. Teaching (5), 48–52(2015). See Chapter 14 in Ref. 9, and Chapter 5 in Ref. 2. In the first case, the theory has a parity symmetry where the φ fieldtransforms like a scalar under parity ( φ ( x , t ) → φ ( − x , t ) ). Thus, φ is necessarily a scalar if it is described by this Lagrangian.Similarly, it is necessarily a pseudo-scalar in the second case( φ ( x , t ) → − φ ( − x , t ) ). The third Lagrangian breaks parity sym-metry because φ changes sign when φ ( x , t ) → − φ ( − x , t ) . So φ is neither a scalar nor a pseudo-scalar in this theory; parity is auseless construct here, and the pseudo-scalar/scalar distinction ismeaningless. This is in part because of the participation credit given for in-classwork. Credit towards final grades should signal what the instructorbelieves are essential components in the course. Students respondto these signals. This does not mean, obviously, that the course is perfect. Thereremain many opportunities for significant improvement. An example is the representation theory for the rotation group,which is needed to build spin- / representations for the Poincar´egroup (resulting in the Dirac equation). Another example is scat-tering theory (in and out states). Participation credit is also an effective way to encourage studentsto keep up with the reading. See: C. E. Heiner, A. I. Banet, andC. Wieman, “Preparing students for class: How to get 80% ofstudents reading the textbook before class,” Am. J. Phys. (10),989–996 (2014). A nonperturbative focus, for example, would spend time on com-posite particles (bound states). QCD’s physical states are all com-posite, but most field theory texts present no treatment of suchstates. D. Kaiser,