This article describes how the author successfully adapted techniques drawn from the literature on active learning for use in a graduate-level course on quantum field theory. Students completed readings and online questions ahead of each class and spent class time working through problems that required them to practice the decisions and skills typical of a theoretical physicist. The instructor monitored these activities and regularly provided timely feedback to guide their thinking. Instructor-student interactions and student enthusiasm were similar to that encountered in one-on-one discussions with advanced graduate students. Course coverage was not compromised. The teaching techniques described here are well suited to other advanced courses.

1.
D. L.
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,
J. M.
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, and
K. P.
Blair
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Norton
,
New York
,
2016
).
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J. D.
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,
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, and
R. R.
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,
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J. M.
Lang
,
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, 1st ed. (
Jossey-Bass
,
San Francisco
,
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).
4.
The website http://www.cwsei.ubc.ca has an extensive collection of resources for instructors, including many two-page “how-to” documents on active learning. It also has a curated collection of education research articles relevant to university-level teaching.
5.
For example:
L.
Deslaurier
,
E.
Schelew
, and
C.
Wieman
, “
Improved learning in a large-enrollment physics class
,”
Science
332
,
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,
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Okoroafor
,
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, and
M. P.
Wenderoth
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Active learning increases student performance in science, engineering, and mathematics
,”
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),
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(
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7.
C.
Wieman
, “
Expertise in university teaching & the implications for teaching effectiveness, evaluation & training
,”
Daedalus
148
(
4
),
47
78
(
2019
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8.
K. A.
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, and
C.
Tesch-Römer
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,”
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9.
The Cambridge Handbook of Expertise and Expert Performance
, 2nd ed., edited by
K. A.
Ericsson
,
R. R.
Hoffman
,
A.
Kozbelt
, and
A. M.
Williams
(
Cambridge U.P
.,
Cambridge
,
2018
).
10.
For example:
D. J.
Jones
,
K. W.
Madison
, and
C. E.
Wieman
, “
Transforming a 4th year modern optics course using a deliberate practice framework
,”
Phys. Rev. ST Phys. Educ. Res.
11
,
020108
(
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).
11.
The textbooks used in this course were:
M. E.
Peskin
and
D. V.
Schroeder
,
An Introduction to Quantum Field Theory
, 1st ed. (
Addison-Wesley
,
Boston
,
1995
); and
M. D.
Schwartz
,
Quantum Field Theory and the Standard Model
, 1st ed. (
Cambridge U.P
.,
Cambridge
,
2014
).
12.

Several of the approaches described in Sec. III D are also useful for formulating online questions.

13.
D. L.
Schwartz
and
J. D.
Bransford
, “
A time for telling
,”
Cognit. Instruction
16
(
4
),
475
522
(
1998
).
14.

For a short overview, see Chapter J in Ref. 1.

15.
More than two or three ideas per lecture overloads students' short-term memories, leading to poor retention. Overloading is surprisingly easy; see for example:
L.
McDonnell
,
M. K.
Barker
, and
C.
Wieman
, “
Concepts first, jargon second improves student articulation of understanding
,”
Mol. Biol. Ed.
44
(
1
),
12
19
(
2015
) and
R. E.
Mayer
,
E.
Griffith
,
I. T. N.
Jurkowitz
, and
D.
Rothman
, “
Increased interestingness of extraneous details in a multimedia science presentation leads to decreased learning
,”
J. Exp. Psychol.
14
(
4
),
329
339
(
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16.
For a review, see
J. D.
Bransford
and
D. L.
Schwartz
, “
Rethinking transfer: A simple proposal with multiple implications
,”
Rev. Res. Ed.
24
,
61
100
(
1999
).
17.
D. L.
Schwartz
and
T.
Martin
, “
Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction
,”
Cognit. Instruction
22
(
2
),
129
184
(
2004
).
18.
For example:
M. K.
Smith
,
W. B.
Wood
,
K.
Krauter
, and
J. K.
Knight
, “
Combining peer discussion with instructor explanation increases student learning from in-class concept questions
,”
CBE-Life Sci. Ed.
10
(
1
),
55
63
(
2011
).
19.
M. K.
Smith
,
W. B.
Wood
,
W. K.
Adams
,
C.
Wieman
,
J. K.
Knight
,
N.
Ġuild
, and
T. T.
Su
, “
Why peer discussion improves student performance on in-class concept questions
,”
Science
323
,
122
124
(
2009
).
20.

There is no need for terms involving extra time derivatives, like 4y/t4, since these can be replaced by terms with only spatial derivatives by using the wave equation (for solutions). This is closely related to the topic of redundant operators in a quantum field theory.

21.

Scalar field theory makes sense as an effective field theory with a finite ultraviolet cutoff Λ of order the scale at which new physics, beyond the scalar theory, appears. Contributions from high-dimension operators (e.g., gϕ6/Λ2) are normally suppressed in applications by powers of p/Λ where p is a momentum typical of the application. The natural size for the mass term, however, is Λ2ϕ2, which means that: the ϕ particle's mass is of order Λ, p/Λ1, and high-dimension operators are not suppressed. A low-3-momentum expansion is still possible, leading to a non-relativistic effective field theory.

22.

The particles are also weakly interacting, because the leading interaction has dimension eight and so is suppressed by 1/Λ4: Lint=g(ϕ·ϕ)2/Λ4.

23.
G. P.
Lepage
, “
What is renormalization
?,” in
Actions to Answers
, edited by
T.
Degrand
and
D.
Toussaint
(
World Scientific
,
Singapore
,
1989
) [arXiv:Hep-ph/0506330].
24.
One of the most important differences between experts and novices is in their mental organizations of subject material. See Chapters 2 and 3 in Ref. 2 for a discussion of expert-novice differences and their relevance to teaching and learning. For a succinct summary, with annotated references to the research literature, see:
W.
Adams
,
C.
Wieman
, and
D. L.
Schwartz
, “
Teaching expert thinking
,” at http://www.cwsei.ubc.ca/resources/files/Teaching_Expert_Thinking.pdf.
25.
This subsection is adapted from a supplement (written by the author) for Ref. 7.
26.
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. Psychol.
103
(
4
),
759
775
(
2011
).
27.
See Chapter C in Ref. 1.
28.

Another example that could have been included is the Majorana spinor field, which is a relativistic theory with a complex field and a first-order field equation, but no new anti-particle. Students were given an opportunity to revisit the thinking used in the present problem, when they were analyzing Majorana fields for a homework problem several weeks later.

29.

Relativistic theories typically have both positive-energy and negative-energy solutions, because E2=p2+m2 has two solutions. Such theories need two terms in the Fourier expansion: exp(ipx) for the positive-energy solutions, and exp(ipx) for the negative-energy solutions, which are associated with the anti-particles. Non-relativistic theories have only positive-energy solutions (because E=p2/2m has only one solution), and so only one term. The last case, like the third case, is linear in itE, but it is a matrix equation whose solution leads immediately to E2=p2+m2.

30.

For the real scalar field, the coefficient of the negative-energy solution must be the conjugate of the coefficient of the positive-energy solution, so that the field is real-valued. This is an example where a particle is its own anti-particle; there is no additional particle. The complex field contains twice as much information, because it has a real and imaginary part, and so needs twice as many Fourier coefficients: ap and bp, where the first is associated with the original particle, and the second with the new anti-particle.

31.

See Chapter 1 in Ref. 3.

32.
For an elaboration on in-class review activities, see:
E. J.
Maxwell
,
L.
McDonnell
, and
C.
Wieman
, “
An improved design for in-class review
,”
J. Coll. Sci. Teach.
44
(
5
),
48
52
(
2015
).
33.

See Chapter 14 in Ref. 9, and Chapter 5 in Ref. 2.

34.

In the first case, the theory has a parity symmetry, where the ϕ field transforms 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 symmetry, because ϕ3 changes sign when ϕ(x,t)ϕ(x,t). So ϕ is neither a scalar nor a pseudo-scalar in this theory; parity is a useless construct here, and the pseudo-scalar/scalar distinction is meaningless.

35.

This is in part because of the participation credit given for in-class work. Credit towards final grades should signal what the instructor believes are essential components in the course. Students respond to these signals.

36.

This does not mean, obviously, that the course is perfect. There remain many opportunities for significant improvement.

37.

An example is the representation theory for the rotation group, which is needed to build spin-1/2 representations for the Poincaré group (resulting in the Dirac equation). Another example is scattering theory (in and out states).

38.
Participation credit is also an effective way to encourage students to keep up with the reading. See
C. E.
Heiner
,
A. I.
Banet
, and
C.
Wieman
, “
Preparing students for class: How to get 80% of students reading the textbook before class
,”
Am. J. Phys.
82
(
10
),
989
996
(
2014
).
39.

A nonperturbative focus, for example, would spend time on composite particles (bound states). QCD's physical states are all composite, but most field theory texts present no treatment of such states.

40.
D.
Kaiser
,
Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics
, 1st ed. (
Chicago U. P
.,
Chicago
,
2005
), pp.
110
111
.
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