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.

## References

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

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

There is no need for terms involving extra time derivatives, like $\u22024y/\u2202t4$, 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.

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., $g\varphi 6/\Lambda 2$) are normally suppressed in applications by powers of $p/\Lambda $ where *p* is a momentum typical of the application. The natural size for the mass term, however, is $\Lambda 2\varphi 2$, which means that: the $\varphi $ particle's mass is of order Λ, $p/\Lambda \u22481$, and high-dimension operators are not suppressed. A low-3-momentum expansion is still possible, leading to a non-relativistic effective field theory.

The particles are also weakly interacting, because the leading interaction has dimension eight and so is suppressed by $1/\Lambda 4$: $Lint=\u2212g(\u2202\varphi \xb7\u2202\varphi )2/\Lambda 4$.

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.

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: $exp\u2009(\u2212ipx)$ for the positive-energy solutions, and $exp\u2009(ipx)$ 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 $i\u2202t\u2194E$, but it is a matrix equation whose solution leads immediately to $E2=p2+m2$.

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 $bp\u2020$, where the first is associated with the original particle, and the second with the new anti-particle.

See Chapter 1 in Ref. 3.

In the first case, the theory has a parity symmetry, where the $\varphi $ field transforms like a scalar under parity ($\varphi (x,t)\u2192\varphi (\u2212x,t)$). Thus, $\varphi $ is necessarily a scalar if it is described by this Lagrangian. Similarly, it is necessarily a pseudo-scalar in the second case ($\varphi (x,t)\u2192\u2212\varphi (\u2212x,t)$). The third Lagrangian breaks parity symmetry, because $\varphi 3$ changes sign when $\varphi (x,t)\u2192\u2212\varphi (\u2212x,t)$. So $\varphi $ 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.

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.

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

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).

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.

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