Different quantum theories with the same equations of motion

Question

In the path integral picture of quantum field theory we can define the theory with the generating functional which, leaving aside all of the unsolved mathematical problems in actually constructing it, is given by $$ Z[J] = \frac{1}{\mathcal{N}}\int \mathcal{D}\phi e^{i S[\phi] + i J \phi}.$$ Now the integration is over all field configurations; not just ones that satisfy the equations of motion. The stationary phase approximation tells us that the leading order behavior (in $\hbar$ I believe) will be the classical equations of motion. I believe the non-classical/non-stationary effects can be quantified in the Schwinger-Dyson differential equation.

Okay, so imagine defining a new action $$ \overline{S}[\phi] =S[\phi] + T[\phi] $$ Where $T[\phi]$ is a term which vanishes when $\phi$ satisfies the classical equation of motion e.g. is a solution to $\frac{\delta S}{\delta \phi}[\phi] = 0$. This would seem to lead to a theory that is quantum mechanically different, but classically the same. I don't think that this is the same thing as a quantum anomaly because there it is that the measure $\mathcal{D}\phi$ is not symmetric under the same symmetries as the classical theory.

Here are my questions:

1. It certainly seems that $\overline{S}$ could lead to different predictions than $S$ (with $T$ chosen appropriately). Is this the case? If so, what is a specific example of two theories that are equivalent classically but not quantum mechanically?

2. How does this connect to the operator view of Quantum field theory? I know that there is an operator ordering ambiguity in defining the Hamiltonian in this approach, and this manifests subtly in how we construct the path integral (I remember reading about this somewhere in Quantization of Gauge Systems by Henneaux and Teitolbom). Is this fundamentally the same ambiguity?

Answer 1

Theta angles.

In many theories you have the freedom to add a topological term to the action. This changes drastically the quantum mechanical picture, but not the classical one.

The simplest simplest example I can think of, is a free particle on a circle, i.e. the quantum mechanics of the coordinate $q(t)$ such that $q(t)\sim q(t)+2\pi$. The Lagrangian is: $\newcommand{\d}{\mathrm{d}}$ $$L_0 = \frac12\dot{q}^2.$$ To this I can add a total derivative term, $\dot{q}$:$^{1}$ $$L_\theta = \frac12\dot{q}^2 + \frac{\theta}{2\pi}\dot{q}.$$ It obviously doesn't affect the classical equations of motion. Quantum mechanically, however the two theories are quite distinct. In this case the example is so simple you can actually completely solve it. Its Hilbert space consists of wavefunctions $$\Psi_n(q) = \left<n|q\right> = \frac{1}{\sqrt{2\pi}} \mathrm{e}^{i n q},\qquad\text{with energy}\qquad E_n(\theta) = \frac12\left(n-\frac{\theta}{2\pi}\right)^2, \qquad n\in\mathbb{Z}.$$ A particularly interesting value of $\theta$ is $\theta=\pi$. You see that in this case the vacuum of the theory is twofold degenerate, namely: $$E_\text{min}(\pi) = E_{0}(\pi) = E_1(\pi).$$ Let's take a step back. All I did is add a total derivative to the action, and for a specific value of its coefficient suddenly the number of ground states of the quantum mechanical theory has changed! This is quite drastic.

The same game can be played in QFT. In a two-dimensional theory of a compact scalar you can add a term of the form $$S^{\theta}_{\text{scalar}} \sim \frac{\theta}{2\pi} \int\d{\phi}\wedge\d{\phi} \sim \frac{\theta}{2\pi} \int\d^2 x \sqrt{g} \epsilon^{\mu\nu} \partial_\mu\phi \partial_\nu\phi.$$ In four-dimensional Maxwell theory you can add (see also this Phys.SE answer) $$S^{\theta}_{\text{Maxwell}} \sim \frac{\theta}{2\pi} \int F\wedge F \sim \frac{\theta}{2\pi} \int\d^4 x \sqrt{g}\epsilon^{\mu\nu\rho\sigma} F_{\mu\nu}F_{\rho\sigma}.$$ And in Yang-Mills theory $$S^{\theta}_{\text{YM}} \sim \frac{\theta}{2\pi} \int\mathrm{tr} F\wedge F.$$

These terms have an immediate observable effect in the quantum mechanical picture. For example magnetically charged particles (in the 2d example) or lines (in the 4d examples) can be shown to also necessarily carry electric charge! [1]. Moreover in Yang-Mills theory there is a beautiful story about the symmetries, anomalies, and the phase diagram of Yang-Mills theory at non-zero theta angle [2] (see also their appendix D for more on the QM example above). Finally, in QCD theta angles are famously connected to the strong CP problem.

I don't have much to say about operator ordering ambiguities, but I don't think it's connected as there is no such ambiguity in the quantum mechanical example I gave above, yet there is a theta angle.

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${^1}$ I'm writing this term as an angle, because $\theta$ leads to the same physics as $\theta+2\pi$, even at the quantum mechanical level.

[1] E. Witten, Dyons of charge $e\theta/2\pi$.

[2] Davide Gaiotto, Anton Kapustin, Zohar Komargodski, Nathan Seiberg, Theta, time reversal, and temperature.